R8-g: sparse supernodal LDLt for the Newton tangent (default off)
rtx_fea::solvers::SparseLdlt: nested-dissection ordering, etree,
fundamental supernodes, multifrontal numeric factorisation (blocked
LDLt, matrixmultiply gemm, rayon over subtrees, bit-deterministic at
any thread count), symbolic analysis reused while the pattern holds.
NonlinearDynamicStepper: TangentSolver::{BandedLu (default, unchanged
float for float), SparseLdlt { reuse }} via with_tangent_solver or
RTX_FEA_TANGENT=sparse[:K]; fixed-pattern CSR assembled from parallel
element evaluations (forces summed in the banded path's order);
optional modified Newton (factor reuse). The per-element kernel is
factored out of assemble() unchanged.
Co-Authored-By: Claude Opus 5.5 (1M context) <[email protected]>
This commit is contained in:
Omar Sobh
co-authored by
Claude Opus 5.5
Claude Opus 5.5
parent
171da41ed1
commit
860f5bb37f
@@ -11,6 +11,7 @@ pub mod flag3d;
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pub mod modal_analysis;
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pub mod nonlinear_analysis;
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pub mod nonlinear_dynamic;
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pub mod sparse_tangent;
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pub mod static_analysis;
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use crate::assembly::AssemblyOptions;
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@@ -26,6 +27,7 @@ pub use dynamic_analysis::*;
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pub use modal_analysis::*;
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pub use nonlinear_analysis::*;
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pub use nonlinear_dynamic::*;
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pub use sparse_tangent::{TangentSolver, TangentStats};
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pub use static_analysis::*;
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/// Base trait for all finite element analyses.
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@@ -45,6 +45,7 @@
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//! change between steps (and between subiterations of one step) through
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//! the stepper.
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use super::sparse_tangent::{SparseTangent, TangentSolver, TangentStats};
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use super::{AnalysisConfig, ConvergenceCriteria};
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use crate::assembly::dof_mapping::{AdvancedDofNumbering, DofComponent, DofMappingStrategy};
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use crate::assembly::SparseMatrix;
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@@ -56,6 +57,7 @@ use crate::materials::{reduced_constitutive, MaterialDatabase};
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use crate::mesh::{Mesh, NodeId};
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use crate::solvers::{BandedLu, LinearSolver, SolverOptions};
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use nalgebra::{DMatrix, DVector, Vector3};
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use rayon::prelude::*;
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/// Time histories and final state of a nonlinear transient run.
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#[derive(Debug, Clone)]
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@@ -118,6 +120,9 @@ pub struct NonlinearDynamicAnalysis {
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body_force: Option<Box<dyn Fn(Vector3<f64>) -> Vector3<f64> + Send + Sync>>,
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nodal_forces: Vec<(NodeId, Vector3<f64>)>,
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tracked_nodes: Vec<NodeId>,
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/// Explicit tangent-solver choice; `None` = `RTX_FEA_TANGENT`, else
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/// the banded LU.
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tangent_solver: Option<TangentSolver>,
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}
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impl NonlinearDynamicAnalysis {
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@@ -145,9 +150,26 @@ impl NonlinearDynamicAnalysis {
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body_force: None,
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nodal_forces: Vec::new(),
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tracked_nodes: Vec::new(),
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tangent_solver: None,
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}
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}
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/// Choose the linear solver of the Newton tangent (default: the
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/// banded LU, or the `RTX_FEA_TANGENT` knob when set). See
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/// [`TangentSolver`].
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#[must_use]
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pub fn with_tangent_solver(mut self, solver: TangentSolver) -> Self {
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self.tangent_solver = Some(solver);
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self
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}
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/// The tangent solver a new stepper will use.
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pub fn tangent_solver(&self) -> TangentSolver {
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self.tangent_solver
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.or_else(TangentSolver::from_env)
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.unwrap_or_default()
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}
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/// Switch to the total-Lagrangian St. Venant–Kirchhoff formulation
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/// (plane strain in 2-D), as on the nonlinear static analysis.
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#[must_use]
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@@ -284,6 +306,8 @@ pub struct NonlinearDynamicStepper<'a> {
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/// Steps carried by step subdivision after both the plain loop and
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/// the line search failed.
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rescued_subdivision: usize,
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/// The sparse tangent path (`None` = the banded LU path).
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sparse: Option<SparseTangent>,
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}
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impl<'a> NonlinearDynamicStepper<'a> {
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@@ -420,6 +444,15 @@ impl<'a> NonlinearDynamicStepper<'a> {
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}
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let added_mass = DVector::zeros(total_dofs);
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let sparse = match analysis.tangent_solver() {
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TangentSolver::BandedLu => None,
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TangentSolver::SparseLdlt { reuse } => Some(SparseTangent::new(
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caches.iter().map(|c| c.dofs.as_slice()),
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&free_index,
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num_free,
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reuse,
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)?),
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};
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let mut stepper = Self {
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added_mass,
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analysis,
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@@ -435,6 +468,7 @@ impl<'a> NonlinearDynamicStepper<'a> {
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solver_options: SolverOptions::default(),
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rescued_line_search: 0,
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rescued_subdivision: 0,
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sparse,
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};
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stepper.set_nodal_forces(&analysis.nodal_forces);
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Ok(stepper)
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@@ -475,9 +509,16 @@ impl<'a> NonlinearDynamicStepper<'a> {
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// inv_beta_dt2 is only read when assembling the tangent.
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let (f_int0, _) = self.assemble(&u, false, 0.0)?;
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let residual0 = &self.external - &f_int0;
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let (a0_free, _) = self
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.solver
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.solve(&self.mass_free, &residual0, &self.solver_options)?;
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let (a0_free, _) = match &mut self.sparse {
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None => self
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.solver
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.solve(&self.mass_free, &residual0, &self.solver_options)?,
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Some(sparse) => {
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sparse
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.mass_solver
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.solve(&self.mass_free, &residual0, &self.solver_options)?
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}
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};
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let mut a = DVector::zeros(self.total_dofs);
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for (k, &dof) in self.free_dofs.iter().enumerate() {
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a[dof] = a0_free[k];
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@@ -602,12 +643,19 @@ impl<'a> NonlinearDynamicStepper<'a> {
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let mut u_iter = u_pred.clone();
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let mut step_converged = false;
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let mut iterations = 0usize;
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// Sparse path only: the previous iteration's residual norm.
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let mut previous_norm = f64::INFINITY;
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for _ in 0..criteria.max_iterations {
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let mut a_new = DVector::zeros(self.total_dofs);
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for &dof in &self.free_dofs {
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a_new[dof] = inv_beta_dt2 * (u_iter[dof] - u_pred[dof]);
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}
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let (f_int, tangent) = self.assemble(&u_iter, true, inv_beta_dt2)?;
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let (f_int, tangent) = if self.sparse.is_some() {
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(self.sparse_forces(&u_iter)?, None)
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} else {
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let (f_int, tangent) = self.assemble(&u_iter, true, inv_beta_dt2)?;
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(f_int, Some(tangent))
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};
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let residual = &self.external - &f_int - self.mass_times(&a_new);
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let residual_norm = residual.norm();
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if line_search && !residual_norm.is_finite() {
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@@ -618,9 +666,28 @@ impl<'a> NonlinearDynamicStepper<'a> {
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break;
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}
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iterations += 1;
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let (delta, _) = self
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.solver
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.solve(&tangent, &residual, &self.solver_options)?;
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let delta = match tangent {
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Some(tangent) => {
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let (delta, _) =
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self.solver
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.solve(&tangent, &residual, &self.solver_options)?;
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delta
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}
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None => {
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let rate = residual_norm / previous_norm;
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previous_norm = residual_norm;
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let sparse = self.sparse.as_mut().expect("sparse path");
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sparse.solve(
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self.caches.iter().map(|c| &c.mass),
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&self.added_mass,
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&self.free_index,
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&residual,
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inv_beta_dt2,
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line_search,
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rate,
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)?
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}
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};
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let alpha = if line_search {
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match self.backtrack(&u_iter, &delta, &u_pred, inv_beta_dt2, residual_norm) {
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Some(alpha) => alpha,
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@@ -732,6 +799,33 @@ impl<'a> NonlinearDynamicStepper<'a> {
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(self.rescued_line_search, self.rescued_subdivision)
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}
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/// Counters and timings of the sparse tangent path (`None` on the
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/// banded path).
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pub fn tangent_stats(&self) -> Option<TangentStats> {
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self.sparse.as_ref().map(SparseTangent::stats)
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}
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/// Sparse path: every element's force and tangent in parallel, the
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/// forces summed in element order (the banded path's order, so the
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/// internal force is the same float for float), the tangents kept for
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/// the factorisation.
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fn sparse_forces(&mut self, solution: &DVector<f64>) -> FeaResult<DVector<f64>> {
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let started = std::time::Instant::now();
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let analysis = self.analysis;
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let results: Vec<(DVector<f64>, DMatrix<f64>)> = self
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.caches
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.par_iter()
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.map(|cache| element_force_and_tangent(analysis, cache, solution))
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.collect::<FeaResult<_>>()?;
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let sparse = self.sparse.as_mut().expect("sparse path");
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Ok(sparse.accumulate(
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results,
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self.caches.iter().map(|c| c.dofs.as_slice()),
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&self.free_index,
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started,
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))
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}
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/// Internal force and (optionally) tangent at a full displacement
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/// vector, reduced to the free DOFs. The tangent includes the Newmark
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/// mass term `M / (β Δt²)` at the caller's `inv_beta_dt2` (the
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@@ -742,41 +836,11 @@ impl<'a> NonlinearDynamicStepper<'a> {
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with_tangent: bool,
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inv_beta_dt2: f64,
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) -> FeaResult<(DVector<f64>, SparseMatrix)> {
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let dim = self.analysis.mesh.spatial_dimension;
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let num_free = self.free_dofs.len();
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let mut internal = DVector::zeros(num_free);
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let mut tangent = SparseMatrix::new(num_free, num_free);
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for cache in &self.caches {
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let material = self
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.analysis
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.materials
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.get_material(cache.material_id)
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.unwrap();
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let fe = StandardFiniteElement::new(cache.element_type, cache.coords.clone());
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let mut element_displacement = DVector::zeros(cache.dofs.len());
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for (local, &dof) in cache.dofs.iter().enumerate() {
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element_displacement[local] = solution[dof];
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}
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let (f_int, k_t) = if self.analysis.total_lagrangian {
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let (lambda, mu) = material.properties().lame_parameters();
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let constitutive = saint_venant_kirchhoff(lambda, mu, dim);
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total_lagrangian::internal_force_and_tangent(
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&fe,
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&cache.coords,
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&element_displacement,
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constitutive.as_ref(),
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None,
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)?
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} else {
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let constitutive = reduced_constitutive(material, dim)?;
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ElementMatrixComputer::compute_internal_force_and_tangent(
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&fe,
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&cache.coords,
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&element_displacement,
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constitutive.as_ref(),
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None,
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)?
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};
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let (f_int, k_t) = element_force_and_tangent(self.analysis, cache, solution)?;
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for (local_row, &dof_row) in cache.dofs.iter().enumerate() {
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let Some(free_row) = self.free_index[dof_row] else {
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continue;
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@@ -834,3 +898,40 @@ impl<'a> NonlinearDynamicStepper<'a> {
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out
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}
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}
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/// One element's internal force and tangent at the full displacement
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/// vector `solution` (total-Lagrangian SVK or small strain, per the
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/// analysis) — shared by the banded and the sparse paths.
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fn element_force_and_tangent(
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analysis: &NonlinearDynamicAnalysis,
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cache: &ElementCache,
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solution: &DVector<f64>,
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) -> FeaResult<(DVector<f64>, DMatrix<f64>)> {
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let dim = analysis.mesh.spatial_dimension;
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let material = analysis.materials.get_material(cache.material_id).unwrap();
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let fe = StandardFiniteElement::new(cache.element_type, cache.coords.clone());
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let mut element_displacement = DVector::zeros(cache.dofs.len());
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for (local, &dof) in cache.dofs.iter().enumerate() {
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element_displacement[local] = solution[dof];
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}
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if analysis.total_lagrangian {
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let (lambda, mu) = material.properties().lame_parameters();
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let constitutive = saint_venant_kirchhoff(lambda, mu, dim);
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total_lagrangian::internal_force_and_tangent(
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&fe,
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&cache.coords,
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&element_displacement,
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constitutive.as_ref(),
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None,
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)
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} else {
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let constitutive = reduced_constitutive(material, dim)?;
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ElementMatrixComputer::compute_internal_force_and_tangent(
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&fe,
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&cache.coords,
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&element_displacement,
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constitutive.as_ref(),
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None,
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)
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}
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}
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@@ -0,0 +1,303 @@
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//! The sparse Newton-tangent path of [`super::NonlinearDynamicStepper`]
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//! (R8-g): a fixed-pattern CSR assembled from per-element blocks through
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//! precomputed positions, factorised by [`SparseLdlt`] with the symbolic
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//! analysis done once (the pattern is fixed by the mesh), optionally
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//! reusing the numeric factor for several Newton iterations (modified
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//! Newton).
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//!
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//! Default off: [`TangentSolver::BandedLu`] is the stepper's original
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//! path, float for float. Select with
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//! [`super::NonlinearDynamicAnalysis::with_tangent_solver`] or the
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//! environment knob `RTX_FEA_TANGENT=sparse` (`sparse:K` = reuse each
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//! factor for up to K Newton solves; `banded` = the default).
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//!
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//! What stays identical to the banded path: the element kernels, the
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//! internal force (summed in the same element/local order), the
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//! residual, the convergence tests, the rescue ladder. What differs: the
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//! linear solve (LDLᵀ in a nested-dissection order vs banded LU with
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//! partial pivoting — rounding-level), and the tangent keeps entries the
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//! old `SparseMatrix` assembly drops below 1e-15.
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use crate::assembly::SparseMatrix;
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use crate::error::FeaResult;
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use crate::solvers::{BandedLu, LinearSolver, SolverOptions, SparseLdlt};
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use nalgebra::{DMatrix, DVector};
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use std::time::Instant;
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/// Which linear solver carries the Newton tangent.
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#[derive(Debug, Clone, Copy, PartialEq, Eq, Default)]
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pub enum TangentSolver {
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/// The original path: `SparseMatrix` assembly + banded LU every
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/// Newton iteration.
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#[default]
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BandedLu,
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/// Fixed-pattern parallel assembly + supernodal sparse LDLᵀ with
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/// symbolic reuse. `reuse` = how many Newton solves one numeric
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/// factor may serve (1 = full Newton; more = modified Newton, the
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/// factor refreshed early when the dt changes, when the residual
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/// contracts by less than half per iteration, and always in the
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/// line-search rescue).
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SparseLdlt {
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/// Newton solves per numeric factorisation (≥ 1).
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reuse: usize,
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},
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}
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impl TangentSolver {
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/// Full-Newton sparse path.
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pub const SPARSE: Self = Self::SparseLdlt { reuse: 1 };
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/// Parse the `RTX_FEA_TANGENT` knob: `banded`, `sparse`, `sparse:K`.
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/// Unset or unparsable → `None` (unparsable values are logged).
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pub fn from_env() -> Option<Self> {
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let value = std::env::var("RTX_FEA_TANGENT").ok()?;
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let value = value.trim();
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let parsed = match value {
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"banded" => Some(Self::BandedLu),
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"sparse" => Some(Self::SPARSE),
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other => other
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.strip_prefix("sparse:")
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.and_then(|k| k.parse::<usize>().ok())
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.filter(|&k| k >= 1)
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.map(|reuse| Self::SparseLdlt { reuse }),
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};
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if parsed.is_none() {
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tracing::warn!("RTX_FEA_TANGENT={value:?} not understood; using the banded LU");
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}
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parsed
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}
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}
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/// Counters and accumulated wall time of the sparse path.
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#[derive(Debug, Clone, Copy, Default, PartialEq)]
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pub struct TangentStats {
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/// Element force/tangent evaluations (parallel) + force summation.
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pub assemblies: usize,
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/// Numeric factorisations.
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pub factorizations: usize,
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/// Linear solves (Newton corrections).
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pub solves: usize,
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/// Symbolic analyses (1 unless the pattern changed).
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pub analyses: usize,
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/// Solves that fell back to the banded LU (failed pivot).
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pub fallbacks: usize,
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/// Seconds in element evaluation + force summation.
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pub assembly_seconds: f64,
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/// Seconds in tangent scatter + numeric factorisation.
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pub factor_seconds: f64,
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/// Seconds in triangular solves.
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pub solve_seconds: f64,
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/// Nonzeros of the factor.
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pub nnz_l: usize,
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}
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/// State of the sparse path, owned by the stepper.
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pub(super) struct SparseTangent {
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num_free: usize,
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row_ptr: Vec<usize>,
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col_idx: Vec<usize>,
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values: Vec<f64>,
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/// Per element, `(local_row · n_e + local_col)` → CSR index, or
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/// `u32::MAX` when either DOF is constrained.
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elem_pos: Vec<Vec<u32>>,
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/// CSR index of each free DOF's diagonal.
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diag_pos: Vec<usize>,
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/// The element tangents of the latest assembly.
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element_tangents: Vec<DMatrix<f64>>,
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solver: SparseLdlt,
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/// Separate instance for the rest state's mass solve.
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pub(super) mass_solver: SparseLdlt,
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reuse: usize,
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uses_left: usize,
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factor_coef: f64,
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factor_valid: bool,
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fallback: BandedLu,
|
||||
stats: TangentStats,
|
||||
}
|
||||
|
||||
impl SparseTangent {
|
||||
/// Build the fixed pattern (every free–free pair sharing an element)
|
||||
/// and the element position maps; analyse it once.
|
||||
pub(super) fn new<'e>(
|
||||
elements: impl Iterator<Item = &'e [usize]> + Clone,
|
||||
free_index: &[Option<usize>],
|
||||
num_free: usize,
|
||||
reuse: usize,
|
||||
) -> FeaResult<Self> {
|
||||
let mut rows: Vec<Vec<usize>> = vec![Vec::new(); num_free];
|
||||
for dofs in elements.clone() {
|
||||
for &r in dofs {
|
||||
let Some(fr) = free_index[r] else { continue };
|
||||
for &c in dofs {
|
||||
if let Some(fc) = free_index[c] {
|
||||
rows[fr].push(fc);
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
for (fr, row) in rows.iter_mut().enumerate() {
|
||||
row.push(fr);
|
||||
row.sort_unstable();
|
||||
row.dedup();
|
||||
}
|
||||
let mut row_ptr = Vec::with_capacity(num_free + 1);
|
||||
row_ptr.push(0);
|
||||
let mut col_idx = Vec::new();
|
||||
for row in &rows {
|
||||
col_idx.extend_from_slice(row);
|
||||
row_ptr.push(col_idx.len());
|
||||
}
|
||||
drop(rows);
|
||||
let find = |r: usize, c: usize| -> usize {
|
||||
let span = &col_idx[row_ptr[r]..row_ptr[r + 1]];
|
||||
row_ptr[r]
|
||||
+ span
|
||||
.binary_search(&c)
|
||||
.expect("pattern covers every element pair")
|
||||
};
|
||||
let mut elem_pos = Vec::new();
|
||||
for dofs in elements {
|
||||
let n_e = dofs.len();
|
||||
let mut pos = vec![u32::MAX; n_e * n_e];
|
||||
for (a, &r) in dofs.iter().enumerate() {
|
||||
let Some(fr) = free_index[r] else { continue };
|
||||
for (b, &c) in dofs.iter().enumerate() {
|
||||
if let Some(fc) = free_index[c] {
|
||||
pos[a * n_e + b] = find(fr, fc) as u32;
|
||||
}
|
||||
}
|
||||
}
|
||||
elem_pos.push(pos);
|
||||
}
|
||||
let diag_pos = (0..num_free).map(|d| find(d, d)).collect();
|
||||
let mut solver = SparseLdlt::new();
|
||||
solver.analyze(num_free, &row_ptr, &col_idx)?;
|
||||
let values = vec![0.0; col_idx.len()];
|
||||
let stats = TangentStats {
|
||||
analyses: 1,
|
||||
nnz_l: solver.stats().map_or(0, |s| s.nnz_l),
|
||||
..TangentStats::default()
|
||||
};
|
||||
Ok(Self {
|
||||
num_free,
|
||||
row_ptr,
|
||||
col_idx,
|
||||
values,
|
||||
elem_pos,
|
||||
diag_pos,
|
||||
element_tangents: Vec::new(),
|
||||
solver,
|
||||
mass_solver: SparseLdlt::new(),
|
||||
reuse: reuse.max(1),
|
||||
uses_left: 0,
|
||||
factor_coef: f64::NAN,
|
||||
factor_valid: false,
|
||||
fallback: BandedLu::new(),
|
||||
stats,
|
||||
})
|
||||
}
|
||||
|
||||
pub(super) fn stats(&self) -> TangentStats {
|
||||
self.stats
|
||||
}
|
||||
|
||||
/// Sum the element internal forces (element order, local order — the
|
||||
/// banded path's order) into the free-DOF vector and keep the element
|
||||
/// tangents for a later factorisation.
|
||||
pub(super) fn accumulate(
|
||||
&mut self,
|
||||
element_results: Vec<(DVector<f64>, DMatrix<f64>)>,
|
||||
element_dofs: impl Iterator<Item = impl AsRef<[usize]>>,
|
||||
free_index: &[Option<usize>],
|
||||
started: Instant,
|
||||
) -> DVector<f64> {
|
||||
let mut internal = DVector::zeros(self.num_free);
|
||||
self.element_tangents.clear();
|
||||
for ((f_e, k_e), dofs) in element_results.into_iter().zip(element_dofs) {
|
||||
for (local, &dof) in dofs.as_ref().iter().enumerate() {
|
||||
if let Some(free) = free_index[dof] {
|
||||
internal[free] += f_e[local];
|
||||
}
|
||||
}
|
||||
self.element_tangents.push(k_e);
|
||||
}
|
||||
self.stats.assemblies += 1;
|
||||
self.stats.assembly_seconds += started.elapsed().as_secs_f64();
|
||||
internal
|
||||
}
|
||||
|
||||
/// The Newton correction `(K_T + coef·(M + M_added)) δ = r`, with the
|
||||
/// tangent of the latest [`Self::accumulate`]. `force_fresh` demands a
|
||||
/// new factor; `rate` is `‖r_i‖/‖r_{i−1}‖` within the step (0 at the
|
||||
/// first iteration).
|
||||
pub(super) fn solve<'m>(
|
||||
&mut self,
|
||||
masses: impl Iterator<Item = &'m DMatrix<f64>>,
|
||||
added_mass: &DVector<f64>,
|
||||
free_index: &[Option<usize>],
|
||||
residual: &DVector<f64>,
|
||||
coef: f64,
|
||||
force_fresh: bool,
|
||||
rate: f64,
|
||||
) -> FeaResult<DVector<f64>> {
|
||||
#[allow(clippy::float_cmp)]
|
||||
let same_coef = coef == self.factor_coef;
|
||||
let refresh =
|
||||
force_fresh || !self.factor_valid || self.uses_left == 0 || !same_coef || rate > 0.5;
|
||||
if refresh {
|
||||
let started = Instant::now();
|
||||
self.values.fill(0.0);
|
||||
for ((k_e, mass), pos) in self.element_tangents.iter().zip(masses).zip(&self.elem_pos) {
|
||||
let n_e = k_e.nrows();
|
||||
for a in 0..n_e {
|
||||
for b in 0..n_e {
|
||||
let p = pos[a * n_e + b];
|
||||
if p != u32::MAX {
|
||||
self.values[p as usize] += k_e[(a, b)] + coef * mass[(a, b)];
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
for (dof, &m) in added_mass.iter().enumerate() {
|
||||
if m != 0.0 {
|
||||
if let Some(free) = free_index[dof] {
|
||||
self.values[self.diag_pos[free]] += coef * m;
|
||||
}
|
||||
}
|
||||
}
|
||||
let outcome = self.solver.factorize_values(&self.values);
|
||||
self.stats.factor_seconds += started.elapsed().as_secs_f64();
|
||||
if let Err(error) = outcome {
|
||||
// A failed pivot: this solve goes to the pivoting banded
|
||||
// LU on the same matrix; the next one tries LDLᵀ again.
|
||||
if self.stats.fallbacks == 0 {
|
||||
tracing::warn!("sparse tangent: {error}; falling back to the banded LU");
|
||||
}
|
||||
self.stats.fallbacks += 1;
|
||||
self.factor_valid = false;
|
||||
let mut triplets = Vec::with_capacity(self.values.len());
|
||||
for row in 0..self.num_free {
|
||||
for idx in self.row_ptr[row]..self.row_ptr[row + 1] {
|
||||
triplets.push((row, self.col_idx[idx], self.values[idx]));
|
||||
}
|
||||
}
|
||||
let matrix = SparseMatrix::from_triplets(self.num_free, self.num_free, &triplets)?;
|
||||
let (x, _) = self
|
||||
.fallback
|
||||
.solve(&matrix, residual, &SolverOptions::default())?;
|
||||
self.stats.solves += 1;
|
||||
return Ok(x);
|
||||
}
|
||||
self.stats.factorizations += 1;
|
||||
self.factor_valid = true;
|
||||
self.factor_coef = coef;
|
||||
self.uses_left = self.reuse;
|
||||
}
|
||||
let started = Instant::now();
|
||||
let x = self.solver.solve_factored(residual)?;
|
||||
self.stats.solve_seconds += started.elapsed().as_secs_f64();
|
||||
self.stats.solves += 1;
|
||||
self.uses_left -= 1;
|
||||
Ok(x)
|
||||
}
|
||||
}
|
||||
@@ -15,6 +15,7 @@ pub mod iterative;
|
||||
#[cfg(all(target_os = "macos", feature = "metal"))]
|
||||
pub mod metal_solvers;
|
||||
pub mod nonlinear;
|
||||
pub mod sparse_ldlt;
|
||||
pub mod time_integration;
|
||||
|
||||
#[cfg(disabled)]
|
||||
@@ -34,6 +35,7 @@ pub use iterative::*;
|
||||
#[cfg(all(target_os = "macos", feature = "metal"))]
|
||||
pub use metal_solvers::*;
|
||||
pub use nonlinear::*;
|
||||
pub use sparse_ldlt::{SparseLdlt, SparseLdltStats};
|
||||
pub use time_integration::*;
|
||||
|
||||
/// Solver configuration and options.
|
||||
|
||||
@@ -0,0 +1,369 @@
|
||||
// Copyright (c) 2024 RustyTorch++ Team
|
||||
// Licensed under the Apache License, Version 2.0
|
||||
|
||||
//! Sparse supernodal LDLᵀ direct solver with symbolic reuse.
|
||||
//!
|
||||
//! Built for the Newton tangent of the 3-D flag (R8-g in omni-cortex):
|
||||
//! `K_T + M/(β Δt²)` is symmetric, its sparsity pattern is fixed by the
|
||||
//! mesh, and the banded LU it replaces does `O(n·b²)` work on a Hex20
|
||||
//! bandwidth of one x-slab (~500 DOFs at 35×2×8). Here:
|
||||
//!
|
||||
//! * [`SparseLdlt::analyze`] — nested-dissection ordering, elimination
|
||||
//! tree, supernodes and every scatter map, once per pattern;
|
||||
//! * [`SparseLdlt::factorize_csr`] — the numeric multifrontal
|
||||
//! factorisation on the analysed pattern (re-analysing only when the
|
||||
//! pattern changes), parallel over independent subtrees and
|
||||
//! bit-deterministic at any thread count;
|
||||
//! * [`SparseLdlt::solve_factored`] — the two triangular solves.
|
||||
//!
|
||||
//! No pivoting: the matrix must be symmetric and have nonzero pivots in
|
||||
//! the chosen order (true of an SPD matrix, and of the mildly indefinite
|
||||
//! Newmark tangents met in practice); a failed pivot is an error the
|
||||
//! caller can answer with a pivoting solver. Only the lower triangle is
|
||||
//! read — the upper is assumed equal.
|
||||
|
||||
mod numeric;
|
||||
mod ordering;
|
||||
mod symbolic;
|
||||
|
||||
use super::{ConvergenceInfo, LinearSolver, SolverCapabilities, SolverOptions};
|
||||
use crate::assembly::SparseMatrix;
|
||||
use crate::error::{FeaResult, SolverError};
|
||||
use nalgebra::DVector;
|
||||
use std::time::Instant;
|
||||
|
||||
/// Pattern and cost statistics of the current analysis.
|
||||
#[derive(Debug, Clone, Copy, PartialEq)]
|
||||
pub struct SparseLdltStats {
|
||||
/// Matrix dimension.
|
||||
pub n: usize,
|
||||
/// Supernodes in the assembly tree.
|
||||
pub supernodes: usize,
|
||||
/// Nonzeros of L including the diagonal.
|
||||
pub nnz_l: usize,
|
||||
/// Largest front dimension.
|
||||
pub max_front: usize,
|
||||
/// Estimated multiply–adds of one numeric factorisation.
|
||||
pub work: f64,
|
||||
/// Symbolic analyses performed so far.
|
||||
pub analyses: usize,
|
||||
/// Numeric factorisations performed so far.
|
||||
pub factorizations: usize,
|
||||
}
|
||||
|
||||
/// Sparse LDLᵀ with symbolic reuse. See the module docs.
|
||||
#[derive(Debug, Default)]
|
||||
pub struct SparseLdlt {
|
||||
symbolic: Option<symbolic::Symbolic>,
|
||||
factor: numeric::Factor,
|
||||
factorized: bool,
|
||||
analyses: usize,
|
||||
factorizations: usize,
|
||||
work: Vec<f64>,
|
||||
}
|
||||
|
||||
impl SparseLdlt {
|
||||
/// A solver with no analysis yet.
|
||||
pub fn new() -> Self {
|
||||
Self::default()
|
||||
}
|
||||
|
||||
/// Symbolic analysis of an `n × n` CSR pattern (structurally
|
||||
/// symmetric; sorted column indices per row are not required).
|
||||
pub fn analyze(&mut self, n: usize, row_ptr: &[usize], col_idx: &[usize]) -> FeaResult<()> {
|
||||
if row_ptr.len() != n + 1 || row_ptr[n] != col_idx.len() {
|
||||
return Err(SolverError::FactorizationFailed {
|
||||
reason: format!(
|
||||
"sparse LDLt: CSR pattern inconsistent with n = {n} ({} row pointers, {} columns)",
|
||||
row_ptr.len(),
|
||||
col_idx.len()
|
||||
),
|
||||
}
|
||||
.into());
|
||||
}
|
||||
if col_idx.len() > u32::MAX as usize {
|
||||
return Err(SolverError::FactorizationFailed {
|
||||
reason: "sparse LDLt: more than 2^32 nonzeros".to_string(),
|
||||
}
|
||||
.into());
|
||||
}
|
||||
let sym = symbolic::Symbolic::analyze(n, row_ptr, col_idx);
|
||||
self.factor.resize(&sym);
|
||||
self.symbolic = Some(sym);
|
||||
self.factorized = false;
|
||||
self.analyses += 1;
|
||||
Ok(())
|
||||
}
|
||||
|
||||
/// Numeric factorisation of the CSR matrix `(row_ptr, col_idx,
|
||||
/// values)`. Reuses the symbolic analysis when the pattern is the
|
||||
/// analysed one, re-analyses otherwise.
|
||||
pub fn factorize_csr(
|
||||
&mut self,
|
||||
row_ptr: &[usize],
|
||||
col_idx: &[usize],
|
||||
values: &[f64],
|
||||
) -> FeaResult<()> {
|
||||
let n = row_ptr.len().saturating_sub(1);
|
||||
let reuse = self
|
||||
.symbolic
|
||||
.as_ref()
|
||||
.is_some_and(|s| s.n == n && s.same_pattern(row_ptr, col_idx));
|
||||
if !reuse {
|
||||
self.analyze(n, row_ptr, col_idx)?;
|
||||
}
|
||||
self.factorize_values(values)
|
||||
}
|
||||
|
||||
/// Numeric factorisation of new values on the analysed pattern
|
||||
/// (the caller guarantees the pattern is unchanged).
|
||||
pub fn factorize_values(&mut self, values: &[f64]) -> FeaResult<()> {
|
||||
let sym = self
|
||||
.symbolic
|
||||
.as_ref()
|
||||
.ok_or(SolverError::FactorizationRequired)?;
|
||||
if values.len() != sym.pattern_col_idx.len() {
|
||||
return Err(SolverError::FactorizationFailed {
|
||||
reason: "sparse LDLt: value count differs from the analysed pattern".to_string(),
|
||||
}
|
||||
.into());
|
||||
}
|
||||
self.factorized = false;
|
||||
// Pivot floor relative to the largest diagonal entry.
|
||||
let mut diag_max = 0.0f64;
|
||||
for row in 0..sym.n {
|
||||
for idx in sym.pattern_row_ptr[row]..sym.pattern_row_ptr[row + 1] {
|
||||
if sym.pattern_col_idx[idx] == row {
|
||||
diag_max = diag_max.max(values[idx].abs());
|
||||
}
|
||||
}
|
||||
}
|
||||
let floor = 1e-14 * diag_max;
|
||||
numeric::factorize(sym, values, &self.factor, floor).map_err(|fail| {
|
||||
SolverError::FactorizationFailed {
|
||||
reason: format!(
|
||||
"sparse LDLt: pivot {:.3e} at factor column {} (floor {floor:.3e})",
|
||||
fail.value, fail.column
|
||||
),
|
||||
}
|
||||
})?;
|
||||
self.factorized = true;
|
||||
self.factorizations += 1;
|
||||
Ok(())
|
||||
}
|
||||
|
||||
/// Solve with the current factors.
|
||||
pub fn solve_factored(&mut self, rhs: &DVector<f64>) -> FeaResult<DVector<f64>> {
|
||||
let sym = self
|
||||
.symbolic
|
||||
.as_ref()
|
||||
.ok_or(SolverError::FactorizationRequired)?;
|
||||
if !self.factorized {
|
||||
return Err(SolverError::FactorizationRequired.into());
|
||||
}
|
||||
if rhs.len() != sym.n {
|
||||
return Err(SolverError::DimensionMismatch {
|
||||
matrix_rows: sym.n,
|
||||
matrix_cols: sym.n,
|
||||
rhs_rows: rhs.len(),
|
||||
rhs_cols: 1,
|
||||
}
|
||||
.into());
|
||||
}
|
||||
self.work.clear();
|
||||
self.work.extend(sym.perm.iter().map(|&old| rhs[old]));
|
||||
numeric::solve_in_place(sym, &self.factor, &mut self.work);
|
||||
let mut x = DVector::zeros(sym.n);
|
||||
for (new, &old) in sym.perm.iter().enumerate() {
|
||||
x[old] = self.work[new];
|
||||
}
|
||||
Ok(x)
|
||||
}
|
||||
|
||||
/// Whether a numeric factor is available.
|
||||
pub fn is_factorized(&self) -> bool {
|
||||
self.factorized
|
||||
}
|
||||
|
||||
/// Statistics of the current analysis (`None` before the first).
|
||||
pub fn stats(&self) -> Option<SparseLdltStats> {
|
||||
self.symbolic.as_ref().map(|s| SparseLdltStats {
|
||||
n: s.n,
|
||||
supernodes: s.num_supernodes(),
|
||||
nnz_l: s.nnz_l,
|
||||
max_front: s.max_front(),
|
||||
work: s.work,
|
||||
analyses: self.analyses,
|
||||
factorizations: self.factorizations,
|
||||
})
|
||||
}
|
||||
}
|
||||
|
||||
impl LinearSolver for SparseLdlt {
|
||||
fn solve(
|
||||
&mut self,
|
||||
matrix: &SparseMatrix,
|
||||
rhs: &DVector<f64>,
|
||||
_options: &SolverOptions,
|
||||
) -> FeaResult<(DVector<f64>, ConvergenceInfo)> {
|
||||
let start_time = Instant::now();
|
||||
if matrix.nrows() != matrix.ncols() || matrix.nrows() != rhs.len() {
|
||||
return Err(SolverError::DimensionMismatch {
|
||||
matrix_rows: matrix.nrows(),
|
||||
matrix_cols: matrix.ncols(),
|
||||
rhs_rows: rhs.len(),
|
||||
rhs_cols: 1,
|
||||
}
|
||||
.into());
|
||||
}
|
||||
let (row_ptr, col_idx) = matrix.structure();
|
||||
if row_ptr.len() != matrix.nrows() + 1 {
|
||||
return Err(SolverError::FactorizationFailed {
|
||||
reason: "sparse LDLt needs a finalized (CSR) matrix".to_string(),
|
||||
}
|
||||
.into());
|
||||
}
|
||||
self.factorize_csr(row_ptr, col_idx, matrix.values())?;
|
||||
let x = self.solve_factored(rhs)?;
|
||||
let mut info = ConvergenceInfo::new();
|
||||
info.set_solve_time(start_time.elapsed());
|
||||
info.set_converged(1, 0.0, 0.0);
|
||||
Ok((x, info))
|
||||
}
|
||||
|
||||
fn name(&self) -> &'static str {
|
||||
"Sparse supernodal LDLt"
|
||||
}
|
||||
|
||||
fn capabilities(&self) -> SolverCapabilities {
|
||||
SolverCapabilities {
|
||||
symmetric: true,
|
||||
positive_definite: false,
|
||||
gpu_acceleration: false,
|
||||
multiple_rhs: true,
|
||||
iterative_refinement: false,
|
||||
memory_efficiency: 5,
|
||||
computational_efficiency: 5,
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
#[cfg(test)]
|
||||
mod tests {
|
||||
use super::*;
|
||||
use crate::solvers::BandedLu;
|
||||
|
||||
/// A 3-D 7-point Laplacian with `dofs` components per node coupled
|
||||
/// by a small symmetric block, shifted by `shift` (negative shifts
|
||||
/// make it indefinite).
|
||||
fn laplacian_3d(nx: usize, ny: usize, nz: usize, dofs: usize, shift: f64) -> SparseMatrix {
|
||||
let nn = nx * ny * nz;
|
||||
let n = nn * dofs;
|
||||
let id = |i: usize, j: usize, k: usize| (k * ny + j) * nx + i;
|
||||
let mut m = SparseMatrix::new(n, n);
|
||||
for k in 0..nz {
|
||||
for j in 0..ny {
|
||||
for i in 0..nx {
|
||||
let p = id(i, j, k);
|
||||
let mut nbrs = Vec::new();
|
||||
if i > 0 {
|
||||
nbrs.push(id(i - 1, j, k));
|
||||
}
|
||||
if i + 1 < nx {
|
||||
nbrs.push(id(i + 1, j, k));
|
||||
}
|
||||
if j > 0 {
|
||||
nbrs.push(id(i, j - 1, k));
|
||||
}
|
||||
if j + 1 < ny {
|
||||
nbrs.push(id(i, j + 1, k));
|
||||
}
|
||||
if k > 0 {
|
||||
nbrs.push(id(i, j, k - 1));
|
||||
}
|
||||
if k + 1 < nz {
|
||||
nbrs.push(id(i, j, k + 1));
|
||||
}
|
||||
for a in 0..dofs {
|
||||
let row = p * dofs + a;
|
||||
m.add_entry(row, row, 6.0 + shift + 0.1 * a as f64).unwrap();
|
||||
for b in 0..dofs {
|
||||
if a != b {
|
||||
m.add_entry(row, p * dofs + b, 0.3).unwrap();
|
||||
}
|
||||
}
|
||||
for &q in &nbrs {
|
||||
m.add_entry(row, q * dofs + a, -1.0).unwrap();
|
||||
for b in 0..dofs {
|
||||
if a != b {
|
||||
m.add_entry(row, q * dofs + b, 0.05).unwrap();
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
m.finalize().unwrap();
|
||||
m
|
||||
}
|
||||
|
||||
fn check(m: &SparseMatrix) {
|
||||
let n = m.nrows();
|
||||
let x_exact = DVector::from_fn(n, |i, _| ((i as f64) * 0.37).sin() + 0.2);
|
||||
let b = m.multiply_vector(&x_exact).unwrap();
|
||||
let opts = SolverOptions::default();
|
||||
let (x, _) = SparseLdlt::new().solve(m, &b, &opts).unwrap();
|
||||
let rel = (&x - &x_exact).norm() / x_exact.norm();
|
||||
assert!(rel < 1e-11, "manufactured rel err {rel:.3e}");
|
||||
let (xb, _) = BandedLu::new().solve(m, &b, &opts).unwrap();
|
||||
let rel_b = (&x - &xb).norm() / xb.norm();
|
||||
assert!(rel_b < 1e-11, "vs banded LU rel err {rel_b:.3e}");
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn spd_3d_block_laplacian() {
|
||||
check(&laplacian_3d(12, 3, 5, 3, 0.0));
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn indefinite_shifted_laplacian() {
|
||||
check(&laplacian_3d(9, 4, 4, 2, -5.3));
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn tiny_and_diagonal() {
|
||||
check(&laplacian_3d(1, 1, 1, 1, 0.0));
|
||||
check(&laplacian_3d(3, 1, 1, 1, 0.0));
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn symbolic_reuse_across_values() {
|
||||
let m1 = laplacian_3d(10, 3, 4, 3, 0.0);
|
||||
let m2 = laplacian_3d(10, 3, 4, 3, 1.5);
|
||||
let mut solver = SparseLdlt::new();
|
||||
let opts = SolverOptions::default();
|
||||
for m in [&m1, &m2, &m1] {
|
||||
let x_exact = DVector::from_fn(m.nrows(), |i, _| (i as f64 * 0.11).cos());
|
||||
let b = m.multiply_vector(&x_exact).unwrap();
|
||||
let (x, _) = solver.solve(m, &b, &opts).unwrap();
|
||||
assert!((&x - &x_exact).norm() / x_exact.norm() < 1e-11);
|
||||
}
|
||||
let stats = solver.stats().unwrap();
|
||||
assert_eq!(stats.analyses, 1);
|
||||
assert_eq!(stats.factorizations, 3);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn zero_pivot_is_an_error() {
|
||||
let mut m = SparseMatrix::new(2, 2);
|
||||
m.add_entry(0, 1, 1.0).unwrap();
|
||||
m.add_entry(1, 0, 1.0).unwrap();
|
||||
m.add_entry(1, 1, 1.0).unwrap();
|
||||
m.finalize().unwrap();
|
||||
let b = DVector::from_vec(vec![1.0, 2.0]);
|
||||
// The (0,0) pivot is structurally absent: zero.
|
||||
let result = SparseLdlt::new().solve(&m, &b, &SolverOptions::default());
|
||||
assert!(result.is_err());
|
||||
}
|
||||
}
|
||||
@@ -0,0 +1,261 @@
|
||||
//! Numeric phase: multifrontal supernodal LDLᵀ (no pivoting) over the
|
||||
//! symbolic assembly tree, and the triangular solves.
|
||||
//!
|
||||
//! Each supernode's front is a dense column-major `m × m` block (lower
|
||||
//! triangle used): the original entries are scattered in, the children's
|
||||
//! update matrices extend-added (in child order), then the `k` pivot
|
||||
//! columns are factorised by a blocked right-looking LDLᵀ whose trailing
|
||||
//! updates are `matrixmultiply` gemms (through nalgebra views). Sibling
|
||||
//! subtrees run on rayon when their estimated work is large; the
|
||||
//! arithmetic of every supernode is independent of the thread schedule,
|
||||
//! so results are bit-identical at any thread count.
|
||||
|
||||
use super::symbolic::Symbolic;
|
||||
use nalgebra::{DMatrixView, DMatrixViewMut};
|
||||
use rayon::prelude::*;
|
||||
use std::sync::Mutex;
|
||||
|
||||
/// Panel width of the blocked partial factorisation.
|
||||
const PANEL: usize = 32;
|
||||
/// Column block of the lower-triangular trailing update.
|
||||
const TRAIL_BLOCK: usize = 64;
|
||||
/// Subtrees below this many multiply–adds are factorised serially.
|
||||
const PARALLEL_WORK: f64 = 2.0e6;
|
||||
|
||||
/// A zero, non-finite or relatively tiny pivot.
|
||||
#[derive(Debug, Clone)]
|
||||
pub(super) struct PivotFailure {
|
||||
pub column: usize,
|
||||
pub value: f64,
|
||||
}
|
||||
|
||||
/// Per-supernode factor blocks: `m × k` column-major, unit-lower L with
|
||||
/// D on the diagonal. Allocations persist across factorisations.
|
||||
#[derive(Debug, Default)]
|
||||
pub(super) struct Factor {
|
||||
pub blocks: Vec<Mutex<Vec<f64>>>,
|
||||
}
|
||||
|
||||
impl Factor {
|
||||
pub fn resize(&mut self, sym: &Symbolic) {
|
||||
let ns = sym.num_supernodes();
|
||||
if self.blocks.len() != ns {
|
||||
self.blocks = (0..ns).map(|_| Mutex::new(Vec::new())).collect();
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
/// Factorise the matrix whose CSR values are `values` (on the analysed
|
||||
/// pattern). `pivot_floor` rejects |d| below it.
|
||||
pub(super) fn factorize(
|
||||
sym: &Symbolic,
|
||||
values: &[f64],
|
||||
factor: &Factor,
|
||||
pivot_floor: f64,
|
||||
) -> Result<(), PivotFailure> {
|
||||
let ctx = Ctx {
|
||||
sym,
|
||||
values,
|
||||
factor,
|
||||
pivot_floor,
|
||||
};
|
||||
let results: Vec<Result<Vec<f64>, PivotFailure>> = if sym.roots.len() > 1 {
|
||||
sym.roots.par_iter().map(|&r| ctx.subtree(r)).collect()
|
||||
} else {
|
||||
sym.roots.iter().map(|&r| ctx.subtree(r)).collect()
|
||||
};
|
||||
for r in results {
|
||||
r?;
|
||||
}
|
||||
Ok(())
|
||||
}
|
||||
|
||||
struct Ctx<'a> {
|
||||
sym: &'a Symbolic,
|
||||
values: &'a [f64],
|
||||
factor: &'a Factor,
|
||||
pivot_floor: f64,
|
||||
}
|
||||
|
||||
impl Ctx<'_> {
|
||||
/// Factorise the subtree rooted at `s`; returns its update matrix
|
||||
/// (`(m − k)²`, column-major).
|
||||
fn subtree(&self, s: usize) -> Result<Vec<f64>, PivotFailure> {
|
||||
let sym = self.sym;
|
||||
let kids = sym.node_children(s);
|
||||
let updates: Vec<Result<Vec<f64>, PivotFailure>> =
|
||||
if kids.len() > 1 && sym.subtree_work[s] > PARALLEL_WORK {
|
||||
kids.par_iter().map(|&c| self.subtree(c)).collect()
|
||||
} else {
|
||||
kids.iter().map(|&c| self.subtree(c)).collect()
|
||||
};
|
||||
|
||||
let rows = sym.front_rows(s);
|
||||
let m = rows.len();
|
||||
let f = sym.start[s];
|
||||
let k = sym.start[s + 1] - f;
|
||||
let mut front = vec![0.0f64; m * m];
|
||||
for &(idx, off) in &sym.amap[sym.amap_ptr[s]..sym.amap_ptr[s + 1]] {
|
||||
front[off as usize] += self.values[idx as usize];
|
||||
}
|
||||
for (&c, update) in kids.iter().zip(updates) {
|
||||
let update = update?;
|
||||
let rel = &sym.relind[sym.relind_ptr[c]..sym.relind_ptr[c + 1]];
|
||||
let u = rel.len();
|
||||
for b in 0..u {
|
||||
let col = rel[b] as usize * m;
|
||||
let src = &update[b * u..(b + 1) * u];
|
||||
for a in b..u {
|
||||
front[rel[a] as usize + col] += src[a];
|
||||
}
|
||||
}
|
||||
}
|
||||
partial_ldlt(&mut front, m, k, self.pivot_floor).map_err(|(j, value)| PivotFailure {
|
||||
column: f + j,
|
||||
value,
|
||||
})?;
|
||||
{
|
||||
let mut block = self.factor.blocks[s].lock().unwrap();
|
||||
block.clear();
|
||||
block.extend_from_slice(&front[..m * k]);
|
||||
}
|
||||
let u = m - k;
|
||||
let mut update = vec![0.0f64; u * u];
|
||||
for b in 0..u {
|
||||
let src = (k + b) * m + k;
|
||||
update[b * u + b..(b + 1) * u].copy_from_slice(&front[src + b..src + u]);
|
||||
}
|
||||
Ok(update)
|
||||
}
|
||||
}
|
||||
|
||||
/// Blocked right-looking LDLᵀ of the first `k` columns of the `m × m`
|
||||
/// column-major front, updating the trailing `(m−k)²` lower block.
|
||||
/// On exit columns `0..k` hold L (unit, strictly below the diagonal) and
|
||||
/// D (on the diagonal).
|
||||
fn partial_ldlt(front: &mut [f64], m: usize, k: usize, floor: f64) -> Result<(), (usize, f64)> {
|
||||
let mut w = Vec::new();
|
||||
let mut p0 = 0;
|
||||
while p0 < k {
|
||||
let p1 = (p0 + PANEL).min(k);
|
||||
// Unblocked factorisation of the panel's columns over rows p0..m.
|
||||
for j in p0..p1 {
|
||||
let d = front[j + j * m];
|
||||
if !d.is_finite() || d.abs() <= floor {
|
||||
return Err((j, d));
|
||||
}
|
||||
let inv_d = 1.0 / d;
|
||||
for c in (j + 1)..p1 {
|
||||
let lcj_d = front[c + j * m] * inv_d;
|
||||
if lcj_d != 0.0 {
|
||||
let (left, right) = front.split_at_mut(c * m);
|
||||
let src = &left[j * m + c..j * m + m];
|
||||
let dst = &mut right[c..m];
|
||||
for (x, &y) in dst.iter_mut().zip(src) {
|
||||
*x -= lcj_d * y;
|
||||
}
|
||||
}
|
||||
}
|
||||
for x in &mut front[j * m + j + 1..j * m + m] {
|
||||
*x *= inv_d;
|
||||
}
|
||||
}
|
||||
// Trailing update of columns p1..m (lower): F -= L_P D_P L_Pᵀ.
|
||||
if p1 < m {
|
||||
let nb = p1 - p0;
|
||||
let r = m - p1;
|
||||
// W = L_P · D_P, rows p1..m, column-major r × nb.
|
||||
w.clear();
|
||||
w.resize(r * nb, 0.0);
|
||||
for (t, j) in (p0..p1).enumerate() {
|
||||
let d = front[j + j * m];
|
||||
for (x, &l) in w[t * r..(t + 1) * r]
|
||||
.iter_mut()
|
||||
.zip(&front[j * m + p1..j * m + m])
|
||||
{
|
||||
*x = l * d;
|
||||
}
|
||||
}
|
||||
let (left, right) = front.split_at_mut(p1 * m);
|
||||
let mut c0 = 0;
|
||||
while c0 < r {
|
||||
let c1 = (c0 + TRAIL_BLOCK).min(r);
|
||||
let rows = r - c0;
|
||||
// L_P rows (p1 + c0)..m: a view into `left` (columns p0..p1).
|
||||
let lp = DMatrixView::from_slice_with_strides_generic(
|
||||
&left[p0 * m + p1 + c0..],
|
||||
nalgebra::Dyn(rows),
|
||||
nalgebra::Dyn(nb),
|
||||
nalgebra::Dyn(1),
|
||||
nalgebra::Dyn(m),
|
||||
);
|
||||
// W rows c0..c1 (the columns being updated), transposed.
|
||||
let wb = DMatrixView::from_slice_with_strides_generic(
|
||||
&w[c0..],
|
||||
nalgebra::Dyn(c1 - c0),
|
||||
nalgebra::Dyn(nb),
|
||||
nalgebra::Dyn(1),
|
||||
nalgebra::Dyn(r),
|
||||
);
|
||||
let mut target = DMatrixViewMut::from_slice_with_strides_generic(
|
||||
&mut right[c0 * m + p1 + c0..],
|
||||
nalgebra::Dyn(rows),
|
||||
nalgebra::Dyn(c1 - c0),
|
||||
nalgebra::Dyn(1),
|
||||
nalgebra::Dyn(m),
|
||||
);
|
||||
target.gemm(-1.0, &lp, &wb.transpose(), 1.0);
|
||||
c0 = c1;
|
||||
}
|
||||
}
|
||||
p0 = p1;
|
||||
}
|
||||
Ok(())
|
||||
}
|
||||
|
||||
/// Solve `L D Lᵀ x = b` in the factor numbering, in place.
|
||||
pub(super) fn solve_in_place(sym: &Symbolic, factor: &Factor, x: &mut [f64]) {
|
||||
let ns = sym.num_supernodes();
|
||||
let blocks: Vec<std::sync::MutexGuard<'_, Vec<f64>>> =
|
||||
factor.blocks.iter().map(|b| b.lock().unwrap()).collect();
|
||||
for s in 0..ns {
|
||||
let rows = sym.front_rows(s);
|
||||
let m = rows.len();
|
||||
let f = sym.start[s];
|
||||
let k = sym.start[s + 1] - f;
|
||||
let block = &blocks[s];
|
||||
for j in 0..k {
|
||||
let xj = x[f + j];
|
||||
if xj != 0.0 {
|
||||
let col = &block[j * m..(j + 1) * m];
|
||||
for i in (j + 1)..m {
|
||||
x[rows[i]] -= col[i] * xj;
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
for s in 0..ns {
|
||||
let m = sym.rows_ptr[s + 1] - sym.rows_ptr[s];
|
||||
let f = sym.start[s];
|
||||
let k = sym.start[s + 1] - f;
|
||||
let block = &blocks[s];
|
||||
for j in 0..k {
|
||||
x[f + j] /= block[j + j * m];
|
||||
}
|
||||
}
|
||||
for s in (0..ns).rev() {
|
||||
let rows = sym.front_rows(s);
|
||||
let m = rows.len();
|
||||
let f = sym.start[s];
|
||||
let k = sym.start[s + 1] - f;
|
||||
let block = &blocks[s];
|
||||
for j in (0..k).rev() {
|
||||
let col = &block[j * m..(j + 1) * m];
|
||||
let mut acc = x[f + j];
|
||||
for i in (j + 1)..m {
|
||||
acc -= col[i] * x[rows[i]];
|
||||
}
|
||||
x[f + j] = acc;
|
||||
}
|
||||
}
|
||||
}
|
||||
@@ -0,0 +1,336 @@
|
||||
//! Fill-reducing ordering: nested dissection on the matrix graph.
|
||||
//!
|
||||
//! The graph is first compressed to supervariables (DOFs with identical
|
||||
//! closed neighbourhoods — the 2 or 3 displacement components of one FE
|
||||
//! node), then dissected recursively: a pseudo-peripheral vertex's BFS
|
||||
//! level structure supplies the candidate separators (one level each,
|
||||
//! trimmed to the vertices that actually touch the far side), the
|
||||
//! lightest one that keeps both halves at least 30 % of the weight is
|
||||
//! taken, and the ordering is `A, B, S`. On a long, thin FE body (the
|
||||
//! flag) the level sets are cross-sections, so the top separators are the
|
||||
//! minimal ones; small subgraphs are left in natural order.
|
||||
//!
|
||||
//! Deterministic: no hashing order, no randomness, no threads.
|
||||
|
||||
use std::collections::HashMap;
|
||||
|
||||
/// Subgraphs at or below this weight (DOFs) are not dissected further.
|
||||
const LEAF_WEIGHT: usize = 96;
|
||||
|
||||
/// Compressed adjacency: vertex `v`'s neighbours are
|
||||
/// `adj[ptr[v]..ptr[v + 1]]` (no self loops), its weight the number of
|
||||
/// DOFs it stands for.
|
||||
struct Graph {
|
||||
ptr: Vec<usize>,
|
||||
adj: Vec<u32>,
|
||||
weight: Vec<usize>,
|
||||
}
|
||||
|
||||
impl Graph {
|
||||
fn neighbours(&self, v: u32) -> &[u32] {
|
||||
&self.adj[self.ptr[v as usize]..self.ptr[v as usize + 1]]
|
||||
}
|
||||
}
|
||||
|
||||
/// A nested-dissection permutation of the `n × n` symmetric pattern
|
||||
/// (`perm[new] = old`). Only the pattern's structure is read; a
|
||||
/// nonsymmetric pattern is symmetrised.
|
||||
pub(super) fn nested_dissection(n: usize, row_ptr: &[usize], col_idx: &[usize]) -> Vec<usize> {
|
||||
if n == 0 {
|
||||
return Vec::new();
|
||||
}
|
||||
// Symmetrised adjacency lists over DOFs, self loops excluded.
|
||||
let mut lists: Vec<Vec<u32>> = vec![Vec::new(); n];
|
||||
for row in 0..n {
|
||||
for &col in &col_idx[row_ptr[row]..row_ptr[row + 1]] {
|
||||
if col != row {
|
||||
lists[row].push(col as u32);
|
||||
lists[col].push(row as u32);
|
||||
}
|
||||
}
|
||||
}
|
||||
for list in &mut lists {
|
||||
list.sort_unstable();
|
||||
list.dedup();
|
||||
}
|
||||
|
||||
// Supervariables: identical closed neighbourhoods.
|
||||
let mut group_of = vec![0u32; n];
|
||||
let mut members: Vec<Vec<u32>> = Vec::new();
|
||||
{
|
||||
let mut index: HashMap<Vec<u32>, u32> = HashMap::new();
|
||||
for v in 0..n {
|
||||
let mut closed = lists[v].clone();
|
||||
let at = closed.partition_point(|&x| x < v as u32);
|
||||
closed.insert(at, v as u32);
|
||||
let next = members.len() as u32;
|
||||
let g = *index.entry(closed).or_insert(next);
|
||||
if g == next {
|
||||
members.push(Vec::new());
|
||||
}
|
||||
members[g as usize].push(v as u32);
|
||||
group_of[v] = g;
|
||||
}
|
||||
}
|
||||
let ng = members.len();
|
||||
let mut ptr = Vec::with_capacity(ng + 1);
|
||||
let mut adj = Vec::new();
|
||||
ptr.push(0);
|
||||
for group in &members {
|
||||
let rep = group[0] as usize;
|
||||
let mut nbrs: Vec<u32> = lists[rep]
|
||||
.iter()
|
||||
.map(|&u| group_of[u as usize])
|
||||
.filter(|&g| g != group_of[rep])
|
||||
.collect();
|
||||
nbrs.sort_unstable();
|
||||
nbrs.dedup();
|
||||
adj.extend_from_slice(&nbrs);
|
||||
ptr.push(adj.len());
|
||||
}
|
||||
drop(lists);
|
||||
let graph = Graph {
|
||||
ptr,
|
||||
adj,
|
||||
weight: members.iter().map(Vec::len).collect(),
|
||||
};
|
||||
|
||||
let mut state = Dissector {
|
||||
graph: &graph,
|
||||
stamp: vec![0u32; ng],
|
||||
level: vec![u32::MAX; ng],
|
||||
next_stamp: 0,
|
||||
out: Vec::with_capacity(ng),
|
||||
};
|
||||
let all: Vec<u32> = (0..ng as u32).collect();
|
||||
state.dissect(all);
|
||||
|
||||
let mut perm = Vec::with_capacity(n);
|
||||
for &g in &state.out {
|
||||
for &v in &members[g as usize] {
|
||||
perm.push(v as usize);
|
||||
}
|
||||
}
|
||||
debug_assert_eq!(perm.len(), n);
|
||||
perm
|
||||
}
|
||||
|
||||
struct Dissector<'g> {
|
||||
graph: &'g Graph,
|
||||
/// `stamp[v] == id` marks membership of the subgraph being processed.
|
||||
stamp: Vec<u32>,
|
||||
level: Vec<u32>,
|
||||
next_stamp: u32,
|
||||
out: Vec<u32>,
|
||||
}
|
||||
|
||||
impl Dissector<'_> {
|
||||
fn mark(&mut self, verts: &[u32]) -> u32 {
|
||||
self.next_stamp += 1;
|
||||
let id = self.next_stamp;
|
||||
for &v in verts {
|
||||
self.stamp[v as usize] = id;
|
||||
}
|
||||
id
|
||||
}
|
||||
|
||||
fn weight(&self, verts: &[u32]) -> usize {
|
||||
verts.iter().map(|&v| self.graph.weight[v as usize]).sum()
|
||||
}
|
||||
|
||||
/// BFS inside the stamped subgraph from `start`; fills `level` for
|
||||
/// the reached vertices and returns them grouped by level.
|
||||
fn bfs(&mut self, id: u32, start: u32) -> Vec<Vec<u32>> {
|
||||
let mut levels: Vec<Vec<u32>> = vec![vec![start]];
|
||||
self.level[start as usize] = 0;
|
||||
let mut visited = vec![start];
|
||||
loop {
|
||||
let depth = levels.len() as u32;
|
||||
let mut next = Vec::new();
|
||||
for &v in levels.last().unwrap() {
|
||||
for &u in self.graph.neighbours(v) {
|
||||
if self.stamp[u as usize] == id && self.level[u as usize] == u32::MAX {
|
||||
self.level[u as usize] = depth;
|
||||
next.push(u);
|
||||
visited.push(u);
|
||||
}
|
||||
}
|
||||
}
|
||||
if next.is_empty() {
|
||||
break;
|
||||
}
|
||||
next.sort_unstable();
|
||||
levels.push(next);
|
||||
}
|
||||
for &v in &visited {
|
||||
self.level[v as usize] = u32::MAX;
|
||||
}
|
||||
levels
|
||||
}
|
||||
|
||||
fn dissect(&mut self, verts: Vec<u32>) {
|
||||
if self.weight(&verts) <= LEAF_WEIGHT || verts.len() < 4 {
|
||||
self.out.extend_from_slice(&verts);
|
||||
return;
|
||||
}
|
||||
let id = self.mark(&verts);
|
||||
// Connected components first.
|
||||
let first = self.bfs(id, verts[0]);
|
||||
let reached: usize = first.iter().map(Vec::len).sum();
|
||||
if reached < verts.len() {
|
||||
let mut component: Vec<u32> = first.into_iter().flatten().collect();
|
||||
component.sort_unstable();
|
||||
let comp_id = self.mark(&component);
|
||||
let rest: Vec<u32> = verts
|
||||
.iter()
|
||||
.copied()
|
||||
.filter(|&v| self.stamp[v as usize] != comp_id)
|
||||
.collect();
|
||||
self.dissect(component);
|
||||
self.dissect(rest);
|
||||
return;
|
||||
}
|
||||
|
||||
// Pseudo-peripheral start: min-degree vertex of the last level,
|
||||
// repeated while the eccentricity grows.
|
||||
let degree = |g: &Graph, v: u32| g.ptr[v as usize + 1] - g.ptr[v as usize];
|
||||
let mut start = *verts
|
||||
.iter()
|
||||
.min_by_key(|&&v| (degree(self.graph, v), v))
|
||||
.unwrap();
|
||||
let mut levels = self.bfs(id, start);
|
||||
for _ in 0..8 {
|
||||
let far = *levels
|
||||
.last()
|
||||
.unwrap()
|
||||
.iter()
|
||||
.min_by_key(|&&v| (degree(self.graph, v), v))
|
||||
.unwrap();
|
||||
let trial = self.bfs(id, far);
|
||||
if trial.len() > levels.len() {
|
||||
start = far;
|
||||
levels = trial;
|
||||
} else {
|
||||
break;
|
||||
}
|
||||
}
|
||||
let _ = start;
|
||||
if levels.len() < 3 {
|
||||
self.out.extend_from_slice(&verts);
|
||||
return;
|
||||
}
|
||||
|
||||
for (depth, level) in levels.iter().enumerate() {
|
||||
for &v in level {
|
||||
self.level[v as usize] = depth as u32;
|
||||
}
|
||||
}
|
||||
let total = self.weight(&verts);
|
||||
let level_weight: Vec<usize> = levels.iter().map(|l| self.weight(l)).collect();
|
||||
// Candidate separators around the weighted median level.
|
||||
let mut best: Option<(usize, usize)> = None; // (separator weight, level)
|
||||
let mut before = 0usize;
|
||||
for l in 0..levels.len() {
|
||||
let after = total - before - level_weight[l];
|
||||
if l >= 1 && l + 1 < levels.len() {
|
||||
let sep: usize = levels[l]
|
||||
.iter()
|
||||
.filter(|&&v| {
|
||||
self.graph.neighbours(v).iter().any(|&u| {
|
||||
self.stamp[u as usize] == id && self.level[u as usize] == l as u32 + 1
|
||||
})
|
||||
})
|
||||
.map(|&v| self.graph.weight[v as usize])
|
||||
.sum();
|
||||
let side_a = before + level_weight[l] - sep;
|
||||
let balanced = 10 * side_a.min(after) >= 3 * total;
|
||||
if balanced && best.is_none_or(|(w, _)| sep < w) {
|
||||
best = Some((sep, l));
|
||||
}
|
||||
}
|
||||
before += level_weight[l];
|
||||
}
|
||||
let cut = match best {
|
||||
Some((_, l)) => l,
|
||||
None => {
|
||||
// No balanced level: take the weighted median.
|
||||
let mut acc = 0usize;
|
||||
let mut pick = levels.len() / 2;
|
||||
for (l, w) in level_weight.iter().enumerate() {
|
||||
acc += w;
|
||||
if 2 * acc >= total {
|
||||
pick = l;
|
||||
break;
|
||||
}
|
||||
}
|
||||
pick.clamp(1, levels.len() - 2)
|
||||
}
|
||||
};
|
||||
let mut part_a = Vec::new();
|
||||
let mut part_b = Vec::new();
|
||||
let mut separator = Vec::new();
|
||||
for &v in &verts {
|
||||
let l = self.level[v as usize] as usize;
|
||||
if l < cut {
|
||||
part_a.push(v);
|
||||
} else if l > cut {
|
||||
part_b.push(v);
|
||||
} else if self
|
||||
.graph
|
||||
.neighbours(v)
|
||||
.iter()
|
||||
.any(|&u| self.stamp[u as usize] == id && self.level[u as usize] == cut as u32 + 1)
|
||||
{
|
||||
separator.push(v);
|
||||
} else {
|
||||
part_a.push(v);
|
||||
}
|
||||
}
|
||||
for &v in &verts {
|
||||
self.level[v as usize] = u32::MAX;
|
||||
}
|
||||
self.dissect(part_a);
|
||||
self.dissect(part_b);
|
||||
self.out.extend_from_slice(&separator);
|
||||
}
|
||||
}
|
||||
|
||||
#[cfg(test)]
|
||||
mod tests {
|
||||
use super::*;
|
||||
|
||||
fn grid_pattern(nx: usize, ny: usize) -> (usize, Vec<usize>, Vec<usize>) {
|
||||
let n = nx * ny;
|
||||
let mut row_ptr = vec![0];
|
||||
let mut col_idx = Vec::new();
|
||||
for j in 0..ny {
|
||||
for i in 0..nx {
|
||||
let mut cols = Vec::new();
|
||||
for (di, dj) in [(-1i64, 0i64), (1, 0), (0, -1), (0, 1), (0, 0)] {
|
||||
let (a, b) = (i as i64 + di, j as i64 + dj);
|
||||
if a >= 0 && b >= 0 && (a as usize) < nx && (b as usize) < ny {
|
||||
cols.push(b as usize * nx + a as usize);
|
||||
}
|
||||
}
|
||||
cols.sort_unstable();
|
||||
col_idx.extend(cols);
|
||||
row_ptr.push(col_idx.len());
|
||||
}
|
||||
}
|
||||
(n, row_ptr, col_idx)
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn nested_dissection_is_a_permutation() {
|
||||
let (n, rp, ci) = grid_pattern(40, 7);
|
||||
let perm = nested_dissection(n, &rp, &ci);
|
||||
let mut seen = vec![false; n];
|
||||
for &p in &perm {
|
||||
assert!(!seen[p]);
|
||||
seen[p] = true;
|
||||
}
|
||||
assert_eq!(perm.len(), n);
|
||||
// Deterministic.
|
||||
assert_eq!(perm, nested_dissection(n, &rp, &ci));
|
||||
}
|
||||
}
|
||||
@@ -0,0 +1,375 @@
|
||||
//! Symbolic analysis: ordering, elimination tree, postorder, column
|
||||
//! structures, fundamental supernodes, the supernodal assembly tree, and
|
||||
//! the maps the numeric phase scatters through. Computed once per
|
||||
//! sparsity pattern and reused for every numeric factorisation.
|
||||
|
||||
use super::ordering::nested_dissection;
|
||||
|
||||
/// Everything the numeric factorisation needs that depends only on the
|
||||
/// pattern. Column indices below are in the *factor* numbering
|
||||
/// (`perm[new] = old`).
|
||||
#[derive(Debug)]
|
||||
pub(super) struct Symbolic {
|
||||
pub n: usize,
|
||||
pub perm: Vec<usize>,
|
||||
/// Supernode `s` owns columns `start[s]..start[s + 1]`.
|
||||
pub start: Vec<usize>,
|
||||
/// Front row indices of supernode `s` (its own columns first, then the
|
||||
/// sorted off-diagonal structure): `rows[rows_ptr[s]..rows_ptr[s+1]]`.
|
||||
pub rows_ptr: Vec<usize>,
|
||||
pub rows: Vec<usize>,
|
||||
/// Children in the assembly tree, increasing (postorder) order.
|
||||
pub children_ptr: Vec<usize>,
|
||||
pub children: Vec<usize>,
|
||||
/// Roots of the assembly forest.
|
||||
pub roots: Vec<usize>,
|
||||
/// For a non-root supernode: position of each of its update rows
|
||||
/// (`rows[k..m]`) in its parent's front — `relind[relind_ptr[s]..]`.
|
||||
pub relind_ptr: Vec<usize>,
|
||||
pub relind: Vec<u32>,
|
||||
/// Per supernode, the lower-triangle entries of A it assembles:
|
||||
/// `(CSR index, column-major offset in the front)`.
|
||||
pub amap_ptr: Vec<usize>,
|
||||
pub amap: Vec<(u32, u32)>,
|
||||
/// The pattern this analysis was built for (for reuse checks).
|
||||
pub pattern_row_ptr: Vec<usize>,
|
||||
pub pattern_col_idx: Vec<usize>,
|
||||
/// Estimated multiply–adds of one numeric factorisation, per subtree
|
||||
/// (used to decide where to fork threads).
|
||||
pub subtree_work: Vec<f64>,
|
||||
/// Nonzeros in L (including the diagonal) and total work.
|
||||
pub nnz_l: usize,
|
||||
pub work: f64,
|
||||
}
|
||||
|
||||
impl Symbolic {
|
||||
pub fn num_supernodes(&self) -> usize {
|
||||
self.start.len() - 1
|
||||
}
|
||||
|
||||
pub fn front_rows(&self, s: usize) -> &[usize] {
|
||||
&self.rows[self.rows_ptr[s]..self.rows_ptr[s + 1]]
|
||||
}
|
||||
|
||||
pub fn node_children(&self, s: usize) -> &[usize] {
|
||||
&self.children[self.children_ptr[s]..self.children_ptr[s + 1]]
|
||||
}
|
||||
|
||||
pub fn max_front(&self) -> usize {
|
||||
(0..self.num_supernodes())
|
||||
.map(|s| self.rows_ptr[s + 1] - self.rows_ptr[s])
|
||||
.max()
|
||||
.unwrap_or(0)
|
||||
}
|
||||
|
||||
/// Full analysis of an `n × n` CSR pattern (symmetric structure
|
||||
/// expected; only the lower triangle is factorised).
|
||||
pub fn analyze(n: usize, row_ptr: &[usize], col_idx: &[usize]) -> Self {
|
||||
let nd = nested_dissection(n, row_ptr, col_idx);
|
||||
// Elimination tree under the ND order, then postorder it.
|
||||
let iperm_nd = inverse(&nd);
|
||||
let rowpat = lower_rows(n, row_ptr, col_idx, &iperm_nd);
|
||||
let parent_nd = etree(n, &rowpat);
|
||||
let post = postorder(&parent_nd);
|
||||
let perm: Vec<usize> = post.iter().map(|&k| nd[k]).collect();
|
||||
drop(rowpat);
|
||||
|
||||
let iperm = inverse(&perm);
|
||||
let rowpat = lower_rows(n, row_ptr, col_idx, &iperm);
|
||||
let parent = etree(n, &rowpat);
|
||||
// Column patterns of the strictly lower triangle of PAPᵀ.
|
||||
let mut colpat: Vec<Vec<usize>> = vec![Vec::new(); n];
|
||||
for (i, cols) in rowpat.iter().enumerate() {
|
||||
for &j in cols {
|
||||
colpat[j].push(i);
|
||||
}
|
||||
}
|
||||
drop(rowpat);
|
||||
let mut nchild = vec![0usize; n];
|
||||
let mut child_lists: Vec<Vec<usize>> = vec![Vec::new(); n];
|
||||
for j in 0..n {
|
||||
if parent[j] != NONE {
|
||||
nchild[parent[j]] += 1;
|
||||
child_lists[parent[j]].push(j);
|
||||
}
|
||||
}
|
||||
|
||||
// Column structures by merging children (postorder: children of j
|
||||
// precede j). A structure is kept only while it may still be
|
||||
// needed: until its parent is processed, and for supernode heads.
|
||||
let mut structs: Vec<Option<Vec<usize>>> = vec![None; n];
|
||||
let mut counts = vec![0usize; n];
|
||||
let mut start = vec![0usize];
|
||||
let mut heads_struct: Vec<Vec<usize>> = Vec::new();
|
||||
let mut mark = vec![NONE; n];
|
||||
for j in 0..n {
|
||||
let mut s: Vec<usize> = Vec::new();
|
||||
mark[j] = j;
|
||||
for &i in &colpat[j] {
|
||||
if mark[i] != j {
|
||||
mark[i] = j;
|
||||
s.push(i);
|
||||
}
|
||||
}
|
||||
for &c in &child_lists[j] {
|
||||
if let Some(cs) = structs[c].take() {
|
||||
for i in cs {
|
||||
if i != j && mark[i] != j {
|
||||
mark[i] = j;
|
||||
s.push(i);
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
s.sort_unstable();
|
||||
counts[j] = s.len();
|
||||
// Does j extend the previous supernode?
|
||||
let joins =
|
||||
j > 0 && parent[j - 1] == j && nchild[j] == 1 && counts[j - 1] == counts[j] + 1;
|
||||
if !joins {
|
||||
if j > 0 {
|
||||
start.push(j);
|
||||
}
|
||||
heads_struct.push(s.clone());
|
||||
}
|
||||
structs[j] = Some(s);
|
||||
}
|
||||
start.push(n);
|
||||
drop(structs);
|
||||
drop(colpat);
|
||||
let ns = start.len() - 1;
|
||||
|
||||
let mut col_to_sn = vec![0usize; n];
|
||||
for s in 0..ns {
|
||||
for c in start[s]..start[s + 1] {
|
||||
col_to_sn[c] = s;
|
||||
}
|
||||
}
|
||||
let mut rows_ptr = vec![0usize];
|
||||
let mut rows = Vec::new();
|
||||
for s in 0..ns {
|
||||
rows.push(start[s]);
|
||||
rows.extend_from_slice(&heads_struct[s]);
|
||||
rows_ptr.push(rows.len());
|
||||
}
|
||||
drop(heads_struct);
|
||||
|
||||
let mut sn_parent = vec![NONE; ns];
|
||||
for s in 0..ns {
|
||||
let last = start[s + 1] - 1;
|
||||
if parent[last] != NONE {
|
||||
sn_parent[s] = col_to_sn[parent[last]];
|
||||
}
|
||||
}
|
||||
let mut children_lists: Vec<Vec<usize>> = vec![Vec::new(); ns];
|
||||
let mut roots = Vec::new();
|
||||
for s in 0..ns {
|
||||
if sn_parent[s] == NONE {
|
||||
roots.push(s);
|
||||
} else {
|
||||
children_lists[sn_parent[s]].push(s);
|
||||
}
|
||||
}
|
||||
let mut children_ptr = vec![0usize];
|
||||
let mut children = Vec::new();
|
||||
for list in &children_lists {
|
||||
children.extend_from_slice(list);
|
||||
children_ptr.push(children.len());
|
||||
}
|
||||
|
||||
// Relative indices of each update row in the parent front.
|
||||
let mut relind_ptr = vec![0usize];
|
||||
let mut relind = Vec::new();
|
||||
for s in 0..ns {
|
||||
let k = start[s + 1] - start[s];
|
||||
let own = &rows[rows_ptr[s]..rows_ptr[s + 1]];
|
||||
if sn_parent[s] != NONE {
|
||||
let p = sn_parent[s];
|
||||
let prow = &rows[rows_ptr[p]..rows_ptr[p + 1]];
|
||||
let mut q = 0usize;
|
||||
for &r in &own[k..] {
|
||||
while prow[q] != r {
|
||||
q += 1;
|
||||
}
|
||||
relind.push(q as u32);
|
||||
}
|
||||
}
|
||||
relind_ptr.push(relind.len());
|
||||
}
|
||||
|
||||
let mut sym = Self {
|
||||
n,
|
||||
perm,
|
||||
start,
|
||||
rows_ptr,
|
||||
rows,
|
||||
children_ptr,
|
||||
children,
|
||||
roots,
|
||||
relind_ptr,
|
||||
relind,
|
||||
amap_ptr: Vec::new(),
|
||||
amap: Vec::new(),
|
||||
pattern_row_ptr: Vec::new(),
|
||||
pattern_col_idx: Vec::new(),
|
||||
subtree_work: Vec::new(),
|
||||
nnz_l: 0,
|
||||
work: 0.0,
|
||||
};
|
||||
sym.build_amap(row_ptr, col_idx, &iperm, &col_to_sn);
|
||||
sym.estimate_work();
|
||||
sym
|
||||
}
|
||||
|
||||
fn build_amap(
|
||||
&mut self,
|
||||
row_ptr: &[usize],
|
||||
col_idx: &[usize],
|
||||
iperm: &[usize],
|
||||
col_to_sn: &[usize],
|
||||
) {
|
||||
let ns = self.num_supernodes();
|
||||
let mut per_sn: Vec<Vec<(u32, u32)>> = vec![Vec::new(); ns];
|
||||
for old_row in 0..self.n {
|
||||
let i = iperm[old_row];
|
||||
for idx in row_ptr[old_row]..row_ptr[old_row + 1] {
|
||||
let j = iperm[col_idx[idx]];
|
||||
if i < j {
|
||||
continue;
|
||||
}
|
||||
let s = col_to_sn[j];
|
||||
let front = self.front_rows(s);
|
||||
let m = front.len();
|
||||
let a = front
|
||||
.binary_search(&i)
|
||||
.expect("A entry outside the symbolic structure");
|
||||
let b = j - self.start[s];
|
||||
per_sn[s].push((idx as u32, (a + b * m) as u32));
|
||||
}
|
||||
}
|
||||
self.amap_ptr = vec![0];
|
||||
for list in per_sn {
|
||||
self.amap.extend(list);
|
||||
self.amap_ptr.push(self.amap.len());
|
||||
}
|
||||
self.pattern_row_ptr = row_ptr.to_vec();
|
||||
self.pattern_col_idx = col_idx.to_vec();
|
||||
}
|
||||
|
||||
fn estimate_work(&mut self) {
|
||||
let ns = self.num_supernodes();
|
||||
let mut own = vec![0.0f64; ns];
|
||||
let mut nnz = 0usize;
|
||||
let mut total = 0.0;
|
||||
for (s, slot) in own.iter_mut().enumerate() {
|
||||
let k = (self.start[s + 1] - self.start[s]) as f64;
|
||||
let m = (self.rows_ptr[s + 1] - self.rows_ptr[s]) as f64;
|
||||
let ku = k as usize;
|
||||
let mu = m as usize;
|
||||
nnz += ku * mu - ku * (ku - 1) / 2;
|
||||
// Partial LDLᵀ of k pivots in an m-front (plus the extend-add).
|
||||
let w = k * (m * m - m * k + k * k / 3.0) + (m - k) * (m - k);
|
||||
*slot = w;
|
||||
total += w;
|
||||
}
|
||||
// Subtree sums in postorder (children precede parents).
|
||||
let mut sub = own;
|
||||
for s in 0..ns {
|
||||
for &c in self.node_children(s).to_vec().iter() {
|
||||
sub[s] += sub[c];
|
||||
}
|
||||
}
|
||||
self.subtree_work = sub;
|
||||
self.nnz_l = nnz;
|
||||
self.work = total;
|
||||
}
|
||||
|
||||
/// Whether `(row_ptr, col_idx)` is exactly the analysed pattern.
|
||||
pub fn same_pattern(&self, row_ptr: &[usize], col_idx: &[usize]) -> bool {
|
||||
self.pattern_row_ptr == row_ptr && self.pattern_col_idx == col_idx
|
||||
}
|
||||
}
|
||||
|
||||
const NONE: usize = usize::MAX;
|
||||
|
||||
fn inverse(perm: &[usize]) -> Vec<usize> {
|
||||
let mut inv = vec![0usize; perm.len()];
|
||||
for (new, &old) in perm.iter().enumerate() {
|
||||
inv[old] = new;
|
||||
}
|
||||
inv
|
||||
}
|
||||
|
||||
/// Row `i`'s strictly-lower columns `{j < i}` of the permuted,
|
||||
/// symmetrised pattern.
|
||||
fn lower_rows(n: usize, row_ptr: &[usize], col_idx: &[usize], iperm: &[usize]) -> Vec<Vec<usize>> {
|
||||
let mut rows: Vec<Vec<usize>> = vec![Vec::new(); n];
|
||||
for old_row in 0..n {
|
||||
let a = iperm[old_row];
|
||||
for &old_col in &col_idx[row_ptr[old_row]..row_ptr[old_row + 1]] {
|
||||
let b = iperm[old_col];
|
||||
if a > b {
|
||||
rows[a].push(b);
|
||||
} else if b > a {
|
||||
rows[b].push(a);
|
||||
}
|
||||
}
|
||||
}
|
||||
for r in &mut rows {
|
||||
r.sort_unstable();
|
||||
r.dedup();
|
||||
}
|
||||
rows
|
||||
}
|
||||
|
||||
/// Liu's elimination tree with path compression.
|
||||
fn etree(n: usize, rowpat: &[Vec<usize>]) -> Vec<usize> {
|
||||
let mut parent = vec![NONE; n];
|
||||
let mut ancestor = vec![NONE; n];
|
||||
for i in 0..n {
|
||||
for &j in &rowpat[i] {
|
||||
let mut k = j;
|
||||
while ancestor[k] != NONE && ancestor[k] != i {
|
||||
let next = ancestor[k];
|
||||
ancestor[k] = i;
|
||||
k = next;
|
||||
}
|
||||
if ancestor[k] == NONE {
|
||||
ancestor[k] = i;
|
||||
parent[k] = i;
|
||||
}
|
||||
}
|
||||
}
|
||||
parent
|
||||
}
|
||||
|
||||
/// Postorder of a forest (`post[k]` = the k-th node visited), children
|
||||
/// in increasing order, roots in increasing order; iterative DFS.
|
||||
fn postorder(parent: &[usize]) -> Vec<usize> {
|
||||
let n = parent.len();
|
||||
let mut kids: Vec<Vec<usize>> = vec![Vec::new(); n];
|
||||
let mut roots = Vec::new();
|
||||
for j in 0..n {
|
||||
if parent[j] == NONE {
|
||||
roots.push(j);
|
||||
} else {
|
||||
kids[parent[j]].push(j);
|
||||
}
|
||||
}
|
||||
let mut post = Vec::with_capacity(n);
|
||||
let mut stack: Vec<(usize, usize)> = Vec::new();
|
||||
for &r in &roots {
|
||||
stack.push((r, 0));
|
||||
while let Some(&mut (v, ref mut next)) = stack.last_mut() {
|
||||
if *next < kids[v].len() {
|
||||
let c = kids[v][*next];
|
||||
*next += 1;
|
||||
stack.push((c, 0));
|
||||
} else {
|
||||
post.push(v);
|
||||
stack.pop();
|
||||
}
|
||||
}
|
||||
}
|
||||
post
|
||||
}
|
||||
Reference in New Issue
Block a user