Goal-oriented Error Estimation Research Articles

A hp-adaptive finite element approach for 3D controlled source electromagnetic forward modeling

Accurate and efficient 3D controlled-source electromagnetic (CSEM) forward modeling is crucial for deciphering CSEM responses in complex geological contexts and for assessing the efficacy of survey designs. It also serves as the foundation for 3D inversion, a critical technique for interpreting CSEM data. We present a novel hp-adaptive finite element method for 3D CSEM forward modeling. This method enhances the traditional h-adaptive approach, which only refines the mesh while keeping the polynomial degree of the basis functions constant, by enabling simultaneous refining mesh and increasing polynomial degree of basis functions. High-order elements yield superior approximations when the solution exhibits sufficient smoothness, while mesh refinement enhances accuracy in regions with less smooth solutions. The adaptive refinement process is guided by a goal-oriented error estimator that relies on the residual of the governing equation and the continuity of electromagnetic fields. The hp-adaptive strategy, which determines whether to refine elements or increase their polynomial degree, is based on the refinement levels of the elements. Constraints are applied to hanging nodes, introduced during the refinement process, to ensure the continuity of the finite element space. We validate the accuracy, efficiency, and convergence of the hp-adaptive method through tests on two layered models. Further simulations on a topographic model and a realistic model corroborate its effectiveness. Our findings indicate that the hp-adaptive method outperforms the traditional h-adaptive method in terms of accuracy, convergence rate and computational cost, making it a promising technique for 3D CSEM forward modeling.

Goal-oriented error estimation based on equilibrated flux reconstruction for the approximation of the harmonic formulations in eddy current problems

Abstract In this work, we propose an a posteriori goal-oriented error estimator for the harmonic $\textbf {A}$-$\varphi $ formulation arising in the modeling of eddy current problems, approximated by nonconforming finite element methods. It is based on the resolution of an adjoint problem associated with the initial one. For each of these two problems, a guaranteed equilibrated estimator is developed using some flux reconstructions. These fluxes also allow to obtain a goal-oriented error estimator that is fully computable and can be split in a principal part and a remainder one. Our theoretical results are illustrated by numerical experiments.

Inverse radiative transfer with goal-oriented hp-adaptive mesh refinement: adaptive-mesh inversion

The inverse problem for radiative transfer is important in many applications, such as optical tomography and remote sensing. Major challenges include large memory requirements and computational expense, which arise from high-dimensionality and the need for iterations in solving the inverse problem. Here, to alleviate these issues, we propose adaptive-mesh inversion: a goal-oriented hp-adaptive mesh refinement method for solving inverse radiative transfer problems. One novel aspect here is that the two optimizations (one for inversion, and one for mesh adaptivity) are treated simultaneously and blended together. By exploiting the connection between duality-based mesh adaptivity and adjoint-based inversion techniques, we propose a goal-oriented error estimator, which is cheap to compute, and can efficiently guide the mesh-refinement to numerically solve the inverse problem. We use discontinuous Galerkin spectral element methods to discretize the forward and the adjoint problems. Then, based on the goal-oriented error estimator, we propose an hp-adaptive algorithm to refine the meshes. Numerical experiments are presented at the end and show convergence speed-up and reduced memory occupation by the goal-oriented mesh adaptive method.

Goal-oriented error estimation based on equilibrated flux and potential reconstruction for the approximation of elliptic and parabolic problems

Goal-oriented error estimation based on equilibrated flux and potential reconstruction for the approximation of elliptic and parabolic problems

Linearization errors in discrete goal-oriented error estimation

This paper is concerned with goal-oriented a posteriori error estimation for nonlinear functionals in the context of nonlinear variational problems solved with continuous Galerkin finite element discretizations. A two-level, or discrete, adjoint-based approach for error estimation is considered. The traditional method to derive an error estimate in this context requires linearizing both the nonlinear variational form and the nonlinear functional of interest which introduces linearization errors into the error estimate. In this paper, we investigate these linearization errors. In particular, we develop a novel discrete goal-oriented error estimate that accounts for traditionally neglected nonlinear terms at the expense of greater computational cost. We demonstrate how this error estimate can be used to drive mesh adaptivity. We show that accounting for linearization errors in the error estimate can improve its effectivity for several nonlinear model problems and quantities of interest. We also demonstrate that an adaptive strategy based on the newly proposed estimate can lead to more accurate approximations of the nonlinear functional with fewer degrees of freedom when compared to uniform refinement and traditional adjoint-based approaches.

EMFEM: A parallel 3D modeling code for frequency-domain electromagnetic method using goal-oriented adaptive finite element method

We have developed an open-source parallel 3D forward modeling code called EMFEM for the frequency-domain electromagnetic method (FDEM). This code is based on the adaptive finite element method and uses the total-field approach to solve the time-harmonic Maxwell equations. The use of tetrahedral meshes enables modeling of problems with complex geometry and topography. We utilize an iterative solver and an optimal block-diagonal preconditioner and auxiliary Maxwell solver to accelerate the convergence rate. We also use goal-oriented error estimation to adaptively refine the mesh and improve the accuracy of the solution. In addition, we develop a parallel algorithm based on the mesh partitioning approach. The effectiveness of the adaptive mesh refinement is demonstrated through a 1D layered model for marine controlled-source electromagnetic (CSEM) problems. We further present two 3D models for CSEM and magnetotelluric (MT) problems to study the scalability of the parallel algorithm, and the accuracy of the solutions is cross-validated with other open-source codes. The code is implemented in C++ and follows the modular design principle, making it easy to extend and maintain. The code is capable of modeling large-scale 3D problems and utilizing high-performance computing platforms. Furthermore, as the code is freely available, it can serve as a cross-validation tool for other 3D modeling codes and can also be utilized to implement 3D inversion routines.

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.

Where to drill next? A dual-weighted approach to adaptive optimal design of groundwater surveys

We present a novel approach to adaptive optimal design of groundwater surveys – a methodology for choosing the location of the next monitoring well. Our dual-weighted approach borrows ideas from Bayesian Optimisation and goal-oriented error estimation to propose the next monitoring well, given that some data is already available from existing wells. Our method is distinct from other optimal design strategies in that it does not rely on Fisher Information and it instead directly exploits the posterior uncertainty and the expected solution to a dual (or adjoint) problem to construct an acquisition function that optimally reduces the uncertainty in the model as a whole and some engineering quantity of interest in particular. We demonstrate our approach in the context of 2D groundwater flow example and show that the dual-weighted approach outperforms the baseline approach with respect to reducing the error in the posterior estimate of the quantity of interest.

Efficient Model Predictive Control for Parabolic PDEs with Goal Oriented Error Estimation

We show how a posteriori goal oriented error estimation can be used to efficiently solve the subproblems occurring in a model predictive control (MPC) algorithm. In MPC, only an initial part of a computed solution is implemented as a feedback, which motivates grid refinement particularly tailored to this context. To this end, we present a truncated cost functional as an objective for goal oriented adaptivity and prove under stabilizability assumptions that error indicators decay exponentially outside the support of this quantity. This leads to very efficient time and space discretizations for MPC, which we will illustrate by means of various numerical examples.

Neural network guided adjoint computations in dual weighted residual error estimation

In this work, we are concerned with neural network guided goal-oriented a posteriori error estimation and adaptivity using the dual weighted residual method. The primal problem is solved using classical Galerkin finite elements. The adjoint problem is solved in strong form with a feedforward neural network using two or three hidden layers. The main objective of our approach is to explore alternatives for solving the adjoint problem with greater potential of a numerical cost reduction. The proposed algorithm is based on the general goal-oriented error estimation theorem including both linear and nonlinear stationary partial differential equations and goal functionals. Our developments are substantiated with some numerical experiments that include comparisons of neural network computed adjoints and classical finite element solutions of the adjoints. In the programming software, the open-source library deal.II is successfully coupled with LibTorch, the PyTorch C++ application programming interface.Article HighlightsAdjoint approximation with feedforward neural network in dual-weighted residual error estimation.Side-by-side comparisons for accuracy and computational cost with classical finite element computations.Numerical experiments for linear and nonlinear problems yielding excellent effectivity indices.

Goal-Oriented Error Estimation and Mesh Adaptation for Tracer Transport Modelling

This paper applies metric-based mesh adaptation methods to advection-dominated tracer transport modelling problems in two and three dimensions, using the finite element package Firedrake. In particular, the mesh adaptation methods considered are built upon goal-oriented estimates for the error incurred in evaluating a diagnostic quantity of interest (QoI). In the motivating example of modelling to support desalination plant outfall design, such a QoI could be the salinity at the plant inlet, which could be negatively impacted by the transport of brine from the plant’s outfall. Four approaches are considered, one of which yields isotropic meshes. The focus on advection-dominated problems means that flows are often anisotropic; thus, three anisotropic approaches are also considered. Meshes resulting from each of the four approaches yield solutions to the tracer transport problem which give better approximations to QoI values than uniform meshing, for a given mesh size. The methodology is validated using an existing 2D tracer transport test case with a known analytical solution. Goal-oriented meshes for an idealised time-dependent desalination outfall scenario are also presented.

Goal oriented error control for stationary incompressible flow coupled to a heat equation

AbstractIn this work, we apply goal oriented error estimation to a stationary Navier‐Stokes benchmark problem coupled with the heat equation. Furthermore, we compare three different methods for the sensitivity weight recovery.

Periodic adjoints and anisotropic mesh adaptation in rotating frame for high-fidelity RANS turbomachinery applications

The scope of this paper is to demonstrate the viability and efficiency of metric-based unstructured anisotropic mesh adaptation techniques to turbomachinery applications. The main difficulty in turbomachinery is the periodicity of the domain that must be taken into account in the mesh-adaptive solution process. The periodicity is strongly enforced in the flow solver using ghost entities to minimize the impact on the source code. For the mesh adaptation, the local remeshing is done in two steps. First, the inner domain is remeshed with frozen periodic frontiers, and, second, the periodic surfaces are remeshed after moving geometric entities from one side of the domain to the other. One of the main goal of this work is to demonstrate that mesh-independent certified numerical solutions can be obtained thanks to anisotropic mesh adaptation and that it is possible to run high-fidelity CFD on unstructured adapted meshes composed only of tetrahedra. This paper demonstrates how mesh adaptation, thanks to its automation, is able to generate meshes that are extremely difficult to envision and almost impossible to generate manually, leading to highly accurate numerical solutions. This study considers feature-based error estimate based on the standard multi-scale Lp interpolation error estimate and goal-oriented error estimate using an adjoint state to control the error on turbomachinery output functionals. A description of the flow solver and the adjoint solver is given in this work, as they are very different from what is encountered in the turbomachinery community. We also present all the specific modifications that have been introduced in the adaptive process to deal with periodic simulations used for turbomachinery applications. The periodic mesh adaptation strategy is then tested and validated on the LS89 high pressure axial turbine vane and the NASA Rotor 37 test cases.

Goal-oriented mesh adaptivity for inverse problems in linear micromorphic elasticity

In this work, we extend goal-oriented mesh adaptivity to parameter identification for a class of linear micromorphic elasticity problems. Starting from a compact formulation in our previous work (Ju and Mahnkhen, 2017), we propose a two-level optimization framework based on goal-oriented error estimation. By means of a sensitivity analysis of the generalized constitutive relations, we establish a gradient-based solver for the inverse problem, where parameters are optimized within an inner optimization loop for a given mesh. Exact error representations are derived from a Lagrange method, aiming at a quantity of interest as a user-defined functional of the parameters. By using a patch recovery technique for enhanced solutions, a computable error estimator is presented and used to drive an adaptive refinement algorithm, which forms an outer optimization loop. For a numerical study, we investigate the performance of the resulting adaptive procedure in case of perfect, incomplete and perturbed data. The results confirm the effectiveness of the proposed adaptive procedure.

Correction to: Goal-oriented error estimation and adaptivity in MsFEM computations

Correction to: Goal-oriented error estimation and adaptivity in MsFEM computations

Goal-oriented mesh adaptation method for nonlinear problems including algebraic errors

We deal with the goal-oriented error estimates and mesh adaptation for nonlinear partial differential equations. The setting of the adjoint problem and the resulting estimates are not based on a differentiation of the primal problem but on a suitable linearization which guarantees the adjoint consistency of the numerical scheme. Furthermore, we develop an efficient adaptive algorithm which balances the errors arising from the discretization and the use of nonlinear as well as linear iterative solvers. Several numerical examples demonstrate the efficiency of this algorithm.

Goal-Oriented Anisotropic hp-Adaptive Discontinuous Galerkin Method for the Euler Equations

We deal with the numerical solution of the compressible Euler equations with the aid of the discontinuous Galerkin (DG) method with focus on the goal-oriented error estimates and adaptivity. We analyse the adjoint consistency of the DG scheme where the adjoint problem is not formulated by the differentiation of the DG form and the target functional but using a suitable linearization of the nonlinear forms. Furthermore, we present the goal-oriented anisotropic hp-mesh adaptation method for the Euler equations. The theoretical results are supported by numerical experiments.

Forward modelling of gravity data on unstructured grids using an adaptive mimetic finite-difference method

One of the techniques to optimize the gradient-based inversion of geophysical gravity data is to deal with the large and dense Jacobian matrices implicitly. This is made possible by solving the normal systems iteratively and to use pseudo-forward problems to calculate the effect of the Jacobian matrices. For this, numerical forward solvers should be used instead of analytical solvers. In this study, we propose mimetic finite-difference schemes for the solution of the forward gravity problem on unstructured grids. While the mimetic finite-difference method shares many traits with the finite-element and finite-volume methods, it has the advantage of naturally accommodating grids with arbitrary polyhedral elements. We use cell-based and vertex-based mimetic finite-difference schemes to solve a diffusion problem in mixed form, and Poisson's problem, respectively. The cell-based scheme is solved for the gravitational potential and attraction at the centroids and the facets of the elements, respectively, and the vertex-based scheme is solved for the potential at the vertices of the elements. We compare these mimetic finite-difference schemes with cell-based and vertex-based finite-volume schemes and a vertex-based finite-element scheme, in terms of accuracy of the potential. We then utilize the versatility of the mimetic finite-difference method, in accommodating arbitrary elements, and develop an adaptive mesh refinement tool. The mesh adaptivity is performed by an iterative h-refinement where goal-oriented error estimates are used to mark the elements for refinement. The marked elements are decomposed into new tetrahedra by regular subdivision. Since arbitrary polyhedra are naturally permitted in the mimetic finite-difference method, the added nodes are not regarded as hanging nodes, and therefore, any modification of the scheme or extra refinement is avoided. This characteristic allows preserving the quality of the initial grid and simplifying the refinement procedure. We demonstrate the practicality of the presented mesh adaptivity for the forward modelling of gravity and gravity gradient data using realistic geological models. We show that the cell-based mimetic scheme is more efficient than the vertex-based scheme in terms of computational resources. We also demonstrate that in the presence of irregular interfaces in the input domain, the regular refinement approach, presented in this study, is more reliable than a standard mesh regeneration technique.

Guaranteed Quantity of Interest Error Estimate Based on Equilibrated Flux Reconstruction

The quality of a local physical quantity obtained by the numerical method, such as the finite element method (FEM), attracts more and more attention in computational electromagnetism. Inspired by the idea of goal-oriented error estimate given for the Laplace problem, this work is devoted to a guaranteed a posteriori error estimate adapted for the quantity of interest (QOI) linked to magnetostatic problems, in particular, to the value of the magnetic flux density. The development is principally based on an equilibrated flux construction, which ensures fully computable estimators without any unknown constant. The main steps of the mathematical development are given in detail with the physical interpretation. An academic example using an analytical solution is considered to illustrate the performance of the approach, and a discussion about different aspects related to the practical point of view is proposed.

Forward modelling of geophysical electromagnetic data on unstructured grids using an adaptive mimetic finite-difference method

While the mimetic finite-difference method shares many similarities with the finite-element and finite-volume methods, it has the advantage of naturally accommodating grids with arbitrary polyhedral elements. In this study, we use this attribute to develop an adaptive scheme for the solution of the geophysical electromagnetic modelling problem on unstructured grids. Starting with an initial tetrahedral grid, our mesh adaptivity implements an iterative h-refinement where a residual- and jump-based goal-oriented error estimator is used to mark a certain portion of the elements. The marked elements are decomposed into new tetrahedra by regular subdivision, creating an octree-like unstructured grid. Since arbitrary polyhedra are naturally permitted in the mimetic finite-difference method, the added nodes are not regarded as hanging nodes and hence any level of non-conformity can be implemented without a modification to the mimetic scheme. In this study, we consider 2-irregularity where two levels of non-conformity between the adjacent elements is permitted. We use a total field approach where the electric field is defined at the edges of the polyhedral elements and the electromagnetic source may have an arbitrary shape and location. The accuracy of the mimetic scheme and the effectiveness of the proposed mesh adaptivity are verified using benchmark and realistic examples that represent various magnetotelluric and controlled-source survey scenarios. The mesh adaptivity generates grids with refinement generally concentrated at the transmitter and receiver locations and the interfaces of materials with contrasting conductivities, and the mimetic finite-difference solutions have good agreement with the reference numerical and real data. We also demonstrate the practicality of our method using an example with an analytical solution and comparison with a standard mesh regeneration technique. The results show that our mesh adaptivity procedure can result in a higher accuracy, with similar numbers of elements, when compared with the mesh regeneration approach. Also, using a generic sparse direct solver, our method is found to be more efficient than the mesh regeneration approach in terms of computation time and memory usage. Moreover, a comparison between 1- and 2-irregularity shows the higher efficiency of the latter in terms of the number of elements required to reach a certain level of accuracy.