Model Reduction With MapReduce-enabled Tall and Skinny Singular Value Decomposition

Abstract

Abstract. We present a method for computing reduced-order models of parameterized partial differential equation solutions. The key analytical tool is the singular value expansion of the parameterized solution, which we approximate with a singular value decomposition of a parameter snapshot matrix. To evaluate the reduced-order model at a new parameter, we interpolate a subset of the right singular vectors to generate the reduced-order model's coefficients. We employ a novel method to select this subset that uses the parameter gradient of the right singular vectors to split the terms in the expansion yielding a mean prediction and a prediction covariance-similar to a Gaussian process approximation. The covariance serves as a confidence measure for the reduce order model.We demonstrate the efficacy of the reduced-order model using a parameter study of heat transfer in random media. The high-fidelity simulations produce more than 4TB of data; we compute the singular value decomposition and evaluate the reduced-order model using scalable MapReduce/Hadoop implementations. We compare the accuracy of our method with a scalar response surface on a set of temperature profile measurements and find that our model better captures sharp, local features in the parameter space. Key words. model reduction, simulation informatics, MapReduce, Hadoop, tall and skinny SVD 1. Introduction & motivation. High-fidelity simulations of partial differential equations are typically too expensive for design optimization and uncertainty quantification, where many independent runs are necessary. Cheaper reduced-order models (ROMs) that approximate the map from simulation inputs to quantities of interest may replace expensive simulations to enable such parameter studies. These ROMs are constructed with a relatively small set of high-fidelity runs chosen to cover a range of input parameter values. Each evaluation of the ROM is a linear combination of basis functions derived from the outputs of the high-fidelity runs; ROM constructions differ in their choice of basis functions and method for computing the coefficients of the linear combination. Projection-based methods project the residual of the governing equations (i.e., a Galerkin projection) to create a relatively small system of equations for the coefficients; see the recent preprint [4] for a survey of projection-based techniques. Alternatively, one may derive a closely related optimization problem to compute the coefficients [9,10,14]. These two formulations can provide a measure of confidence along with the ROM. However, they are often difficult to implement in existing solvers since they need access to the equation's operators or residual.To bypass the implementation difficulties, one may use response surfaces-e.g., collocation [3,32] or Gaussian process regression [29]-which are essentially interpolation methods applied to the high-fidelity outputs. They do not need access to the differential operators or residuals and are therefore relatively easy to implement. However, measures of co...

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

“…For some problems, the state dimension may become sufficiently large-scale that storage becomes an issue. In those cases one would have to use algorithms that trade additional computations for storage (e.g., incremental SVD algorithms that could process data in batches) and that use scalable implementations (e.g., for recent work in this area see [69], which uses MapReduce/Hadoop for scalable parametric model reduction). Given the current research focus on "big data," much attention is being given to such algorithms; parametric model reduction is one area that might benefit from these advances.…”

“…Another set of nonintrusive approaches represents the parametric solution in a reduced subspace (usually using POD) and then interpolates those solutions without recourse to the underlying full system. Interpolation can be achieved using polynomial or spline interpolation [60,163,69], least squares fitting [59,178], or radial basis function models [20].…”

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

“…I n , the truncated tensor Z Ð Xˆ1 U (1)T yields a smaller unfolding matrix Z (2) P R I 2ˆR1 I 3¨¨¨IN , so that the multiplication Z (2) Z T (2) can be faster in the next iterations [5,212]. Furthermore, since the unfolding matrices Y T (n) are typically very "tall and skinny", a huge-scale truncated SVD and other constrained low-rank matrix factorizations can be computed efficiently based on the Hadoop / MapReduce paradigm [20,48,49].…”

“…For the HOOI algorithms, see Algorithm 4 and Algorithm 5. For more sophisticated algorithms for Tucker decompositions with orthogonality and nonnegativity constraints, suitable for large-scale data tensors, see [49,104,169,236].…”

“…A simple SVD/QR based algorithm for the PVD with orthogonality constraints is presented in Algorithm 7 [49,51,219]. However, it should be noted that this algorithm does not provide an optimal solution in the sense…”