Approximating Computationally Intensive Functions and Sampling Design in High Dimensions - Methodology UQ

This is a core problem in constructing surrogates, with application to uncertainty propagation, inference, prediction, and design.

Relevant research questions:

Cross-examination of different methods:

Projection, regression, interpolation, L1 minimization, Gaussian process/kriging, etc

Appropriate measures of performance/accuracy and their dependence on the intended use of the surrogate model.

Error analysis and convergence properties

Sparse representations: l1 minimization, pursuit algorithms, low-rank approximation

“Optimal” choices of nodes/design points for different methods

Adaptive approaches: a posteriori error estimates; derivative properties; dimension reduction; additional optimization of nodes; ANOVA; relation to sequential design of experiments

Interpreting and combining uncertainty information from stochastic surrogates (e.g., Gaussian process variance) and deterministic error bounds

Deriving optimality criteria and search algorithms that are good for high dimensions

Borrow existing theoretical results in high dimensional statistics (Donoho, etc.) to shed light on the structure of “optimal” designs in high dimension.

Additional issues—
Lack of regularity: discovering and approximating discontinuities in high dimensions

Incorporating gradient information

Enforcing constraints on output and input domains

Fault tolerance and missing samples