SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page B254-B279, June 2024. Abstract. This work presents a novel bound-preserving and positivity-preserving direct arbitrary Lagrangian–Eulerian discontinuous Galerkin (ALE-DG) method for compressible multimedium flows by solving the five-equation transport model. The proposed method satisfies the discrete geometric conservation law (D-GCL) which indicates that uniform flow is precisely preserved during the simulation. More importantly, based on the D-GCL condition, we present a theoretical analysis on designing an efficient bound-preserving and positivity-preserving limiting strategy, which is able to maintain the boundedness of the volume fraction and the positivity of the partial density and internal energy, with the aim of avoiding the occurrence of inadmissible solutions and meanwhile improving the computational robustness. The accuracy and robustness of the proposed method are demonstrated by various one- and two-dimensional benchmark test cases. The numerical results verify the well capacity of the proposed high-order ALE-DG method for compressible multimedium flows with both the ideal and stiffened gas equation of state.
{"title":"A Bound-Preserving and Positivity-Preserving High-Order Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Method for Compressible Multi-Medium Flows","authors":"Fan Zhang, Jian Cheng","doi":"10.1137/23m1588810","DOIUrl":"https://doi.org/10.1137/23m1588810","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page B254-B279, June 2024. <br/>Abstract. This work presents a novel bound-preserving and positivity-preserving direct arbitrary Lagrangian–Eulerian discontinuous Galerkin (ALE-DG) method for compressible multimedium flows by solving the five-equation transport model. The proposed method satisfies the discrete geometric conservation law (D-GCL) which indicates that uniform flow is precisely preserved during the simulation. More importantly, based on the D-GCL condition, we present a theoretical analysis on designing an efficient bound-preserving and positivity-preserving limiting strategy, which is able to maintain the boundedness of the volume fraction and the positivity of the partial density and internal energy, with the aim of avoiding the occurrence of inadmissible solutions and meanwhile improving the computational robustness. The accuracy and robustness of the proposed method are demonstrated by various one- and two-dimensional benchmark test cases. The numerical results verify the well capacity of the proposed high-order ALE-DG method for compressible multimedium flows with both the ideal and stiffened gas equation of state.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140829924","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}
Christian Bayer, Chiheb Ben Hammouda, Raúl Tempone
SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1514-A1548, June 2024. Abstract. The multilevel Monte Carlo (MLMC) method is highly efficient for estimating expectations of a functional of a solution to a stochastic differential equation (SDE). However, MLMC estimators may be unstable and have a poor (noncanonical) complexity in the case of low regularity of the functional. To overcome this issue, we extend our previously introduced idea of numerical smoothing in [Quant. Finance, 23 (2023), pp. 209–227], in the context of deterministic quadrature methods to the MLMC setting. The numerical smoothing technique is based on root-finding methods combined with one-dimensional numerical integration with respect to a single well-chosen variable. This study is motivated by the computation of probabilities of events, pricing options with a discontinuous payoff, and density estimation problems for dynamics where the discretization of the underlying stochastic processes is necessary. The analysis and numerical experiments reveal that the numerical smoothing significantly improves the strong convergence and, consequently, the complexity and robustness (by making the kurtosis at deep levels bounded) of the MLMC method. In particular, we show that numerical smoothing enables recovering the MLMC complexities obtained for Lipschitz functionals due to the optimal variance decay rate when using the Euler–Maruyama scheme. For the Milstein scheme, numerical smoothing recovers the canonical MLMC complexity, even for the nonsmooth integrand mentioned above. Finally, our approach efficiently estimates univariate and multivariate density functions.
{"title":"Multilevel Monte Carlo with Numerical Smoothing for Robust and Efficient Computation of Probabilities and Densities","authors":"Christian Bayer, Chiheb Ben Hammouda, Raúl Tempone","doi":"10.1137/22m1495718","DOIUrl":"https://doi.org/10.1137/22m1495718","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1514-A1548, June 2024. <br/>Abstract. The multilevel Monte Carlo (MLMC) method is highly efficient for estimating expectations of a functional of a solution to a stochastic differential equation (SDE). However, MLMC estimators may be unstable and have a poor (noncanonical) complexity in the case of low regularity of the functional. To overcome this issue, we extend our previously introduced idea of numerical smoothing in [Quant. Finance, 23 (2023), pp. 209–227], in the context of deterministic quadrature methods to the MLMC setting. The numerical smoothing technique is based on root-finding methods combined with one-dimensional numerical integration with respect to a single well-chosen variable. This study is motivated by the computation of probabilities of events, pricing options with a discontinuous payoff, and density estimation problems for dynamics where the discretization of the underlying stochastic processes is necessary. The analysis and numerical experiments reveal that the numerical smoothing significantly improves the strong convergence and, consequently, the complexity and robustness (by making the kurtosis at deep levels bounded) of the MLMC method. In particular, we show that numerical smoothing enables recovering the MLMC complexities obtained for Lipschitz functionals due to the optimal variance decay rate when using the Euler–Maruyama scheme. For the Milstein scheme, numerical smoothing recovers the canonical MLMC complexity, even for the nonsmooth integrand mentioned above. Finally, our approach efficiently estimates univariate and multivariate density functions.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140829820","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 3, Page A1373-A1396, June 2024. Abstract. We investigate additive Schwarz methods for semilinear elliptic problems with convex energy functionals, which have wide scientific applications. A key observation is that the convergence rates of both one- and two-level additive Schwarz methods have bounds independent of the nonlinear term in the problem. That is, the convergence rates do not deteriorate by the presence of nonlinearity, so that solving a semilinear problem requires no more iterations than a linear problem. Moreover, the two-level method is scalable in the sense that the convergence rate of the method depends on [math] and [math] only, where [math] and [math] are the typical diameters of an element and a subdomain, respectively, and [math] measures the overlap among the subdomains. Numerical results are provided to support our theoretical findings.
{"title":"Additive Schwarz Methods for Semilinear Elliptic Problems with Convex Energy Functionals: Convergence Rate Independent of Nonlinearity","authors":"Jongho Park","doi":"10.1137/23m159545x","DOIUrl":"https://doi.org/10.1137/23m159545x","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1373-A1396, June 2024. <br/>Abstract. We investigate additive Schwarz methods for semilinear elliptic problems with convex energy functionals, which have wide scientific applications. A key observation is that the convergence rates of both one- and two-level additive Schwarz methods have bounds independent of the nonlinear term in the problem. That is, the convergence rates do not deteriorate by the presence of nonlinearity, so that solving a semilinear problem requires no more iterations than a linear problem. Moreover, the two-level method is scalable in the sense that the convergence rate of the method depends on [math] and [math] only, where [math] and [math] are the typical diameters of an element and a subdomain, respectively, and [math] measures the overlap among the subdomains. Numerical results are provided to support our theoretical findings.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140829822","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 3, Page A1441-A1460, June 2024. Abstract.The classic exact simulation scheme for the Ornstein–Uhlenbeck driven stochastic volatility model is designed for the single volatility factor case. Extension to the multifactor case results in a cumbersome procedure requiring multiple numerical inversions of Laplace transforms and subsequent random sampling through numerical methods, resulting in it being perceptively slow to run. Moreover, for each volatility factor, the error is controlled by two parameters, ensuring difficult control of the bias. In this paper, we propose a new exact simulation scheme for the multifactor Ornstein–Uhlenbeck driven stochastic volatility model that is easier to implement, faster to run, and allows for an improved control of the error, which, in contrast to the existing method, is controlled by only one parameter, regardless of the number of volatility factors. Numerical results show that the proposed approach is three times faster than the original approach when one volatility factor is considered and 11 times faster when four volatility factors are considered, while still being theoretically exact.
{"title":"Exact Simulation of the Multifactor Ornstein–Uhlenbeck Driven Stochastic Volatility Model","authors":"Riccardo Brignone","doi":"10.1137/23m1595102","DOIUrl":"https://doi.org/10.1137/23m1595102","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1441-A1460, June 2024. <br/>Abstract.The classic exact simulation scheme for the Ornstein–Uhlenbeck driven stochastic volatility model is designed for the single volatility factor case. Extension to the multifactor case results in a cumbersome procedure requiring multiple numerical inversions of Laplace transforms and subsequent random sampling through numerical methods, resulting in it being perceptively slow to run. Moreover, for each volatility factor, the error is controlled by two parameters, ensuring difficult control of the bias. In this paper, we propose a new exact simulation scheme for the multifactor Ornstein–Uhlenbeck driven stochastic volatility model that is easier to implement, faster to run, and allows for an improved control of the error, which, in contrast to the existing method, is controlled by only one parameter, regardless of the number of volatility factors. Numerical results show that the proposed approach is three times faster than the original approach when one volatility factor is considered and 11 times faster when four volatility factors are considered, while still being theoretically exact.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140829960","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}
Enrico Facca, Gabriele Todeschi, Andrea Natale, Michele Benzi
SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1397-A1422, June 2024. Abstract. In this paper, we address the numerical solution of the quadratic optimal transport problem in its dynamical form, the so-called Benamou-Brenier formulation. When solved using interior point methods, the main computational bottleneck is the solution of large saddle point linear systems arising from the associated Newton-Raphson scheme. The main purpose of this paper is to design efficient preconditioners to solve these linear systems via iterative methods. Among the proposed preconditioners, we introduce one based on the partial commutation of the operators that compose the dual Schur complement of these saddle point linear systems, which we refer to as the [math]-preconditioner. A series of numerical tests show that the [math]-preconditioner is the most efficient among those presented, despite a performance deterioration in the last steps of the interior point method. It is in fact the only one having a CPU time that scales only slightly worse than linearly with respect to the number of unknowns used to discretize the problem.
{"title":"Efficient Preconditioners for Solving Dynamical Optimal Transport via Interior Point Methods","authors":"Enrico Facca, Gabriele Todeschi, Andrea Natale, Michele Benzi","doi":"10.1137/23m1570430","DOIUrl":"https://doi.org/10.1137/23m1570430","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1397-A1422, June 2024. <br/> Abstract. In this paper, we address the numerical solution of the quadratic optimal transport problem in its dynamical form, the so-called Benamou-Brenier formulation. When solved using interior point methods, the main computational bottleneck is the solution of large saddle point linear systems arising from the associated Newton-Raphson scheme. The main purpose of this paper is to design efficient preconditioners to solve these linear systems via iterative methods. Among the proposed preconditioners, we introduce one based on the partial commutation of the operators that compose the dual Schur complement of these saddle point linear systems, which we refer to as the [math]-preconditioner. A series of numerical tests show that the [math]-preconditioner is the most efficient among those presented, despite a performance deterioration in the last steps of the interior point method. It is in fact the only one having a CPU time that scales only slightly worse than linearly with respect to the number of unknowns used to discretize the problem.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140830093","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}
Ana Budiša, Xiaozhe Hu, Miroslav Kuchta, Kent-Andre Mardal, Ludmil Zikatanov
SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1461-A1486, June 2024. Abstract. We develop multilevel methods for interface-driven multiphysics problems that can be coupled across dimensions and where complexity and strength of the interface coupling deteriorates the performance of standard methods. We focus on aggregation-based algebraic multigrid methods with custom smoothers that preserve the coupling information on each coarse level. We prove that, with the proper choice of subspace splitting, we obtain uniform convergence in discretization and physical parameters in the two-level setting. Additionally, we show parameter robustness and scalability with regard to the number of the degrees of freedom of the system on several numerical examples related to the biophysical processes in the brain, namely, the electric signaling in excitable tissue modeled by bidomain, the extracellular-membrane-intracellular (EMI) model, and reduced EMI equations. 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/anabudisa/metric-amg-examples and in the supplementary materials (metric-amg-examples-master.zip [30KB]).
{"title":"Algebraic Multigrid Methods for Metric-Perturbed Coupled Problems","authors":"Ana Budiša, Xiaozhe Hu, Miroslav Kuchta, Kent-Andre Mardal, Ludmil Zikatanov","doi":"10.1137/23m1572076","DOIUrl":"https://doi.org/10.1137/23m1572076","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1461-A1486, June 2024. <br/>Abstract. We develop multilevel methods for interface-driven multiphysics problems that can be coupled across dimensions and where complexity and strength of the interface coupling deteriorates the performance of standard methods. We focus on aggregation-based algebraic multigrid methods with custom smoothers that preserve the coupling information on each coarse level. We prove that, with the proper choice of subspace splitting, we obtain uniform convergence in discretization and physical parameters in the two-level setting. Additionally, we show parameter robustness and scalability with regard to the number of the degrees of freedom of the system on several numerical examples related to the biophysical processes in the brain, namely, the electric signaling in excitable tissue modeled by bidomain, the extracellular-membrane-intracellular (EMI) model, and reduced EMI equations. 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/anabudisa/metric-amg-examples and in the supplementary materials (metric-amg-examples-master.zip [30KB]).","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140829821","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 3, Page A1423-A1440, June 2024. Abstract. A modification of Newton’s method for solving systems of [math] nonlinear equations is presented. The new matrix-free method is exact as opposed to a range of inexact Newton methods in the sense that both the Jacobians and the solutions to the linear Newton systems are computed without truncation. It relies on a given decomposition of a structurally dense invertible Jacobian of the residual into a product of [math] structurally sparse invertible elemental Jacobians according to the chain rule of differentiation. Inspired by the adjoint mode of algorithmic differentiation, explicit accumulation of the Jacobian of the residual is avoided. Prospective, generally applicable implementations of the new method can be based on similar ideas. Sparsity is exploited for the direct solution of the linear Newton systems. Optimal exploitation of sparsity yields various well-known computationally intractable combinatorial optimization problems in sparse linear algebra such as Bandwidth or Directed Elimination Ordering. The method is motivated in the context of a decomposition into elemental Jacobians with bandwidth [math] for [math]. In the likely scenario of [math], the computational cost of the standard Newton algorithm is dominated by the cost of accumulating the Jacobian of the residual. It can be estimated as [math], thus exceeding the cost of [math] for the direct solution of the linear Newton system. The new method reduces this cost to [math], yielding a potential improvement by a factor of [math]. Supporting run time measurements are presented for the tridiagonal case showing a reduction of the computational cost by [math]. Generalization yields the combinatorial Matrix-Free Exact Newton Step problem. We prove NP-completeness, and we present algorithmic components for building methods for the approximate solution. Potential applications of the matrix-free exact Newton method in machine learning of surrogates for computationally expensive nonlinear residuals are touched on briefly as part of various conclusions to be drawn.
{"title":"A Matrix-Free Exact Newton Method","authors":"Uwe Naumann","doi":"10.1137/23m157017x","DOIUrl":"https://doi.org/10.1137/23m157017x","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1423-A1440, June 2024. <br/> Abstract. A modification of Newton’s method for solving systems of [math] nonlinear equations is presented. The new matrix-free method is exact as opposed to a range of inexact Newton methods in the sense that both the Jacobians and the solutions to the linear Newton systems are computed without truncation. It relies on a given decomposition of a structurally dense invertible Jacobian of the residual into a product of [math] structurally sparse invertible elemental Jacobians according to the chain rule of differentiation. Inspired by the adjoint mode of algorithmic differentiation, explicit accumulation of the Jacobian of the residual is avoided. Prospective, generally applicable implementations of the new method can be based on similar ideas. Sparsity is exploited for the direct solution of the linear Newton systems. Optimal exploitation of sparsity yields various well-known computationally intractable combinatorial optimization problems in sparse linear algebra such as Bandwidth or Directed Elimination Ordering. The method is motivated in the context of a decomposition into elemental Jacobians with bandwidth [math] for [math]. In the likely scenario of [math], the computational cost of the standard Newton algorithm is dominated by the cost of accumulating the Jacobian of the residual. It can be estimated as [math], thus exceeding the cost of [math] for the direct solution of the linear Newton system. The new method reduces this cost to [math], yielding a potential improvement by a factor of [math]. Supporting run time measurements are presented for the tridiagonal case showing a reduction of the computational cost by [math]. Generalization yields the combinatorial Matrix-Free Exact Newton Step problem. We prove NP-completeness, and we present algorithmic components for building methods for the approximate solution. Potential applications of the matrix-free exact Newton method in machine learning of surrogates for computationally expensive nonlinear residuals are touched on briefly as part of various conclusions to be drawn.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140829928","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 3, Page B205-B228, June 2024. Abstract. This work introduces a parallel and rank-adaptive matrix integrator for dynamical low-rank approximation. The method is related to the previously proposed rank-adaptive basis update and Galerkin (BUG) integrator but differs significantly in that all arising differential equations, both for the basis and the Galerkin coefficients, are solved in parallel. Moreover, this approach eliminates the need for a potentially costly coefficient update with augmented basis matrices. The integrator also incorporates a new step rejection strategy that enhances the robustness of both the parallel integrator and the BUG integrator. By construction, the parallel integrator inherits the robust error bound of the BUG and projector-splitting integrators. Comparisons of the parallel and BUG integrators are presented by a series of numerical experiments which demonstrate the efficiency of the proposed method, for problems from radiative transfer and radiation therapy.
{"title":"A Parallel Rank-Adaptive Integrator for Dynamical Low-Rank Approximation","authors":"Gianluca Ceruti, Jonas Kusch, Christian Lubich","doi":"10.1137/23m1565103","DOIUrl":"https://doi.org/10.1137/23m1565103","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page B205-B228, June 2024. <br/> Abstract. This work introduces a parallel and rank-adaptive matrix integrator for dynamical low-rank approximation. The method is related to the previously proposed rank-adaptive basis update and Galerkin (BUG) integrator but differs significantly in that all arising differential equations, both for the basis and the Galerkin coefficients, are solved in parallel. Moreover, this approach eliminates the need for a potentially costly coefficient update with augmented basis matrices. The integrator also incorporates a new step rejection strategy that enhances the robustness of both the parallel integrator and the BUG integrator. By construction, the parallel integrator inherits the robust error bound of the BUG and projector-splitting integrators. Comparisons of the parallel and BUG integrators are presented by a series of numerical experiments which demonstrate the efficiency of the proposed method, for problems from radiative transfer and radiation therapy.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140829921","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}
Ahsan Ali, James J. Brannick, Karsten Kahl, Oliver A. Krzysik, Jacob B. Schroder, Ben S. Southworth
SIAM Journal on Scientific Computing, Ahead of Print. Abstract. This paper focuses on developing a reduction-based algebraic multigrid (AMG) method that is suitable for solving general (non)symmetric linear systems and is naturally robust from pure advection to pure diffusion. Initial motivation comes from a new reduction-based AMG approach, [math] (local approximate ideal restriction), that was developed for solving advection-dominated problems. Though this new solver is very effective in the advection-dominated regime, its performance degrades in cases where diffusion becomes dominant. This is consistent with the fact that in general, reduction-based AMG methods tend to suffer from growth in complexity and/or convergence rates as the problem size is increased, especially for diffusion-dominated problems in two or three dimensions. Motivated by the success of [math] in the advective regime, our aim in this paper is to generalize the AIR framework with the goal of improving the performance of the solver in diffusion-dominated regimes. To do so, we propose a novel way to combine mode constraints as used commonly in energy-minimization AMG methods with the local approximation of ideal operators used in [math]. The resulting constrained [math] algorithm is able to achieve fast scalable convergence on advective and diffusive problems. In addition, it is able to achieve standard low complexity hierarchies in the diffusive regime through aggressive coarsening, something that was previously difficult for reduction-based methods.
{"title":"Constrained Local Approximate Ideal Restriction for Advection-Diffusion Problems","authors":"Ahsan Ali, James J. Brannick, Karsten Kahl, Oliver A. Krzysik, Jacob B. Schroder, Ben S. Southworth","doi":"10.1137/23m1583442","DOIUrl":"https://doi.org/10.1137/23m1583442","url":null,"abstract":"SIAM Journal on Scientific Computing, Ahead of Print. <br/> Abstract. This paper focuses on developing a reduction-based algebraic multigrid (AMG) method that is suitable for solving general (non)symmetric linear systems and is naturally robust from pure advection to pure diffusion. Initial motivation comes from a new reduction-based AMG approach, [math] (local approximate ideal restriction), that was developed for solving advection-dominated problems. Though this new solver is very effective in the advection-dominated regime, its performance degrades in cases where diffusion becomes dominant. This is consistent with the fact that in general, reduction-based AMG methods tend to suffer from growth in complexity and/or convergence rates as the problem size is increased, especially for diffusion-dominated problems in two or three dimensions. Motivated by the success of [math] in the advective regime, our aim in this paper is to generalize the AIR framework with the goal of improving the performance of the solver in diffusion-dominated regimes. To do so, we propose a novel way to combine mode constraints as used commonly in energy-minimization AMG methods with the local approximation of ideal operators used in [math]. The resulting constrained [math] algorithm is able to achieve fast scalable convergence on advective and diffusive problems. In addition, it is able to achieve standard low complexity hierarchies in the diffusive regime through aggressive coarsening, something that was previously difficult for reduction-based methods.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140829834","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}
Konstantin Althaus, Iason Papaioannou, Elisabeth Ullmann
SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1487-A1513, June 2024. Abstract. In this paper, we introduce a new algorithm for rare event estimation based on adaptive importance sampling. We consider a smoothed version of the optimal importance sampling density, which is approximated by an ensemble of interacting particles. The particle dynamics is governed by a McKean–Vlasov stochastic differential equation, which was introduced and analyzed in [Carrillo et al., Stud. Appl. Math., 148 (2022), pp. 1069–1140] for consensus-based sampling and optimization of posterior distributions arising in the context of Bayesian inverse problems. We develop automatic updates for the internal parameters of our algorithm. This includes a novel time step size controller for the exponential Euler method, which discretizes the particle dynamics. The behavior of all parameter updates depends on easy to interpret accuracy criteria specified by the user. We show in numerical experiments that our method is competitive to state-of-the-art adaptive importance sampling algorithms for rare event estimation, namely a sequential importance sampling method and the ensemble Kalman filter for rare event estimation. 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/AlthausKonstantin/rareeventestimation/tree/master/docs/figures_paper and in the supplementary materials (rareeventestimation-0.3.0.zip [9.66MB]).
{"title":"Consensus-Based Rare Event Estimation","authors":"Konstantin Althaus, Iason Papaioannou, Elisabeth Ullmann","doi":"10.1137/23m1565966","DOIUrl":"https://doi.org/10.1137/23m1565966","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1487-A1513, June 2024. <br/> Abstract. In this paper, we introduce a new algorithm for rare event estimation based on adaptive importance sampling. We consider a smoothed version of the optimal importance sampling density, which is approximated by an ensemble of interacting particles. The particle dynamics is governed by a McKean–Vlasov stochastic differential equation, which was introduced and analyzed in [Carrillo et al., Stud. Appl. Math., 148 (2022), pp. 1069–1140] for consensus-based sampling and optimization of posterior distributions arising in the context of Bayesian inverse problems. We develop automatic updates for the internal parameters of our algorithm. This includes a novel time step size controller for the exponential Euler method, which discretizes the particle dynamics. The behavior of all parameter updates depends on easy to interpret accuracy criteria specified by the user. We show in numerical experiments that our method is competitive to state-of-the-art adaptive importance sampling algorithms for rare event estimation, namely a sequential importance sampling method and the ensemble Kalman filter for rare event estimation. 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/AlthausKonstantin/rareeventestimation/tree/master/docs/figures_paper and in the supplementary materials (rareeventestimation-0.3.0.zip [9.66MB]).","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140829958","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}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号