SIAM Journal on Scientific Computing, Ahead of Print. Abstract. Block floating point (BFP) arithmetic is currently seeing a resurgence in interest because it requires less power and less chip area and is less complicated to implement in hardware than standard floating point arithmetic. This paper explores the application of BFP to mixed- and progressive-precision multigrid methods, enabling the solution of linear elliptic partial differential equations (PDEs) in energy- and hardware-efficient integer arithmetic. While most existing applications of BFP arithmetic tend to use small block sizes, the block size here is chosen to be maximal such that matrices and vectors share a single exponent for all entries. This is sometimes also referred to as a scaled fixed point format. We provide algorithms for BLAS-like routines for BFP arithmetic that ensure exact vector-vector and matrix-vector computations up to a specified precision. Using these algorithms, we study the asymptotic precision requirements for achieving discretization-error-accuracy. We demonstrate that some computations can be performed using only 4-bit integers, while the number of bits required to attain a certain target accuracy is similar to that of standard floating point arithmetic. Finally, we present a heuristic for full multigrid in BFP arithmetic based on saturation and truncation that still achieves discretization-error-accuracy without the need for expensive normalization steps of intermediate results.
{"title":"Multigrid Methods Using Block Floating Point Arithmetic","authors":"Nils Kohl, Stephen F. McCormick, Rasmus Tamstorf","doi":"10.1137/23m1581819","DOIUrl":"https://doi.org/10.1137/23m1581819","url":null,"abstract":"SIAM Journal on Scientific Computing, Ahead of Print. <br/> Abstract. Block floating point (BFP) arithmetic is currently seeing a resurgence in interest because it requires less power and less chip area and is less complicated to implement in hardware than standard floating point arithmetic. This paper explores the application of BFP to mixed- and progressive-precision multigrid methods, enabling the solution of linear elliptic partial differential equations (PDEs) in energy- and hardware-efficient integer arithmetic. While most existing applications of BFP arithmetic tend to use small block sizes, the block size here is chosen to be maximal such that matrices and vectors share a single exponent for all entries. This is sometimes also referred to as a scaled fixed point format. We provide algorithms for BLAS-like routines for BFP arithmetic that ensure exact vector-vector and matrix-vector computations up to a specified precision. Using these algorithms, we study the asymptotic precision requirements for achieving discretization-error-accuracy. We demonstrate that some computations can be performed using only 4-bit integers, while the number of bits required to attain a certain target accuracy is similar to that of standard floating point arithmetic. Finally, we present a heuristic for full multigrid in BFP arithmetic based on saturation and truncation that still achieves discretization-error-accuracy without the need for expensive normalization steps of intermediate results.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611969","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}
Nick Wulbusch, Reinhild Roden, Matthias Blau, Alexey Chernov
SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page B422-B447, August 2024. Abstract. We consider the problem of identifying the acoustic impedance of a wall surface from noisy pressure measurements in a closed room using a Bayesian approach. The room acoustics are modeled by the interior Helmholtz equation with impedance boundary conditions. The aim is to compute moments of the acoustic impedance to estimate a suitable density function of the impedance coefficient. For the computation of moments we use ratio estimators and Monte Carlo sampling. We consider two different experimental scenarios. In the first scenario, the noisy measurements correspond to a wall modeled by impedance boundary conditions. In this case, the Bayesian algorithm uses a model that is (up to the noise) consistent with the measurements and our algorithm is able to identify acoustic impedance with high accuracy. In the second scenario, the noisy measurements come from a coupled acoustic-structural problem, modeling a wall made of glass, whereas the Bayesian algorithm still uses a model with impedance boundary conditions. In this case, the parameter identification model is inconsistent with the measurements and therefore is not capable to represent them well. Nonetheless, for particular frequency bands the Bayesian algorithm identifies estimates with high likelihood. Outside these frequency bands the algorithm fails. We discuss the results of both examples and possible reasons for the failure of the latter case for particular frequency values.
{"title":"Bayesian Parameter Identification in Impedance Boundary Conditions for Helmholtz Problems","authors":"Nick Wulbusch, Reinhild Roden, Matthias Blau, Alexey Chernov","doi":"10.1137/23m1591517","DOIUrl":"https://doi.org/10.1137/23m1591517","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page B422-B447, August 2024. <br/> Abstract. We consider the problem of identifying the acoustic impedance of a wall surface from noisy pressure measurements in a closed room using a Bayesian approach. The room acoustics are modeled by the interior Helmholtz equation with impedance boundary conditions. The aim is to compute moments of the acoustic impedance to estimate a suitable density function of the impedance coefficient. For the computation of moments we use ratio estimators and Monte Carlo sampling. We consider two different experimental scenarios. In the first scenario, the noisy measurements correspond to a wall modeled by impedance boundary conditions. In this case, the Bayesian algorithm uses a model that is (up to the noise) consistent with the measurements and our algorithm is able to identify acoustic impedance with high accuracy. In the second scenario, the noisy measurements come from a coupled acoustic-structural problem, modeling a wall made of glass, whereas the Bayesian algorithm still uses a model with impedance boundary conditions. In this case, the parameter identification model is inconsistent with the measurements and therefore is not capable to represent them well. Nonetheless, for particular frequency bands the Bayesian algorithm identifies estimates with high likelihood. Outside these frequency bands the algorithm fails. We discuss the results of both examples and possible reasons for the failure of the latter case for particular frequency values.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141585759","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}
Walter Boscheri, Michael Dumbser, Pierre-Henri Maire
SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2224-A2247, August 2024. Abstract. The equations of Lagrangian gas dynamics fall into the larger class of overdetermined hyperbolic and thermodynamically compatible (HTC) systems of partial differential equations. They satisfy an entropy inequality (second principle of thermodynamics) and conserve total energy (first principle of thermodynamics). The aim of this work is to construct a novel thermodynamically compatible cell-centered Lagrangian finite volume scheme on unstructured meshes. Unlike in existing schemes, we choose to directly discretize the entropy inequality, hence obtaining total energy conservation as a consequence of the new thermodynamically compatible discretization of the other equations. First, the governing equations are written in fluctuation form. Next, the noncompatible centered numerical fluxes are corrected according to the approach recently introduced by Abgrall et al. using a scalar correction factor that is defined at the nodes of the grid. This perfectly fits into the formalism of nodal solvers which is typically adopted in cell-centered Lagrangian finite volume methods. Semidiscrete entropy conservative and entropy stable Lagrangian schemes are devised, and they are adequately blended together via a convex combination based on either a priori or a posteriori detectors of discontinuous solutions. The nonlinear stability in the energy norm is rigorously demonstrated, and the new schemes are provably positivity preserving for density and pressure. Furthermore, they exhibit zero numerical diffusion for isentropic flows while still being nonlinearly stable. The new schemes are tested against classical benchmarks for Lagrangian hydrodynamics, assessing their convergence and robustness and comparing their numerical dissipation with classical Lagrangian finite volume methods.
{"title":"A New Thermodynamically Compatible Finite Volume Scheme for Lagrangian Gas Dynamics","authors":"Walter Boscheri, Michael Dumbser, Pierre-Henri Maire","doi":"10.1137/23m1580863","DOIUrl":"https://doi.org/10.1137/23m1580863","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2224-A2247, August 2024. <br/> Abstract. The equations of Lagrangian gas dynamics fall into the larger class of overdetermined hyperbolic and thermodynamically compatible (HTC) systems of partial differential equations. They satisfy an entropy inequality (second principle of thermodynamics) and conserve total energy (first principle of thermodynamics). The aim of this work is to construct a novel thermodynamically compatible cell-centered Lagrangian finite volume scheme on unstructured meshes. Unlike in existing schemes, we choose to directly discretize the entropy inequality, hence obtaining total energy conservation as a consequence of the new thermodynamically compatible discretization of the other equations. First, the governing equations are written in fluctuation form. Next, the noncompatible centered numerical fluxes are corrected according to the approach recently introduced by Abgrall et al. using a scalar correction factor that is defined at the nodes of the grid. This perfectly fits into the formalism of nodal solvers which is typically adopted in cell-centered Lagrangian finite volume methods. Semidiscrete entropy conservative and entropy stable Lagrangian schemes are devised, and they are adequately blended together via a convex combination based on either a priori or a posteriori detectors of discontinuous solutions. The nonlinear stability in the energy norm is rigorously demonstrated, and the new schemes are provably positivity preserving for density and pressure. Furthermore, they exhibit zero numerical diffusion for isentropic flows while still being nonlinearly stable. The new schemes are tested against classical benchmarks for Lagrangian hydrodynamics, assessing their convergence and robustness and comparing their numerical dissipation with classical Lagrangian finite volume methods.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141585761","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 4, Page B403-B421, August 2024. Abstract. Quantum dynamics, typically expressed in the form of a time-dependent Schrödinger equation with a Hermitian Hamiltonian, is a natural application for quantum computing. However, when simulating quantum dynamics that involves the emission of electrons, it is necessary to use artificial boundary conditions (ABCs) to confine the computation within a fixed domain. The introduction of ABCs alters the Hamiltonian structure of the dynamics, and existing quantum algorithms cannot be directly applied since the evolution is no longer unitary. The current paper utilizes a recently introduced Schrödingerization method that converts non-Hermitian dynamics into a Schrödinger form for the artificial boundary problems [S. Jin, N. Liu, and Y. Yu, Quantum Simulation of Partial Differential Equations via Schrödingerisation, preprint, arXiv:2212.13969, 2022], [S. Jin, N. Liu, and Y. Yu, Phys. Rev. A, 108 (2023), 032603]. We implement this method for three types of ABCs, including the complex absorbing potential technique, perfectly matched layer methods, and Dirichlet-to-Neumann approach. We analyze the query complexity of these algorithms and perform numerical experiments to demonstrate the validity of this approach. This helps to bridge the gap between available quantum algorithms and computational models for quantum dynamics in unbounded domains.
SIAM 科学计算期刊》，第 46 卷第 4 期，第 B403-B421 页，2024 年 8 月。 摘要量子动力学通常以具有赫米特哈密顿的时变薛定谔方程的形式表示，是量子计算的自然应用。然而，在模拟涉及电子发射的量子动力学时，有必要使用人工边界条件（ABC）将计算限制在一个固定的域内。ABC 的引入改变了动力学的哈密顿结构，现有的量子算法无法直接应用，因为演化不再是单一的。本文利用最近引入的薛定谔化方法，将非赫米态动力学转换为薛定谔形式，用于人工边界问题 [S. Jin, N. Liu, and Y. J., J., J., J., J., J., J., J., J., J., J., J., J., J., J., J., J., J.S. Jin, N. Liu, and Y. Yu, Quantum Simulation of Partial Differential Equations via Schrödingerisation, preprint, arXiv:2212.13969, 2022]，[S. Jin, N. Liu, and Y. Yu, Phys. Rev. A, 108 (2023), 032603]。我们针对三种 ABC 实现了这种方法，包括复杂吸收势技术、完全匹配层方法和 Dirichlet 到 Neumann 方法。我们分析了这些算法的查询复杂度，并通过数值实验证明了这种方法的有效性。这有助于缩小现有量子算法与无界域量子动力学计算模型之间的差距。
{"title":"Quantum Simulation for Quantum Dynamics with Artificial Boundary Conditions","authors":"Shi Jin, Xiantao Li, Nana Liu, Yue Yu","doi":"10.1137/23m1563451","DOIUrl":"https://doi.org/10.1137/23m1563451","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page B403-B421, August 2024. <br/> Abstract. Quantum dynamics, typically expressed in the form of a time-dependent Schrödinger equation with a Hermitian Hamiltonian, is a natural application for quantum computing. However, when simulating quantum dynamics that involves the emission of electrons, it is necessary to use artificial boundary conditions (ABCs) to confine the computation within a fixed domain. The introduction of ABCs alters the Hamiltonian structure of the dynamics, and existing quantum algorithms cannot be directly applied since the evolution is no longer unitary. The current paper utilizes a recently introduced Schrödingerization method that converts non-Hermitian dynamics into a Schrödinger form for the artificial boundary problems [S. Jin, N. Liu, and Y. Yu, Quantum Simulation of Partial Differential Equations via Schrödingerisation, preprint, arXiv:2212.13969, 2022], [S. Jin, N. Liu, and Y. Yu, Phys. Rev. A, 108 (2023), 032603]. We implement this method for three types of ABCs, including the complex absorbing potential technique, perfectly matched layer methods, and Dirichlet-to-Neumann approach. We analyze the query complexity of these algorithms and perform numerical experiments to demonstrate the validity of this approach. This helps to bridge the gap between available quantum algorithms and computational models for quantum dynamics in unbounded domains.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141585762","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 4, Page C297-C322, August 2024. Abstract. We present a physics-informed machine learning (PIML) approach for the approximation of slow invariant manifolds of singularly perturbed systems, providing functionals in an explicit form that facilitates the construction and numerical integration of reduced-order models (ROMs). The proposed scheme solves the partial differential equation corresponding to the invariance equation (IE) within the geometric singular perturbation theory (GSPT) framework. For the solution of the IE, we used two neural network structures, namely, feedforward neural networks and random projection neural networks, with symbolic differentiation for the computation of the gradients required for the learning process. The efficiency of our PIML method is assessed via three benchmark problems, namely, the Michaelis–Menten, the target-mediated drug disposition reaction mechanism, and the 3D Sel’kov model. We show that the proposed PIML scheme provides approximations of equivalent or even higher accuracy than those provided by other traditional GSPT-based methods, and importantly, for any practical purposes, it is not affected by the magnitude of the perturbation parameter. This is of particular importance because there are many systems for which the gap between the fast and slow timescales is not that big, but still, ROMs can be constructed. A comparison of the computational costs between symbolic, automatic, and numerical approximation of the required derivatives in the learning process is also provided. 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://epubs.siam.org/doi/suppl/10.1137/23M1602991/suppl_file/131735_1_supp_551502_s5k7wy_sc.pdf and https://epubs.siam.org/doi/suppl/10.1137/23M1602991/suppl_file/SISC_PIML_SIMs_SP-main.zip.
{"title":"Slow Invariant Manifolds of Singularly Perturbed Systems via Physics-Informed Machine Learning","authors":"Dimitrios Patsatzis, Gianluca Fabiani, Lucia Russo, Constantinos Siettos","doi":"10.1137/23m1602991","DOIUrl":"https://doi.org/10.1137/23m1602991","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page C297-C322, August 2024. <br/> Abstract. We present a physics-informed machine learning (PIML) approach for the approximation of slow invariant manifolds of singularly perturbed systems, providing functionals in an explicit form that facilitates the construction and numerical integration of reduced-order models (ROMs). The proposed scheme solves the partial differential equation corresponding to the invariance equation (IE) within the geometric singular perturbation theory (GSPT) framework. For the solution of the IE, we used two neural network structures, namely, feedforward neural networks and random projection neural networks, with symbolic differentiation for the computation of the gradients required for the learning process. The efficiency of our PIML method is assessed via three benchmark problems, namely, the Michaelis–Menten, the target-mediated drug disposition reaction mechanism, and the 3D Sel’kov model. We show that the proposed PIML scheme provides approximations of equivalent or even higher accuracy than those provided by other traditional GSPT-based methods, and importantly, for any practical purposes, it is not affected by the magnitude of the perturbation parameter. This is of particular importance because there are many systems for which the gap between the fast and slow timescales is not that big, but still, ROMs can be constructed. A comparison of the computational costs between symbolic, automatic, and numerical approximation of the required derivatives in the learning process is also provided. 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://epubs.siam.org/doi/suppl/10.1137/23M1602991/suppl_file/131735_1_supp_551502_s5k7wy_sc.pdf and https://epubs.siam.org/doi/suppl/10.1137/23M1602991/suppl_file/SISC_PIML_SIMs_SP-main.zip.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141575185","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}
Peter Ohm, Jesus Bonilla, Edward Phillips, John N. Shadid, Michael Crockatt, Ray S. Tuminaro, Jonathan Hu, Xian-Zhu Tang
SIAM Journal on Scientific Computing, Ahead of Print. Abstract. This study investigates multiphysics block preconditioners that are critical in devising scalable Newton–Krylov iterative solvers for longer time-scale fully implicit fluid plasma models. The specific model of interest is the visco-resistive, low Mach number, compressible magnetohydrodynamics (MHD) model. This model describes the dynamics of conducting fluids in the presence of electromagnetic fields and can be used to study aspects of astrophysical phenomena, important science and technology applications, and basic plasma physics. The specific application of interest that motivates this study is the macroscopic simulation of longer time-scale stability and disruptions of magnetic confinement fusion devices, specifically the ITER Tokamak. The computational solution of the governing balance equations for mass, momentum, heat transfer, and magnetic induction for resistive MHD systems can be extremely challenging. These difficulties arise from both the strong nonlinear, nonsymmetric coupling of fluid and electromagnetic phenomena as well as the significant range of time and length scales that the interactions of these physical mechanisms produce. To handle the range of time and spatial scales of interest, a fully implicit unstructured variational multiscale finite element formulation is employed. For the scalable solution of the Newton linearized systems, fully coupled block preconditioners are designed to leverage algebraic multigrid subsolves. Results are presented for the strong and weak scaling of the method as well as the robustness of these techniques for a large range of Lundquist numbers.
{"title":"Scalable Multiphysics Block Preconditioning for Low Mach Number Compressible Resistive MHD with Application to Magnetic Confinement Fusion","authors":"Peter Ohm, Jesus Bonilla, Edward Phillips, John N. Shadid, Michael Crockatt, Ray S. Tuminaro, Jonathan Hu, Xian-Zhu Tang","doi":"10.1137/23m1582667","DOIUrl":"https://doi.org/10.1137/23m1582667","url":null,"abstract":"SIAM Journal on Scientific Computing, Ahead of Print. <br/> Abstract. This study investigates multiphysics block preconditioners that are critical in devising scalable Newton–Krylov iterative solvers for longer time-scale fully implicit fluid plasma models. The specific model of interest is the visco-resistive, low Mach number, compressible magnetohydrodynamics (MHD) model. This model describes the dynamics of conducting fluids in the presence of electromagnetic fields and can be used to study aspects of astrophysical phenomena, important science and technology applications, and basic plasma physics. The specific application of interest that motivates this study is the macroscopic simulation of longer time-scale stability and disruptions of magnetic confinement fusion devices, specifically the ITER Tokamak. The computational solution of the governing balance equations for mass, momentum, heat transfer, and magnetic induction for resistive MHD systems can be extremely challenging. These difficulties arise from both the strong nonlinear, nonsymmetric coupling of fluid and electromagnetic phenomena as well as the significant range of time and length scales that the interactions of these physical mechanisms produce. To handle the range of time and spatial scales of interest, a fully implicit unstructured variational multiscale finite element formulation is employed. For the scalable solution of the Newton linearized systems, fully coupled block preconditioners are designed to leverage algebraic multigrid subsolves. Results are presented for the strong and weak scaling of the method as well as the robustness of these techniques for a large range of Lundquist numbers.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141575181","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}
Elisa Baioni, Antoine Lejay, Géraldine Pichot, Giovanni Michele Porta
SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2202-A2223, August 2024. Abstract. Diffusive transport in media with discontinuous properties is a challenging problem that arises in many applications. This paper focuses on one dimensional discontinuous media with generalized permeable boundary conditions at the discontinuity interface. It presents novel analytical expressions from the method of images to simulate diffusive processes, such as mass or thermal transport. The analytical expressions are used to formulate a generalization of the existing Skew Brownian Motion, HYMLA, and Uffink’s method, here named as GSBM, GHYMLA, and GUM, respectively, to handle generic interface conditions. The algorithms rely upon the random walk method and are tested by simulating transport in a bimaterial and in a multilayered medium with piecewise constant properties. The results indicate that the GUM algorithm provides the best performance in terms of accuracy and computational cost. The methods proposed can be applied for simulation of a wide range of differential problems.
{"title":"Modeling Diffusion in One Dimensional Discontinuous Media under Generalized Permeable Interface Conditions: Theory and Algorithms","authors":"Elisa Baioni, Antoine Lejay, Géraldine Pichot, Giovanni Michele Porta","doi":"10.1137/23m1590846","DOIUrl":"https://doi.org/10.1137/23m1590846","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2202-A2223, August 2024. <br/> Abstract. Diffusive transport in media with discontinuous properties is a challenging problem that arises in many applications. This paper focuses on one dimensional discontinuous media with generalized permeable boundary conditions at the discontinuity interface. It presents novel analytical expressions from the method of images to simulate diffusive processes, such as mass or thermal transport. The analytical expressions are used to formulate a generalization of the existing Skew Brownian Motion, HYMLA, and Uffink’s method, here named as GSBM, GHYMLA, and GUM, respectively, to handle generic interface conditions. The algorithms rely upon the random walk method and are tested by simulating transport in a bimaterial and in a multilayered medium with piecewise constant properties. The results indicate that the GUM algorithm provides the best performance in terms of accuracy and computational cost. The methods proposed can be applied for simulation of a wide range of differential problems.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141551534","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 4, Page A2178-A2201, August 2024. Abstract. Small parameters in partial differential equations can give rise to solutions with sharp inner layers that evolve over time. However, the standard model reduction method becomes inefficient when applied to these problems due to the slow decaying Kolmogorov [math]-width of the solution manifold. To address this issue, a natural approach is to transform the equation in such a way that the transformed solution manifold exhibits a fast decaying Kolmogorov [math]-width. In this paper, we focus on the Allen–Cahn equation as a model problem. We employ asymptotic analysis to identify slow variables and perform a transformation of the partial differential equations accordingly. Subsequently, we apply the proper orthogonal decomposition method and a QR discrete empirical interpolation method (qDEIM) technique to the transformed equation with the slow variables. Numerical experiments demonstrate that the new model reduction method yields significantly improved results compared to direct model reduction applied to the original equation. Furthermore, this approach can be extended to other equations, such as the convection equation and the Burgers equation. 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/toreanony/TransformedModelReduction and in the supplementary materials (TransformedModelReduction-master.zip [19.1KB]).
{"title":"Transformed Model Reduction for Partial Differential Equations with Sharp Inner Layers","authors":"Tianyou Tang, Xianmin Xu","doi":"10.1137/23m1589980","DOIUrl":"https://doi.org/10.1137/23m1589980","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2178-A2201, August 2024. <br/> Abstract. Small parameters in partial differential equations can give rise to solutions with sharp inner layers that evolve over time. However, the standard model reduction method becomes inefficient when applied to these problems due to the slow decaying Kolmogorov [math]-width of the solution manifold. To address this issue, a natural approach is to transform the equation in such a way that the transformed solution manifold exhibits a fast decaying Kolmogorov [math]-width. In this paper, we focus on the Allen–Cahn equation as a model problem. We employ asymptotic analysis to identify slow variables and perform a transformation of the partial differential equations accordingly. Subsequently, we apply the proper orthogonal decomposition method and a QR discrete empirical interpolation method (qDEIM) technique to the transformed equation with the slow variables. Numerical experiments demonstrate that the new model reduction method yields significantly improved results compared to direct model reduction applied to the original equation. Furthermore, this approach can be extended to other equations, such as the convection equation and the Burgers equation. 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/toreanony/TransformedModelReduction and in the supplementary materials (TransformedModelReduction-master.zip [19.1KB]).","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141553221","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 4, Page B377-B402, August 2024. Abstract. This paper describes a class of shape optimization problems for optical metamaterials comprised of periodic microscale inclusions composed of a dielectric, low-dimensional material suspended in a nonmagnetic bulk dielectric. The shape optimization approach is based on a homogenization theory for time-harmonic Maxwell’s equations that describes effective material parameters for the propagation of electromagnetic waves through the metamaterial. The control parameter of the optimization is a deformation field representing the deviation of the microscale geometry from a reference configuration of the cell problem. This allows for describing the homogenized effective permittivity tensor as a function of the deformation field. We show that the underlying deformed cell problem is well-posed and regular. This, in turn, proves that the shape optimization problem is well-posed. In addition, a numerical scheme is formulated that utilizes an adjoint formulation with either gradient descent or BFGS as optimization algorithms. The developed algorithm is tested numerically on a number of prototypical shape optimization problems with a prescribed effective permittivity tensor as the target. 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://zenodo.org/records/10459309.
{"title":"Shape Optimization of Optical Microscale Inclusions","authors":"Manaswinee Bezbaruah, Matthias Maier, Winnifried Wollner","doi":"10.1137/23m158262x","DOIUrl":"https://doi.org/10.1137/23m158262x","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page B377-B402, August 2024. <br/> Abstract. This paper describes a class of shape optimization problems for optical metamaterials comprised of periodic microscale inclusions composed of a dielectric, low-dimensional material suspended in a nonmagnetic bulk dielectric. The shape optimization approach is based on a homogenization theory for time-harmonic Maxwell’s equations that describes effective material parameters for the propagation of electromagnetic waves through the metamaterial. The control parameter of the optimization is a deformation field representing the deviation of the microscale geometry from a reference configuration of the cell problem. This allows for describing the homogenized effective permittivity tensor as a function of the deformation field. We show that the underlying deformed cell problem is well-posed and regular. This, in turn, proves that the shape optimization problem is well-posed. In addition, a numerical scheme is formulated that utilizes an adjoint formulation with either gradient descent or BFGS as optimization algorithms. The developed algorithm is tested numerically on a number of prototypical shape optimization problems with a prescribed effective permittivity tensor as the target. 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://zenodo.org/records/10459309.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141551535","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 4, Page A2150-A2177, August 2024. Abstract. In this paper, we present a fast divergence-free spectral algorithm for the curl-curl problem. Divergence-free bases in two and three dimensions are constructed by using the generalized Jacobi polynomials. An accurate spectral method with exact preservation of the divergence-free constraint pointwisely is then proposed, and its corresponding error estimate is established. We then present a highly efficient solution algorithm based on a combination of the matrix-free preconditioned Krylov subspace iterative method and a fully diagonalizable auxiliary problem, which is derived from the spectral discretizations of generalized eigenvalue problems of Laplace and biharmonic operators. We rigorously prove that the dimensions of the invariant subspace of the preconditioned linear system resulting from the divergence-free spectral method with respect to the dominant eigenvalue 1 are [math] and [math] for two- and three-dimensional problems with [math] and [math] unknowns, respectively. Thus, the proposed method usually takes only several iterations to converge, and, astonishingly, as the problem size (polynomial order) increases, the number of iterations will decrease, even for highly indefinite system and oscillatory solutions. As a result, the computational cost of the solution algorithm is only a small multiple of [math] and [math] floating number operations for two- and three-dimensional problems, respectively. Plenty of numerical examples for solving the curl-curl problem with both constant and variable coefficients in two and three dimensions are presented to demonstrate the accuracy and efficiency of the proposed method.
{"title":"A Highly Efficient and Accurate Divergence-Free Spectral Method for the Curl-Curl Equation in Two and Three Dimensions","authors":"Lechang Qin, Changtao Sheng, Zhiguo Yang","doi":"10.1137/23m1587038","DOIUrl":"https://doi.org/10.1137/23m1587038","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2150-A2177, August 2024. <br/> Abstract. In this paper, we present a fast divergence-free spectral algorithm for the curl-curl problem. Divergence-free bases in two and three dimensions are constructed by using the generalized Jacobi polynomials. An accurate spectral method with exact preservation of the divergence-free constraint pointwisely is then proposed, and its corresponding error estimate is established. We then present a highly efficient solution algorithm based on a combination of the matrix-free preconditioned Krylov subspace iterative method and a fully diagonalizable auxiliary problem, which is derived from the spectral discretizations of generalized eigenvalue problems of Laplace and biharmonic operators. We rigorously prove that the dimensions of the invariant subspace of the preconditioned linear system resulting from the divergence-free spectral method with respect to the dominant eigenvalue 1 are [math] and [math] for two- and three-dimensional problems with [math] and [math] unknowns, respectively. Thus, the proposed method usually takes only several iterations to converge, and, astonishingly, as the problem size (polynomial order) increases, the number of iterations will decrease, even for highly indefinite system and oscillatory solutions. As a result, the computational cost of the solution algorithm is only a small multiple of [math] and [math] floating number operations for two- and three-dimensional problems, respectively. Plenty of numerical examples for solving the curl-curl problem with both constant and variable coefficients in two and three dimensions are presented to demonstrate the accuracy and efficiency of the proposed method.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141517777","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}
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