Adjoint-based optimization of PDE systems with alternative gradients

Abstract

In this work we investigate a technique for accelerating convergence of adjoint-based optimization of PDE systems based on a nonlinear change of variables in the control space. This change of variables is accomplished in the differentiate-then-discretize approach by constructing the descent directions in a control space not equipped with the Hilbert structure. We show how such descent directions can be computed in general Lebesgue and Besov spaces, and argue that in the Besov space case determination of descent directions can be interpreted as nonlinear wavelet filtering of the adjoint field. The freedom involved in choosing parameters characterizing the spaces in which the steepest descent directions are constructed can be leveraged to accelerate convergence of iterations. Our computational examples involving state estimation problems for the 1D Kuramoto-Sivashinsky and 3D Navier-Stokes equations indeed show significantly improved performance of the proposed method as compared to the standard approaches.

Space-Dependent Sobolev Gradients as a Regularization for Inverse Radiative Transfer Problems

“…As reported by Protas [39], the gradient is identified in a given space for which the metric had been selected. To do so, the two following definitions are needed.…”

“…This way for writing down the cost function gradient is the ordinary one as first suggested by Lions [46] and Céa [47]. However, according to Protas [39], it is suggested to use other inner product definitions due to the poor scaling of the corresponding discrete optimization problem. As reported in [48], the incorporation of such derivatives into the inner product, instead of using the 2 inner product, has the effect of scale-dependent filtering and allows one to extract smoother gradients, thereby preconditioning the optimization process.…”

“…As a matter of fact, several inner products are to be defined when considering the inverse optimization problem. According to Protas [39], each of them (cost function definition, adjoint identity, and gradient extraction after defining the space used in the definition relationship between the cost function derivative and its gradient) may somehow yield to a particular regularization opportunity. The noise present in the measurements data propagates through the adjoint problem and then to the cost function gradient from which the directions of descent are computed to supply the optimizer.…”

Two‐objective optimization strategies using the adjoint method and game theory in inverse natural convection problems

“…For measuring the effects of the noise level, we used D 0.5, 0.3, and 0.1 with a fixed iteration as a stopping rule, as shown in Figure 6. Two heat source functions are used: a symmetric sinusoidal curve-like function: The noise removal in the measured data can be accomplished by the so-called Sobolev gradient method [37]. The Sobolev gradient contribution rJ S 2 H 1 0 OE0, 1 can be obtained from the corresponding Euclidean contribution rJ 2 L 2 OE0, 1 (see Equation (20) or Equation (21)) by solving the following elliptic boundary-value (in time) problem with homogeneous Neumann boundary condition:…”

“…where n denotes the unit outward normal vector on @OE0, 1 and 2 R is a regularization parameter representing the cut-off length-scale later, which the gradient information is essentially filtered out as result of solving Equation (26) for rJ S ; see [37]. For spatially discretizing Equation (26), a 1D piecewise linear finite element is used.…”