Pub Date : 2022-11-10DOI: 10.48550/arXiv.2211.05496
K. Pentland, M. Tamborrino, Timothy John Sullivan
Stochastic parareal (SParareal) is a probabilistic variant of the popular parallel-in-time algorithm known as parareal. Similarly to parareal, it combines fine- and coarse-grained solutions to an ordinary differential equation (ODE) using a predictor-corrector (PC) scheme. The key difference is that carefully chosen random perturbations are added to the PC to try to accelerate the location of a stochastic solution to the ODE. In this paper, we derive superlinear and linear mean-square error bounds for SParareal applied to nonlinear systems of ODEs using different types of perturbations. We illustrate these bounds numerically on a linear system of ODEs and a scalar nonlinear ODE, showing a good match between theory and numerics.
{"title":"Error bound analysis of the stochastic parareal algorithm","authors":"K. Pentland, M. Tamborrino, Timothy John Sullivan","doi":"10.48550/arXiv.2211.05496","DOIUrl":"https://doi.org/10.48550/arXiv.2211.05496","url":null,"abstract":"Stochastic parareal (SParareal) is a probabilistic variant of the popular parallel-in-time algorithm known as parareal. Similarly to parareal, it combines fine- and coarse-grained solutions to an ordinary differential equation (ODE) using a predictor-corrector (PC) scheme. The key difference is that carefully chosen random perturbations are added to the PC to try to accelerate the location of a stochastic solution to the ODE. In this paper, we derive superlinear and linear mean-square error bounds for SParareal applied to nonlinear systems of ODEs using different types of perturbations. We illustrate these bounds numerically on a linear system of ODEs and a scalar nonlinear ODE, showing a good match between theory and numerics.","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80737796","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-10DOI: 10.1007/s10915-022-02043-y
Huoyuan Duan, Can Wang, Zhijie Du
{"title":"A Div FOSLS Method Suitable for Quadrilateral RT and Hexahedral RTNH(rmdiv)-elements","authors":"Huoyuan Duan, Can Wang, Zhijie Du","doi":"10.1007/s10915-022-02043-y","DOIUrl":"https://doi.org/10.1007/s10915-022-02043-y","url":null,"abstract":"","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82792101","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Construction and Evaluation of Pythagorean Hodograph Curves in Exponential-Polynomial Spaces","authors":"Lucia Romani, Alberto Viscardi","doi":"10.1137/21m1455711","DOIUrl":"https://doi.org/10.1137/21m1455711","url":null,"abstract":"","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80711556","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Linearly Implicit Multistep Methods for Time Integration","authors":"S. R. Glandon, M. Narayanamurthi, Adrian Sandu","doi":"10.1137/20m133748x","DOIUrl":"https://doi.org/10.1137/20m133748x","url":null,"abstract":"","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83350785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
9 Under the Strong Exponential-Time Hypothesis, the diameter of general unweighted graphs 10 cannot be computed in truly subquadratic time (in the size n + m of the input), as shown 11 by Roditty and Williams. Nevertheless there are several graph classes for which this can be 12 done such as bounded-treewidth graphs, interval graphs and planar graphs, to name a few. We 13 propose to study unweighted graphs of constant distance VC-dimension as a broad generalization 14 of many such classes – where the distance VC-dimension of a graph G is defined as the VC-15 dimension of its ball hypergraph: whose hyperedges are the balls of all possible radii and centers 16 in G . In particular for any fixed H , the class of H -minor free graphs has distance VC-dimension 17 at most | V ( H ) | − 1. 18 • Our first main result is a Monte Carlo algorithm that on graphs of distance VC-dimension 19 at most d , for any fixed k , either computes the diameter or concludes that it is larger than 20 k in time ˜ O ( k · mn 1 − ε d ), where ε d ∈ (0; 1) only depends on d 1 . We thus obtain a truly 21 subquadratic-time parameterized algorithm for computing the diameter on such graphs. 22 • Then as a byproduct of our approach, we get a truly subquadratic-time randomized algo-23 rithm for constant diameter computation on all the nowhere dense graph classes. The latter 24 classes include all proper minor-closed graph classes, bounded-degree graphs and graphs of 25 bounded expansion. Before our work, the only known such algorithm was resulting from 26 an application of Courcelle’s theorem, see Grohe et al. [47]. 27
{"title":"Diameter, Eccentricities and Distance Oracle Computations on H-Minor Free Graphs and Graphs of Bounded (Distance) Vapnik-Chervonenkis Dimension","authors":"G. Ducoffe, M. Habib, L. Viennot","doi":"10.1137/20m136551x","DOIUrl":"https://doi.org/10.1137/20m136551x","url":null,"abstract":"9 Under the Strong Exponential-Time Hypothesis, the diameter of general unweighted graphs 10 cannot be computed in truly subquadratic time (in the size n + m of the input), as shown 11 by Roditty and Williams. Nevertheless there are several graph classes for which this can be 12 done such as bounded-treewidth graphs, interval graphs and planar graphs, to name a few. We 13 propose to study unweighted graphs of constant distance VC-dimension as a broad generalization 14 of many such classes – where the distance VC-dimension of a graph G is defined as the VC-15 dimension of its ball hypergraph: whose hyperedges are the balls of all possible radii and centers 16 in G . In particular for any fixed H , the class of H -minor free graphs has distance VC-dimension 17 at most | V ( H ) | − 1. 18 • Our first main result is a Monte Carlo algorithm that on graphs of distance VC-dimension 19 at most d , for any fixed k , either computes the diameter or concludes that it is larger than 20 k in time ˜ O ( k · mn 1 − ε d ), where ε d ∈ (0; 1) only depends on d 1 . We thus obtain a truly 21 subquadratic-time parameterized algorithm for computing the diameter on such graphs. 22 • Then as a byproduct of our approach, we get a truly subquadratic-time randomized algo-23 rithm for constant diameter computation on all the nowhere dense graph classes. The latter 24 classes include all proper minor-closed graph classes, bounded-degree graphs and graphs of 25 bounded expansion. Before our work, the only known such algorithm was resulting from 26 an application of Courcelle’s theorem, see Grohe et al. [47]. 27","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74761085","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-25DOI: 10.48550/arXiv.2210.14333
N. Sharon, Rafael Sherbu Cohen, H. Wendland
We address the problem of approximating an unknown function from its discrete samples given at arbitrarily scattered sites. This problem is essential in numerical sciences, where modern applications also highlight the need for a solution to the case of functions with manifold values. In this paper, we introduce and analyze a combination of kernel-based quasi-interpolation and multiscale approximations for both scalar- and manifold-valued functions. While quasi-interpolation provides a powerful tool for approximation problems if the data is defined on infinite grids, the situation is more complicated when it comes to scattered data. Here, higher-order quasi-interpolation schemes either require derivative information or become numerically unstable. Hence, this paper principally studies the improvement achieved by combining quasi-interpolation with a multiscale technique. The main contributions of this paper are as follows. First, we introduce the multiscale quasi-interpolation technique for scalar-valued functions. Second, we show how this technique can be carried over using moving least-squares operators to the manifold-valued setting. Third, we give a mathematical proof that converging quasi-interpolation will also lead to converging multiscale quasi-interpolation. Fourth, we provide ample numerical evidence that multiscale quasi-interpolation has superior convergence to quasi-interpolation. In addition, we will provide examples showing that the multiscale quasi-interpolation approach offers a powerful tool for many data analysis tasks, such as denoising and anomaly detection. It is especially attractive for cases of massive data points and high dimensionality.
{"title":"On multiscale quasi-interpolation of scattered scalar- and manifold-valued functions","authors":"N. Sharon, Rafael Sherbu Cohen, H. Wendland","doi":"10.48550/arXiv.2210.14333","DOIUrl":"https://doi.org/10.48550/arXiv.2210.14333","url":null,"abstract":"We address the problem of approximating an unknown function from its discrete samples given at arbitrarily scattered sites. This problem is essential in numerical sciences, where modern applications also highlight the need for a solution to the case of functions with manifold values. In this paper, we introduce and analyze a combination of kernel-based quasi-interpolation and multiscale approximations for both scalar- and manifold-valued functions. While quasi-interpolation provides a powerful tool for approximation problems if the data is defined on infinite grids, the situation is more complicated when it comes to scattered data. Here, higher-order quasi-interpolation schemes either require derivative information or become numerically unstable. Hence, this paper principally studies the improvement achieved by combining quasi-interpolation with a multiscale technique. The main contributions of this paper are as follows. First, we introduce the multiscale quasi-interpolation technique for scalar-valued functions. Second, we show how this technique can be carried over using moving least-squares operators to the manifold-valued setting. Third, we give a mathematical proof that converging quasi-interpolation will also lead to converging multiscale quasi-interpolation. Fourth, we provide ample numerical evidence that multiscale quasi-interpolation has superior convergence to quasi-interpolation. In addition, we will provide examples showing that the multiscale quasi-interpolation approach offers a powerful tool for many data analysis tasks, such as denoising and anomaly detection. It is especially attractive for cases of massive data points and high dimensionality.","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79964811","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. Numerically solving the Liouville equation in classical mechanics with a discontinuous potential often leads to the 4 challenges of how to preserve the Hamiltonian across the potential barrier and a severe time step constraint according to the CFL 5 condition. Motivated by the Hamiltonian-preserving finite volume schemes by Jin and Wen [19], we introduce a Hamiltonian- 6 preserving discontinuous Galerkin (DG) scheme for the Liouville equation with discontinuous potential in this paper. The DG 7 method can be designed with arbitrary order of accuracy, and offers many advantages including easy adaptivity, compact stencils 8 and the ability of handling complicated boundary condition and interfaces. We propose to carefully design the numerical fluxes 9 of the DG methods to build the behavior of a classical particle at the potential barrier into the numerical scheme, which ensures 10 the continuity of the Hamiltonian across the potential barrier and the correct transmission and reflection condition. Our scheme 11 is proved to be positive and stable in L 1 norm if the positivity-preserving limiter is applied. Numerical examples are provided to 12 illustrate the accuracy and effectiveness of the proposed numerical scheme. 13 results show 2D2V test discontinuity of HPDG
{"title":"Hamiltonian-Preserving Discontinuous Galerkin Methods for the Liouville Equation With Discontinuous Potential","authors":"Boyang Ye, Shi Jin, Y. Xing, Xinghui Zhong","doi":"10.1137/22m147952x","DOIUrl":"https://doi.org/10.1137/22m147952x","url":null,"abstract":". Numerically solving the Liouville equation in classical mechanics with a discontinuous potential often leads to the 4 challenges of how to preserve the Hamiltonian across the potential barrier and a severe time step constraint according to the CFL 5 condition. Motivated by the Hamiltonian-preserving finite volume schemes by Jin and Wen [19], we introduce a Hamiltonian- 6 preserving discontinuous Galerkin (DG) scheme for the Liouville equation with discontinuous potential in this paper. The DG 7 method can be designed with arbitrary order of accuracy, and offers many advantages including easy adaptivity, compact stencils 8 and the ability of handling complicated boundary condition and interfaces. We propose to carefully design the numerical fluxes 9 of the DG methods to build the behavior of a classical particle at the potential barrier into the numerical scheme, which ensures 10 the continuity of the Hamiltonian across the potential barrier and the correct transmission and reflection condition. Our scheme 11 is proved to be positive and stable in L 1 norm if the positivity-preserving limiter is applied. Numerical examples are provided to 12 illustrate the accuracy and effectiveness of the proposed numerical scheme. 13 results show 2D2V test discontinuity of HPDG","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82158992","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Invariant-Domain-Preserving High-Order Time Stepping: I. Explicit Runge-Kutta Schemes","authors":"A. Ern, J. Guermond","doi":"10.1137/21m145793x","DOIUrl":"https://doi.org/10.1137/21m145793x","url":null,"abstract":"","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75153503","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Frequency Extraction for BEM Matrices Arising From the 3D Scalar Helmholtz Equation","authors":"Simon Dirckx, D. Huybrechs, K. Meerbergen","doi":"10.1137/20m1382957","DOIUrl":"https://doi.org/10.1137/20m1382957","url":null,"abstract":"","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84254976","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Space-Split Algorithm for Sensitivity Analysis of Discrete Chaotic Systems With Multidimensional Unstable Manifolds","authors":"Adam A. Śliwiak, Qiqi Wang","doi":"10.1137/21m1452135","DOIUrl":"https://doi.org/10.1137/21m1452135","url":null,"abstract":"","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84857778","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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