Sebastian Acosta, Tahsin Khajah, Benjamin Palacios
This paper introduces a new pseudodifferential preconditioner for the Helmholtz equation in variable media with absorption. The pseudodifferential operator is associated with the multiplicative inverse to the symbol of the Helmholtz operator. This approach is well suited for the intermediate and high-frequency regimes. The main novel idea for the fast evaluation of the preconditioner is to interpolate its symbol, not as a function of the (high-dimensional) phase-space variables, but as a function of the wave speed itself. Since the wave speed is a real-valued function, this approach allows us to interpolate in a univariate setting even when the original problem is posed in a multidimensional physical space. As a result, the needed number of interpolation points is small, and the interpolation coefficients can be computed using the fast Fourier transform. The overall computational complexity is log-linear with respect to the degrees of freedom as inherited from the fast Fourier transform. We present some numerical experiments to illustrate the effectiveness of the preconditioner to solve the discrete Helmholtz equation using the GMRES iterative method. The implementation of an absorbing layer for scattering problems using a complex-valued wave speed is also developed. Limitations and possible extensions are also discussed.
{"title":"A NEW INTERPOLATED PSEUDODIFFERENTIAL PRECONDITIONER FOR THE HELMHOLTZ EQUATION IN HETEROGENEOUS MEDIA.","authors":"Sebastian Acosta, Tahsin Khajah, Benjamin Palacios","doi":"10.1137/24M1642184","DOIUrl":"10.1137/24M1642184","url":null,"abstract":"<p><p>This paper introduces a new pseudodifferential preconditioner for the Helmholtz equation in variable media with absorption. The pseudodifferential operator is associated with the multiplicative inverse to the symbol of the Helmholtz operator. This approach is well suited for the intermediate and high-frequency regimes. The main novel idea for the fast evaluation of the preconditioner is to interpolate its symbol, not as a function of the (high-dimensional) phase-space variables, but as a function of the wave speed itself. Since the wave speed is a real-valued function, this approach allows us to interpolate in a <i>univariate</i> setting even when the original problem is posed in a multidimensional physical space. As a result, the needed number of interpolation points is small, and the interpolation coefficients can be computed using the fast Fourier transform. The overall computational complexity is log-linear with respect to the degrees of freedom as inherited from the fast Fourier transform. We present some numerical experiments to illustrate the effectiveness of the preconditioner to solve the discrete Helmholtz equation using the GMRES iterative method. The implementation of an absorbing layer for scattering problems using a complex-valued wave speed is also developed. Limitations and possible extensions are also discussed.</p>","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"47 2","pages":"A1017-A1039"},"PeriodicalIF":2.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12987680/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147470131","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Joe Kileel, Nicholas F Marshall, Oscar Mickelin, Amit Singer
We devise fast and provably accurate algorithms to transform between an Cartesian voxel representation of a three-dimensional function and its expansion into the ball harmonics, that is, the eigenbasis of the Dirichlet Laplacian on the unit ball in . Given , our algorithms achieve relative accuracy in time , while the naive direct application of the expansion operators has time complexity . We illustrate our methods on numerical examples.
我们设计了一种快速且可证明准确的算法，将三维函数的N × N × N笛卡尔体素表示转换为球谐波，即r3中单位球上狄利克雷拉普拉斯算子的特征基。在给定ε >的情况下，我们的算法在时间变量（n3 (l o g N) 2 + n3 | l o g ε | 2）上实现了相对的1 -∞精度ε，而单纯直接应用展开算子的时间变量复杂度为n6。我们用数值例子来说明我们的方法。
{"title":"FAST EXPANSION INTO HARMONICS ON THE BALL.","authors":"Joe Kileel, Nicholas F Marshall, Oscar Mickelin, Amit Singer","doi":"10.1137/24m1668159","DOIUrl":"10.1137/24m1668159","url":null,"abstract":"<p><p>We devise fast and provably accurate algorithms to transform between an <math><mi>N</mi> <mo>×</mo> <mi>N</mi> <mo>×</mo> <mi>N</mi></math> Cartesian voxel representation of a three-dimensional function and its expansion into the ball harmonics, that is, the eigenbasis of the Dirichlet Laplacian on the unit ball in <math> <msup><mrow><mi>R</mi></mrow> <mrow><mn>3</mn></mrow> </msup> </math> . Given <math><mi>ε</mi> <mo>></mo> <mn>0</mn></math> , our algorithms achieve relative <math> <msup><mrow><mo>ℓ</mo></mrow> <mrow><mn>1</mn></mrow> </msup> <mo>-</mo> <msup><mrow><mo>ℓ</mo></mrow> <mrow><mi>∞</mi></mrow> </msup> </math> accuracy <math><mi>ε</mi></math> in time <math><mi>𝒪</mi> <mo>(</mo> <mrow> <msup><mrow><mi>N</mi></mrow> <mrow><mn>3</mn></mrow> </msup> <mo>(</mo> <mi>l</mi> <mi>o</mi> <mi>g</mi> <mspace></mspace> <mi>N</mi> <msup><mrow><mo>)</mo></mrow> <mrow><mn>2</mn></mrow> </msup> <mo>+</mo> <msup><mrow><mi>N</mi></mrow> <mrow><mn>3</mn></mrow> </msup> <mo>|</mo> <mi>l</mi> <mi>o</mi> <mi>g</mi> <mspace></mspace> <mi>ε</mi> <msup><mrow><mo>|</mo></mrow> <mrow><mn>2</mn></mrow> </msup> </mrow> <mo>)</mo></math> , while the naive direct application of the expansion operators has time complexity <math><mi>𝒪</mi> <mfenced> <mrow> <msup><mrow><mi>N</mi></mrow> <mrow><mn>6</mn></mrow> </msup> </mrow> </mfenced> </math> . We illustrate our methods on numerical examples.</p>","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"47 2","pages":"A1117-A1144"},"PeriodicalIF":2.6,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12974893/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147437018","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Scientific Computing, Volume 46, Issue 5, Page A2972-A2998, October 2024. Abstract. In this paper, the numerical approximation of the generalized Burgers–Huxley equation (GBHE) with weakly singular kernels using nonconforming methods will be presented. Specifically, we discuss two new formulations. The first formulation is based on the nonconforming finite element method. The other formulation is based on discontinuous Galerkin finite element methods. The wellposedness results for both formulations are proved. Then, a priori error estimates for both the semidiscrete and fully discrete schemes are derived. Specific numerical examples, including some applications for the GBHE with a weakly singular model, are discussed to validate the theoretical results.
{"title":"Finite Element Approximation for the Delayed Generalized Burgers–Huxley Equation with Weakly Singular Kernel: Part II Nonconforming and DG Approximation","authors":"Sumit Mahahjan, Arbaz Khan","doi":"10.1137/23m1612196","DOIUrl":"https://doi.org/10.1137/23m1612196","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 5, Page A2972-A2998, October 2024. <br/> Abstract. In this paper, the numerical approximation of the generalized Burgers–Huxley equation (GBHE) with weakly singular kernels using nonconforming methods will be presented. Specifically, we discuss two new formulations. The first formulation is based on the nonconforming finite element method. The other formulation is based on discontinuous Galerkin finite element methods. The wellposedness results for both formulations are proved. Then, a priori error estimates for both the semidiscrete and fully discrete schemes are derived. Specific numerical examples, including some applications for the GBHE with a weakly singular model, are discussed to validate the theoretical results.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"10 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142262881","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Scientific Computing, Volume 46, Issue 5, Page A2951-A2971, October 2024. Abstract. This paper addresses the problem of solving nonlinear systems in the context of symmetric quantum signal processing (QSP), a powerful technique for implementing matrix functions on quantum computers. Symmetric QSP focuses on representing target polynomials as products of matrices in SU(2) that possess symmetry properties. We present a novel Newton’s method tailored for efficiently solving the nonlinear system involved in determining the phase factors within the symmetric QSP framework. Our method demonstrates rapid and robust convergence in all parameter regimes, including the challenging scenario with ill-conditioned Jacobian matrices, using standard double precision arithmetic operations. For instance, solving symmetric QSP for a highly oscillatory target function [math] (polynomial degree [math]) takes 6 iterations to converge to machine precision when [math], and the number of iterations only increases to 18 iterations when [math] with a highly ill-conditioned Jacobian matrix. Leveraging the matrix product state structure of symmetric QSP, the computation of the Jacobian matrix incurs a computational cost comparable to a single function evaluation. Moreover, we introduce a reformulation of symmetric QSP using real-number arithmetics, further enhancing the method’s efficiency. Extensive numerical tests validate the effectiveness and robustness of our approach, which has been implemented in the QSPPACK software package.
{"title":"Robust Iterative Method for Symmetric Quantum Signal Processing in All Parameter Regimes","authors":"Yulong Dong, Lin Lin, Hongkang Ni, Jiasu Wang","doi":"10.1137/23m1598192","DOIUrl":"https://doi.org/10.1137/23m1598192","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 5, Page A2951-A2971, October 2024. <br/> Abstract. This paper addresses the problem of solving nonlinear systems in the context of symmetric quantum signal processing (QSP), a powerful technique for implementing matrix functions on quantum computers. Symmetric QSP focuses on representing target polynomials as products of matrices in SU(2) that possess symmetry properties. We present a novel Newton’s method tailored for efficiently solving the nonlinear system involved in determining the phase factors within the symmetric QSP framework. Our method demonstrates rapid and robust convergence in all parameter regimes, including the challenging scenario with ill-conditioned Jacobian matrices, using standard double precision arithmetic operations. For instance, solving symmetric QSP for a highly oscillatory target function [math] (polynomial degree [math]) takes 6 iterations to converge to machine precision when [math], and the number of iterations only increases to 18 iterations when [math] with a highly ill-conditioned Jacobian matrix. Leveraging the matrix product state structure of symmetric QSP, the computation of the Jacobian matrix incurs a computational cost comparable to a single function evaluation. Moreover, we introduce a reformulation of symmetric QSP using real-number arithmetics, further enhancing the method’s efficiency. Extensive numerical tests validate the effectiveness and robustness of our approach, which has been implemented in the QSPPACK software package.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"3 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217589","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Lars Simon, Holger Eble, Hagen-Henrik Kowalski, Manuel Radons
SIAM Journal on Scientific Computing, Volume 46, Issue 5, Page B600-B620, October 2024. Abstract. This article focuses on developing classical surrogates for parametrized quantum circuits using interpolation via (trigonometric) polynomials. We develop two algorithms for the construction of such surrogates and prove performance guarantees. The constructions are based on circuit evaluations which are blackbox in the sense that no structural specifics of the circuits are exploited. While acknowledging the limitations of the blackbox approach compared to whitebox evaluations, which exploit specific circuit properties, we demonstrate scenarios in which the blackbox approach might prove beneficial. Sample applications include but are not restricted to the approximation of variational quantum eigensolvers and the alleviaton of the barren plateau problem.
{"title":"Interpolating Parametrized Quantum Circuits Using Blackbox Queries","authors":"Lars Simon, Holger Eble, Hagen-Henrik Kowalski, Manuel Radons","doi":"10.1137/23m1609543","DOIUrl":"https://doi.org/10.1137/23m1609543","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 5, Page B600-B620, October 2024. <br/> Abstract. This article focuses on developing classical surrogates for parametrized quantum circuits using interpolation via (trigonometric) polynomials. We develop two algorithms for the construction of such surrogates and prove performance guarantees. The constructions are based on circuit evaluations which are blackbox in the sense that no structural specifics of the circuits are exploited. While acknowledging the limitations of the blackbox approach compared to whitebox evaluations, which exploit specific circuit properties, we demonstrate scenarios in which the blackbox approach might prove beneficial. Sample applications include but are not restricted to the approximation of variational quantum eigensolvers and the alleviaton of the barren plateau problem.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"10 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217588","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Scientific Computing, Volume 46, Issue 5, Page A2925-A2950, October 2024. Abstract. The adaptive spectral Koopman (ASK) method was introduced to numerically solve autonomous dynamical systems that laid the foundation for numerous applications across different fields in science and engineering. Although ASK achieves high accuracy, it is computationally more expensive for multidimensional systems compared with conventional time integration schemes like Runge–Kutta. In this work, we combine the sparse grid and ASK to accelerate the computation for multidimensional systems. This sparse-grid-based ASK (SASK) method uses the Smolyak structure to construct multidimensional collocation points as well as associated polynomials that are used to approximate eigenfunctions of the Koopman operator of the system. In this way, the number of collocation points is reduced compared with using the tensor product rule. We demonstrate that SASK can be used to solve ordinary differential equations (ODEs) and partial differential equations (PDEs) based on their semidiscrete forms. Numerical experiments are illustrated to compare the performance of SASK and state-of-the-art ODE solvers.
{"title":"The Sparse-Grid-Based Adaptive Spectral Koopman Method","authors":"Bian Li, Yue Yu, Xiu Yang","doi":"10.1137/23m1578292","DOIUrl":"https://doi.org/10.1137/23m1578292","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 5, Page A2925-A2950, October 2024. <br/> Abstract. The adaptive spectral Koopman (ASK) method was introduced to numerically solve autonomous dynamical systems that laid the foundation for numerous applications across different fields in science and engineering. Although ASK achieves high accuracy, it is computationally more expensive for multidimensional systems compared with conventional time integration schemes like Runge–Kutta. In this work, we combine the sparse grid and ASK to accelerate the computation for multidimensional systems. This sparse-grid-based ASK (SASK) method uses the Smolyak structure to construct multidimensional collocation points as well as associated polynomials that are used to approximate eigenfunctions of the Koopman operator of the system. In this way, the number of collocation points is reduced compared with using the tensor product rule. We demonstrate that SASK can be used to solve ordinary differential equations (ODEs) and partial differential equations (PDEs) based on their semidiscrete forms. Numerical experiments are illustrated to compare the performance of SASK and state-of-the-art ODE solvers.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"92 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Scientific Computing, Volume 46, Issue 5, Page A2899-A2924, October 2024. Abstract. Central-upwind (CU) schemes are Riemann-problem-solver-free finite-volume methods widely applied to a variety of hyperbolic systems of PDEs. Exact solutions of these systems typically satisfy certain bounds, and it is highly desirable and even crucial for the numerical schemes to preserve these bounds. In this paper, we develop and analyze bound-preserving (BP) CU schemes for general hyperbolic systems of conservation laws. Unlike many other Godunov-type methods, CU schemes cannot, in general, be recast as convex combinations of first-order BP schemes. Consequently, standard BP analysis techniques are invalidated. We address these challenges by establishing a novel framework for analyzing the BP property of CU schemes. To this end, we discover that the CU schemes can be decomposed as a convex combination of several intermediate solution states. Thanks to this key finding, the goal of designing BPCU schemes is simplified to the enforcement of four more accessible BP conditions, each of which can be achieved with the help of a minor modification of the CU schemes. We employ the proposed approach to construct provably BPCU schemes for the Euler equations of gas dynamics. The robustness and effectiveness of the BPCU schemes are validated by several demanding numerical examples, including high-speed jet problems, flow past a forward-facing step, and a shock diffraction problem.
SIAM 科学计算期刊》，第 46 卷第 5 期，第 A2899-A2924 页，2024 年 10 月。 摘要中央上风（CU）方案是一种无黎曼问题求解器的有限体积方法，广泛应用于各种双曲型 PDE 系统。这些系统的精确解通常满足一定的边界，而数值方案保持这些边界是非常理想的，甚至是至关重要的。在本文中，我们开发并分析了一般双曲守恒律系统的保界（BP）CU 方案。与许多其他戈杜诺夫型方法不同，CU 方案一般不能被重塑为一阶 BP 方案的凸组合。因此，标准的 BP 分析技术就失效了。我们通过建立一个分析 CU 方案 BP 特性的新框架来应对这些挑战。为此，我们发现 CU 方案可以分解为多个中间解状态的凸组合。有了这一关键发现，设计 BPCU 方案的目标就简化为执行四个更易实现的 BP 条件，其中每个条件都可以通过对 CU 方案稍加修改来实现。我们采用所提出的方法为气体动力学欧拉方程构建了可证明的 BPCU 方案。BPCU 方案的稳健性和有效性通过几个苛刻的数值示例得到了验证，包括高速射流问题、流过前向台阶问题和冲击衍射问题。
{"title":"Bound-Preserving Framework for Central-Upwind Schemes for General Hyperbolic Conservation Laws","authors":"Shumo Cui, Alexander Kurganov, Kailiang Wu","doi":"10.1137/23m1628024","DOIUrl":"https://doi.org/10.1137/23m1628024","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 5, Page A2899-A2924, October 2024. <br/> Abstract. Central-upwind (CU) schemes are Riemann-problem-solver-free finite-volume methods widely applied to a variety of hyperbolic systems of PDEs. Exact solutions of these systems typically satisfy certain bounds, and it is highly desirable and even crucial for the numerical schemes to preserve these bounds. In this paper, we develop and analyze bound-preserving (BP) CU schemes for general hyperbolic systems of conservation laws. Unlike many other Godunov-type methods, CU schemes cannot, in general, be recast as convex combinations of first-order BP schemes. Consequently, standard BP analysis techniques are invalidated. We address these challenges by establishing a novel framework for analyzing the BP property of CU schemes. To this end, we discover that the CU schemes can be decomposed as a convex combination of several intermediate solution states. Thanks to this key finding, the goal of designing BPCU schemes is simplified to the enforcement of four more accessible BP conditions, each of which can be achieved with the help of a minor modification of the CU schemes. We employ the proposed approach to construct provably BPCU schemes for the Euler equations of gas dynamics. The robustness and effectiveness of the BPCU schemes are validated by several demanding numerical examples, including high-speed jet problems, flow past a forward-facing step, and a shock diffraction problem.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"24 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217591","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Scientific Computing, Volume 46, Issue 5, Page A2874-A2898, October 2024. Abstract. We propose a novel class of temporal high-order parametric finite element methods for solving a wide range of geometric flows of curves and surfaces. By incorporating the backward differentiation formula (BDF) for time discretization into the BGN formulation, originally proposed by Barrett, Garcke, and Nürnberg (J. Comput. Phys., 222 (2007), pp. 441–467), we successfully develop high-order BGN/BDF[math] schemes. The proposed BGN/BDF[math] schemes not only retain almost all the advantages of the classical first-order BGN scheme such as computational efficiency and good mesh quality, but also exhibit the desired [math]th-order temporal accuracy in terms of shape metrics, ranging from second-order to fourth-order accuracy. Furthermore, we validate the performance of our proposed BGN/BDF[math] schemes through extensive numerical examples, demonstrating their high-order temporal accuracy for various types of geometric flows while maintaining good mesh quality throughout the evolution.
{"title":"Stable Backward Differentiation Formula Time Discretization of BGN-Based Parametric Finite Element Methods for Geometric Flows","authors":"Wei Jiang, Chunmei Su, Ganghui Zhang","doi":"10.1137/23m1625597","DOIUrl":"https://doi.org/10.1137/23m1625597","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 5, Page A2874-A2898, October 2024. <br/> Abstract. We propose a novel class of temporal high-order parametric finite element methods for solving a wide range of geometric flows of curves and surfaces. By incorporating the backward differentiation formula (BDF) for time discretization into the BGN formulation, originally proposed by Barrett, Garcke, and Nürnberg (J. Comput. Phys., 222 (2007), pp. 441–467), we successfully develop high-order BGN/BDF[math] schemes. The proposed BGN/BDF[math] schemes not only retain almost all the advantages of the classical first-order BGN scheme such as computational efficiency and good mesh quality, but also exhibit the desired [math]th-order temporal accuracy in terms of shape metrics, ranging from second-order to fourth-order accuracy. Furthermore, we validate the performance of our proposed BGN/BDF[math] schemes through extensive numerical examples, demonstrating their high-order temporal accuracy for various types of geometric flows while maintaining good mesh quality throughout the evolution.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"6 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217592","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ru Huang, Kai Chang, Huan He, Ruipeng Li, Yuanzhe Xi
SIAM Journal on Scientific Computing, Ahead of Print. Abstract. We propose a data-driven and machine-learning-based approach to compute non-Galerkin coarse-grid operators in multigrid (MG) methods, addressing the well-known issue of increasing operator complexity. Guided by the MG theory on spectrally equivalent coarse-grid operators, we have developed novel machine learning algorithms that utilize neural networks combined with smooth test vectors from multigrid eigenvalue problems. The proposed method demonstrates promise in reducing the complexity of coarse-grid operators while maintaining overall MG convergence for solving parametric partial differential equation problems. Numerical experiments on anisotropic rotated Laplacian and linear elasticity problems are provided to showcase the performance and comparison with existing methods for computing non-Galerkin coarse-grid operators. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/liruipeng/SparseCoarseOperator.
{"title":"Reducing Operator Complexity of Galerkin Coarse-grid Operators with Machine Learning","authors":"Ru Huang, Kai Chang, Huan He, Ruipeng Li, Yuanzhe Xi","doi":"10.1137/23m1583533","DOIUrl":"https://doi.org/10.1137/23m1583533","url":null,"abstract":"SIAM Journal on Scientific Computing, Ahead of Print. <br/> Abstract. We propose a data-driven and machine-learning-based approach to compute non-Galerkin coarse-grid operators in multigrid (MG) methods, addressing the well-known issue of increasing operator complexity. Guided by the MG theory on spectrally equivalent coarse-grid operators, we have developed novel machine learning algorithms that utilize neural networks combined with smooth test vectors from multigrid eigenvalue problems. The proposed method demonstrates promise in reducing the complexity of coarse-grid operators while maintaining overall MG convergence for solving parametric partial differential equation problems. Numerical experiments on anisotropic rotated Laplacian and linear elasticity problems are provided to showcase the performance and comparison with existing methods for computing non-Galerkin coarse-grid operators. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/liruipeng/SparseCoarseOperator.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"10 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142217599","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Matthias Bolten, Misha E. Kilmer, Scott MacLachlan
SIAM Journal on Scientific Computing, Ahead of Print. Abstract. In this paper, we are concerned with efficiently solving the sequences of regularized linear least-squares problems associated with employing Tikhonov-type regularization with regularization operators designed to enforce edge recovery. An optimal regularization parameter, which balances the fidelity to the data with the edge-enforcing constraint term, is typically not known a priori. This adds to the total number of regularized linear least-squares problems that must be solved before the final image can be recovered. Therefore, in this paper, we determine effective multigrid preconditioners for these sequences of systems. We focus our approach on the sequences that arise as a result of the edge-preserving method introduced in [S. Gazzola et al., Inverse Problems, 36 (2020), 124004], where we can exploit an interpretation of the regularization term as a diffusion operator; however, our methods are also applicable in other edge-preserving settings, such as iteratively reweighted least-squares problems. Particular attention is paid to the selection of components of the multigrid preconditioner in order to achieve robustness for different ranges of the regularization parameter value. In addition, we present a parameter trimming approach that, when used with the L-curve heuristic, reduces the total number of solves required. We demonstrate our preconditioning and parameter trimming routines on examples in computed tomography and image deblurring.
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