Yanchen He, Paul Houston, Christoph Schwab, Thomas P Wihler
We study the fully explicit numerical approximation of a semilinear elliptic boundary value model problem, which features a monomial reaction and analytic forcing, in a bounded polygon $varOmega subset{mathbb{R}}^{2}$ with a finite number of straight edges. In particular, we analyse the convergence of $hp$-type iterative linearized Galerkin ($hp$-ILG) solvers. Our convergence analysis is carried out for conforming $hp$-finite element (FE) Galerkin discretizations on sequences of regular, simplicial partitions of $varOmega $, with geometric corner refinement, with polynomial degrees increasing in tandem with the geometric mesh refinement towards the corners of $varOmega $. For a sequence of discrete solutions generated by the ILG solver with a stopping criterion that is consistent with the exponential convergence of the exact $hp$-FE Galerkin solution we prove exponential convergence in $text{H}^{1}(varOmega )$ to the unique weak solution of the boundary value problem. Numerical experiments illustrate the exponential convergence of the numerical approximations obtained from the proposed scheme in terms of the number of degrees of freedom as well as of the computational complexity involved.
{"title":"Exponential convergence of hp-ILGFEM for semilinear elliptic boundary value problems with monomial reaction","authors":"Yanchen He, Paul Houston, Christoph Schwab, Thomas P Wihler","doi":"10.1093/imanum/draf030","DOIUrl":"https://doi.org/10.1093/imanum/draf030","url":null,"abstract":"We study the fully explicit numerical approximation of a semilinear elliptic boundary value model problem, which features a monomial reaction and analytic forcing, in a bounded polygon $varOmega subset{mathbb{R}}^{2}$ with a finite number of straight edges. In particular, we analyse the convergence of $hp$-type iterative linearized Galerkin ($hp$-ILG) solvers. Our convergence analysis is carried out for conforming $hp$-finite element (FE) Galerkin discretizations on sequences of regular, simplicial partitions of $varOmega $, with geometric corner refinement, with polynomial degrees increasing in tandem with the geometric mesh refinement towards the corners of $varOmega $. For a sequence of discrete solutions generated by the ILG solver with a stopping criterion that is consistent with the exponential convergence of the exact $hp$-FE Galerkin solution we prove exponential convergence in $text{H}^{1}(varOmega )$ to the unique weak solution of the boundary value problem. Numerical experiments illustrate the exponential convergence of the numerical approximations obtained from the proposed scheme in terms of the number of degrees of freedom as well as of the computational complexity involved.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"26 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144500473","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}
Zvonimir Bujanović, Luka Grubišić, Daniel Kressner, Hei Yin Lam
Various iterative eigenvalue solvers have been developed to compute parts of the spectrum for a large sparse matrix, including the power method, Krylov subspace methods, contour integral methods and preconditioned solvers such as the so-called locally optimally block preconditioned conjugate gradient (LOBPCG) method. All of these solvers rely on random matrices to determine, e.g., starting vectors that have, with high probability, a non-negligible overlap with the eigenvectors of interest. For this purpose, a safe and common choice are unstructured Gaussian random matrices. In this work, we investigate the use of random Khatri–Rao products in eigenvalue solvers. On the one hand, we establish a novel subspace embedding property that provides theoretical justification for the use of such structured random matrices. On the other hand, we highlight the potential algorithmic benefits when solving eigenvalue problems with Kronecker product structure, as they arise frequently from the discretization of eigenvalue problems for differential operators on tensor product domains. In particular, we consider the use of random Khatri–Rao products within a contour integral method and LOBPCG. Numerical experiments indicate that the gains for the contour integral method strongly depend on the ability to efficiently and accurately solve (shifted) matrix equations with low-rank right-hand side. The flexibility of LOBPCG to directly employ preconditioners makes it easier to benefit from Khatri–Rao product structure, at the expense of having less theoretical justification.
{"title":"Subspace embedding with random Khatri–Rao products and its application to eigensolvers","authors":"Zvonimir Bujanović, Luka Grubišić, Daniel Kressner, Hei Yin Lam","doi":"10.1093/imanum/draf043","DOIUrl":"https://doi.org/10.1093/imanum/draf043","url":null,"abstract":"Various iterative eigenvalue solvers have been developed to compute parts of the spectrum for a large sparse matrix, including the power method, Krylov subspace methods, contour integral methods and preconditioned solvers such as the so-called locally optimally block preconditioned conjugate gradient (LOBPCG) method. All of these solvers rely on random matrices to determine, e.g., starting vectors that have, with high probability, a non-negligible overlap with the eigenvectors of interest. For this purpose, a safe and common choice are unstructured Gaussian random matrices. In this work, we investigate the use of random Khatri–Rao products in eigenvalue solvers. On the one hand, we establish a novel subspace embedding property that provides theoretical justification for the use of such structured random matrices. On the other hand, we highlight the potential algorithmic benefits when solving eigenvalue problems with Kronecker product structure, as they arise frequently from the discretization of eigenvalue problems for differential operators on tensor product domains. In particular, we consider the use of random Khatri–Rao products within a contour integral method and LOBPCG. Numerical experiments indicate that the gains for the contour integral method strongly depend on the ability to efficiently and accurately solve (shifted) matrix equations with low-rank right-hand side. The flexibility of LOBPCG to directly employ preconditioners makes it easier to benefit from Khatri–Rao product structure, at the expense of having less theoretical justification.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"36 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144500767","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}
We study the two-dimensional Navier–Stokes equation with transport noise subject to periodic boundary conditions. Our main result is an error estimate for the time discretization showing a convergence rate of order (up to) 1/2. It holds with respect to mean square error convergence, whereas previously such a rate for the stochastic Navier–Stokes equations was only known with respect to convergence in probability. Our result is based on uniform-in-probability estimates for the continuous as well as the time-discrete solution exploiting the particular structure of the noise. Eventually, we perform numerical simulations for the corresponding problem on bounded domains with no-slip boundary conditions. They suggest the same convergence rate as proved for the periodic problem hinging sensitively on the compatibility of the data. We also compare the energy profiles with those for corresponding problems with additive or multiplicative Itô-type noise.
{"title":"Mean square temporal error estimates for the two-dimensional stochastic Navier–Stokes equations with transport noise","authors":"D Breit, T C Moyo, A Prohl, J Wichmann","doi":"10.1093/imanum/draf042","DOIUrl":"https://doi.org/10.1093/imanum/draf042","url":null,"abstract":"We study the two-dimensional Navier–Stokes equation with transport noise subject to periodic boundary conditions. Our main result is an error estimate for the time discretization showing a convergence rate of order (up to) 1/2. It holds with respect to mean square error convergence, whereas previously such a rate for the stochastic Navier–Stokes equations was only known with respect to convergence in probability. Our result is based on uniform-in-probability estimates for the continuous as well as the time-discrete solution exploiting the particular structure of the noise. Eventually, we perform numerical simulations for the corresponding problem on bounded domains with no-slip boundary conditions. They suggest the same convergence rate as proved for the periodic problem hinging sensitively on the compatibility of the data. We also compare the energy profiles with those for corresponding problems with additive or multiplicative Itô-type noise.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"26 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144288199","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}
We theoretically explore boundary conditions for lattice Boltzmann methods, focusing on a toy two-velocities scheme to tackle a linear one-dimensional advection equation. By mapping lattice Boltzmann schemes to Finite Difference schemes, we facilitate rigorous consistency and stability analyses. We develop kinetic boundary conditions for inflows and outflows, highlighting the trade-off between accuracy and stability, which we successfully overcome. Consistency analysis relies on modified equations, whereas stability is assessed using the GKS (Gustafsson, Kreiss and Sundström) theory and—when this approach fails on coarse meshes—spectral and pseudo-spectral analyses of the scheme’s matrix that explain effects germane to low resolutions.
{"title":"Consistency and stability of boundary conditions for a two-velocities lattice Boltzmann scheme","authors":"Thomas Bellotti","doi":"10.1093/imanum/draf039","DOIUrl":"https://doi.org/10.1093/imanum/draf039","url":null,"abstract":"We theoretically explore boundary conditions for lattice Boltzmann methods, focusing on a toy two-velocities scheme to tackle a linear one-dimensional advection equation. By mapping lattice Boltzmann schemes to Finite Difference schemes, we facilitate rigorous consistency and stability analyses. We develop kinetic boundary conditions for inflows and outflows, highlighting the trade-off between accuracy and stability, which we successfully overcome. Consistency analysis relies on modified equations, whereas stability is assessed using the GKS (Gustafsson, Kreiss and Sundström) theory and—when this approach fails on coarse meshes—spectral and pseudo-spectral analyses of the scheme’s matrix that explain effects germane to low resolutions.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"48 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144268846","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}
The main challenge in the analysis of numerical methods for the Zakharov system (ZS) originates from the presence of derivatives in the nonlinearity. In this paper, we present a novel reformulation of the ZS, which allows us to construct second-order time symmetric methods and higher-order numerical methods for the ZS even with generalized nonlinear terms. By considering exponential wave integrators (EWIs) for this reformulation, a new time symmetric EWI is formulated and its properties are rigorously studied. The proposed method is proved to have two conservation laws at the discrete level. The second-order convergence in time is rigorously shown under a time-step restriction that is independent of the spatial discretization. Moreover, by the strategy presented in this paper, higher-order methods are obtained for the ZS with generalized nonlinear terms. Numerical explorations confirm the theoretical results and superiority of the proposed integrators.
{"title":"A new framework for the construction and analysis of exponential wave integrators for the Zakharov system","authors":"Jiyong Li, Bin Wang","doi":"10.1093/imanum/draf016","DOIUrl":"https://doi.org/10.1093/imanum/draf016","url":null,"abstract":"The main challenge in the analysis of numerical methods for the Zakharov system (ZS) originates from the presence of derivatives in the nonlinearity. In this paper, we present a novel reformulation of the ZS, which allows us to construct second-order time symmetric methods and higher-order numerical methods for the ZS even with generalized nonlinear terms. By considering exponential wave integrators (EWIs) for this reformulation, a new time symmetric EWI is formulated and its properties are rigorously studied. The proposed method is proved to have two conservation laws at the discrete level. The second-order convergence in time is rigorously shown under a time-step restriction that is independent of the spatial discretization. Moreover, by the strategy presented in this paper, higher-order methods are obtained for the ZS with generalized nonlinear terms. Numerical explorations confirm the theoretical results and superiority of the proposed integrators.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"20 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144268625","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}
To obtain strong convergence rates of numerical schemes, an overwhelming majority of existing works impose a global monotonicity condition on coefficients of stochastic differential equations (SDEs). Nevertheless, there are still many SDEs from applications that do not have globally monotone coefficients. As a recent breakthrough, the authors of (Hutzenthaler and Jentzen 2020, Ann. Prob., 48, 53–93) originally presented a perturbation theory for SDEs, which is crucial to recovering strong convergence rates of numerical schemes in a non-globally monotone setting. However, only a convergence rate of order $1/2$ was obtained there for time-stepping schemes such as a stopped increment-tamed Euler–Maruyama (SITEM) method. An interesting question arises, also raised by the aforementioned work, as to whether a higher convergence rate than $1/2$ can be obtained when higher order schemes are used. The present work attempts to give a positive answer to this question. To this end, we develop some new perturbation estimates that are able to reveal the order-one strong convergence of numerical methods. As the first application of the newly developed estimates, we identify the expected order-one pathwise uniformly strong convergence of the SITEM method for additive noise driven SDEs and multiplicative noise driven second-order SDEs with non-globally monotone coefficients. As the other application, we propose and analyze a positivity preserving explicit Milstein-type method for Lotka–Volterra competition model driven by multi-dimensional noise, with a pathwise uniformly strong convergence rate of order one recovered under mild assumptions. These obtained results are completely new and significantly improve the existing theory. Numerical experiments are also provided to confirm the theoretical findings.
{"title":"Perturbation estimates for order-one strong approximations of SDEs without globally monotone coefficients","authors":"Lei Dai, Xiaojie Wang","doi":"10.1093/imanum/draf034","DOIUrl":"https://doi.org/10.1093/imanum/draf034","url":null,"abstract":"To obtain strong convergence rates of numerical schemes, an overwhelming majority of existing works impose a global monotonicity condition on coefficients of stochastic differential equations (SDEs). Nevertheless, there are still many SDEs from applications that do not have globally monotone coefficients. As a recent breakthrough, the authors of (Hutzenthaler and Jentzen 2020, Ann. Prob., 48, 53–93) originally presented a perturbation theory for SDEs, which is crucial to recovering strong convergence rates of numerical schemes in a non-globally monotone setting. However, only a convergence rate of order $1/2$ was obtained there for time-stepping schemes such as a stopped increment-tamed Euler–Maruyama (SITEM) method. An interesting question arises, also raised by the aforementioned work, as to whether a higher convergence rate than $1/2$ can be obtained when higher order schemes are used. The present work attempts to give a positive answer to this question. To this end, we develop some new perturbation estimates that are able to reveal the order-one strong convergence of numerical methods. As the first application of the newly developed estimates, we identify the expected order-one pathwise uniformly strong convergence of the SITEM method for additive noise driven SDEs and multiplicative noise driven second-order SDEs with non-globally monotone coefficients. As the other application, we propose and analyze a positivity preserving explicit Milstein-type method for Lotka–Volterra competition model driven by multi-dimensional noise, with a pathwise uniformly strong convergence rate of order one recovered under mild assumptions. These obtained results are completely new and significantly improve the existing theory. Numerical experiments are also provided to confirm the theoretical findings.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"26 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144252258","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}
This paper discusses the error and cost aspects of ill-posed integral equations when given discrete noisy point evaluations on a fine grid. Standard solution methods usually employ discretization schemes that are directly induced by the measurement points. Thus, they may scale unfavourably with the number of evaluation points, which can result in computational inefficiency. To address this issue we propose an algorithm that achieves the same level of accuracy while significantly reducing computational costs. Our approach involves an initial averaging procedure to sparsify the underlying grid. To keep the exposition simple we focus on regularization via the truncated singular value decomposition of one-dimensional ill-posed integral equations that have sufficient smoothness. However, the approach can be generalized to other popular regularization methods and more complicated two- and three-dimensional problems with appropriate modifications.
{"title":"Efficient solution of ill-posed integral equations through averaging","authors":"Michael Griebel, Tim Jahn","doi":"10.1093/imanum/draf038","DOIUrl":"https://doi.org/10.1093/imanum/draf038","url":null,"abstract":"This paper discusses the error and cost aspects of ill-posed integral equations when given discrete noisy point evaluations on a fine grid. Standard solution methods usually employ discretization schemes that are directly induced by the measurement points. Thus, they may scale unfavourably with the number of evaluation points, which can result in computational inefficiency. To address this issue we propose an algorithm that achieves the same level of accuracy while significantly reducing computational costs. Our approach involves an initial averaging procedure to sparsify the underlying grid. To keep the exposition simple we focus on regularization via the truncated singular value decomposition of one-dimensional ill-posed integral equations that have sufficient smoothness. However, the approach can be generalized to other popular regularization methods and more complicated two- and three-dimensional problems with appropriate modifications.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"10 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144228438","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}
Giovanni S Alberti, Luca Ratti, Matteo Santacesaria, Silvia Sciutto
In inverse problems it is widely recognized that the incorporation of a sparsity prior yields a regularization effect on the solution. This approach is grounded on the a priori assumption that the unknown can be appropriately represented in a basis with a limited number of significant components while most coefficients are close to zero. This occurrence is frequently observed in real-world scenarios, such as with piecewise smooth signals. In this study we propose a probabilistic sparsity prior formulated as a mixture of degenerate Gaussians, capable of modelling sparsity with respect to a generic basis. Under this premise we design a neural network that can be interpreted as the Bayes estimator for linear inverse problems. Additionally, we put forth both a supervised and an unsupervised training strategy to estimate the parameters of this network. To evaluate the effectiveness of our approach we conduct a numerical comparison with commonly employed sparsity-promoting regularization techniques, namely Least Absolute Shrinkage and Selection Operator (LASSO), group LASSO, iterative hard thresholding and sparse coding/dictionary learning. Notably, our reconstructions consistently exhibit lower mean square error values across all one-dimensional datasets utilized for the comparisons, even in cases where the datasets significantly deviate from a Gaussian mixture model.
{"title":"Learning a Gaussian mixture for sparsity regularization in inverse problems","authors":"Giovanni S Alberti, Luca Ratti, Matteo Santacesaria, Silvia Sciutto","doi":"10.1093/imanum/draf037","DOIUrl":"https://doi.org/10.1093/imanum/draf037","url":null,"abstract":"In inverse problems it is widely recognized that the incorporation of a sparsity prior yields a regularization effect on the solution. This approach is grounded on the a priori assumption that the unknown can be appropriately represented in a basis with a limited number of significant components while most coefficients are close to zero. This occurrence is frequently observed in real-world scenarios, such as with piecewise smooth signals. In this study we propose a probabilistic sparsity prior formulated as a mixture of degenerate Gaussians, capable of modelling sparsity with respect to a generic basis. Under this premise we design a neural network that can be interpreted as the Bayes estimator for linear inverse problems. Additionally, we put forth both a supervised and an unsupervised training strategy to estimate the parameters of this network. To evaluate the effectiveness of our approach we conduct a numerical comparison with commonly employed sparsity-promoting regularization techniques, namely Least Absolute Shrinkage and Selection Operator (LASSO), group LASSO, iterative hard thresholding and sparse coding/dictionary learning. Notably, our reconstructions consistently exhibit lower mean square error values across all one-dimensional datasets utilized for the comparisons, even in cases where the datasets significantly deviate from a Gaussian mixture model.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"39 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144228439","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}
In this paper we propose a systematic study of classical and multilevel variants of the Uzawa and Arrow–Hurwicz methods. The multilevel methods are obtained from the classical ones by the introduction of multilevel inner iterations to calculate the solution of the first equation instead of its exact calculation as in the classical Uzawa or Arrow–Hurwicz methods. In our study, an essential role is played by the LBB condition. For classical and multilevel variants of the Uzawa and Arrow–Hurwicz algorithms, we prove theorems which give the convergence conditions of the methods and explicit formulas of the convergence rates. On the basis of these results we compare the convergence conditions and the convergence rates of the classical methods with those of their corresponding ones in the multilevel methods. Concerning the Uzawa methods, we prove that, the limit of the convergence condition and the convergence rate of the multilevel method, when the number of the inner iterations tends to infinity, coincide with those of the classical one. Also, from the dependence of the convergence rate on the number of inner iterations of the multilevel method, we conclude that, the multilevel method with a small number of inner iterations converges better than the classical one. For the Arrow–Hurwicz methods we found that for a large number of inner iterations of the multilevel algorithm, the convergence condition of the multilevel method coincides with that of the classical method and the convergence rate of the multilevel method is equal to or smaller than that of the classical method. Finally, the behavior of the introduced methods is investigated by numerical experiments carried out for the driven-cavity Stokes problem and they confirm the theoretical results.
{"title":"On the convergence of classical and multilevel variants of the Uzawa and Arrow–Hurwicz algorithms related to the LBB condition","authors":"Lori Badea","doi":"10.1093/imanum/draf040","DOIUrl":"https://doi.org/10.1093/imanum/draf040","url":null,"abstract":"In this paper we propose a systematic study of classical and multilevel variants of the Uzawa and Arrow–Hurwicz methods. The multilevel methods are obtained from the classical ones by the introduction of multilevel inner iterations to calculate the solution of the first equation instead of its exact calculation as in the classical Uzawa or Arrow–Hurwicz methods. In our study, an essential role is played by the LBB condition. For classical and multilevel variants of the Uzawa and Arrow–Hurwicz algorithms, we prove theorems which give the convergence conditions of the methods and explicit formulas of the convergence rates. On the basis of these results we compare the convergence conditions and the convergence rates of the classical methods with those of their corresponding ones in the multilevel methods. Concerning the Uzawa methods, we prove that, the limit of the convergence condition and the convergence rate of the multilevel method, when the number of the inner iterations tends to infinity, coincide with those of the classical one. Also, from the dependence of the convergence rate on the number of inner iterations of the multilevel method, we conclude that, the multilevel method with a small number of inner iterations converges better than the classical one. For the Arrow–Hurwicz methods we found that for a large number of inner iterations of the multilevel algorithm, the convergence condition of the multilevel method coincides with that of the classical method and the convergence rate of the multilevel method is equal to or smaller than that of the classical method. Finally, the behavior of the introduced methods is investigated by numerical experiments carried out for the driven-cavity Stokes problem and they confirm the theoretical results.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"7 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144228464","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}
We consider a stable unique continuation problem for the wave equation that has been discretized so far using fairly sophisticated space-time methods. Here, we propose to solve this problem using a standard discontinuous Galerkin method for the temporal discretization. Error estimates are established under a geometric control condition. We also investigate two preconditioning strategies that can be used to solve the arising globally coupled space-time system by means of simple time-stepping procedures. Our numerical experiments test the performance of these strategies and highlight the importance of the geometric control condition for reconstructing the solution beyond the data domain.
{"title":"Unique continuation for the wave equation based on a discontinuous Galerkin time discretization","authors":"Erik Burman, Janosch Preuss","doi":"10.1093/imanum/draf036","DOIUrl":"https://doi.org/10.1093/imanum/draf036","url":null,"abstract":"We consider a stable unique continuation problem for the wave equation that has been discretized so far using fairly sophisticated space-time methods. Here, we propose to solve this problem using a standard discontinuous Galerkin method for the temporal discretization. Error estimates are established under a geometric control condition. We also investigate two preconditioning strategies that can be used to solve the arising globally coupled space-time system by means of simple time-stepping procedures. Our numerical experiments test the performance of these strategies and highlight the importance of the geometric control condition for reconstructing the solution beyond the data domain.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"3 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144184114","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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