Numerical Linear Algebra with Applications最新文献

{"title":"Quasi‐Newton variable preconditioning for nonlinear elasticity systems in 3D","authors":"J. Karátson, S. Sysala, M. Béreš","doi":"10.1002/nla.2537","DOIUrl":"https://doi.org/10.1002/nla.2537","url":null,"abstract":"Summary Quasi‐Newton iterations are constructed for the finite element solution of small‐strain nonlinear elasticity systems in 3D. The linearizations are based on spectral equivalence and hence considered as variable preconditioners arising from proper simplifications in the differential operator. Convergence is proved, providing bounds uniformly w.r.t. the FEM discretization. Convenient iterative solvers for linearized systems are also proposed. Numerical experiments in 3D confirm that the suggested quasi‐Newton methods are competitive with Newton's method.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":"84 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135367516","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A tensor bidiagonalization method for higher‐order singular value decomposition with applications","authors":"A. El Hachimi, K. Jbilou, A. Ratnani, L. Reichel","doi":"10.1002/nla.2530","DOIUrl":"https://doi.org/10.1002/nla.2530","url":null,"abstract":"Abstract The need to know a few singular triplets associated with the largest singular values of a third‐order tensor arises in data compression and extraction. This paper describes a new method for their computation using the t‐product. Methods for determining a couple of singular triplets associated with the smallest singular values also are presented. The proposed methods generalize available restarted Lanczos bidiagonalization methods for computing a few of the largest or smallest singular triplets of a matrix. The methods of this paper use Ritz and harmonic Ritz lateral slices to determine accurate approximations of the largest and smallest singular triplets, respectively. Computed examples show applications to data compression and face recognition.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":"153 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135458577","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Conditioning of hybrid variational data assimilation","authors":"Shaerdan Shataer, Amos S. Lawless, Nancy K. Nichols","doi":"10.1002/nla.2534","DOIUrl":"https://doi.org/10.1002/nla.2534","url":null,"abstract":"Abstract In variational assimilation, the most probable state of a dynamical system under Gaussian assumptions for the prior and likelihood can be found by solving a least‐squares minimization problem. In recent years, we have seen the popularity of hybrid variational data assimilation methods for Numerical Weather Prediction. In these methods, the prior error covariance matrix is a weighted sum of a climatological part and a flow‐dependent ensemble part, the latter being rank deficient. The nonlinear least squares problem of variational data assimilation is solved using iterative numerical methods, and the condition number of the Hessian is a good proxy for the convergence behavior of such methods. In this article, we study the conditioning of the least squares problem in a hybrid four‐dimensional variational data assimilation (Hybrid 4D‐Var) scheme by establishing bounds on the condition number of the Hessian. In particular, we consider the effect of the ensemble component of the prior covariance on the conditioning of the system. Numerical experiments show that the bounds obtained can be useful in predicting the behavior of the true condition number and the convergence speed of an iterative algorithm","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134958103","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A family of inertial‐based derivative‐free projection methods with a correction step for constrained nonlinear equations and their applications","authors":"Pengjie Liu, Hu Shao, Zihang Yuan, Jianhao Zhou","doi":"10.1002/nla.2533","DOIUrl":"https://doi.org/10.1002/nla.2533","url":null,"abstract":"Abstract Numerous attempts have been made to develop efficient methods for solving the system of constrained nonlinear equations due to its widespread use in diverse engineering applications. In this article, we present a family of inertial‐based derivative‐free projection methods with a correction step for solving such system, in which the selection of the derivative‐free search direction is flexible. This family does not require the computation of corresponding Jacobian matrix or approximate matrix at every iteration and possess the following theoretical properties: (i) the inertial‐based corrected direction framework always automatically satisfies the sufficient descent and trust region properties without specific search directions, and is independent of any line search; (ii) the global convergence of the proposed family is proven under a weaker monotonicity condition on the mapping , without the typical monotonicity or pseudo‐monotonicity assumption; (iii) the results about convergence rate of the proposed family are established under slightly stronger assumptions. Furthermore, we propose two effective inertial‐based derivative‐free projection methods, each embedding a specific search direction into the proposed family. We present preliminary numerical experiments on certain test problems to demonstrate the effectiveness and superiority of the proposed methods in comparison with existing ones. Additionally, we utilize these methods for solving sparse signal restorations and image restorations in compressive sensing applications.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":"91 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136060602","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Stage‐parallel preconditioners for implicit Runge–Kutta methods of arbitrarily high order, linear problems","authors":"Owe Axelsson, Ivo Dravins, Maya Neytcheva","doi":"10.1002/nla.2532","DOIUrl":"https://doi.org/10.1002/nla.2532","url":null,"abstract":"Abstract Fully implicit Runge–Kutta methods offer the possibility to use high order accurate time discretization to match space discretization accuracy, an issue of significant importance for many large scale problems of current interest, where we may have fine space resolution with many millions of spatial degrees of freedom and long time intervals. In this work, we consider strongly A‐stable implicit Runge–Kutta methods of arbitrary order of accuracy, based on Radau quadratures. For the arising large algebraic systems we introduce efficient preconditioners, that (1) use only real arithmetic, (2) demonstrate robustness with respect to problem and discretization parameters, and (3) allow for fully stage‐parallel solution. The preconditioners are based on the observation that the lower‐triangular part of the coefficient matrices in the Butcher tableau has larger in magnitude values, compared to the corresponding strictly upper‐triangular part. We analyze the spectrum of the corresponding preconditioned systems and illustrate their performance with numerical experiments. Even though the observation has been made some time ago, its impact on constructing stage‐parallel preconditioners has not yet been done and its systematic study constitutes the novelty of this article.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135063385","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Impact of correlated observation errors on the conditioning of variational data assimilation problems","authors":"O. Goux, S. Gürol, A. Weaver, Y. Diouane, Oliver Guillet","doi":"10.1002/nla.2529","DOIUrl":"https://doi.org/10.1002/nla.2529","url":null,"abstract":"An important class of nonlinear weighted least‐squares problems arises from the assimilation of observations in atmospheric and ocean models. In variational data assimilation, inverse error covariance matrices define the weighting matrices of the least‐squares problem. For observation errors, a diagonal matrix (i.e., uncorrelated errors) is often assumed for simplicity even when observation errors are suspected to be correlated. While accounting for observation‐error correlations should improve the quality of the solution, it also affects the convergence rate of the minimization algorithms used to iterate to the solution. If the minimization process is stopped before reaching full convergence, which is usually the case in operational applications, the solution may be degraded even if the observation‐error correlations are correctly accounted for. In this article, we explore the influence of the observation‐error correlation matrix () on the convergence rate of a preconditioned conjugate gradient (PCG) algorithm applied to a one‐dimensional variational data assimilation (1D‐Var) problem. We design the idealized 1D‐Var system to include two key features used in more complex systems: we use the background error covariance matrix () as a preconditioner (B‐PCG); and we use a diffusion operator to model spatial correlations in and . Analytical and numerical results with the 1D‐Var system show a strong sensitivity of the convergence rate of B‐PCG to the parameters of the diffusion‐based correlation models. Depending on the parameter choices, correlated observation errors can either speed up or slow down the convergence. In practice, a compromise may be required in the parameter specifications of and between staying close to the best available estimates on the one hand and ensuring an adequate convergence rate of the minimization algorithm on the other.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":" ","pages":""},"PeriodicalIF":4.3,"publicationDate":"2023-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45145878","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Rank‐structured approximation of some Cauchy matrices with sublinear complexity","authors":"Mikhail Lepilov, J. Xia","doi":"10.1002/nla.2526","DOIUrl":"https://doi.org/10.1002/nla.2526","url":null,"abstract":"In this article, we consider the rank‐structured approximation of one important type of Cauchy matrix. This approximation plays a key role in some structured matrix methods such as stable and efficient direct solvers and other algorithms for Toeplitz matrices and certain kernel matrices. Previous rank‐structured approximations (specifically hierarchically semiseparable, or HSS, approximations) for such a matrix of size cost at least complexity. Here, we show how to construct an HSS approximation with sublinear (specifically, ) complexity. The main ideas include extensive computation reuse and an analytical far‐field compression strategy. Low‐rank compression at each hierarchical level is restricted to just a single off‐diagonal block row, and a resulting basis matrix is then reused for other off‐diagonal block rows as well as off‐diagonal block columns. The relationships among the off‐diagonal blocks are rigorously analyzed. The far‐field compression uses an analytical proxy point method where we optimize the choice of some parameters so as to obtain accurate low‐rank approximations. Both the basis reuse ideas and the resulting analytical hierarchical compression scheme can be generalized to some other kernel matrices and are useful for accelerating relevant rank‐structured approximations (though not subsequent operations like matrix‐vector multiplications).","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":" ","pages":""},"PeriodicalIF":4.3,"publicationDate":"2023-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42911576","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Volume‐based subset selection","authors":"Alexander Osinsky","doi":"10.1002/nla.2525","DOIUrl":"https://doi.org/10.1002/nla.2525","url":null,"abstract":"This paper provides a fast algorithm for the search of a dominant (locally maximum volume) submatrix, generalizing the existing algorithms from n⩽r$$ nleqslant r $$ to n>r$$ n>r $$ submatrix columns, where r$$ r $$ is the number of searched rows. We prove the bound on the number of steps of the algorithm, which allows it to outperform the existing subset selection algorithms in either the bounds on the norm of the pseudoinverse of the found submatrix, or the bounds on the complexity, or both.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":" ","pages":""},"PeriodicalIF":4.3,"publicationDate":"2023-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46946721","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Total positivity and accurate computations with Gram matrices of Said‐Ball bases","authors":"E. Mainar, J. M. Pena, B. Rubio","doi":"10.1002/nla.2521","DOIUrl":"https://doi.org/10.1002/nla.2521","url":null,"abstract":"In this article, it is proved that Gram matrices of totally positive bases of the space of polynomials of a given degree on a compact interval are totally positive. Conditions to guarantee computations to high relative accuracy with those matrices are also obtained. Furthermore, a fast and accurate algorithm to compute the bidiagonal factorization of Gram matrices of the Said‐Ball bases is obtained and used to compute to high relative accuracy their singular values and inverses, as well as the solution of some linear systems associated with these matrices. Numerical examples are included.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":" ","pages":""},"PeriodicalIF":4.3,"publicationDate":"2023-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47051257","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Nonlinear approximation of functions based on nonnegative least squares solver","authors":"Petr N. Vabishchevich","doi":"10.1002/nla.2522","DOIUrl":"https://doi.org/10.1002/nla.2522","url":null,"abstract":"In computational practice, most attention is paid to rational approximations of functions and approximations by the sum of exponents. We consider a wide enough class of nonlinear approximations characterized by a set of two required parameters. The approximating function is linear in the first parameter; these parameters are assumed to be positive. The individual terms of the approximating function represent a fixed function that depends nonlinearly on the second parameter. A numerical approximation minimizes the residual functional by approximating function values at individual points. The second parameter's value is set on a more extensive set of points of the interval of permissible values. The proposed approach's key feature consists in determining the first parameter on each separate iteration of the classical nonnegative least squares method. The computational algorithm is used to rational approximate the function <math altimg=\"urn:x-wiley:nla:media:nla2522:nla2522-math-0001\" display=\"inline\" location=\"graphic/nla2522-math-0001.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<msup>\u0000<mrow>\u0000<mi>x</mi>\u0000</mrow>\u0000<mrow>\u0000<mo form=\"prefix\">−</mo>\u0000<mi>α</mi>\u0000</mrow>\u0000</msup>\u0000<mo>,</mo>\u0000<mspace width=\"0.3em\"></mspace>\u0000<mn>0</mn>\u0000<mo>&lt;</mo>\u0000<mi>α</mi>\u0000<mo>&lt;</mo>\u0000<mn>1</mn>\u0000<mo>,</mo>\u0000<mspace width=\"0.3em\"></mspace>\u0000<mi>x</mi>\u0000<mo>≥</mo>\u0000<mn>1</mn>\u0000</mrow>\u0000$$ {x}^{-alpha },kern0.3em 0&lt;alpha &lt;1,kern0.3em xge 1 $$</annotation>\u0000</semantics></math>. The second example concerns the approximation of the stretching exponential function <math altimg=\"urn:x-wiley:nla:media:nla2522:nla2522-math-0002\" display=\"inline\" location=\"graphic/nla2522-math-0002.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>exp</mi>\u0000<mo stretchy=\"false\">(</mo>\u0000<mo form=\"prefix\">−</mo>\u0000<msup>\u0000<mrow>\u0000<mi>x</mi>\u0000</mrow>\u0000<mrow>\u0000<mi>α</mi>\u0000</mrow>\u0000</msup>\u0000<mo stretchy=\"false\">)</mo>\u0000<mo>,</mo>\u0000<mspace width=\"0.0em\"></mspace>\u0000<mspace width=\"0.0em\"></mspace>\u0000<mspace width=\"0.2em\"></mspace>\u0000<mn>0</mn>\u0000<mo>&lt;</mo>\u0000<mi>α</mi>\u0000<mo>&lt;</mo>\u0000<mn>1</mn>\u0000</mrow>\u0000$$ exp left(-{x}^{alpha}right),0&lt;alpha &lt;1 $$</annotation>\u0000</semantics></math> at <math altimg=\"urn:x-wiley:nla:media:nla2522:nla2522-math-0003\" display=\"inline\" location=\"graphic/nla2522-math-0003.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>x</mi>\u0000<mo>≥</mo>\u0000<mn>0</mn>\u0000</mrow>\u0000$$ xge 0 $$</annotation>\u0000</semantics></math> by the sum of exponents.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":"147 3","pages":""},"PeriodicalIF":4.3,"publicationDate":"2023-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138502938","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}