The Anchored Separated Representation for High Dimensional Problems

Although topics in science and engineering that involve dimensions beyond x-y-z appear obscure, in truth numerous examples abound. For instance, uncertainty quantification requires approximating and integrating functions with many inputs, while the study of non-linear random vibrations involve PDEs whose dimensionality scales with the system's degrees of freedom. Such problems are numerically difficult since the application of classical mesh based techniques incur computational demands that scale exponentially with the dimension; this is the curse of dimensionality. In this thesis we formulate two tractable numerical methods that circumvent this curse by assuming sparsity in the model representation. The first method, the anchored separated representation, is a provably minimal decomposition for functions that are sparse in a multiplicative sense. The second method is an iterative extension for approximating functions that are sparse in rank. We illustrate with numerical examples the application of both methods to a broad class of high dimensional problems.M.A.S