arXiv subjects

Hou-Biao Li

Publications and source records attributed to Hou-Biao Li.

11 recordsLinked to original sources

arXiv2025

OBK-RCM: Accelerated Orthogonal Block Kaczmarz Algorithm via RCM Reordering and Dynamic Grouping for Sparse Linear Systems

Existing block Kaczmarz methods face challenges in balancing computational efficiency and convergence for large sparse linear systems with scattered nonzero patterns, due to costly partitioning strategies and non-orthogonal projections. In this paper, we propose the orthogonal block Kaczmarz (OBK-RCM) algorithm with the Reverse Cuthill-McKee (RCM), which integrates the RCM reordering with a novel orthogonal block partitioning strategy. RCM transforms sparse matrices into banded structures to enhance inter-block orthogonality, while dynamic grouping of mutually orthogonal blocks based on angle cosine thresholds reduces iterative complexity. In addition, two extended versions (SOBK-RCM and UOBK-RCM) are proposed to deal with non-square systems by constructing extended matrices without sacrificing sparsity. This work offers a practical framework for efficient sparse linear algebra solvers. Experiments on 33 real-world and synthetic matrices show that OBK-RCM achieves 10-50 times faster CPU time (up to several hundred) and 50-90% fewer iterations than state-of-the-art methods (RBK,RBK(k),GREBK(k),aRBK), especially for scattered sparse structures in most cases. Theoretical analysis confirms linear convergence, driven by hyperplane orthogonality.

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arXiv2025

U-WNO: U-Net Enhanced Wavelet Neural Operator for Solving Parametric Partial Differential Equations

High-frequency features are critical in multiscale phenomena such as turbulent flows and phase transitions, since they encode essential physical information. The recently proposed Wavelet Neural Operator (WNO) utilizes wavelets' time-frequency localization to capture spatial manifolds effectively. While its factorization strategy improves noise robustness, it suffers from high-frequency information loss caused by finite-scale wavelet decomposition. In this study, a new U-WNO network architecture is proposed. It incorporates the U-Net path and residual shortcut into the wavelet layer to enhance the extraction of high-frequency features and improve the learning of spatial manifolds. Furthermore, we introduce an adaptive activation mechanism to mitigate spectral bias through trainable slope parameters. Extensive benchmarks across seven PDE families (Burgers, Darcy flow, Navier-Stokes, etc.) show that U-WNO achieves 45--83% error reduction compared to baseline WNO, with mean $L_2$ relative errors ranging from 0.043% to 1.56%. This architecture establishes a framework combining multiresolution analysis with deep feature learning, addressing the spectral-spatial tradeoff in operator learning. Code and data used are available on https://github.com/WeiminLei/U-WNO.git.

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arXiv2025

MgFNO: Multi-grid Architecture Fourier Neural Operator for Parametric Partial Differential Equations

Neural operators are a new type of models that can map between function spaces, allowing trained models to emulate the solution operators of partial differential equations (PDEs). This paper proposes a multigrid Fourier neural operator (MgFNO) that accelerates the training of traditional Fourier neural operators through a novel three-level hierarchical architecture. The key innovation of MgFNO lies in its decoupled training strategy employing three distinct networks at different resolution levels: a coarse-level network first learns low-resolution approximations, an intermediate network refines the solution, and a fine-level network achieves high-resolution accuracy. By combining the frequency principle of deep neural networks with multigrid methodology, MgFNO effectively bridges the complementary learning patterns of neural networks (low-to-high frequency) and multigrid methods (high-to-low frequency error reduction).Experimental results demonstrate that MgFNO achieves relative errors of 0.17%, 0.28%, and 0.22% on the Burgers' equation, Darcy flow, and Navier-Stokes equations, respectively, representing reductions of 89%, 71%, and 83% compared to the conventional FNO. Furthermore, MgFNO supports zero-shot super-resolution prediction, enabling direct application to high-resolution scenarios after training on coarse grids. This study establishes an efficient and high-accuracy new paradigm for solving complex PDEs dominated by high-frequency dynamics. Code and data used are available on https://github.com/guozihao-hub/MgFNO/tree/master.

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arXiv2022

Greedy double subspaces coordinate descent method via orthogonalization

The coordinate descent method is an effective iterative method for solving large linear least-squares problems. In this paper, for the highly coherent columns case, we construct an effective coordinate descent method which iteratively projects the estimate onto a solution space formed by two greedily selected hyperplanes via Gram-Schmidt orthogonalization. Our methods may be regarded as a simple block version of coordinate descent method which involves two active columns. The convergence analysis of this method is provided and numerical simulations also confirm the effectiveness for matrices with highly coherent columns.

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arXiv2020

Two CSCS-based iteration methods for solving absolute value equations

Recently, two families of HSS-based iteration methods are constructed for solving the system of absolute value equations (AVEs), which is a class of non-differentiable NP-hard problems. In this study, we establish the Picard-CSCS iteration method and the nonlinear CSCS-like iteration method for AVEs involving the Toeplitz matrix. Then, we analyze the convergence of the Picard-CSCS iteration method for solving AVEs. By using the theory about nonsmooth analysis, we particularly prove the convergence of the nonlinear CSCS-like iterationsolver for AVEs. The advantage of these methods is that they do not require the storage of coefficient matrices at all, and the sub-system of linear equations can be solved efficiently via the fast Fourier transforms (FFTs). Therefore, computational cost and storage can be saved in practical implementations. Numerical examples including numerical solutions of nonlinear fractional diffusion equations are reported to show the effectiveness of the proposed methods in comparison with some existing methods.

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arXiv2015

Numerical Gradient Schemes for Heat Equations Based on the Collocation Polynomial and Hermite Interpolation

As is well-known, the advantage of the high-order compact difference scheme (H-OCD) is unconditionally stable and convergent with the order $O(τ^2+h^4)$ under the maximum norm. In this article, a new numerical gradient scheme based on the collocation polynomial and Hermite interpolation is presented. Moreover, the convergence order of this kind of method is also $O(τ^2+h^4)$ under the discrete maximum norm when the space step size is just twice the one of H-OCD method, which accelerates the computational process and makes the result much smoother to some extent. In addition, some corresponding analyses are made and the Richardson extrapolation technique is also considered in time direction. The results of numerical experiments are also consistent with these theoretical analysis.

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arXiv2014

Recycling BiCG for families of shifted linear systems

Many problems in science and engineering fields require the solution of shifted linear systems. To solve such systems efficiently, the recycling BiCG (RBiCG) algorithm in [SIAM J. SCI. COMPUT, 34 (2012) 1925-1949] is extended in this paper. However, the shift-invariant property could no longer hold over the augmented Krylov subspace due to adding the recycling spaces. To remedy this situation, a strategy to enforce the collinearity condition on the shifted system is adopted and then a short term recurrence for the solution update of the shifted system is derived when the seed system is solving. The new method not only improves the convergence but also has a potential to simultaneously compute approximate solutions for shifted linear systems using only as many matrix-vector multiplications as the solution of a single system requires. In addition, some numerical experiments also confirm the efficiency of our method.

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arXiv2014

Some new bounds for the Hadamard product and the Fan product of matrices

If $A$ and $B$ are nonnegative matrices, a sharp upper bound on the spectral radius $ρ(A\circ B)$ for the Hadamard product of two nonnegative matrices is given, and the minimum eigenvalue $τ(A\star B)$ of the Fan product of two $M$-matrices $A$ and $B$ is discussed. In addition, we also give a sharp lower bound on $τ(A\circ B^{-1})$ for the Hadamard product of $A$ and $B^{-1}$. Several examples, illustrating that the given bound is stronger than the existing bounds, are also given.

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arXiv2013

Strang-type preconditioners for solving fractional diffusion equations by boundary value methods

The finite difference scheme with the shifted Grünwarld formula is employed to semi-discrete the fractional diffusion equations. This spatial discretization can reduce to the large system of ordinary differential equations (ODEs) with initial values. Recently, boundary value method (BVM) was developed as a popular algorithm for solving large systems of ODEs. This method requires the solutions of one or more nonsymmetric, large and sparse linear systems. In this paper, the GMRES method with the block circulant preconditioner is proposed for solving these linear systems. One of the main results is that if an $A_{ν_1,ν_2}$-stable boundary value method is used for an m-by-m system of ODEs, then the preconditioner is invertible and the preconditioned matrix can be decomposed as I+L, where I is the identity matrix and the rank of L is at most $2m(ν_1+ν_2)$. It means that when the GMRES method is applied to solve the preconditioned linear systems, the method will converge in at most $2m(ν_1+ν_2)+1$ iterations.Finally, extensive numerical experiments are reported to illustrate the effectiveness of our methods for solving the fractional diffusion equations.

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arXiv2013

A SQMRCGstab Algorithm for Families of Shifted Linear Systems

This study is mainly focused on iterative solutions to shifted linear systems arising from a Quantum Chromodynamics (QCD) problem. To solve such system efficiently, we explore a kind of shifted QMRCGstab (SQMRCGstab) methods, which is derived by extending the quasi-minimum residual to the shifted BiCGstab. The shifted QMRCGstab method takes advantage of the shifted structure, so that the number of matrix-vector products and the number of inner products are the same as a single linear system. Moreover, the SQMRCGstab achieves a smoothing of the residual compared to the shifted BiCGstab, and is more competitive than the MS-QMRIDR(s) and the shifted BiCGstab on the QCD problem. Numerical examples show also the efficiency of the method when one applies it to the real problems.

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arXiv2013

A Symmetric Rank-one Quasi Newton Method for Non-negative Matrix Factorization

As we all known, the nonnegative matrix factorization (NMF) is a dimension reduction method that has been widely used in image processing, text compressing and signal processing etc. In this paper, an algorithm for nonnegative matrix approximation is proposed. This method mainly bases on the active set and the quasi-Newton type algorithm, by using the symmetric rank-one and negative curvature direction technologies to approximate the Hessian matrix. Our method improves the recent results of those methods in [Pattern Recognition, 45(2012)3557-3565; SIAM J. Sci. Comput., 33(6)(2011)3261-3281; Neural Computation, 19(10)(2007)2756-2779, etc.]. Moreover, the object function decreases faster than many other NMF methods. In addition, some numerical experiments are presented in the synthetic data, imaging processing and text clustering. By comparing with the other six nonnegative matrix approximation methods, our experiments confirm to our analysis.

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