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Gradient-based optimization for regression in the functional tensor-train format

Journal of Computational Physics

Gorodetsky, Alex A.; Jakeman, John D.

Predictive analysis of complex computational models, such as uncertainty quantification (UQ), must often rely on using an existing database of simulation runs. In this paper we consider the task of performing low-multilinear-rank regression on such a database. Specifically we develop and analyze an efficient gradient computation that enables gradient-based optimization procedures, including stochastic gradient descent and quasi-Newton methods, for learning the parameters of a functional tensor-train (FT). We compare our algorithms with 22 other nonparametric and parametric regression methods on 10 real-world data sets and show that for many physical systems, exploiting low-rank structure facilitates efficient construction of surrogate models. We use a number of synthetic functions to build insight into behavior of our algorithms, including the rank adaptation and group-sparsity regularization procedures that we developed to reduce overfitting. Finally we conclude the paper by building a surrogate of a physical model of a propulsion plant on a naval vessel.

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TYPE Journal Article YEAR 2018
ScopusOSTIDOI

Neural Networks as Surrogates of Nonlinear High-Dimensional Parameter-to-Prediction Maps

Jakeman, John D.; Perego, Mauro P.; Severa, William M.

We present a preliminary investigation of the use of Multi-Layer Perceptrons (MLP) and Recurrent Neural Networks (RNNs) as surrogates of parameter-to-prediction maps of com- putational expensive dynamical models. In particular, we target the approximation of Quan- tities of Interest (QoIs) derived from the solution of a Partial Differential Equations (PDEs) at different time instants. In order to limit the scope of our study while targeting a rele- vant application, we focus on the problem of computing variations in the ice sheets mass (our QoI), which is a proxy for global mean sea-level changes. We present a number of neural network formulations and compare their performance with that of Polynomial Chaos Expansions (PCE) constructed on the same data.

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TYPE SAND Report YEAR 2018
OSTIDOI

Generation and application of multivariate polynomial quadrature rules

Computer Methods in Applied Mechanics and Engineering

Jakeman, John D.; Narayan, Akil

The search for multivariate quadrature rules of minimal size with a specified polynomial accuracy has been the topic of many years of research. Finding such a rule allows accurate integration of moments, which play a central role in many aspects of scientific computing with complex models. The contribution of this paper is twofold. First, we provide novel mathematical analysis of the polynomial quadrature problem that provides a lower bound for the minimal possible number of nodes in a polynomial rule with specified accuracy. We give concrete but simplistic multivariate examples where a minimal quadrature rule can be designed that achieves this lower bound, along with situations that showcase when it is not possible to achieve this lower bound. Our second contribution is the formulation of an algorithm that is able to efficiently generate multivariate quadrature rules with positive weights on non-tensorial domains. Our tests show success of this procedure in up to 20 dimensions. We test our method on applications to dimension reduction and chemical kinetics problems, including comparisons against popular alternatives such as sparse grids, Monte Carlo and quasi Monte Carlo sequences, and Stroud rules. The quadrature rules computed in this paper outperform these alternatives in almost all scenarios.

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TYPE Journal Article YEAR 2018
ScopusOSTIDOI

Compressed sensing with sparse corruptions: Fault-tolerant sparse collocation approximations

Adcock, Ben A.; Bao, Anyi B.; Jakeman, John D.; Naryan, Akil N.

The recovery of approximately sparse or compressible coefficients in a polynomial chaos expansion is a common goal in many modern parametric uncertainty quantification (UQ) problems. However, relatively little effort in UQ has been directed toward theoretical and computational strategies for addressing the sparse corruptions problem, where a small number of measurements are highly corrupted. Such a situation has become pertinent today since modern computational frameworks are sufficiently complex with many interdependent components that may introduce hardware and software failures, some of which can be difficult to detect and result in a highly polluted simulation result. In this paper we present a novel compressive sampling-based theoretical analysis for a regularized t1 minimization algorithm that aims to recover sparse expansion coefficients in the presence of measurement corruptions. Our recovery results are uniform (the theoretical guarantees hold for all compressible signals and compressible corruptions vectors), and prescribe algorithmic regularization parameters in terms of a user-defined a priori estimate on the ratio of measurements that are believed to be corrupted. We also propose an iteratively reweighted optimization algorithm that automatically refines the value of the regularization parameter, and empirically produces superior results. Our numerical results test our framework on several medium-to-high dimensional examples of solutions to parameterized differential equations, and demonstrate the effectiveness of our approach.

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TYPE Other Report YEAR 2018
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Results 51–75 of 147
Results 51–75 of 147