Space–time adaptive solution of inverse problems with the discrete adjoint method

Abstract

Abstract. Adaptivity in both space and time has become the norm for solving problems modeled by partial differential equations. The size of the discretized problem makes uniformly refined grids computationally prohibitive. Adaptive refinement of meshes and time steps allows to capture the phenomena of interest while keeping the cost of a simulation tractable on the current hardware. Many fields in science and engineering require the solution of inverse problems where parameters for a given model are estimated based on available measurement information. In contrast to forward (regular) simulations, inverse problems have not extensively benefited from the adaptive solver technology. Previous research in inverse problems has focused mainly on the continuous approach to calculate sensitivities, and has typically employed fixed time and space meshes in the solution process. Inverse problem solvers that make exclusive use of uniform or static meshes avoid complications such as the differentiation of mesh motion equations, or inconsistencies in the sensitivity equations between subdomains with different refinement levels. However, this comes at the cost of low computational efficiency. More efficient computations are possible through judicious use of adaptive mesh refinement, adaptive time steps, and the discrete adjoint method. This paper develops a framework for the construction and analysis of discrete adjoint sensitivities in the context of time dependent, adaptive grid, adaptive step models. Discrete adjoints are attractive in practice since they can be generated with low effort using automatic differentiation. However, this approach brings several important challenges. The adjoint of the forward numerical scheme may be inconsistent with the continuous adjoint equations. A reduction in accuracy of the discrete adjoint sensitivities may appear due to the intergrid transfer operators. Moreover, the optimization algorithm may need to accommodate state and gradient vectors whose dimensions change between iterations. This work shows that several of these potential issues can be avoided for the discontinuous Galerkin (DG) method. The adjoint model development is considerably simplified by decoupling the adaptive mesh refinement mechanism from the forward model solver, and by selectively applying automatic differentiation on individual algorithms.In forward models discontinuous Galerkin discretizations can efficiently handle high orders of accuracy, h/p-refinement, and parallel computation. The analysis reveals that this approach, paired with Runge Kutta time stepping, is well suited for the adaptive solutions of inverse problems. The usefulness of discrete discontinuous Galerkin adjoints is illustrated on a two-dimensional adaptive data assimilation problem. Space-time adaptive solution of inverse problems with the discrete adjoint method2

Discretely Exact Derivatives for Hyperbolic PDE-Constrained Optimization Problems Discretized by the Discontinuous Galerkin Method

“…Existence and (local) uniqueness of solutions to (2) and (3) depend on the form of the cost functionJ , properties of the solution and parameter spaces and of the hyperbolic system and have to be studied on a case-to-case basis (we refer, for instance, to [3,17,30,36]). Our main focus is not the solution of the optimization problem (3), but the computation of derivatives of J with respect to c, and the interplay of this derivative computation with the spatial and temporal discretization of the hyperbolic system (1). Gradients (and second derivatives) of J are important to solve (3) efficiently, and can be used for studying parameter sensitivities or quantifying the uncertainty in the solution of inverse problems [6].…”

“…Moreover, applying AD to parallel implementations can be challenging [37]. The notion of adjoint consistency for dG (see [1,20,21,32,34]) is related to the discussion in this paper. Adjoint consistency refers to the fact that the exact solution of the dual (or adjoint) problem satisfies the discrete adjoint equation.…”

“…The focus of this paper goes beyond adjoint consistency to consider consistency of the gradient expressions, and considers in what sense the discrete gradient is a discretization of the continuous gradient. For discontinuous Galerkin discretization, the latter aspect is called dual consistency in [1]. A systematic study presented in [29] compares different dG methods for linear-quadratic optimal control problems subject to advection-diffusion-reaction equations.…”

Discontinuous Galerkin unsteady discrete adjoint method for real-time efficient tsunami simulations

“…While it is possible to implement this feature in the model (see e.g. ( [38])), additional projection operators to transfer the data when an element is modified need to be taken into account when constructing the adjoint. The different states of the mesh during the adjoint and forward runs need to match.…”

Generalized adjoint consistent treatment of wall boundary conditions for compressible flows

“…Here, typically the same numerical flux (e.g., the Roe flux [18] or the local Lax-Friedrichs flux, among many others) is employed like on interior faces. In contrast to the normal boundary flux (1a) the numerical flux function involved in (2) introduces some numerical dissipation at the boundary.…”