Khoa Toán - Cơ - Tin học

Thời gian: 
T2, 29/05/2017 - 09:30
Địa điểm: 
Người báo cáo: 
Assistant Professor Tan Bui-Thanh
Tên báo cáo: 
A data-scalable randomized misfit approach for solving large-scale PDE-constrained inverse problems
Nội dung: 

For the ICES PhD Fellowship
1) Introduction to the Institute for Computational Engineering and Sciences: mission, structure, and key research centers/groups
2) PhD fellowship: admission process: What is an excellent application? What do Vietnamese students need to pay attention to?

A randomized misfit approach is presented for the efficient solution of large-scale PDE-constrained inverse problems with high-dimensional data. The purpose  is to offer a theory-based framework for random projections in this inverse problem setting. The stochastic approximation to the misfit is analyzed using random projection theory. By expanding beyond mean estimator convergence, a practical characterization of randomized misfit convergence can be achieved. The theoretical results developed hold with any valid random projection in the literature. The class of feasible distributions is broad yet simple to characterize compared to previous stochastic misfit methods. This class includes very sparse random projections which provide additional computational benefit. A different proof for a variant of the Johnson–Lindenstrauss lemma is also provided. This leads to a different intuition for the $O(\epsilon^{-2}$ factor in bounds for Johnson–Lindenstrauss results. Our main contribution is a theoretical result showing the method guarantees a valid solution for small reduced misfit dimensions. The interplay between Johnson–Lindenstrauss theory and Morozov's discrepancy principle is shown to be essential to the result. The computational cost savings for large-scale PDE-constrained problems with high-dimensional data is discussed. Numerical verification of the developed theory is presented for model problems of estimating a distributed parameter in an elliptic partial differential equation. Results with different random projections are presented to demonstrate the viability and accuracy of the proposed approach.