Numerical construction of Green’s functions in high dimensional elliptic problems with variable coefficients and analysis of renewable energy data via sparse and separable approximations

摘要

This thesis consists of two parts. In Part I, we describe an algorithm for approximating the Greenu27s function for elliptic problems with variable coefficients in arbitrary dimension. The basis for our approach is the separated representation, which appears as a way of approximating functions of many variables by sums of products of univariate functions. While the differential operator we wish to invert is typically ill-conditioned, its conditioning may be improved by first applying the Greenu27s function for the constant coefficient problem. This function may be computed either numerically or, in some case, analytically in a separated format. The variable coefficient Greenu27s function is then computed using a quadratically convergent iteration on the preconditioned operator, with sparsity maintained via representation in a wavelet basis. Of particular interest is that the method scales linearly in the number of dimensions, a feature that very desirable in high dimensional problems in which the curse of dimensionality must be reckoned with. As a corollary to this work, we described a randomized algorithm for maintaining low separation rank of the functions used in the construction of the Greenu27s function. For certain functions of practical interest, one can avoid the cost of using standard methods such as alternating least squares (ALS) to reduce the separation rank. Instead, terms from the separated representation may be selected using a randomized approach based on matrix skeletonization and the interpolative decomposition. The use of random projections can greatly reduce the cost of rank reduction, as well as calculation of the Frobenius norm and term-wise Gram matrices. In Part II of the thesis, we highlight three practical applications of sparse and separable approximations to the analysis of renewable energy data. In the first application, error estimates gleaned from repeated measurements are incorporated into sparse regression algorithms (LASSO and the Dantzig selector) to minimize the statistical uncertainty of the resulting model. Applied to real biomass data, this approach leads to sparser regression coefficients corresponding to improved accuracy as measured by k-fold cross validation error. In the second application, a regression model based on separated representations is fit to reliability data for cadmium telluride (CdTe) thin-film solar cells. The data is inherently multi-way, and our approach avoids artificial matricization that would typically be performed for use with standard regression algorithms. Two distinct modes of degradation, corresponding to short- and long-term decrease in cell efficiency, are identified. In the third application, some theoretical properties of a popular chemometrics algorithm called orthogonal projections to latent structures (O-PLS) are derived.