{"title":"A Wavelet-Based Approach for the Simulation and Optimal Control of NonLocal Operator Equations","authors":"S. Dahlke, H. Harbrecht, T. Surowiec","doi":"10.1137/20m1350790","DOIUrl":"https://doi.org/10.1137/20m1350790","url":null,"abstract":"","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":"12 1","pages":"2691-"},"PeriodicalIF":0.0,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90446743","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":"BDDC Deluxe Algorithms for Two-Dimensional H($curl$) Isogeometric Analysis","authors":"O. Widlund, S. Scacchi, L. Pavarino","doi":"10.1137/21m1438839","DOIUrl":"https://doi.org/10.1137/21m1438839","url":null,"abstract":"","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":"240 1","pages":"2349-"},"PeriodicalIF":0.0,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73363566","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":"Piecewise-Global Nonlinear Model Order Reduction for PDE-Constrained Optimization in High-Dimensional Parameter Spaces","authors":"Gabriele Boncoraglio, C. Farhat","doi":"10.1137/21m1435343","DOIUrl":"https://doi.org/10.1137/21m1435343","url":null,"abstract":"","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":"1 1","pages":"2176-"},"PeriodicalIF":0.0,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79793453","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 : 2022-07-27DOI: 10.48550/arXiv.2207.13674
Nicholas F. Marshall, Oscar Mickelin, A. Singer
We present a fast and numerically accurate method for expanding digitized $L times L$ images representing functions on $[-1,1]^2$ supported on the disk ${x in mathbb{R}^2 : |x|<1}$ in the harmonics (Dirichlet Laplacian eigenfunctions) on the disk. Our method, which we refer to as the Fast Disk Harmonics Transform (FDHT), runs in $O(L^2 log L)$ operations. This basis is also known as the Fourier-Bessel basis, and it has several computational advantages: it is orthogonal, ordered by frequency, and steerable in the sense that images expanded in the basis can be rotated by applying a diagonal transform to the coefficients. Moreover, we show that convolution with radial functions can also be efficiently computed by applying a diagonal transform to the coefficients.
{"title":"Fast expansion into harmonics on the disk: a steerable basis with fast radial convolutions","authors":"Nicholas F. Marshall, Oscar Mickelin, A. Singer","doi":"10.48550/arXiv.2207.13674","DOIUrl":"https://doi.org/10.48550/arXiv.2207.13674","url":null,"abstract":"We present a fast and numerically accurate method for expanding digitized $L times L$ images representing functions on $[-1,1]^2$ supported on the disk ${x in mathbb{R}^2 : |x|<1}$ in the harmonics (Dirichlet Laplacian eigenfunctions) on the disk. Our method, which we refer to as the Fast Disk Harmonics Transform (FDHT), runs in $O(L^2 log L)$ operations. This basis is also known as the Fourier-Bessel basis, and it has several computational advantages: it is orthogonal, ordered by frequency, and steerable in the sense that images expanded in the basis can be rotated by applying a diagonal transform to the coefficients. Moreover, we show that convolution with radial functions can also be efficiently computed by applying a diagonal transform to the coefficients.","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":"1 1","pages":"2431-"},"PeriodicalIF":0.0,"publicationDate":"2022-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90718936","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":"Eulerian-Lagrangian Runge-Kutta Discontinuous Galerkin Method for Transport Simulations on Unstructured Meshes","authors":"Xiaofeng Cai, Jing-Mei Qiu","doi":"10.1137/21m1456753","DOIUrl":"https://doi.org/10.1137/21m1456753","url":null,"abstract":"","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":"5 1","pages":"2037-"},"PeriodicalIF":0.0,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75739700","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}
Reza MohammadiArani, M. Dehghan, Mostafa Abbaszadeh
{"title":"Proper Orthogonal Decomposition-Lattice Boltzmann Method: Simulating the Air Pollutant Problem in Street Canyon Areas","authors":"Reza MohammadiArani, M. Dehghan, Mostafa Abbaszadeh","doi":"10.1137/21m1405733","DOIUrl":"https://doi.org/10.1137/21m1405733","url":null,"abstract":"","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":"111 1","pages":"885-"},"PeriodicalIF":0.0,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79345844","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 : 2022-07-21DOI: 10.48550/arXiv.2207.10321
E. E. Arar, D. Sohier, P. D. O. Castro, E. Petit
Stochastic rounding (SR) offers an alternative to the deterministic IEEE-754 floating-point rounding modes. In some applications such as PDEs, ODEs and neural networks, SR empirically improves the numerical behavior and convergence to accurate solutions while no sound theoretical background has been provided. Recent works by Ipsen, Zhou, Higham, and Mary have computed SR probabilistic error bounds for basic linear algebra kernels. For example, the inner product SR probabilistic bound of the forward error is proportional to $sqrt$ nu instead of nu for the default rounding mode. To compute the bounds, these works show that the errors accumulated in computation form a martingale. This paper proposes an alternative framework to characterize SR errors based on the computation of the variance. We pinpoint common error patterns in numerical algorithms and propose a lemma that bounds their variance. For each probability and through Bienaym{'e}-Chebyshev inequality, this bound leads to better probabilistic error bound in several situations. Our method has the advantage of providing a tight probabilistic bound for all algorithms fitting our model. We show how the method can be applied to give SR error bounds for the inner product and Horner polynomial evaluation.
{"title":"Stochastic rounding variance and probabilistic bounds: A new approach","authors":"E. E. Arar, D. Sohier, P. D. O. Castro, E. Petit","doi":"10.48550/arXiv.2207.10321","DOIUrl":"https://doi.org/10.48550/arXiv.2207.10321","url":null,"abstract":"Stochastic rounding (SR) offers an alternative to the deterministic IEEE-754 floating-point rounding modes. In some applications such as PDEs, ODEs and neural networks, SR empirically improves the numerical behavior and convergence to accurate solutions while no sound theoretical background has been provided. Recent works by Ipsen, Zhou, Higham, and Mary have computed SR probabilistic error bounds for basic linear algebra kernels. For example, the inner product SR probabilistic bound of the forward error is proportional to $sqrt$ nu instead of nu for the default rounding mode. To compute the bounds, these works show that the errors accumulated in computation form a martingale. This paper proposes an alternative framework to characterize SR errors based on the computation of the variance. We pinpoint common error patterns in numerical algorithms and propose a lemma that bounds their variance. For each probability and through Bienaym{'e}-Chebyshev inequality, this bound leads to better probabilistic error bound in several situations. Our method has the advantage of providing a tight probabilistic bound for all algorithms fitting our model. We show how the method can be applied to give SR error bounds for the inner product and Horner polynomial evaluation.","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":"96 1","pages":"255-"},"PeriodicalIF":0.0,"publicationDate":"2022-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85732749","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 : 2022-07-15DOI: 10.48550/arXiv.2207.07615
J. Brust, M. Saunders
We propose iterative projection methods for solving square or rectangular consistent linear systems Ax = b. Existing projection methods use sketching matrices (possibly randomized) to generate a sequence of small projected subproblems, but even the smaller systems can be costly. We develop a process that appends one column to the sketching matrix each iteration and converges in a finite number of iterations whether the sketch is random or deterministic. In general, our process generates orthogonal updates to the approximate solution xk. By choosing the sketch to be the set of all previous residuals, we obtain a simple recursive update and convergence in at most rank(A) iterations (in exact arithmetic). By choosing a sequence of identity columns for the sketch, we develop a generalization of the Kaczmarz method. In experiments on large sparse systems, our method (PLSS) with residual sketches is competitive with LSQR and LSMR, and with residual and identity sketches compares favorably with state-of-the-art randomized methods.
{"title":"PLSS: A Projected Linear Systems Solver","authors":"J. Brust, M. Saunders","doi":"10.48550/arXiv.2207.07615","DOIUrl":"https://doi.org/10.48550/arXiv.2207.07615","url":null,"abstract":"We propose iterative projection methods for solving square or rectangular consistent linear systems Ax = b. Existing projection methods use sketching matrices (possibly randomized) to generate a sequence of small projected subproblems, but even the smaller systems can be costly. We develop a process that appends one column to the sketching matrix each iteration and converges in a finite number of iterations whether the sketch is random or deterministic. In general, our process generates orthogonal updates to the approximate solution xk. By choosing the sketch to be the set of all previous residuals, we obtain a simple recursive update and convergence in at most rank(A) iterations (in exact arithmetic). By choosing a sequence of identity columns for the sketch, we develop a generalization of the Kaczmarz method. In experiments on large sparse systems, our method (PLSS) with residual sketches is competitive with LSQR and LSMR, and with residual and identity sketches compares favorably with state-of-the-art randomized methods.","PeriodicalId":21812,"journal":{"name":"SIAM J. Sci. Comput.","volume":"66 1","pages":"1012-"},"PeriodicalIF":0.0,"publicationDate":"2022-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89821924","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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