Using COMSOL with the Adjoint Method for Inverse Problems

A. Emery
University of Washington, Seattle, WA, USA

The standard way of estimating parameters in solving inverse problems is based upon least squares fitting which minimizes the sum of the residuals. Another way of minimizing is based upon the adjoint approach in which the sum of the residuals is augmented by the constraint that the solution satisfies the appropriate algebraic or differential equations. This method involves solving the adjoint equation backwards in time in contrast to the LSQ approach in which the equations are solved forward in time. The adjoint method is particularly well suited when several parameters are to be estimated.

The paper describes using COMSOL Multiphysics PDE Solver to solve the adjoint equations for several heat transfer problems in which a time varying flux is applied to the boundaries of a fluid/thermal problem. The solution is compared to the LSQ approach based upon the forward solution of the finite element problem using COMSOL.