arXiv subjects

Anna Yesypenko

Publications and source records attributed to Anna Yesypenko.

7 recordsLinked to original sources

arXiv2025

An overlapping domain decomposition method based on solution-transfer operators

An overlapping domain decomposition method is described for variable-coefficient elliptic boundary value problems on domains that can be decomposed into slabs or shells. The method represents the global solution through its traces on internal interfaces, coupled by local Dirichlet solution transfer operators posed on overlapping double slab domains. The key observation is that these interface maps act between separated interfaces, and as such can be written as smooth-kernel integral operators. The resulting global equilibrium system is Fredholm second kind and, unlike non-overlapping formulations, requires no same-interface Dirichlet-to-Neumann or other interface maps with singular kernels. This makes its off-diagonal blocks highly amenable to hierarchical low-rank compression. The formulation admits complementary continuum and discrete interpretations. At fixed slab width, the method is stable under discretization, sufficiently accurate local solves and compression. At the discrete level, the system can be interpreted as a block Jacobi preconditioned Schur complement system. For compatible SPD discretizations satisfying a standard stable-splitting assumption, the symmetrically scaled interface matrix satisfies an energy-norm condition-number bound that depends on the slab width but is uniform with respect to the local resolution. The formulation is implemented using high-order local solvers and hierarchical compression based on randomized sampling. Numerical experiments report iteration counts, accuracy, and compressibility for 2D and 3D elliptic, nonsymmetric, and oscillatory problems with as many as 28 million degrees of freedom.

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arXiv2025

A Two-Level Direct Solver for the Hierarchical Poincaré-Steklov Method

We introduce a two-level direct solver for the Hierarchical Poincaré-Steklov (HPS) method for solving linear elliptic PDEs. HPS combines multidomain spectral collocation with a direct solver, enabling high-order discretizations for highly oscillatory solutions while preserving computational efficiency. Our method employs batched linear algebra routines with GPU acceleration to reduce the problem to subdomain interfaces, yielding a block-sparse linear system. This system is then factorized using a sparse direct solver that employs pivoting to achieve better numerical stability than the original HPS scheme. For a discretization of local order $p$ involving a total of $N$ degrees of freedom, the initial reduction step has asymptotic complexity $O(N p^6)$ in three dimensions. Nevertheless, the high efficiency of batched GPU routines makes the overall cost for practical purposes independent of polynomial order (for order $p=20$ or even higher). Additionally, the cost of the sparse direct solver is independent of the polynomial order. We present a description and justification of our method, along with numerical experiments on three-dimensional problems to evaluate its accuracy and performance.

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arXiv2025

SlabLU: A Two-Level Sparse Direct Solver for Elliptic PDEs

The paper describes a sparse direct solver for the linear systems that arise from the discretization of an elliptic PDE on a two dimensional domain. The scheme decomposes the domain into thin subdomains, or ``slabs'' and uses a two-level approach that is designed with parallelization in mind. The scheme takes advantage of $\mathcal H^2$-matrix structure emerging during factorization and utilizes randomized algorithms to efficiently recover this structure. As opposed to multi-level nested dissection schemes that incorporate the use of $\mathcal H$ or $\mathcal H^2$ matrices for a hierarchy of front sizes, SlabLU is a two-level scheme which only uses $\mathcal H^2$-matrix algebra for fronts of roughly the same size. The simplicity allows the scheme to be easily tuned for performance on modern architectures and GPUs. The solver described is compatible with a range of different local discretizations, and numerical experiments demonstrate its performance for regular discretizations of rectangular and curved geometries. The technique becomes particularly efficient when combined with very high-order accurate multi-domain spectral collocation schemes. With this discretization, a Helmholtz problem on a domain of size $1000 λ\times 1000 λ$ (for which $N=100 \rm{M}$) is solved in 15 minutes to 6 correct digits on a high-powered desktop with GPU acceleration.

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arXiv2025

A Simplified Fast Multipole Method Based on Strong Recursive Skeletonization

This work introduces a kernel-independent, multilevel, adaptive algorithm for efficiently evaluating a discrete convolution kernel with a given source distribution. The method is based on linear algebraic tools such as low rank approximation and ``skeleton representations'' to approximate far-field interactions. While this work is related to previous linear algebraic formulations of the fast multipole method, the proposed algorithm is distinguished by relying on simpler data structures. The proposed algorithm eliminates the need for explicit interaction lists by restructuring computations to operate exclusively on the near-neighbor list at each level of the tree, thereby simplifying both implementation and data structures. This work also introduces novel translation operators that significantly simplify the handling of adaptive point distributions. As a kernel-independent approach, it only requires evaluation of the kernel function, making it easily adaptable to a variety of kernels. By using operations on the neighbor list (of size at most 27 in 3D) rather than the interaction list (of size up to 189 in 3D), the algorithm is particularly well-suited for parallel implementation on modern hardware. Numerical experiments on uniform and non-uniform point distributions in 2D and 3D demonstrate the effectiveness of the proposed parallel algorithm for Laplace and (low-frequency) Helmholtz kernels. The algorithm constructs a tailored skeleton representation for the given geometry during a precomputation stage. After precomputation, the fast summation achieves high efficiency on the GPU using batched linear algebra operations.

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arXiv2025

GPU Optimizations for the Hierarchical Poincaré-Steklov Scheme

This manuscript presents GPU optimizations for the 2D Hierarchical Poincaré-Steklov (HPS) discretization scheme. HPS is a multi-domain spectral collocation method that combines high-order discretizations with direct solvers to accurately resolve highly oscillatory solutions. The domain decomposition approach of HPS connects domains directly via a sparse direct solver. The proposed optimizations exploit batched linear algebra on modern hybrid architectures, are straightforward to implement, and improve the solver's practical speed. The manuscript demonstrates that GPU optimizations can significantly reduce the traditionally high cost of performing local static condensation for discretizations with very high local order $p$. Numerical experiments for the Helmholtz equation with high wavenumbers on curved and rectangular domains confirm the high accuracy achieved by the HPS discretization and the significant reduction in computation time achieved with GPU optimizations.

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arXiv2025

Randomized Block Low-Rank Matrix Compression by Tagging

In this work, we present randomized compression algorithms for flat rank-structured matrices with shared bases, termed uniform Block Low-Rank (BLR) matrices. Our main contribution is a technique called tagging, which improves upon the efficiency of existing algorithms for basis matrix computation while preserving accuracy. Tagging operates on the matrix using matrix-vector products of the matrix and its adjoint, making it suitable for black-box environments where accessing individual matrix entries is computationally expensive or infeasible. We show tagging requires a constant number of matrix-vector products coupled with linear post-processing; crucially, the asymptotic pre-factors in tagging depend only on the rank parameter and the underlying problem geometry. We also establish a theoretical connection between the optimal construction of tagging matrices and projective varieties in algebraic geometry, suggesting a hybrid numeric-symbolic avenue of future work. To validate our approach, we apply tagging to compress uniform BLR matrices arising from the discretization of integral and partial differential equations. Empirical results show that tagging outperforms alternative compression techniques, significantly reducing both the number of required matrix-vector products and overall computational time. These findings highlight the practicality and scalability of tagging as an efficient method for flat rank-structured matrices in scientific computing.

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arXiv2023

Randomized Strong Recursive Skeletonization: Simultaneous Compression and LU Factorization of Hierarchical Matrices using Matrix-Vector Products

The hierarchical matrix framework partitions matrices into subblocks that are either small or of low numerical rank, enabling linear storage complexity and efficient matrix-vector multiplication. This work focuses on the $H^2$-matrix format constructed under the strong admissibility condition, which has two key properties: (1) a compressed representation that approximates far-field interactions with low-rank blocks while near-field interactions are stored densely, and (2) a nested basis structure that reuses basis matrices across hierarchy levels. Although these matrices support fast Cholesky and LU factorizations, implementing them - especially for 3D PDE discretizations - remains challenging due to the nested recursions and recompressions involved. This paper introduces an algorithm that simultaneously compresses and factorizes a general ${H}^{2}$ matrix, using only the action of the matrix and its adjoint on vectors. The number of required matrix-vector products is independent of the matrix size, and depends only on the problem geometry and a rank parameter. The resulting LU factorization is invertible and can serve as an approximate direct solver, with accuracy influenced by the spectral properties of the matrix. To achieve competitive sample complexity, the method employs dense Gaussian test matrices without explicitly encoding structured sparsity. Samples are drawn only once at the start of the algorithm; as the factorization proceeds, structure is dynamically introduced into the test matrices through efficient linear algebraic operations. Numerical experiments demonstrate robustness to indefiniteness and ill-conditioning, as well as the efficiency of the method for integral and differential equations in 2D and 3D.

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