We apply a new industrial strength numerical approximation, called the “mesh-free adjoint method,” to solve the nonlinear filtering problem. This algorithm exploits the smoothness of the problem, unlike particle filters, and hence we expect that mesh-free adjoints are superior to particle filters for many practical applications. The nonlinear filter problem is equivalent to solving the Fokker-P...

In this article we deal with the determination of a diffusion coefficient function in a quasi-linear parabolic system. The motivation comes from a metallurgy setting. The solution method is based on the output least-squares approach (ols) with minimization applying the adjoint equation. As the diffusion coefficient has an a-priori unknown form, we apply freeform determination in which the coeff...

This tutorial describes the classic method of conjugate directions: the generalization of the conjugate-gradient method in iterative least-square inversion. I derive the algebraic equations of the conjugate-direction method from general optimization principles. The derivation explains the “magic” properties of conjugate gradients. It also justifies the use of conjugate directions in cases when ...

Abstract. This paper is concerned with the computation of numerical discretization error for uncertainty quantification. An a posteriori error formula is described for a functional measurement of the solution to a scalar advection equation that is estimated by finite volume approximations. An exact error formula and computable error estimate are derived based on an abstractly defined approximat...