CAREER: Optimal Approximation Algorithms in High Dimensions

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The increasing power of modern computational hardware has enabled computer-based simulation of sophisticated mathematical models that resolve important physical phenomena in great detail. With the advent of these computational abilities has come an increased demand to include more complex physical interactions in the models, and thus an increased strain on computational resources. Modern engineering design utilizes such models, and these design problems typically involve (1) numerous tunable parameters that affect reliability, cost, and failure, (2) uncertainty about external influences manifesting as randomness in the model, and (3) epistemic ignorance involving model form uncertainty. In realistic applications, the collection of these effects leads to predictions that depend on a cumulatively high-dimensional parameter. This project focuses on development and deployment of novel, near-optimal experimental design and sampling algorithms for the accurate and efficient simulation of physical models parameterized by high-dimensional inputs. The work of this project involves the application of recently developed approximation theory results in the computational arena, targeted advances that extend theoretical mathematics for computational purposes, and the development and implementation of algorithms for large-scale computations.The technical aspects of this project are designed to provide feasible computational algorithms and concrete mathematical guarantees for tasks in high-dimensional approximation. The three major core components for the completion of this task involve the design, implementation, and analysis of algorithms that leverage optimality characteristics of (1) random and deterministic experimental and sampling design, (2) computational algorithms for identifying efficient sampling schemes, and (3) strategies and techniques for emerging approximation paradigms such as sparse approximation and dimension reduction. A crosscutting theme is application of these methods to problems of modern interest in scientific computing. This project involves fundamental contributions to the fields of applied approximation theory and computational approximation methods through the development of applications-oriented sampling designs with provable near-optimality. Theoretical investigations of this project connect classical techniques in approximation and linear algebra with emerging algorithms in data reduction and reduced order modeling. The implementation of these algorithms will significantly enhance theoretical understanding and computational feasibility for goal-oriented design, parameter study and reduction, sparse and compressive representations, model verification and calibration, and data-driven simulations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.