A bilevel learning approach for optimal observation placement in variational data assimilation

Abstract

In this paper we propose a bilevel optimization approach for the placement of space and time observations in variational data assimilation problems. Within the framework of supervised learning, we consider a bilevel problem where the lower-level task is the variational reconstruction of the initial condition of a semilinear system, and the upper-level problem solves the optimal placement with help of a sparsity inducing function. Due to the pointwise nature of the observations, an optimality system with regular Borel measures is obtained as necessary optimality condition for the lower-level problem. The latter is then considered as constraint for the upper-level instance, yielding an optimization problem constrained by a multi-state system with measures. We demonstrate existence of Lagrange multipliers and derive a necessary optimality system characterizing the optimal solutions of the bilevel problem. The numerical solution is carried out also on two levels. The lower-level problem is solved using a standard BFGS method, while the upper-level one is solved by means of a projected BFGS algorithm, based on the estimation of ǫ-active sets. Finally some numerical experiments are presented to illustrate the main features of our approach.

Optimal experimental design for infinite-dimensional Bayesian inverse problems governed by PDEs: a review

“…Optimal design of experiments for inverse problems governed by ordinary differential equations or differential-algebraic equations appear in numerous works; examples include [11,13,17,28,67]. Recent works also include OED for inverse problems constrained by PDEs; see e.g., [4,25,42,55,57,82,105,108]. Bayesian approaches to OED for nonlinear inverse problems governed by computationally challenging models were presented in [58,59]; these articles use polynomial chaos expansions and Monte Carlo to estimate the OED objective, and use a stochastic optimization approach to compute the optimal design.…”

Optimal Experimental Design for Infinite-dimensional Bayesian Inverse Problems Governed by PDEs: A Review

“…Optimal design of experiments for inverse problems governed by ordinary differential equations or differential-algebraic equations appear in numerous works; examples include [9,10,12,21,46]. Recent works also include OED for inverse problems constrained by PDEs; see e.g., [3,18,31,38,39,56,74]. Bayesian approaches to OED for nonlinear inverse problems governed by computationally challenging models were presented in [40,41]; these articles use polynomial chaos expansions and Monte Carlo to estimate the OED objective, and use a stochastic optimization approach to compute the optimal design.…”