A computational framework for the regularization of adjoint analysis in multiscale PDE systems

This paper examines the regularization opportunities available in the adjoint analysis and optimization of multiscale PDE systems. Regularization may be introduced into such optimization problems by modifying the form of the evolution equation and the forms of the norms and inner products used to frame the adjoint analysis. Typically, L2 brackets are used in the definition of the cost functional, the adjoint operator, and the cost functional gradient. If instead we adopt the more general Sobolev brackets, the various fields involved in the adjoint analysis may be made smoother and therefore easier to resolve numerically. The present paper identifies several relationships which illustrate how the different regularization options fit together to form a general framework. The regularization strategies proposed are exemplified using a 1D Kuramoto–Sivashinsky forecasting problem, and computational examples are provided which exhibit their utility. A multiscale preconditioning algorithm is also proposed that noticeably accelerates convergence of the optimization procedure. Application of the proposed regularization strategies to more complex optimization problems of physical and engineering relevance is also discussed.