A Tensor-Train accelerated solver for integral equations in complex geometries

Corona, Eduardo; Rahimian, Abtin; Zorin, Denis

doi:10.1016/j.jcp.2016.12.051

Cited by 24 publications

(19 citation statements)

References 76 publications

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“…Moreover, tensor networks have the ability to efficiently parameterize, through structured compact representations, very general high-dimensional spaces which arise in modern applications [19,39,50,116,121,136,229]. Tensor networks also naturally account for intrinsic multidimensional and distributed patterns present in data, and thus provide the opportunity to develop very sophisticated models for capturing multiple interactions and couplings in data -these are more physically insightful and interpretable than standard pair-wise interactions.…”

Section: Advantages Of Multiway Analysis Via Tensor Networkmentioning

confidence: 99%

Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions

Cichocki

Lee

Oseledets

et al. 2016

FNT in Machine Learning

371

342

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Modern applications in engineering and data science are increasingly based on multidimensional data of exceedingly high volume, variety, and structural richness. However, standard machine learning algorithms typically scale exponentially with data volume and complexity of cross-modal couplings - the so called curse of dimensionality - which is prohibitive to the analysis of large-scale, multi-modal and multi-relational datasets. Given that such data are often efficiently represented as multiway arrays or tensors, it is therefore timely and valuable for the multidisciplinary machine learning and data analytic communities to review low-rank tensor decompositions and tensor networks as emerging tools for dimensionality reduction and large scale optimization problems. Our particular emphasis is on elucidating that, by virtue of the underlying low-rank approximations, tensor networks have the ability to alleviate the curse of dimensionality in a number of applied areas. In Part 1 of this monograph we provide innovative solutions to low-rank tensor network decompositions and easy to interpret graphical representations of the mathematical operations on tensor networks. Such a conceptual insight allows for seamless migration of ideas from the flat-view matrices to tensor network operations and vice versa, and provides a platform for further developments, practical applications, and non-Euclidean extensions. It also permits the introduction of various tensor network operations without an explicit notion of mathematical expressions, which may be beneficial for many research communities that do not directly rely on multilinear algebra. Our focus is on the Tucker and tensor train (TT) decompositions and their extensions, and on demonstrating the ability of tensor networks to provide linearly or even super-linearly (e.g., logarithmically) scalable solutions, as illustrated in detail in Part 2 of this monograph

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Section: Advantages Of Multiway Analysis Via Tensor Networkmentioning

confidence: 99%

Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions

Cichocki

Lee

Oseledets

et al. 2016

FNT in Machine Learning

371

342

View full text Add to dashboard Cite

show abstract

“…Because AMEN Cross and related TT rank revealing approaches proceed by enriching low-TT-rank approximations, all computations are performed on matrices of size r k−1 n k ×r k or less. In [CRZ15], it is shown that the complexity for this algorithm is thus bounded by O(r 3 d) or equivalently O(r 3 log N ), where r = max(r k ) is the maximal TT-rank that may be a function of sample size N and accuracy ε. Fig.…”

Section: Algorithm 1 Tt Decompositionmentioning

confidence: 99%

“…Further details about variants of this approach can be found in [OD12,DS13a,DS13b]. The complexity of this algorithm for a maximum TT rank r for both A and A −1 is bound by O(r 4 log N ), as shown in [CRZ15].…”

Section: Algorithm 1 Tt Decompositionmentioning

confidence: 99%

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Tensor Train Accelerated Solvers for Nonsmooth Rigid Body Dynamics

Corona

Gorsich

Jayakumar

et al. 2019

Applied Mechanics Reviews

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In the last two decades, increased need for high-fidelity simulations of the time evolution and propagation of forces in granular media has spurred a renewed interest in the discrete element method (DEM) modeling of frictional contact. Force penalty methods, while economic and widely accessible, introduce artificial stiffness, requiring small time steps to retain numerical stability. Optimization-based methods, which enforce contacts geometrically through complementarity constraints leading to a differential variational inequality problem (DVI), allow for the use of larger time steps at the expense of solving a nonlinear complementarity problem (NCP) each time step. We review the latest efforts to produce solvers for this NCP, focusing on its relaxation to a cone complementarity problem (CCP) and solution via an equivalent quadratic optimization problem with conic constraints. We distinguish between first order methods, which use only gradient information and are thus linearly convergent and second order methods, which rely on a Newton type step to gain quadratic convergence and are typically more robust and problem-independent. However, they require the approximate solution of large sparse linear systems, thus losing their competitive advantages in large scale problems due to computational cost.In this work, we propose a novel acceleration for the solution of Newton step linear systems in second order methods using low-rank compression based fast direct solvers, leveraging on recent direct solver techniques for structured linear systems arising from differential and integral equations. We employ the Quantized Tensor Train (QTT) decomposition to produce efficient approximate representations of the system matrix and its inverse. This provides a versatile and robust framework to accelerate its solution using this inverse in a direct or a preconditioned iterative method. We demonstrate compressibility of the Newton step matrices in Primal Dual Interior Point (PDIP) methods as applied to the multibody dynamics problem. Using a number of numerical tests, we demonstrate that this approach displays sublinear scaling of precomputation costs, may be efficiently updated across Newton iterations as well as across simulation time steps, and leads to a fast, optimal complexity solution of the Newton step. This allows our method to gain an order of magnitude speedups over state-of-the-art preconditioning techniques for moderate to large-scale systems, hence mitigating the computational bottleneck of second order methods.

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“…Finally, extending the framework of low-rank compression, [9] uses tensor-train compression to re-write K (X, Y ) as a tensor with one dimension per coordinate, i.e., K (x 1 , . .…”

mentioning

confidence: 99%

Fast Low-Rank Kernel Matrix Factorization Using Skeletonized Interpolation

Cambier

Darve

2019

SIAM J. Sci. Comput.

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Integral equations are commonly encountered when solving complex physical problems. Their discretization leads to a dense kernel matrix that is block or hierarchically low-rank. This paper proposes a new way to build a low-rank factorization of those low-rank blocks at a nearly optimal cost of O(nr) for a n × n block submatrix of rank r. This is done by first sampling the kernel function at new interpolation points, then selecting a subset of those using a CUR decomposition and finally using this reduced set of points as pivots for a RRLU-type factorization. We also explain how this implicitly builds an optimal interpolation basis for the Kernel under consideration. We show the asymptotic convergence of the algorithm, explain its stability and demonstrate on numerical examples that it performs very well in practice, allowing to obtain rank nearly equal to the optimal rank at a fraction of the cost of the naive algorithm.

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A Tensor-Train accelerated solver for integral equations in complex geometries

Cited by 24 publications

References 76 publications

Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions

Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions

Tensor Train Accelerated Solvers for Nonsmooth Rigid Body Dynamics

Fast Low-Rank Kernel Matrix Factorization Using Skeletonized Interpolation

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