This seminar will introduce 2D-MoSub, a derivative-free optimization method based on the subspace method and quadratic models, specifically tackling large-scale derivative-free problems. 2D-MoSub combines 2-dimensional quadratic interpolation models and trust-region techniques to update the points and explore the 2-dimensional subspace iteratively. Its framework includes constructing the interpolation set, building the quadratic interpolation model, performing trust-region trial steps, and updating the trust-region radius and subspace. Computation details and theoretical properties will be discussed. Numerical results demonstrate the advantage of 2D-MoSub.

Short Bio:
Pengcheng Xie, PhD (Chinese Academy of Sciences), is joining Lawrence Berkeley National Laboratory as a postdoctoral scholar specializing in mathematical optimization and numerical analysis. He has developed optimization methods, including 2D-MoSub and SUSD-TR. Pengcheng has published in major journals and presented at ISMP 2024 (upcoming), ICIAM 2023, and CSIAM 2022. He received the Hua Loo-keng scholarship in 2019 and the CAS-AMSS Presidential scholarship in 2023.
 

In this talk, we consider the problem of approximating failure regions. More specifically, given a costly computational model with random parameters and a failure condition, our objective is to determine the parameter region in which the failure condition is likely to not be satisfied. In mathematical terms, this problem can be cast as approximating the level set of a probability density function. We solve this problem by dividing it into two: 1) The design of an efficient Monte Carlo strategy for probability estimation. 2) The construction of an efficient algorithm for level-set approximation. Following this structure, this talk is comprised of two parts:

In the first part, we present a new multi-output multilevel best linear unbiased estimator (MLBLUE) for approximating expectations. The advantage of this estimator is in its convenience and optimality: Given any set of computational models with known covariance structure, MLBLUE automatically constructs a provenly optimal estimator for any (finite) number of quantities of interest. Nevertheless, the optimality of MLBLUE is tied to its optimal set-up, which requires the solution of a nonlinear optimization problem. We show how the latter can be reformulated as a semi-definite program and thus be solved reliably and efficiently.

In the second part, we construct an adaptive level-set approximation algorithm for smooth functions corrupted by noise in $\mathbb{R}^d$. This algorithm only requires point value data and is thus compatible with Monte Carlo estimators. The algorithm is comprised of a criterion for level-set adaptivity combined with an a posteriori error estimator. Under suitable assumptions, we can prove that our algorithm will correctly capture the target level set at the same cost complexity of uniformly approximating a $(d-1)$-dimensional function.