Adaptive Algebraic Reuse of Reordering in Cholesky Factorization with Dynamic Sparsity Pattern

2024-12-16

arXiv.org

AI Summary

Cholesky solvers face performance bottlenecks in elasticity dynamics (e.g., IPC) and geometry processing (e.g., remeshing) due to frequent symbolic factorization triggered by dynamically changing sparsity patterns. To address this, we propose an adaptive local ordering reuse framework: (1) a novel hierarchical graph decomposition algorithm enabling fine-grained, fill-aware reordering reuse at the subgraph level; (2) Parth, a method that incrementally updates ordering vectors only where local connectivity changes; and (3) a dual-graph–based hierarchical structure for adaptive sparsity pattern tracking. Our approach integrates seamlessly into mainstream libraries with minimal overhead—just three lines of code. Experiments show up to 255× speedup in reordering for IPC and 13× for remeshing; 6.85–10.7× acceleration in symbolic analysis; and end-to-end solver speedups of up to 5.89×.

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• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
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Abstract

Cholesky linear solvers are a critical bottleneck in challenging applications within computer graphics and scientific computing. These applications include but are not limited to elastodynamic barrier methods such as Incremental Potential Contact (IPC), and geometric operations such as remeshing and morphology. In these contexts, the sparsity patterns of the linear systems frequently change across successive calls to the Cholesky solver, necessitating repeated symbolic analyses that dominate the overall solver runtime. To address this bottleneck, we evaluate our method on over 150,000 linear systems generated from diverse nonlinear problems with dynamic sparsity changes in Incremental Potential Contact (IPC) and patch remeshing on a wide range of triangular meshes of various sizes. Our analysis using three leading sparse Cholesky libraries, Intel MKL Pardiso, SuiteSparse CHOLMOD, and Apple Accelerate, reveals that the primary performance constraint lies in the symbolic re-ordering phase of the solver. Recognizing this, we introduce Parth, an innovative re-ordering method designed to update ordering vectors only where local connectivity changes occur adaptively. Parth employs a novel hierarchical graph decomposition algorithm to break down the dual graph of the input matrix into fine-grained subgraphs, facilitating the selective reuse of fill-reducing orderings when sparsity patterns exhibit temporal coherence. Our extensive evaluation demonstrates that Parth achieves up to a 255x and 13x speedup in fill-reducing ordering for our IPC and remeshing benchmark and a 6.85x and 10.7x acceleration in symbolic analysis. These enhancements translate to up to 2.95x and 5.89x reduction in overall solver runtime. Additionally, Parth's integration requires only three lines of code, resulting in significant computational savings without the requirement of changes to the computational stack.

Problem

• Reduces symbolic analysis time in Cholesky solvers with dynamic sparsity
• Improves fill-reducing ordering efficiency for frequently changing sparsity patterns
• Accelerates overall solver runtime via adaptive algebraic reuse of reordering

Innovation

• Adaptive algebraic reuse of reordering vectors
• Hierarchical graph decomposition for subgraphs
• Selective fill-reducing ordering reuse