{"title":"Preconditioners for Krylov subspace methods: An overview","authors":"John W. Pearson, Jennifer Pestana","doi":"10.1002/gamm.202000015","DOIUrl":"10.1002/gamm.202000015","url":null,"abstract":"<p>When simulating a mechanism from science or engineering, or an industrial process, one is frequently required to construct a mathematical model, and then resolve this model numerically. If accurate numerical solutions are necessary or desirable, this can involve solving large-scale systems of equations. One major class of solution methods is that of preconditioned iterative methods, involving preconditioners which are computationally cheap to apply while also capturing information contained in the linear system. In this article, we give a short survey of the field of preconditioning. We introduce a range of preconditioners for partial differential equations, followed by optimization problems, before discussing preconditioners constructed with less standard objectives in mind.</p>","PeriodicalId":53634,"journal":{"name":"GAMM Mitteilungen","volume":"43 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1002/gamm.202000015","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73816556","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Iterative and doubling algorithms for Riccati-type matrix equations: A comparative introduction","authors":"Federico Poloni","doi":"10.1002/gamm.202000018","DOIUrl":"10.1002/gamm.202000018","url":null,"abstract":"<p>We review a family of algorithms for Lyapunov- and Riccati-type equations which are all related to each other by the idea of <i>doubling</i>: they construct the iterate <math>\u0000 <mrow>\u0000 <msub>\u0000 <mrow>\u0000 <mi>Q</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </msub>\u0000 <mo>=</mo>\u0000 <msub>\u0000 <mrow>\u0000 <mi>X</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <msup>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 </msub>\u0000 </mrow></math> of another naturally-arising fixed-point iteration <span>(<i>X</i><sub><i>h</i></sub>)</span> via a sort of repeated squaring. The equations we consider are Stein equations <span><i>X</i> − <i>A</i><sup>∗</sup> <i>X A</i> = <i>Q</i></span>, Lyapunov equations <span><i>A</i><sup>∗</sup> <i>X</i> + <i>X A</i> + <i>Q</i> = 0</span>, discrete-time algebraic Riccati equations <span><i>X</i> = <i>Q</i> + <i>A</i><sup>∗</sup> <i>X</i>(<i>I</i> + <i>G X</i>)<sup>−1</sup><i>A</i></span>, continuous-time algebraic Riccati equations <span><i>Q</i> + <i>A</i><sup>∗</sup> <i>X</i> + <i>X A</i> − <i>X G X</i> = 0</span>, palindromic quadratic matrix equations <span><i>A</i> + <i>Q Y</i> + <i>A</i><sup>∗</sup><i>Y</i><sup>2</sup> = 0</span>, and nonlinear matrix equations <span><i>X</i> + <i>A</i><sup>∗</sup> <i>X</i><sup>−1</sup><i>A</i> = <i>Q</i></span>. We draw comparisons among these algorithms, highlight the connections between them and to other algorithms such as subspace iteration, and discuss open issues in their theory.</p>","PeriodicalId":53634,"journal":{"name":"GAMM Mitteilungen","volume":"43 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1002/gamm.202000018","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88337416","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":"Krylov methods for inverse problems: Surveying classical, and introducing new, algorithmic approaches","authors":"Silvia Gazzola, Malena Sabaté Landman","doi":"10.1002/gamm.202000017","DOIUrl":"10.1002/gamm.202000017","url":null,"abstract":"<p>Large-scale linear systems coming from suitable discretizations of linear inverse problems are challenging to solve. Indeed, since they are inherently ill-posed, appropriate regularization should be applied; since they are large-scale, well-established direct regularization methods (such as Tikhonov regularization) cannot often be straightforwardly employed, and iterative linear solvers should be exploited. Moreover, every regularization method crucially depends on the choice of one or more regularization parameters, which should be suitably tuned. The aim of this paper is twofold: (a) survey some well-established regularizing projection methods based on Krylov subspace methods (with a particular emphasis on methods based on the Golub-Kahan bidiagonalization algorithm), and the so-called hybrid approaches (which combine Tikhonov regularization and projection onto Krylov subspaces of increasing dimension); (b) introduce a new principled and adaptive algorithmic approach for regularization similar to specific instances of hybrid methods. In particular, the new strategy provides reliable parameter choice rules by leveraging the framework of bilevel optimization, and the links between Gauss quadrature and Golub-Kahan bidiagonalization. Numerical tests modeling inverse problems in imaging illustrate the performance of existing regularizing Krylov methods, and validate the new algorithms.</p>","PeriodicalId":53634,"journal":{"name":"GAMM Mitteilungen","volume":"43 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1002/gamm.202000017","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76182732","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Kirk M. Soodhalter, Eric de Sturler, Misha E. Kilmer
{"title":"A survey of subspace recycling iterative methods","authors":"Kirk M. Soodhalter, Eric de Sturler, Misha E. Kilmer","doi":"10.1002/gamm.202000016","DOIUrl":"10.1002/gamm.202000016","url":null,"abstract":"<p>This survey concerns <i>subspace recycling methods</i>, a popular class of iterative methods that enable effective reuse of subspace information in order to speed up convergence and find good initial vectors over a sequence of linear systems with slowly changing coefficient matrices, multiple right-hand sides, or both. The subspace information that is recycled is usually generated during the run of an iterative method (usually a Krylov subspace method) on one or more of the systems. Following introduction of definitions and notation, we examine the history of early augmentation schemes along with deflation preconditioning schemes and their influence on the development of recycling methods. We then discuss a general residual constraint framework through which many augmented Krylov and recycling methods can both be viewed. We review several augmented and recycling methods within this framework. We then discuss some known effective strategies for choosing subspaces to recycle before taking the reader through more recent developments that have generalized recycling for (sequences of) shifted linear systems, some of them with multiple right-hand sides in mind. We round out our survey with a brief review of application areas that have seen benefit from subspace recycling methods.</p>","PeriodicalId":53634,"journal":{"name":"GAMM Mitteilungen","volume":"43 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1002/gamm.202000016","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76431675","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":"Limited-memory polynomial methods for large-scale matrix functions","authors":"Stefan Güttel, Daniel Kressner, Kathryn Lund","doi":"10.1002/gamm.202000019","DOIUrl":"10.1002/gamm.202000019","url":null,"abstract":"<p>Matrix functions are a central topic of linear algebra, and problems requiring their numerical approximation appear increasingly often in scientific computing. We review various limited-memory methods for the approximation of the action of a large-scale matrix function on a vector. Emphasis is put on polynomial methods, whose memory requirements are known or prescribed a priori. Methods based on explicit polynomial approximation or interpolation, as well as restarted Arnoldi methods, are treated in detail. An overview of existing software is also given, as well as a discussion of challenging open problems.</p>","PeriodicalId":53634,"journal":{"name":"GAMM Mitteilungen","volume":"43 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78119560","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":"A literature survey of matrix methods for data science","authors":"Martin Stoll","doi":"10.1002/gamm.202000013","DOIUrl":"10.1002/gamm.202000013","url":null,"abstract":"<p>Efficient numerical linear algebra is a core ingredient in many applications across almost all scientific and industrial disciplines. With this survey we want to illustrate that numerical linear algebra has played and is playing a crucial role in enabling and improving data science computations with many new developments being fueled by the availability of data and computing resources. We highlight the role of various different factorizations and the power of changing the representation of the data as well as discussing topics such as randomized algorithms, functions of matrices, and high-dimensional problems. We briefly touch upon the role of techniques from numerical linear algebra used within deep learning.</p>","PeriodicalId":53634,"journal":{"name":"GAMM Mitteilungen","volume":"43 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1002/gamm.202000013","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72476180","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Matrix functions in network analysis","authors":"Michele Benzi, Paola Boito","doi":"10.1002/gamm.202000012","DOIUrl":"10.1002/gamm.202000012","url":null,"abstract":"<p>We review the recent use of functions of matrices in the analysis of graphs and networks, with special focus on centrality and communicability measures and diffusion processes. Both undirected and directed networks are considered, as well as dynamic (temporal) networks. Computational issues are also addressed.</p>","PeriodicalId":53634,"journal":{"name":"GAMM Mitteilungen","volume":"43 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79600590","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":"Numerical linear algebra in data assimilation","authors":"Melina A. Freitag","doi":"10.1002/gamm.202000014","DOIUrl":"10.1002/gamm.202000014","url":null,"abstract":"<p>Data assimilation is a method that combines observations (ie, real world data) of a state of a system with model output for that system in order to improve the estimate of the state of the system and thereby the model output. The model is usually represented by a discretized partial differential equation. The data assimilation problem can be formulated as a large scale Bayesian inverse problem. Based on this interpretation we will derive the most important variational and sequential data assimilation approaches, in particular three-dimensional and four-dimensional variational data assimilation (3D-Var and 4D-Var) and the Kalman filter. We will then consider more advanced methods which are extensions of the Kalman filter and variational data assimilation and pay particular attention to their advantages and disadvantages. The data assimilation problem usually results in a very large optimization problem and/or a very large linear system to solve (due to inclusion of time and space dimensions). Therefore, the second part of this article aims to review advances and challenges, in particular from the numerical linear algebra perspective, within the various data assimilation approaches.</p>","PeriodicalId":53634,"journal":{"name":"GAMM Mitteilungen","volume":"43 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/gamm.202000014","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85018547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Modeling and simulation of human induced pluripotent stem cell-derived cardiac tissue","authors":"Alexander Jung, Manfred Staat","doi":"10.1002/gamm.202000011","DOIUrl":"10.1002/gamm.202000011","url":null,"abstract":"<p> </p><p>In the discussion section of Jung and Staat<span><sup>1</sup></span>, the statement “a factor of 1000 when the ventricular-like” should be corrected to “a factor of 10 if the ventricular-like”.</p><p>The online version has been corrected.</p>","PeriodicalId":53634,"journal":{"name":"GAMM Mitteilungen","volume":"43 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1002/gamm.202000011","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90943111","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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