SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page C249-C271, June 2024. Abstract. In this work, we introduce a new definition of observability for dynamical systems, formulated on the principles of dynamic optimization. This definition gives rise to the concept of an effective region, specifically designed for partial differential equations (PDEs). The usefulness of these concepts is demonstrated through examples of state estimation using observational information for PDEs in a limited area. The findings empower a more efficient analysis of PDE observability. By confining computations to an effective region significantly smaller than the overall region in which the PDE is defined, we demonstrate a substantial reduction in computational demand of evaluating observability. As an application of observability and effective region, we propose a learning-based surrogate data assimilation (DA) model for efficient state estimation in a limited area. Our model employs a feedforward neural network for online computation, eliminating the need for integrating high-dimensional limited-area models. This approach offers significant computational advantages over traditional DA algorithms. Furthermore, our method avoids the requirement of lateral boundary conditions for the limited-area model in both online and offline computations.
{"title":"Observability and Effective Region of Partial Differential Equations with Application to Data Assimilation","authors":"Wei Kang, Liang Xu, Hong Zhou","doi":"10.1137/23m1586690","DOIUrl":"https://doi.org/10.1137/23m1586690","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page C249-C271, June 2024. <br/> Abstract. In this work, we introduce a new definition of observability for dynamical systems, formulated on the principles of dynamic optimization. This definition gives rise to the concept of an effective region, specifically designed for partial differential equations (PDEs). The usefulness of these concepts is demonstrated through examples of state estimation using observational information for PDEs in a limited area. The findings empower a more efficient analysis of PDE observability. By confining computations to an effective region significantly smaller than the overall region in which the PDE is defined, we demonstrate a substantial reduction in computational demand of evaluating observability. As an application of observability and effective region, we propose a learning-based surrogate data assimilation (DA) model for efficient state estimation in a limited area. Our model employs a feedforward neural network for online computation, eliminating the need for integrating high-dimensional limited-area models. This approach offers significant computational advantages over traditional DA algorithms. Furthermore, our method avoids the requirement of lateral boundary conditions for the limited-area model in both online and offline computations.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"18 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140938910","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}
Benjamin Boutin, Anaïs Crestetto, Nicolas Crouseilles, Josselin Massot
SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1574-A1598, June 2024. Abstract. In this work, Lawson type numerical methods are studied to solve Vlasov type equations on a phase space grid. These time integrators are known to satisfy enhanced stability properties in this context since they do not suffer from the stability condition induced from the linear part. We introduce here a class of modified Lawson integrators in which the linear part is approximated in such a way that some geometric properties of the underlying model are preserved, which has important consequences for the analysis of the scheme. Several Vlasov–Maxwell examples are presented to illustrate the good behavior of the approach.
{"title":"Modified Lawson Methods for Vlasov Equations","authors":"Benjamin Boutin, Anaïs Crestetto, Nicolas Crouseilles, Josselin Massot","doi":"10.1137/22m154301x","DOIUrl":"https://doi.org/10.1137/22m154301x","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1574-A1598, June 2024. <br/>Abstract. In this work, Lawson type numerical methods are studied to solve Vlasov type equations on a phase space grid. These time integrators are known to satisfy enhanced stability properties in this context since they do not suffer from the stability condition induced from the linear part. We introduce here a class of modified Lawson integrators in which the linear part is approximated in such a way that some geometric properties of the underlying model are preserved, which has important consequences for the analysis of the scheme. Several Vlasov–Maxwell examples are presented to illustrate the good behavior of the approach.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"67 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140938893","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 A1549-A1573, June 2024. Abstract. The Riesz maps of the [math] de Rham complex frequently arise as subproblems in the construction of fast preconditioners for more complicated problems. In this work, we present multigrid solvers for high-order finite-element discretizations of these Riesz maps with the same time and space complexity as sum-factorized operator application, i.e., with optimal complexity in polynomial degree in the context of Krylov methods. The key idea of our approach is to build new finite elements for each space in the de Rham complex with orthogonality properties in both the [math]- and [math]-inner products ([math] on the reference hexahedron. The resulting sparsity enables the fast solution of the patch problems arising in the Pavarino, Arnold–Falk–Winther, and Hiptmair space decompositions in the separable case. In the nonseparable case, the method can be applied to an auxiliary operator that is sparse by construction. With exact Cholesky factorizations of the sparse patch problems, the application complexity is optimal, but the setup costs and storage are not. We overcome this with the finer Hiptmair space decomposition and the use of incomplete Cholesky factorizations imposing the sparsity pattern arising from static condensation, which applies whether static condensation is used for the solver or not. This yields multigrid relaxations with time and space complexity that are both optimal in the polynomial degree. 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://doi.org/10.5281/zenodo.7358044 and in the supplementary materials (pmg_de_rham.zip [61.2KB]).
{"title":"Multigrid Solvers for the de Rham Complex with Optimal Complexity in Polynomial Degree","authors":"Pablo D. Brubeck, Patrick E. Farrell","doi":"10.1137/22m1537370","DOIUrl":"https://doi.org/10.1137/22m1537370","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1549-A1573, June 2024. <br/> Abstract. The Riesz maps of the [math] de Rham complex frequently arise as subproblems in the construction of fast preconditioners for more complicated problems. In this work, we present multigrid solvers for high-order finite-element discretizations of these Riesz maps with the same time and space complexity as sum-factorized operator application, i.e., with optimal complexity in polynomial degree in the context of Krylov methods. The key idea of our approach is to build new finite elements for each space in the de Rham complex with orthogonality properties in both the [math]- and [math]-inner products ([math] on the reference hexahedron. The resulting sparsity enables the fast solution of the patch problems arising in the Pavarino, Arnold–Falk–Winther, and Hiptmair space decompositions in the separable case. In the nonseparable case, the method can be applied to an auxiliary operator that is sparse by construction. With exact Cholesky factorizations of the sparse patch problems, the application complexity is optimal, but the setup costs and storage are not. We overcome this with the finer Hiptmair space decomposition and the use of incomplete Cholesky factorizations imposing the sparsity pattern arising from static condensation, which applies whether static condensation is used for the solver or not. This yields multigrid relaxations with time and space complexity that are both optimal in the polynomial degree. 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://doi.org/10.5281/zenodo.7358044 and in the supplementary materials (pmg_de_rham.zip [61.2KB]).","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"112 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140883383","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}
Tianpei Cheng, Haijian Yang, Jizu Huang, Chao Yang
SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page B306-B330, June 2024. Abstract. This paper proposes an adaptive space-time algorithm based on domain decomposition for the large-scale simulation of a recently developed thermodynamically consistent reservoir problem. In the approach, the bound constraints are represented by means of a minimum-type complementarity function to enforce the positivity of the reservoir model, and a space-time mixed finite element method is applied for the parallel-in-time monolithic discretization. In particular, we propose a time-adaptive strategy using the improved backward differencing formula of second order, to take full advantage of the high degree of space-time parallelism. Moreover, the complicated dynamics with higher nonlinearity of space-time discretization require some innovative nonlinear and linear solution strategies. Therefore, we present a class of modified semismooth Newton algorithms to enhance the convergence rate of nonlinear iterations. Multilevel space-time restricted additive Schwarz algorithms, whose subdomains cover both space and time variables, are also studied for domain decomposition-based preconditioning. Numerical experiments demonstrate the robustness and parallel scalability of the proposed adaptive space-time algorithm on a supercomputer with tens of thousands of processor cores.
{"title":"Adaptive Space-Time Domain Decomposition for Multiphase Flow in Porous Media with Bound Constraints","authors":"Tianpei Cheng, Haijian Yang, Jizu Huang, Chao Yang","doi":"10.1137/23m1578139","DOIUrl":"https://doi.org/10.1137/23m1578139","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page B306-B330, June 2024. <br/> Abstract. This paper proposes an adaptive space-time algorithm based on domain decomposition for the large-scale simulation of a recently developed thermodynamically consistent reservoir problem. In the approach, the bound constraints are represented by means of a minimum-type complementarity function to enforce the positivity of the reservoir model, and a space-time mixed finite element method is applied for the parallel-in-time monolithic discretization. In particular, we propose a time-adaptive strategy using the improved backward differencing formula of second order, to take full advantage of the high degree of space-time parallelism. Moreover, the complicated dynamics with higher nonlinearity of space-time discretization require some innovative nonlinear and linear solution strategies. Therefore, we present a class of modified semismooth Newton algorithms to enhance the convergence rate of nonlinear iterations. Multilevel space-time restricted additive Schwarz algorithms, whose subdomains cover both space and time variables, are also studied for domain decomposition-based preconditioning. Numerical experiments demonstrate the robustness and parallel scalability of the proposed adaptive space-time algorithm on a supercomputer with tens of thousands of processor cores.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"20 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140883469","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 B229-B253, June 2024. Abstract. In this paper we present a framework for the construction and implementation of general virtual element spaces based on projections built from constrained least squares problems. Building on the triples used for finite element spaces, we introduce the concept of a virtual element method (VEM) tuple which encodes the necessary building blocks to construct these projections. Using this approach, a wide range of virtual element spaces can be defined. We discuss [math]-conforming spaces for [math] as well as divergence and curl free spaces. This general framework has the advantage of being easily integrated into any existing finite element package, and we demonstrate this within the open source software package Dune. 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://gitlab.dune-project.org/dune-fem/dune-vem-paper and in the supplementary materials (128492_2_supp_546442_s3hsrj.zip [22KB]).
{"title":"A Framework for Implementing General Virtual Element Spaces","authors":"Andreas Dedner, Alice Hodson","doi":"10.1137/23m1573653","DOIUrl":"https://doi.org/10.1137/23m1573653","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page B229-B253, June 2024. <br/> Abstract. In this paper we present a framework for the construction and implementation of general virtual element spaces based on projections built from constrained least squares problems. Building on the triples used for finite element spaces, we introduce the concept of a virtual element method (VEM) tuple which encodes the necessary building blocks to construct these projections. Using this approach, a wide range of virtual element spaces can be defined. We discuss [math]-conforming spaces for [math] as well as divergence and curl free spaces. This general framework has the advantage of being easily integrated into any existing finite element package, and we demonstrate this within the open source software package Dune. 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://gitlab.dune-project.org/dune-fem/dune-vem-paper and in the supplementary materials (128492_2_supp_546442_s3hsrj.zip [22KB]).","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"82 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140829929","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 B280-B305, June 2024. Abstract. We consider a vibrational system control problem over a finite time horizon. The performance measure of the system is taken to be a [math]-mixed [math] norm which generalizes the standard [math] norm. We present an algorithm for efficient calculation of this norm in the case when the system is parameter dependent and the number of inputs or outputs of the system is significantly smaller than the order of the system. Our approach is based on a novel procedure which is not based on solving Lyapunov equations and which takes into account the structure of the system. We use a characterization of the [math] norm given in terms of integrals which we solve using adaptive quadrature rules. This enables us to use recycling strategies as well as parallelization. The efficiency of the new algorithm allows for an analysis of the influence of various system parameters and different finite time horizons on the value of the [math]-mixed [math] norm. We illustrate our approach by numerical examples concerning an [math]-mass oscillator with one damper.
{"title":"Finite Time Horizon Mixed Control of Vibrational Systems","authors":"Ivica Nakić, Marinela Pilj Vidaković, Zoran Tomljanović","doi":"10.1137/22m1488648","DOIUrl":"https://doi.org/10.1137/22m1488648","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page B280-B305, June 2024. <br/> Abstract. We consider a vibrational system control problem over a finite time horizon. The performance measure of the system is taken to be a [math]-mixed [math] norm which generalizes the standard [math] norm. We present an algorithm for efficient calculation of this norm in the case when the system is parameter dependent and the number of inputs or outputs of the system is significantly smaller than the order of the system. Our approach is based on a novel procedure which is not based on solving Lyapunov equations and which takes into account the structure of the system. We use a characterization of the [math] norm given in terms of integrals which we solve using adaptive quadrature rules. This enables us to use recycling strategies as well as parallelization. The efficiency of the new algorithm allows for an analysis of the influence of various system parameters and different finite time horizons on the value of the [math]-mixed [math] norm. We illustrate our approach by numerical examples concerning an [math]-mass oscillator with one damper.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":"11 1","pages":""},"PeriodicalIF":3.1,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140829835","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 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":"17 1","pages":""},"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":"23 1","pages":""},"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":"60 1","pages":""},"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":"12 1","pages":""},"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}
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