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Parallel algorithms for tensor train arithmetic

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arXiv 2020年

作者： Al Daas, Hussam Ballard, Grey Benner, Peter Department of Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg Germany Computer Science Department Wake Forest University Winston SalemNC United States

We present efficient and scalable parallel algorithms for performing mathematical operations for low-rank tensors represented in the tensor train (TT) format. We consider algorithms for addition, elementwise multiplication, computing norms and inner products, orthonormalization, and rounding (rank truncation). These are the kernel operations for applications such as iterative Krylov solvers that exploit the TT structure. The parallel algorithms are designed for distributed- memory computation, and we propose a data distribution and strategy that parallelizes computations for individual cores within the TT format. We analyze the computation and communication costs of the proposed algorithms to show their scalability, and we present numerical experiments that demonstrate their efficiency on both shared-memory and distributed-memory parallel systems. For example, we observe better single-core performance than the existing MATLAB TT-Toolbox in rounding a 2GB TT tensor, and our implementation achieves a 34× speedup using all 40 cores of a single node. We also show nearly linear parallel scaling on larger TT tensors up to over 10,000 cores for all mathematical operations. Copyright © 2020, The Authors. All rights reserved.

关键词： Tensors

GR decompositions and their relations to Cholesky-like factorizations

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PAMM 2021年第1期20卷

作者： Peter Benner Carolin Penke Max Planck Institute for Dynamics of Complex Technical Systems: Computational Methods in Systems and Control Theory Sandtorstr. 1 39106 Magdeburg

For a given matrix, we are interested in computing GR decompositions A = GR , where G is an isometry with respect to given scalar products. The orthogonal QR decomposition is the representative for the Euclidian scalar product. For a signature matrix, a respective factorization is given as the hyperbolic QR decomposition. Considering a skew-symmetric matrix leads to the symplectic QR decomposition. The standard approach for computing GR decompositions is based on the successive elimination of subdiagonal matrix entries. For the hyperbolic and symplectic case, this approach does in general not lead to a satisfying numerical accuracy. An alternative approach computes the QR decomposition via a Cholesky factorization, but also has bad stability. It is improved by repeating the procedure a second time. In the same way, the hyperbolic and the symplectic QR decomposition are related to the LDL T and a skew-symmetric Cholesky-like factorization. We show that methods exploiting this connection can provide better numerical stability than elimination-based approaches.

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Efficient structure-preserving support tensor train machine

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arXiv 2020年

作者： Kour, Kirandeep Benner, Peter Dolgov, Sergey Stoll, Martin Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems MagdeburgD-39106 Germany Department of Mathematical Sciences University of Bath BathBA2 7AY United Kingdom Faculty of Mathematics Technische Universität Chemnitz ChemnitzD-09107 Germany

An increasing amount of collected data are high-dimensional multi-way arrays (tensors), and it is crucial for efficient learning algorithms to exploit this tensorial structure as much as possible. The ever present curse of dimensionality for high dimensional data and the loss of structure when vectorizing the data motivates the use of tailored low-rank tensor classification methods. In the presence of small amounts of training data, kernel methods offer an attractive choice as they provide the possibility for a nonlinear decision boundary. We develop the Tensor Train Multi-way Multi-level Kernel (TT-MMK), which combines the simplicity of the Canonical Polyadic decomposition, the classification power of the Dual Structure-preserving Support Vector Machine, and the reliability of the Tensor Train (TT) approximation. We show by experiments that the TT-MMK method is usually more reliable computationally, less sensitive to tuning parameters, and gives higher prediction accuracy in the SVM classification when benchmarked against other state-of-the-art techniques. Copyright © 2020, The Authors. All rights reserved.

关键词： Tensors

On an invariance principle for the solution space of the differential riccati equation

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On an invariance principle for the solution space of the dif...

89th Annual Meeting on the International Association of Applied Mathematics and Mechanics, GAMM 2018

作者： Behr, Maximilian Benner, Peter Heiland, Jan Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany

The differential Riccati equation appears in different fields of applied mathematics like control theory and systems theory. For large-scale systems the numerical solution comes with a large amount of storage requirements. This motivates the use of Krylov subspace and projection based methods [1–3]. In the present paper we apply an invariance theorem for ODEs to the differential Riccati Equation. We show that the solution is contained in a Krylov like subspace and extend our results to certain time-varying cases. © 2018 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.

关键词： Riccati equations

Sustainable research software hand-over

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arXiv 2019年

作者： Fehr, Jörg Himpe, Christian Rave, Stephan Saak, Jens Institute of Engineering and Computational Mechanics University of Stuttgart Pfaffenwaldring 9 Stuttgart70569 Germany Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Sandtorstr. 1 Magdeburg39106 Germany Institute for Computational and Applied Mathematics University of Münster Einsteinstrasse 62 Münster48149 Germany

Scientific software projects evolve rapidly in their initial development phase, yet at the end of a funding period, the completion of a research project, thesis, or publication, further engagement in the project may slow down or cease completely. To retain the invested effort for the sciences, this software needs to be preserved or handed over to a succeeding developer or team, such as the next generation of (PhD) students. Comparable guides provide top-down recommendations for project leads. This paper intends to be a bottom-up approach for sustainable hand-over processes from a developer's perspective. An important characteristic in this regard is the project's size, by which this guideline is structured. Furthermore, checklists are provided, which can serve as a practical guide for implementing the proposed *** Codes 68N30, 97N80, 97P99 Copyright © 2019, The Authors. All rights reserved.

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Exact Optimization: Part I

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arXiv 2019年

作者： Lin, Li-Gang Liang, Yew-Wen The Department of Mechanical Engineering National Central University Taoyuan32001 Taiwan The Department of Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg39106 Germany The Institute of Electrical Control Engineering National Yang Ming Chiao Tung University Hsinchu30010 Taiwan

Nonlinear programming is explicitly analyzed via a novel perspective/method and from a bottom-up manner. The philosophy is based on the recent findings on convex quadratic equation (CQE), which help clarify a geometric interpretation that relates CQE to convex quadratic function (CQF). More specifically, regarding the solvability of CQE, its necessary and sufficient condition as well as a unified parameterization of all the solutions has recently been analytically formulated. Moving forward, the understanding of CQE is utilized to describe the geometric structure of CQF, and the CQE-CQF relation. All these results are shown closely related to a basis in the optimization literature, namely quadratic programming (QP). For the first time from this viewpoint, the QPs subject to equality, inequality, equality-and-inequality, and extended constraints can be algebraically solved in derivative-free closed formulae, respectively. All the results are derived without knowing a feasible point, a priori and any time during the *** Codes Primary: 15A18, 52A41, 90C20, 90C25, and 90C46 Copyright © 2019, The Authors. All rights reserved.

关键词： Matrix algebra

High performance solution of skew-symmetric eigenvalue problems with applications in solving the Bethe-Salpeter Eigenvalue Problem

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arXiv 2019年

作者： Penke, Carolin Marek, Andreas Vorwerk, Christian Draxl, Claudia Benner, Peter Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Germany Max Planck Computing and Data Facility Garching Germany Institut für Physik and IRIS Adlershof Humboldt-Universität zu Berlin Berlin Germany

We present a high-performance solver for dense skew-symmetric matrix eigenvalue problems. Our work is motivated by applications in computational quantum physics, where one solution approach to solve the so-called Bethe-Salpeter equation involves the solution of a large, dense, skew-symmetric eigenvalue problem. The computed eigenpairs can be used to compute the optical absorption spectrum of molecules and crystalline systems. One state-of-the art high-performance solver package for symmetric matrices is the ELPA (Eigenvalue SoLvers for Petascale Applications) library. We exploit a link between tridiagonal skew-symmetric and symmetric matrices in order to extend the methods available in ELPA to skew-symmetric matrices. This way, the presented solution method can benefit from the optimizations available in ELPA that make it a well-established, efficient and scalable library. The solution strategy is to reduce a matrix to tridiagonal form, solve the tridiagonal eigenvalue problem and perform a back-transformation for eigenvectors of interest. ELPA employs a one-step or a two-step approach for the tridiagonalization of symmetric matrices. We adapt these to suit the skew-symmetric case. The two-step approach is generally faster as memory locality is exploited better. If all eigenvectors are required, the performance improvement is compensated by the additional back transformation step. We exploit the symmetry in the spectrum of skew-symmetric matrices, such that only half of the eigenpairs need to be computed, making the two-step approach the favorable method. We compare performance and scalability of our method to the only available high-performance approach for skew-symmetric matrices, an indirect route involving complex arithmetic. In total, we achieve a performance that is up to 3.67 higher than the reference method using Intel’s ScaLAPACK implementation. Our method is freely available in the current release of the ELPA *** Codes 65Y05, 15B57 Copyright © 2019

关键词： Eigenvalues and eigenfunctions

Residual minimizing shift parameters for the low-rank ADI iteration

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arXiv 2018年

作者： Kürschner, Patrick Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory Magdeburg Germany

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.15A06, 65F10, 65F30 Copyright © 2018, The Authors. All rights reserved.

关键词： Lyapunov functions

Improving the performance of numerical algorithms for the bethe-salpeter eigenvalue problem

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Improving the performance of numerical algorithms for the be...

89th Annual Meeting on the International Association of Applied Mathematics and Mechanics, GAMM 2018

作者： Benner, Peter Marek, Andreas Penke, Carolin Max Planck Institute for Dynamics of Complex Technical Systems: Computational Methods in Systems and Control Theory Sandtorstr. 1 Magdeburg39106 Germany Max Planck Computing and Data Facility Gießenbachstraße 2 Garching85748 Germany

The Bethe-Salpeter eigenvalue problem arises in the computation of the electronic structure of many-body physical systems. The resulting matrix is complex, admits a certain block structure and can become extremely large. This raises the need for structure-preserving algorithms running in parallel on high performance compute clusters. In this paper we examine how a recently proposed direct method given in the BSEPACK library can be improved using eigenvalue solvers from the ELPA library. For large matrices a runtime reduction of up to 80% is achieved. © 2018 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.

关键词： Electronic structure