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Shifted and extrapolated power methods for tensor $ell^p$-eigenpairs 张量$ well ^p$-特征对的移位和外推幂方法
ETNA - Electronic Transactions on Numerical Analysis Pub Date : 2019-09-26 DOI: 10.1553/etna_vol53s1
S. Cipolla, M. Redivo-Zaglia, Francesco Tudisco
{"title":"Shifted and extrapolated power methods for tensor $ell^p$-eigenpairs","authors":"S. Cipolla, M. Redivo-Zaglia, Francesco Tudisco","doi":"10.1553/etna_vol53s1","DOIUrl":"https://doi.org/10.1553/etna_vol53s1","url":null,"abstract":"This work is concerned with the computation of $ell^p$-eigenvalues and eigenvectors of square tensors with $d$ modes. In the first part we propose two possible shifted variants of the popular (higher-order) power method %for the computation of $ell^p$-eigenpairs proving the convergence of both the schemes to the Perron $ell^p$-eigenvector of the tensor, and the maximal corresponding $ell^p$-eigenvalue, when the tensor is entrywise nonnegative and $p$ is strictly larger than the number of modes. Then, motivated by the slow rate of convergence that the proposed methods achieve for certain real-world tensors, when $papprox d$, the number of modes, in the second part we introduce an extrapolation framework based on the simplified topological $varepsilon$-algorithm to efficiently accelerate the shifted power sequences. Numerical results on synthetic and real world problems show the improvements gained by the introduction of the shifting parameter and the efficiency of the acceleration technique.","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"158 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124725445","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}
引用次数: 14
Preconditioning the coarse problem of BDDC methods ‐ three-level, algebraic multigrid, and vertex-based preconditioners 预处理BDDC方法的粗问题——三级、代数多网格和基于顶点的预处理
ETNA - Electronic Transactions on Numerical Analysis Pub Date : 2019-06-18 DOI: 10.1553/ETNA_VOL51S432
A. Klawonn, M. Lanser, O. Rheinbach, J. Weber
{"title":"Preconditioning the coarse problem of BDDC methods ‐ three-level, algebraic multigrid, and vertex-based preconditioners","authors":"A. Klawonn, M. Lanser, O. Rheinbach, J. Weber","doi":"10.1553/ETNA_VOL51S432","DOIUrl":"https://doi.org/10.1553/ETNA_VOL51S432","url":null,"abstract":"A fair comparison of three Balancing Domain Decomposition by Constraints (BDDC) methods with an approximate coarse space solver is attempted for the first time. The comparison is made for a BDDC method with an algebraic multigrid preconditioner for the coarse problem, a three-level BDDC method, and a BDDC method with a vertex-based coarse preconditioner which was recently introduced by Clark Dohrmann, Kendall Pierson, and Olof Widlund. For the first time, all methods are presented and discussed in a common framework. Condition number bounds are provided for all approaches. All methods are implemented in a common highly parallel scalable BDDC software package based on PETSc, to allow for a fair comparison. Numerical results showing the parallel scalability are presented for the equations of linear elasticity. For the first time, this includes parallel scalability tests for the vertex-based approximate BDDC method.","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121742766","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}
引用次数: 3
Bouligand-Levenberg-Marquardt iteration for a non-smooth ill-posed inverse problem 非光滑不适定逆问题的Bouligand-Levenberg-Marquardt迭代
ETNA - Electronic Transactions on Numerical Analysis Pub Date : 2019-02-27 DOI: 10.1553/etna_vol51s274
Christian Clason, V. H. Nhu
{"title":"Bouligand-Levenberg-Marquardt iteration for a non-smooth ill-posed inverse problem","authors":"Christian Clason, V. H. Nhu","doi":"10.1553/etna_vol51s274","DOIUrl":"https://doi.org/10.1553/etna_vol51s274","url":null,"abstract":"In this paper, we consider a modified Levenberg--Marquardt method for solving an ill-posed inverse problem where the forward mapping is not G^ateaux differentiable. By relaxing the standard assumptions for the classical smooth setting, we derive asymptotic stability estimates that are then used to prove the convergence of the proposed method. This method can be applied to an inverse source problem for a non-smooth semilinear elliptic PDE where a Bouligand subdifferential can be used in place of the non-existing Fr'echet derivative, and we show that the corresponding Bouligand-Levenberg-Marquardt iteration is an iterative regularization scheme. Numerical examples illustrate the advantage over the corresponding Bouligand-Landweber iteration.","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"388 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126277708","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}
引用次数: 9
ADMM-Softmax: an ADMM approach for multinomial logistic regression ADMM- softmax:一种用于多项逻辑回归的ADMM方法
ETNA - Electronic Transactions on Numerical Analysis Pub Date : 2019-01-27 DOI: 10.1553/etna_vol52s214
Samy Wu Fung, Sanna Tyrväinen, Lars Ruthotto, E. Haber
{"title":"ADMM-Softmax: an ADMM approach for multinomial logistic regression","authors":"Samy Wu Fung, Sanna Tyrväinen, Lars Ruthotto, E. Haber","doi":"10.1553/etna_vol52s214","DOIUrl":"https://doi.org/10.1553/etna_vol52s214","url":null,"abstract":"We present ADMM-Softmax, an alternating direction method of multipliers (ADMM) for solving multinomial logistic regression (MLR) problems. Our method is geared toward supervised classification tasks with many examples and features. It decouples the nonlinear optimization problem in MLR into three steps that can be solved efficiently. In particular, each iteration of ADMM-Softmax consists of a linear least-squares problem, a set of independent small-scale smooth, convex problems, and a trivial dual variable update. Solution of the least-squares problem can be be accelerated by pre-computing a factorization or preconditioner, and the separability in the smooth, convex problem can be easily parallelized across examples. For two image classification problems, we demonstrate that ADMM-Softmax leads to improved generalization compared to a Newton-Krylov, a quasi Newton, and a stochastic gradient descent method.","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115142926","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}
引用次数: 9
Numerical evaluation of special power series including the numbers of Lyndon words: an approach to interpolation functions for Apostol-type numbers and polynomials 包含Lyndon词数的特殊幂级数的数值计算:apostoll型数和多项式的插值函数的一种方法
ETNA - Electronic Transactions on Numerical Analysis Pub Date : 2019-01-10 DOI: 10.1553/etna_vol50s98
Irem Kucukoglu, Y. Simsek
{"title":"Numerical evaluation of special power series including the numbers of Lyndon words: an approach to interpolation functions for Apostol-type numbers and polynomials","authors":"Irem Kucukoglu, Y. Simsek","doi":"10.1553/etna_vol50s98","DOIUrl":"https://doi.org/10.1553/etna_vol50s98","url":null,"abstract":"Because the Lyndon words and their numbers have practical applications in many different disciplines such as mathematics, probability, statistics, computer programming, algorithms, etc., it is known that not only mathematicians but also statisticians, computer programmers, and other scientists have studied them using different methods. Contrary to other studies, in this paper we use methods associated with zeta-type functions, which interpolate the family of Apostol-type numbers and polynomials of order k. Therefore, the main purpose of this paper is not only to give a special power series including the numbers of Lyndon words and binomial coefficients but also to construct new computational algorithms in order to simulate these series by numerical evaluations and plots. By using these algorithms, we provide novel computational methods to the area of combinatorics on words including Lyndon words. We also define new functions related to these power series, Lyndon words counting numbers, and the Apostol-type numbers and polynomials. Furthermore, we present some illustrations and observations on approximations of functions by rational functions associated with Apostol-type numbers that can provide ideas on the reduction of the algorithmic complexity of these algorithms.","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"134 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127591038","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}
引用次数: 0
Multiple Hermite polynomials and simultaneous Gaussian quadrature 多重厄米特多项式和同时高斯正交
ETNA - Electronic Transactions on Numerical Analysis Pub Date : 2018-12-04 DOI: 10.1553/etna_vol50s182
W. Assche, A. Vuerinckx
{"title":"Multiple Hermite polynomials and simultaneous Gaussian quadrature","authors":"W. Assche, A. Vuerinckx","doi":"10.1553/etna_vol50s182","DOIUrl":"https://doi.org/10.1553/etna_vol50s182","url":null,"abstract":"Multiple Hermite polynomials are an extension of the classical Hermite polynomials for which orthogonality conditions are imposed with respect to $r>1$ normal (Gaussian) weights $w_j(x)=e^{-x^2+c_jx}$ with different means $c_j/2$, $1 leq j leq r$. These polynomials have a number of properties, such as a Rodrigues formula, recurrence relations (connecting polynomials with nearest neighbor multi-indices), a differential equation, etc. The asymptotic distribution of the (scaled) zeros is investigated and an interesting new feature happens: depending on the distance between the $c_j$, $1 leq j leq r$, the zeros may accumulate on $s$ disjoint intervals, where $1 leq s leq r$. We will use the zeros of these multiple Hermite polynomials to approximate integrals of the form $displaystyle int_{-infty}^{infty} f(x) exp(-x^2 + c_jx), dx$ simultaneously for $1 leq j leq r$ for the case $r=3$ and the situation when the zeros accumulate on three disjoint intervals. We also give some properties of the corresponding quadrature weights.","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116222393","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}
引用次数: 3
Adaptive discontinuous Galerkin approximation of optimal control problems governed by transient convection-diffusion equations 瞬态对流扩散方程最优控制问题的自适应间断伽辽金逼近
ETNA - Electronic Transactions on Numerical Analysis Pub Date : 2018-11-22 DOI: 10.1553/ETNA_VOL48S407
Hamdullah Yücel, M. Stoll, P. Benner
{"title":"Adaptive discontinuous Galerkin approximation of optimal control problems governed by transient convection-diffusion equations","authors":"Hamdullah Yücel, M. Stoll, P. Benner","doi":"10.1553/ETNA_VOL48S407","DOIUrl":"https://doi.org/10.1553/ETNA_VOL48S407","url":null,"abstract":"In this paper, we investigate a posteriori error estimates of a control-constrained optimal control problem governed by a time-dependent convection diffusion equation. The control constraints are handled by using the primal-dual active set algorithm as a semi-smooth Newton method and by adding a Moreau-Yosida-type penalty function to the cost functional. Residual-based error estimators are proposed for both approaches. The derived error estimators are used as error indicators to guide the mesh refinements. A symmetric interior penalty Galerkin method in space and a backward Euler method in time are applied in order to discretize the optimization problem. Numerical results are presented, which illustrate the performance of the proposed error estimators.","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123362841","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}
引用次数: 1
Topological derivative for the nonlinear magnetostatic problem 非线性静磁问题的拓扑导数
ETNA - Electronic Transactions on Numerical Analysis Pub Date : 2018-11-21 DOI: 10.1553/etna_vol51s169
P. Gangl, S. Amstutz
{"title":"Topological derivative for the nonlinear magnetostatic problem","authors":"P. Gangl, S. Amstutz","doi":"10.1553/etna_vol51s169","DOIUrl":"https://doi.org/10.1553/etna_vol51s169","url":null,"abstract":"The topological derivative represents the sensitivity of a domain-dependent functional with respect to a local perturbation of the domain and is a valuable tool in topology optimization. Motivated by an application from electrical engineering, we derive the topological derivative for an optimization problem which is constrained by the quasilinear equation of two-dimensional magnetostatics. Here, the main ingredient is to establish a sufficiently fast decay of the variation of the direct state at scale 1 as $|x|rightarrow infty$. In order to apply the method in a bi-directional topology optimization algorithm, we derive both the sensitivity for introducing air inside ferromagnetic material and the sensitivity for introducing material inside an air region. We explicitly compute the arising polarization matrices and introduce a way to efficiently evaluate the obtained formulas. Finally, we employ the derived formulas in a level-set based topology optimization algorithm and apply it to the design optimization of an electric motor.","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"112 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117286769","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}
引用次数: 13
Approximate residual-minimizing shift parameters for the low-rank ADI iteration 低秩ADI迭代的近似残差最小化移位参数
ETNA - Electronic Transactions on Numerical Analysis Pub Date : 2018-11-13 DOI: 10.1553/etna_vol51s240
Patrick Kurschner
{"title":"Approximate residual-minimizing shift parameters for the low-rank ADI iteration","authors":"Patrick Kurschner","doi":"10.1553/etna_vol51s240","DOIUrl":"https://doi.org/10.1553/etna_vol51s240","url":null,"abstract":"The low-rank alternating directions implicit (LR-ADI) iteration is a frequently employed method for efficiently computing low-rank approximate solutions of large-scale Lyapunov equations. In order to achieve a rapid error reduction, the iteration requires shift parameters whose selection and generation is often a difficult task, especially for nonsymmetric coefficients in the Lyapunov equation. This article represents a follow up of Benner et al. [ETNA, 43 (2014-2015), pp. 142-162] and investigates self-generating shift parameters based on a minimization principle for the Lyapunov residual norm. Since the involved objective functions are too expensive to evaluate and, hence, intractable, compressed objective functions are introduced which are efficiently constructed from the available data generated by the LR-ADI iteration. Several numerical experiments indicate that these residual minimizing shifts using approximated objective functions outperform existing precomputed and dynamic shift parameter selection techniques, although their generation is more involved.","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114378745","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}
引用次数: 6
A note on parallel preconditioning for all-at-once evolutionary PDEs 关于一次性进化偏微分方程并行预处理的注解
ETNA - Electronic Transactions on Numerical Analysis Pub Date : 2018-10-01 DOI: 10.1553/ETNA_VOL51S135
A. Goddard, A. Wathen
{"title":"A note on parallel preconditioning for all-at-once evolutionary PDEs","authors":"A. Goddard, A. Wathen","doi":"10.1553/ETNA_VOL51S135","DOIUrl":"https://doi.org/10.1553/ETNA_VOL51S135","url":null,"abstract":"McDonald, Pestana and Wathen (SIAM J. Sci. Comput. 40(2), pp. A2012-A1033, 2018) present a method for preconditioning of time-dependent PDEs via approximation by a nearby time-periodic problem, that is, they employ circulant-related matrices as preconditioners for the non-symmetric block Toeplitz matrices which arise from an all-at-once formulation. They suggest that such an approach might be efficiently implemented in parallel. \u0000In this short article, we present parallel numerical results for their preconditioner which exhibit strong scaling. We also extend their preconditioner via a Neumann series approach, which also allows for efficient parallel execution. Our simple implementation (in C++ and MPI) is available at the Git repository PARALAAOMPI. this https URL","PeriodicalId":282695,"journal":{"name":"ETNA - Electronic Transactions on Numerical Analysis","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123168715","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}
引用次数: 33
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