Adjoint-Based Error Estimation and Mesh Adaptation for Hybridized Discontinuous Galerkin Methods

We present a robust and efficient target-based mesh adaptation methodology, building on hybridized discontinuous Galerkin schemes for (nonlinear) convection-diffusion problems, including the compressible Euler and Navier-Stokes equations. Hybridization of finite element discretizations has the main advantage, that the resulting set of algebraic equations has globally coupled degrees of freedom only on the skeleton of the computational mesh. Consequently, solving for these degrees of freedom involves the solution of a potentially much smaller system. This not only reduces storage requirements, but also allows for a faster solution with iterative solvers. The mesh adaptation is driven by an error estimate obtained via a discrete adjoint approach. Furthermore, the computed target functional can be corrected with this error estimate to obtain an even more accurate value. The aim of this paper is twofold: Firstly, to show the superiority of adjoint-based mesh adaptation over uniform and residual-based mesh refinement, and secondly to investigate the efficiency of the global error estimate

Discretely exact derivatives for hyperbolic PDE-constrained optimization problems discretized by the discontinuous Galerkin method

This paper discusses the computation of derivatives for optimization problems governed by linear hyperbolic systems of partial differential equations (PDEs) that are discretized by the discontinuous Galerkin (dG) method. An efficient and accurate computation of these derivatives is important, for instance, in inverse problems and optimal control problems. This computation is usually based on an adjoint PDE system, and the question addressed in this paper is how the discretization of this adjoint system should relate to the dG discretization of the hyperbolic state equation. Adjoint-based derivatives can either be computed before or after discretization; these two options are often referred to as the optimize-then-discretize and discretize-then-optimize approaches. We discuss the relation between these two options for dG discretizations in space and Runge-Kutta time integration. Discretely exact discretizations for several hyperbolic optimization problems are derived, including the advection equation, Maxwell's equations and the coupled elastic-acoustic wave equation. We find that the discrete adjoint equation inherits a natural dG discretization from the discretization of the state equation and that the expressions for the discretely exact gradient often have to take into account contributions from element faces. For the coupled elastic-acoustic wave equation, the correctness and accuracy of our derivative expressions are illustrated by comparisons with finite difference gradients. The results show that a straightforward discretization of the continuous gradient differs from the discretely exact gradient, and thus is not consistent with the discretized objective. This inconsistency may cause difficulties in the convergence of gradient based algorithms for solving optimization problems

A Continuous hp−hp-Mesh Model for Discontinuous Petrov-Galerkin Finite Element Schemes with Optimal Test Functions

We present an anisotropic $hp-$mesh adaptation strategy using a continuous mesh model for discontinuous Petrov-Galerkin (DPG) finite element schemes with optimal test functions, extending our previous work on $h-$adaptation. The proposed strategy utilizes the inbuilt residual-based error estimator of the DPG discretization to compute both the polynomial distribution and the anisotropy of the mesh elements. In order to predict the optimal order of approximation, we solve local problems on element patches, thus making these computations highly parallelizable. The continuous mesh model is formulated either with respect to the error in the solution, measured in a suitable norm, or with respect to certain admissible target functionals. We demonstrate the performance of the proposed strategy using several numerical examples on triangular grids. Keywords: Discontinuous Petrov-Galerkin, Continuous mesh models, $hp-$ adaptations, Anisotrop

Drag Prediction Using Adaptive Discontinuous Finite Elements

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/106502/1/AIAA2013-51.pd

A Statistical Framework for Domain Shape Estimation in Stokes Flows

We develop and implement a Bayesian approach for the estimation of the shape of a two dimensional annular domain enclosing a Stokes flow from sparse and noisy observations of the enclosed fluid. Our setup includes the case of direct observations of the flow field as well as the measurement of concentrations of a solute passively advected by and diffusing within the flow. Adopting a statistical approach provides estimates of uncertainty in the shape due both to the non-invertibility of the forward map and to error in the measurements. When the shape represents a design problem of attempting to match desired target outcomes, this "uncertainty" can be interpreted as identifying remaining degrees of freedom available to the designer. We demonstrate the viability of our framework on three concrete test problems. These problems illustrate the promise of our framework for applications while providing a collection of test cases for recently developed Markov Chain Monte Carlo (MCMC) algorithms designed to resolve infinite dimensional statistical quantities

Towards the efficient calculation of quantity of interest from steady Euler equations II: a CNNs-based automatic implementation

In \cite{wang2023towards}, a dual-consistent dual-weighted residual-based $h$-adaptive method has been proposed based on a Newton-GMG framework, towards the accurate calculation of a given quantity of interest from Euler equations. The performance of such a numerical method is satisfactory, i.e., the stable convergence of the quantity of interest can be observed in all numerical experiments. In this paper, we will focus on the efficiency issue to further develop this method, since efficiency is vital for numerical methods in practical applications such as the optimal design of the vehicle shape. Three approaches are studied for addressing the efficiency issue, i.e., i). using convolutional neural networks as a solver for dual equations, ii). designing an automatic adjustment strategy for the tolerance in the $h$-adaptive process to conduct the local refinement and/or coarsening of mesh grids, and iii). introducing OpenMP, a shared memory parallelization technique, to accelerate the module such as the solution reconstruction in the method. The feasibility of each approach and numerical issues are discussed in depth, and significant acceleration from those approaches in simulations can be observed clearly from a number of numerical experiments. In convolutional neural networks, it is worth mentioning that the dual consistency plays an important role to guarantee the efficiency of the whole method and that unstructured meshes are employed in all simulations.Comment: In this papers, we use the CNNs architecture to solve the dual equations proble

A fully discrete framework for the adaptive solution of inverse problems

We investigate and contrast the differences between the discretize-then-differentiate and differentiate-then-discretize approaches to the numerical solution of parameter estimation problems. The former approach is attractive in practice due to the use of automatic differentiation for the generation of the dual and optimality equations in the first-order KKT system. The latter strategy is more versatile, in that it allows one to formulate efficient mesh-independent algorithms over suitably chosen function spaces. However, it is significantly more difficult to implement, since automatic code generation is no longer an option. The starting point is a classical elliptic inverse problem. An a priori error analysis for the discrete optimality equation shows consistency and stability are not inherited automatically from the primal discretization. Similar to the concept of dual consistency, We introduce the concept of optimality consistency. However, the convergence properties can be restored through suitable consistent modifications of the target functional. Numerical tests confirm the theoretical convergence order for the optimal solution. We then derive a posteriori error estimates for the infinite dimensional optimal solution error, through a suitably chosen error functional. This estimates are constructed using second order derivative information for the target functional. For computational efficiency, the Hessian is replaced by a low order BFGS approximation. The efficiency of the error estimator is confirmed by a numerical experiment with multigrid optimization

Goal-oriented error analysis of iterative Galerkin discretizations for nonlinear problems including linearization and algebraic errors

We consider the goal-oriented error estimates for a linearized iterative solver for nonlinear partial differential equations. For the adjoint problem and iterative solver we consider, instead of the differentiation of the primal problem, a suitable linearization which guarantees the adjoint consistency of the numerical scheme. We derive error estimates and develop an efficient adaptive algorithm which balances the errors arising from the discretization and use of iterative solvers. Several numerical examples demonstrate the efficiency of this algorithm.Comment: submitte