Randomized algorithms for generalized Hermitian eigenvalue problems with application to computing Karhunen–Loève expansion

Saibaba, Arvind K.; Lee, Jonghyun; Kitanidis, Peter K.

doi:10.1002/nla.2026

Cited by 59 publications

(61 citation statements)

References 21 publications

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“…It is important to note that while the mass matrix M is symmetric positive definite and may be sparse, T is symmetric positive semidefinite and dense. Since Q is dense, we applied a data sparse technique to store it with the hierarchical matrix (

scriptH

‐matrix) format . Consequently, the computational cost of matrix‐vector products involving Q is reduced from

scriptO false(n^{2} false)

scriptO false(n \log n false)

, with n the number of discretization points.…”

Section: Models and Methodsmentioning

confidence: 99%

“…Since Q is dense, we applied a data sparse technique to store it with the hierarchical matrix (-matrix) format. 66,67 Consequently, the computational cost of matrix-vector products involving Q is reduced from (n 2 ) to (n log n), with n the number of discretization points. The -matrix technique is a hierarchical division of a given matrix into rectangular blocks and further approximation of these blocks by low-rank matrices.…”

Section: Computing Karhunen-loéve Approximationmentioning

confidence: 99%

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Uncertainty in cardiac myofiber orientation and stiffnesses dominate the variability of left ventricle deformation response

Rodríguez‐Cantano

Sundnes

Rognes

2019

Numer Methods Biomed Eng

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Computational cardiac modelling is a mature area of biomedical computing and is currently evolving from a pure research tool to aiding in clinical decision making. Assessing the reliability of computational model predictions is a key factor for clinical use, and uncertainty quantification (UQ) and sensitivity analysis are important parts of such an assessment. In this study, we apply UQ in computational heart mechanics to study uncertainty both in material parameters characterizing global myocardial stiffness and in the local muscle fiber orientation that governs tissue anisotropy. The uncertainty analysis is performed using the polynomial chaos expansion (PCE) method, which is a nonintrusive meta‐modeling technique that surrogates the original computational model with a series of orthonormal polynomials over the random input parameter space. In addition, in order to study variability in the muscle fiber architecture, we model the uncertainty in orientation of the fiber field as an approximated random field using a truncated Karhunen‐Loéve expansion. The results from the UQ and sensitivity analysis identify clear differences in the impact of various material parameters on global output quantities. Furthermore, our analysis of random field variations in the fiber architecture demonstrate a substantial impact of fiber angle variations on the selected outputs, highlighting the need for accurate assignment of fiber orientation in computational heart mechanics models.

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scriptH

‐matrix) format . Consequently, the computational cost of matrix‐vector products involving Q is reduced from

scriptO false(n^{2} false)

scriptO false(n \log n false)

, with n the number of discretization points.…”

Section: Models and Methodsmentioning

confidence: 99%

Section: Computing Karhunen-loéve Approximationmentioning

confidence: 99%

Uncertainty in cardiac myofiber orientation and stiffnesses dominate the variability of left ventricle deformation response

Rodríguez‐Cantano

Sundnes

Rognes

2019

Numer Methods Biomed Eng

View full text Add to dashboard Cite

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“…After suitable discretization, e.g., by a finite element discretization, the generalized eigenvalue problem (3.18) results in an algebraic eigenproblem of the form Aψ = λBψ with A, B ∈ R n×n and ψ ∈ R n . Algorithm 2 summarizes the so-called double pass randomized algorithm to solve the algebraic generalized eigenvalue problem (see [42,70] for details of the algorithms and [76] for its implementation).…”

Section: Computation Of the Gradient And Hessian Of The Control Objecmentioning

confidence: 99%

“…The mean and variance expressions of the (linear and quadratic) Taylor expansions involve the trace of the covariance-preconditioned Hessian of the control objective with respect to the uncertain parameter. Randomized algorithms for solution of generalized eigenvalue problems [76,70,69] allow for accurate and efficient approximation of this trace and only require computing the action of the covariancepreconditioned Hessian on a number of random directions that depends on its numerical rank. As we showed in the numerical results, this approach is more efficient and accurate than the Gaussian trace estimator when the eigenvalues of the covariancepreconditioned Hessian exhibit fast decay, which is true when the control objective is only sensitive to a limited number of directions in the uncertain parameter space.…”

Section: 2mentioning

confidence: 99%

Taylor approximation and variance reduction for PDE-constrained optimal control under uncertainty

Chen

Villa

Ghattas

2019

Journal of Computational Physics

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In this work we develop a scalable computational framework for the solution of PDE-constrained optimal control under high-dimensional uncertainty. Specifically, we consider a mean-variance formulation of the control objective and employ a Taylor expansion with respect to the uncertain parameter either to directly approximate the control objective or as a control variate for variance reduction. The expressions for the mean and variance of the Taylor approximation are known analytically, although their evaluation requires efficient computation of the trace of the (preconditioned) Hessian of the control objective. We propose to estimate this trace by solving a generalized eigenvalue problem using a randomized algorithm that only requires the action of the Hessian on a small number of random directions. Then, the computational work does not depend on the nominal dimension of the uncertain parameter, but depends only on the effective dimension (i.e., the rank of the preconditioned Hessian), thus ensuring scalability to high-dimensional problems. Moreover, to increase the estimation accuracy of the mean and variance of the control objective by the Taylor approximation, we use it as a control variate for variance reduction, which results in considerable computational savings (several orders of magnitude) compared to a plain Monte Carlo method. In summary, our approach amounts to solving an optimal control constrained by the original PDE and a set of linearized PDEs, which arise from the computation of the gradient and Hessian of the control objective with respect to the uncertain parameter. We use the Lagrangian formalism to derive expressions for the gradient with respect to the control and apply a gradient-based optimization method to solve the problem. We demonstrate the accuracy, efficiency, and scalability of the proposed computational method for two examples with high-dimensional uncertain parameters: subsurface flow in a porous medium modeled as an elliptic PDE, and turbulent jet flow modeled by the Reynolds-averaged Navier-Stokes equations coupled with a nonlinear advection-diffusion equation characterizing model uncertainty. In particular, for the latter more challenging example we show scalability of our algorithm up to one million parameters resulting from discretization of the uncertain parameter field.

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“…Such ridge approximation is obtained by minimizing an upper bound on the Kullback-Leibler distance between the posterior distribution and its approximation.In this paper, we propose dimension reduction directly based on partial (generalized) spectral decomposition of the prior covariance or the covariance of local Gaussian approximation to the posterior (GAP). The intrinsic low-dimensional subspace is identified by r leading eigen-functions, which can be efficiently obtained by randomized linear algebraic algorithms [15][16][17]. Unlike [8], the posterior covariance projected in the subspace is not empirically updated, but rather approximated in a diagonal form which can still capture the most variation of the projected posterior.…”

mentioning

confidence: 99%

Adaptive dimension reduction to accelerate infinite-dimensional geometric Markov Chain Monte Carlo

Lan

2019

Journal of Computational Physics

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Bayesian inverse problems highly rely on efficient and effective inference methods for uncertainty quantification (UQ). Infinite-dimensional MCMC algorithms, directly defined on function spaces, are robust under refinement (through discretization, spectral approximation) of physical models. Recent development of this class of algorithms has started to incorporate the geometry of the posterior informed by data so that they are capable of exploring complex probability structures, as frequently arise in UQ for PDE constrained inverse problems. However, the required geometric quantities, including the Gauss-Newton Hessian operator or Fisher information metric, are usually expensive to obtain in high dimensions. On the other hand, most geometric information of the unknown parameter space in this setting is concentrated in an intrinsic finite-dimensional subspace. To mitigate the computational intensity and scale up the applications of infinite-dimensional geometric MCMC (∞-GMC), we apply geometry-informed algorithms to the intrinsic subspace to probe its complex structure, and simpler methods like preconditioned Crank-Nicolson (pCN) to its geometry-flat complementary subspace.In this work, we take advantage of dimension reduction techniques to accelerate the original ∞-GMC algorithms. More specifically, partial spectral decomposition (e.g. through randomized linear algebra) of the (prior or Gaussian-approximate posterior) covariance operator is used to identify certain number of principal eigen-directions as a basis for the intrinsic subspace. The combination of dimension-independent algorithms, geometric information, and dimension reduction yields more efficient implementation, (adaptive) dimensionreduced infinite-dimensional geometric MCMC. With a small amount of computational overhead, we can achieve over 70 times speed-up compared to pCN using a simulated elliptic inverse problem and an inverse problem involving turbulent combustion with thousands of dimensions after discretization. A number of error bounds comparing various MCMC proposals are presented to predict the asymptotic behavior of the proposed dimension-reduced algorithms.Gaussian. In particular, infinite-dimensional geometric MCMC (∞-GMC) [10] put a series of 'dimensionindependent' MCMC algorithms in the context of increasingly adopting geometry (gradient, Hessian). With the help of such geometric information, [10] show that with the prior-based splitting strategy, ∞-GMC algorithms can achieve up to two orders of magnitude speed up in sampling efficiency compared to vanilla pCN. However, fully computing the required geometric quantities is prohibitive in the discretized parameter space with thousands of dimensions. Therefore, it is natural to consider approximations to the gradient vector and Hessian (Fisher) matrix and compute them in a subspace with reduced dimensions. The key to the dimension reduction in this setting is to identify an intrinsic low-dimensional subspace and apply geometric methods to effectively explore its complex structure; while s...

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Randomized algorithms for generalized Hermitian eigenvalue problems with application to computing Karhunen–Loève expansion

Cited by 59 publications

References 21 publications

Uncertainty in cardiac myofiber orientation and stiffnesses dominate the variability of left ventricle deformation response

Uncertainty in cardiac myofiber orientation and stiffnesses dominate the variability of left ventricle deformation response

Taylor approximation and variance reduction for PDE-constrained optimal control under uncertainty

Adaptive dimension reduction to accelerate infinite-dimensional geometric Markov Chain Monte Carlo

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