تجزیه و تحلیل حساسیت معادلات دیفرانسیل، جبری و معادلات دیفرانسیل با مشتقات جزئی

Sensitivity analysis generates essential information for model development, design optimization, parameter estimation, optimal control, model reduction and experimental design. In this paper we describe the forward and adjoint methods for sensitivity analysis, and outline some of our recent work on theory, algorithms and software for sensitivity analysis of differential-algebraic equation (DAE) and time-dependent partial differential equation (PDE) systems.

In recent years there has been a growing interest in sensitivity analysis for large-scale systems governed by both differential-algebraic equations (DAEs) and partial differential equations (PDEs). The results of sensitivity analysis have wide-ranging applications in science and engineering, including model development, optimization, parameter estimation, model simplification, data assimilation, optimal control, uncertainty analysis and experimental design. Recent work on methods and software for sensitivity analysis of DAE and PDE systems has demonstrated that forward sensitivities can be computed reliably and efficiently. However, for problems which require the sensitivities with respect to a large number of parameters, the forward sensitivity approach is intractable and the adjoint (backward) method is advantageous. Unfortunately, the adjoint problem is quite a bit more complicated both to pose and to solve. Our goal for both DAE and PDE systems has been the development of methods and software in which generation and solution of the adjoint sensitivity system are transparent to the user. This has been largely achieved for DAE systems. We have proposed a solution to this problem for PDE systems solved with adaptive mesh refinement (AMR). This paper has three parts. In the first part we introduce the basic concepts of sensitivity analysis, including the forward and the adjoint method. In the second part we outline the basic problem of sensitivity analysis for DAE systems and examine the recent results on numerical methods and software for DAE sensitivity analysis based on the forward and adjoint methods. The third part of the paper deals with sensitivity analysis for time-dependent PDE systems solved by adaptive mesh refinement.