arXiv subjects

Alex A. Gorodetsky

Publications and source records attributed to Alex A. Gorodetsky.

25 records · Page 2Linked to original sources

arXiv2019

A Generalized Approximate Control Variate Framework for Multifidelity Uncertainty Quantification

We describe and analyze a variance reduction approach for Monte Carlo (MC) sampling that accelerates the estimation of statistics of computationally expensive simulation models using an ensemble of models with lower cost. These lower cost models --- which are typically lower fidelity with unknown statistics --- are used to reduce the variance in statistical estimators relative to a MC estimator with equivalent cost. We derive the conditions under which our proposed approximate control variate framework recovers existing multi-model variance reduction schemes as special cases. We demonstrate that these existing strategies use recursive sampling strategies, and as a result, their maximum possible variance reduction is limited to that of a control variate algorithm that uses only a single low-fidelity model with known mean. This theoretical result holds regardless of the number of low-fidelity models and/or samples used to build the estimator. We then derive new sampling strategies within our framework that circumvent this limitation to make efficient use of all available information sources. In particular, we demonstrate that a significant gap can exist, of orders of magnitude in some cases, between the variance reduction achievable by using a single low-fidelity model and our non-recursive approach. We also present initial sample allocation approaches for exploiting this gap. They yield the greatest benefit when augmenting the high-fidelity model evaluations is impractical because, for instance, they arise from a legacy database. Several analytic examples and an example with a hyperbolic PDE describing elastic wave propagation in heterogeneous media are used to illustrate the main features of the methodology.

stat.CO↗

arXiv2019

Semi-implicit methods for the dynamics of elastic sheets

Recent applications (e.g. active gels and self-assembly of elastic sheets) motivate the need to efficiently simulate the dynamics of thin elastic sheets. We present semi-implicit time stepping algorithms to improve the time step constraints that arise in explicit methods while avoiding much of the complexity of fully-implicit approaches. For a triangular lattice discretization with stretching and bending springs, our semi-implicit approach involves discrete Laplacian and biharmonic operators, and is stable for all time steps in the case of overdamped dynamics. For a more general finite-difference formulation that can allow for general elastic constants, we use the analogous approach on a square grid, and find that the largest stable time step is two to three orders of magnitude greater than for an explicit scheme. For a model problem with a radial traveling wave form of the reference metric, we find transitions from quasi-periodic to chaotic dynamics as the sheet thickness is reduced, wave amplitude is increased, and damping constant is reduced.

physics.comp-ph↗

arXiv2018

A continuous analogue of the tensor-train decomposition

We develop new approximation algorithms and data structures for representing and computing with multivariate functions using the functional tensor-train (FT), a continuous extension of the tensor-train (TT) decomposition. The FT represents functions using a tensor-train ansatz by replacing the three-dimensional TT cores with univariate matrix-valued functions. The main contribution of this paper is a framework to compute the FT that employs adaptive approximations of univariate fibers, and that is not tied to any tensorized discretization. The algorithm can be coupled with any univariate linear or nonlinear approximation procedure. We demonstrate that this approach can generate multivariate function approximations that are several orders of magnitude more accurate, for the same cost, than those based on the conventional approach of compressing the coefficient tensor of a tensor-product basis. Our approach is in the spirit of other continuous computation packages such as Chebfun, and yields an algorithm which requires the computation of "continuous" matrix factorizations such as the LU and QR decompositions of vector-valued functions. To support these developments, we describe continuous versions of an approximate maximum-volume cross approximation algorithm and of a rounding algorithm that re-approximates an FT by one of lower ranks. We demonstrate that our technique improves accuracy and robustness, compared to TT and quantics-TT approaches with fixed parameterizations, of high-dimensional integration, differentiation, and approximation of functions with local features such as discontinuities and other nonlinearities.

math.NA↗

arXiv2018

High-Dimensional Stochastic Optimal Control using Continuous Tensor Decompositions

Motion planning and control problems are embedded and essential in almost all robotics applications. These problems are often formulated as stochastic optimal control problems and solved using dynamic programming algorithms. Unfortunately, most existing algorithms that guarantee convergence to optimal solutions suffer from the curse of dimensionality: the run time of the algorithm grows exponentially with the dimension of the state space of the system. We propose novel dynamic programming algorithms that alleviate the curse of dimensionality in problems that exhibit certain low-rank structure. The proposed algorithms are based on continuous tensor decompositions recently developed by the authors. Essentially, the algorithms represent high-dimensional functions (e.g., the value function) in a compressed format, and directly perform dynamic programming computations (e.g., value iteration, policy iteration) in this format. Under certain technical assumptions, the new algorithms guarantee convergence towards optimal solutions with arbitrary precision. Furthermore, the run times of the new algorithms scale polynomially with the state dimension and polynomially with the ranks of the value function. This approach realizes substantial computational savings in "compressible" problem instances, where value functions admit low-rank approximations. We demonstrate the new algorithms in a wide range of problems, including a simulated six-dimensional agile quadcopter maneuvering example and a seven-dimensional aircraft perching example. In some of these examples, we estimate computational savings of up to ten orders of magnitude over standard value iteration algorithms. We further demonstrate the algorithms running in real time on board a quadcopter during a flight experiment under motion capture.

cs.RO↗

arXiv2018

Gradient-based Optimization for Regression in the Functional Tensor-Train Format

We consider the task of low-multilinear-rank functional regression, i.e., learning a low-rank parametric representation of functions from scattered real-valued data. Our first contribution is the development and analysis of 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). The functional tensor-train uses the tensor-train (TT) representation of low-rank arrays as an ansatz for a class of low-multilinear-rank functions. The FT is represented by a set of matrix-valued functions that contain a set of univariate functions, and the regression task is to learn the parameters of these univariate functions. Our second contribution demonstrates that using nonlinearly parameterized univariate functions, e.g., symmetric kernels with moving centers, within each core can outperform the standard approach of using a linear expansion of basis functions. Our final contributions are new rank adaptation and group-sparsity regularization procedures to minimize overfitting. We use several benchmark problems to demonstrate at least an order of magnitude lower accuracy with gradient-based optimization methods than standard alternating least squares procedures in the low-sample number regime. We also demonstrate an order of magnitude reduction in accuracy on a test problem resulting from using nonlinear parameterizations over linear parameterizations. Finally we compare regression performance with 22 other nonparametric and parametric regression methods on 10 real-world data sets. We achieve top-five accuracy for seven of the data sets and best accuracy for two of the data sets. These rankings are the best amongst parametric models and competetive with the best non-parametric methods.

stat.CO↗

arXiv2016

Mercer kernels and integrated variance experimental design: connections between Gaussian process regression and polynomial approximation

This paper examines experimental design procedures used to develop surrogates of computational models, exploring the interplay between experimental designs and approximation algorithms. We focus on two widely used approximation approaches, Gaussian process (GP) regression and non-intrusive polynomial approximation. First, we introduce algorithms for minimizing a posterior integrated variance (IVAR) design criterion for GP regression. Our formulation treats design as a continuous optimization problem that can be solved with gradient-based methods on complex input domains, without resorting to greedy approximations. We show that minimizing IVAR in this way yields point sets with good interpolation properties, and that it enables more accurate GP regression than designs based on entropy minimization or mutual information maximization. Second, using a Mercer kernel/eigenfunction perspective on GP regression, we identify conditions under which GP regression coincides with pseudospectral polynomial approximation. Departures from these conditions can be understood as changes either to the kernel or to the experimental design itself. We then show how IVAR-optimal designs, while sacrificing discrete orthogonality of the kernel eigenfunctions, can yield lower approximation error than orthogonalizing point sets. Finally, we compare the performance of adaptive Gaussian process regression and adaptive pseudospectral approximation for several classes of target functions, identifying features that are favorable to the GP + IVAR approach.

math.NA↗

arXiv2014

Efficient Localization of Discontinuities in Complex Computational Simulations

Surrogate models for computational simulations are input-output approximations that allow computationally intensive analyses, such as uncertainty propagation and inference, to be performed efficiently. When a simulation output does not depend smoothly on its inputs, the error and convergence rate of many approximation methods deteriorate substantially. This paper details a method for efficiently localizing discontinuities in the input parameter domain, so that the model output can be approximated as a piecewise smooth function. The approach comprises an initialization phase, which uses polynomial annihilation to assign function values to different regions and thus seed an automated labeling procedure, followed by a refinement phase that adaptively updates a kernel support vector machine representation of the separating surface via active learning. The overall approach avoids structured grids and exploits any available simplicity in the geometry of the separating surface, thus reducing the number of model evaluations required to localize the discontinuity. The method is illustrated on examples of up to eleven dimensions, including algebraic models and ODE/PDE systems, and demonstrates improved scaling and efficiency over other discontinuity localization approaches.

cs.CE↗