Randomized matrix-free trace and log-determinant estimators

Saibaba, Arvind K.; Alexanderian, Alen; Ipsen, Ilse C. F.

doi:10.1007/s00211-017-0880-z

Cited by 57 publications

(93 citation statements)

References 49 publications

(79 reference statements)

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“…As shown in Ref. 70, the trace error resulting from the subspace iteration method approaches the sum of neglected small eigenvalues exponentially fast with the number of iterations, which makes valid our error bound of Eq. 24.…”

Section: Effective Basis For the Rpa Energysupporting

confidence: 73%

Random Phase Approximation Applied to Many-Body Noncovalent Systems

Modrzejewski

Yourdkhani

Klimeš

2019

J. Chem. Theory Comput.

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The random phase approximation (RPA) has received a considerable interest in the field of modeling systems where noncovalent interactions are important. Its advantages over widely used density functional theory (DFT) approximations are the exact treatment of exchange and the description of long-range correlation. In this work we address two open questions related to RPA. First, how accurately RPA describes nonadditive interactions encountered in many-body expansion of a binding energy. We consider three-body nonadditive energies in molecular and atomic clusters. Second, how does the accuracy of RPA depend on input provided by different DFT models, without resorting to selfconsistent RPA procedure which is currently impractical for calculations employing periodic boundary conditions. We find that RPA based on the SCAN0 and PBE0 models, i.e., hybrid DFT, achieves an overall accuracy between CCSD and MP3 on a dataset of molecular trimers of Řezáč et al. (J. Chem. Theory. Comput. 2015, 11, 3065) Finally, many-body expansion for molecular clusters and solids often leads to a large number of small contributions that need to be calculated with a high precision. We therefore present a cubic-scaling (or SCF-like) implementation of RPA in atomic basis set, which is designed for calculations with a high numerical precision. rors originating from the exchange functional are on the same order as the errors originating from the missing dispersion energy. [19][20][21] Approximate exchange functionals alone can lead to both strongly overestimated and underestimated noncovalent interactions. 22 For two body systems such issues tend to be masked by adjusting the equilibrium-and short-distance behavior of the dispersion correction. 19 However, the three-body exchange errors cannot be compensated in a similar way by adjusting the pairwise additive dispersion corrections. 19 Overall, the conclusion originating from the existing literature is that no existing semilocal functional can reliably account for many body effects. 21,23 Affordable schemes based on perturbation theory could offer higher and systematically improvable accuracy for calculations of condensed systems compared to standard DFT functionals. 21,[24][25][26][27] Of such schemes, the random phase approximation to the correlation energy (RPA) is promising as it is both compatible with the Hartree-Fock (HF) exchange and it contains terms describing higher-order (nonadditive) correlation effects. 28,29 RPA has been tested for interaction energies of dimers, 30-32 for adsorption, [33][34][35] or for molecular 36-39 and atomic solids 40 and interfaces. 41,42 For the cases involving noncovalent interactions, high accuracy has been achieved with addition of the singles corrections. 43,44 However, its accuracy for predicting nonadditive energies is unknown and this is one of our interests in this work. Moreover, most of the RPA calculations nowadays are performed non-self-consistently, using DFT orbitals and energies in the RPA energy expression. Here we obtain RPA results usi...

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Section: Effective Basis For the Rpa Energysupporting

confidence: 73%

Random Phase Approximation Applied to Many-Body Noncovalent Systems

Modrzejewski

Yourdkhani

Klimeš

2019

J. Chem. Theory Comput.

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“…There are multiple advantages in using Algorithm 2 to approximate the trace, see e.g., [42,69]. In terms of accuracy, the approximation error is bounded by the sum of the remaining eigenvalues, so that the error is small if the eigenvalues decay fast or if the Hessian H has low rank, see [42,69] for more details. In terms of computational efficiency, the 2(k + p) Hessian matrix-vector products, which entail the solution of a pair of linearized state/adjoint equations (as shown in Sec.…”

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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“…In [44], the authors propose an alternate design criterion given by a lower bound of the expected information gain, and use PC representation of the forward model to accelerate objective function evaluations. However, the approaches based on PC representations remain limited in scope to problems with low to moderate parameter dimensions (e.g., parameter dimensions in order of tens).Efficient estimators for the evaluation of D-optimal criterion were developed in [38,39]; however, these works do not discuss the problem of computing D-optimal designs. The mathematical formulation of Bayesian D-optimality for infinite-dimensional Bayesian linear inverse problems was established in [2].…”

mentioning

confidence: 99%

“…Efficient estimators for the evaluation of D-optimal criterion were developed in [38,39]; however, these works do not discuss the problem of computing D-optimal designs. The mathematical formulation of Bayesian D-optimality for infinite-dimensional Bayesian linear inverse problems was established in [2].…”

mentioning

confidence: 99%

Efficient D-Optimal Design of Experiments for Infinite-Dimensional Bayesian Linear Inverse Problems

Alexanderian¹,

Saibaba²

2018

SIAM J. Sci. Comput.

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We develop a computational framework for D-optimal experimental design for PDEbased Bayesian linear inverse problems with infinite-dimensional parameters. We follow a formulation of the experimental design problem that remains valid in the infinite-dimensional limit. The optimal design is obtained by solving an optimization problem that involves repeated evaluation of the logdeterminant of high-dimensional operators along with their derivatives. Forming and manipulating these operators is computationally prohibitive for large-scale problems. Our methods exploit the lowrank structure in the inverse problem in three different ways, yielding efficient algorithms. Our main approach is to use randomized estimators for computing the D-optimal criterion, its derivative, as well as the Kullback-Leibler divergence from posterior to prior. Two other alternatives are proposed based on low-rank approximation of the prior-preconditioned data misfit Hessian, and a fixed low-rank approximation of the prior-preconditioned forward operator. Detailed error analysis is provided for each of the methods, and their effectiveness is demonstrated on a model sensor placement problem for initial state reconstruction in a time-dependent advection-diffusion equation in two space dimensions.we seek sensor placements that maximize the expected information gain in parameter inversion. Our approach, however, is general in that it is applicable to D-optimal experimental design for a broad class of linear inverse problems. ]. In particular, A-optimal experimental design for large-scale applications has been addressed in [3, 4, 17, 20-22, 24, 43]. Computing Aoptimal designs for large-scale applications faces similar challenges to the D-optimal designs. The use of Monte-Carlo trace estimators [7] has been instrumental in making A-optimal design computationally feasible for such applications.Fast algorithms for computing expected information gain, i.e., the D-optimal criterion, for nonlinear inverse problems, via Laplace approximations, were proposed in [31,32]. The focus of the aforementioned works was efficient computation of the D-optimal criterion. The optimal design was then computed by an exhaustive search over prespecified sets of experimental scenarios. We also mention [26-28] that use polynomial chaos (PC) expansions to build easy-to-evaluate surrogates of the forward model. The expected information gain is then evaluated using an appropriate Monte Carlo procedure. In [27,28], the authors use a gradient based approach to obtain the optimal design. In [44], the authors propose an alternate design criterion given by a lower bound of the expected information gain, and use PC representation of the forward model to accelerate objective function evaluations. However, the approaches based on PC representations remain limited in scope to problems with low to moderate parameter dimensions (e.g., parameter dimensions in order of tens).Efficient estimators for the evaluation of D-optimal criterion were developed in [38,39]; however, these works do not...

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Randomized matrix-free trace and log-determinant estimators

Cited by 57 publications

References 49 publications

Random Phase Approximation Applied to Many-Body Noncovalent Systems

Random Phase Approximation Applied to Many-Body Noncovalent Systems

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

Efficient D-Optimal Design of Experiments for Infinite-Dimensional Bayesian Linear Inverse Problems

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