Schenk, Olaf Eftekhari, Aryan 2021-08-04 Fundamental tasks in multivariate and numerical analysis, such as sparse precision matrix estimation via graphical lasso and function approximation, are formulated in ever-increasing dimensions. Consequently, this results in a significant increase in the computational demand that quickly renders standard solution methods intractable. With this motivation, we present two scalable algorithms that mitigate the obstacles faced in high-dimensional settings. First, we build on the current developments of second-order solution methods for the graphical lasso estimator and introduce a performant algorithm that exploits the sparsity and the block structure of the underlying computation. The algorithm is then parallelized, taking advantage of both shared- and distributed-memory architectures. For validation, we present large-scale test results for problems of up to 10 million dimensions (or equivalently, random variables). Second, we propose a highly efficient and generic function approximation framework that leverages dimensional decomposition with adaptive sparse grids. The hallmark of the proposed approach is the decomposition of a high-dimensional function into a nested summation of low-dimensional component functions. We present an efficient parallelization scheme that leverages the intrinsic separability of the formulation. Finally, an economic case study is presented where the framework is deployed on 1,024 nodes at the Swiss National Supercomputing Center. application/pdf https://n2t.net/ark:/12658/srd1319150 https://susi.usi.ch/global/documents/319150 https://susi.usi.ch/documents/319150/files/2021INFO008.pdf eng info:eu-repo/semantics/altIdentifier/urn/urn:nbn:ch:rero-006-119427 info:eu-repo/semantics/altIdentifier/ark/12658/srd1319150 info:eu-repo/semantics/openAccess License undefined Precision matrix Inverse covariance matrix Graphical lasso Function approximation Sparse grid High-dimensional model representation Dynamic stochastic economic models info:eu-repo/classification/udc/004 Scalable algorithms for high-dimensional graphical lasso and function approximation http://purl.org/coar/resource_type/c_db06