Goal-oriented Error Estimation Research Articles

Goal-oriented error control of stochastic system approximations using metric-based anisotropic adaptations

The simulation of complex nonlinear engineering systems such as compressible fluid flows may be targeted to make more efficient and accurate the approximation of a specific (scalar) quantity of interest of the system. Putting aside modeling error and parametric uncertainty, this may be achieved by combining goal-oriented error estimates and adaptive anisotropic spatial mesh refinements. To this end, an elegant and efficient framework is the one of (Riemannian) metric-based adaptation where a goal-based a priori error estimation is used as indicator for adaptivity.This work proposes a novel extension of this approach to the case of aforementioned system approximations bearing a stochastic component. In this case, an optimization problem leading to the best control of the distinct sources of errors is formulated in the continuous framework of the Riemannian metric space. Algorithmic developments are also presented in order to quantify and adaptively adjust the error components in the deterministic and stochastic approximation spaces.The capability of the proposed method is tested on various problems including a supersonic scramjet inlet subject to geometrical and operational parametric uncertainties. It is demonstrated to accurately capture discontinuous features of stochastic compressible flows impacting pressure-related quantities of interest, while balancing computational budget and refinements in both spaces.

English

We deal with a posteriori error control of discontinuous Galerkin approximations for linear boundary value problems. The computational error is estimated in the framework of the Dual Weighted Residual method (DWR) for goal-oriented error estimation which requires to solve an additional (adjoint) problem. We focus on the control of the algebraic errors arising from iterative solutions of algebraic systems corresponding to both the primal and adjoint problems. Moreover, we present two different reconstruction techniques allowing an efficient evaluation of the error estimators. Finally, we propose a complex algorithm which controls discretization and algebraic errors and drives the adaptation of the mesh in the close to optimal manner with respect to the given quantity of interest.

Goal-Oriented Regional Angular Adaptive Algorithm for the SN Equations

Angular discretization errors inherent in the discrete ordinates method are a major problem, especially for localized source problems and problems with strongly absorbing media or large-volume void regions, where angular discretization errors would be totally unacceptable. This paper proposes a regional angular adaptive algorithm together with a goal-oriented error estimate to solve the SN equations. Standard angular adaptive refinement techniques are based on estimated local errors. We compare an interpolated angular flux value against a calculated value to generate local errors. The adaptive quadrature sets can be created by subdividing a spherical quadrilateral into four spherical subquadrilaterals that have positive weights and can be locally refined. Techniques for mapping angular fluxes from one quadrature set to another are developed to transfer angular fluxes on the interfaces of different spatial regions. To provide a better detector response, local errors are weighted by the importance of a given angular region toward the computational goal, providing an appropriate goal-oriented angular adaptivity. First collision source methods are employed to improve adjoint flux calculation accuracy. We tested the performance and accuracy of the proposed goal-oriented regional angular adaptive algorithm within the ARES code for a number of benchmark problems and present the results of a one-region test model and the Kobayashi benchmark problems. The reduction of angular number is at least one order of magnitude for adaptive refinement. The benchmarks demonstrate that the proposed goal-oriented adaptive refinement can achieve the same level of accuracy as the SN method, which has significantly higher computation cost. Thus, adaptive refinement is a viable approach for investigating difficult particle radiation transport problems.

An Equilibrated Fluxes Approach to the Certified Descent Algorithm for Shape Optimization Using Conforming Finite Element and Discontinuous Galerkin Discretizations

The certified descent algorithm (CDA) is a gradient-based method for shape optimization which certifies that the direction computed using the shape gradient is a genuine descent direction for the objective functional under analysis. It relies on the computation of an upper bound of the error introduced by the finite element approximation of the shape gradient. In this paper, we present a goal-oriented error estimator which depends solely on local quantities and is fully-computable. By means of the equilibrated fluxes approach, we construct a unified strategy valid for both conforming finite element approximations and discontinuous Galerkin discretizations. The new variant of the CDA is tested on the inverse identification problem of electrical impedance tomography: both its ability to identify a genuine descent direction at each iteration and its reliable stopping criterion are confirmed.

Stochastic goal-oriented error estimation with memory

We propose a stochastic dual-weighted error estimator for the viscous shallow-water equation with boundaries. For this purpose, previous work on memory-less stochastic dual-weighted error estimation is extended by incorporating memory effects. The memory is introduced by describing the local truncation error as a sum of time-correlated random variables. The random variables itself represent the temporal fluctuations in local truncation errors and are estimated from high-resolution information at near-initial times.The resulting error estimator is evaluated experimentally in two classical ocean-type experiments, the Munk gyre and the flow around an island. In these experiments, the stochastic process is adapted locally to the respective dynamical flow regime. Our stochastic dual-weighted error estimator is shown to provide meaningful error bounds for a range of physically relevant goals. We prove, as well as show numerically, that our approach can be interpreted as a linearized stochastic-physics ensemble.

Strict upper and lower bounds for quantities of interest in static response sensitivity analysis

• A CRE-based technique is proposed for goal-oriented error estimation in sensitivity analysis. • Strict bounds for quantities associated with sensitivity derivatives are obtained. • The bounding property is exemplified by model problems. In this paper, a goal-oriented error estimation technique for static response sensitivity analysis is proposed based on the constitutive relation error (CRE) estimation for finite element analysis (FEA). Strict upper and lower bounds of various quantities of interest (QoI) that are associated with the response sensitivity derivative fields are acquired. Numerical results are presented to assess the strict bounding properties of the proposed technique.

Equivalence of two strictly bounding approaches for goal-oriented error estimation

Equivalence of two strictly bounding approaches for goal-oriented error estimation

A Goal-Oriented Error Estimator for a Class of Homogenization Problems

We present a goal-oriented a posteriori error estimator for finite element approximations of a class of homogenization problems. As a rule, homogenization problems are defined through the coupling of a macroscopic solution and the solution of auxiliary problems. In this work we assume that the homogenized problem is known and that it depends on a finite number of auxiliary problems. The accuracy in the goal functional depends therefore on the discretization error of the macroscopic and the auxiliary solutions. We show that it is possible to compute the error contributions of all solution components separately and use this information to balance the different discretization errors. Additionally, we steer a local mesh refinement for both the macroscopic problem and the auxiliary problems. The high efficiency of this approach is shown by numerical examples. These include the upscaling of a periodic diffusion tensor, the case of a Stokes flow over a porous bed, and the homogenization of a fuel cell model which includes the flow in a gas channel over a porous substrate coupled with a multispecies nonlinear transport equation.

Adaptive mesh relocation refinement on Kim's method: enhanced approximations and upper bounds for American options

This study demonstrates the efficiency of an adaptive mesh relocation refinement (AMrR) to Kim's (1990) American options pricing method. Given the suboptimal uniform time discretisation proposed by Kim, we test an r-adaptive strategy that controls the overall error using an adjoint objective function. Next, we build an a posteriori goal-oriented error estimator as a measure of the global error incurred by the time mesh used. The analytics show the AMrR with n = 30 time steps improves upon Kim's (1990) method root mean squared relative error (RMSRE) by 85.74% for short and by 73.70% for long-term American options. Furthermore, we use the AMrR to obtain tighter upper bounds for the option's theoretical value, built directly from Broadie and Detemple (1996) and Chung et al. (2010) upper bound methods. In respect to both studies, we find substantial improvements in the RMSRE for both short and long American options.

A stochastic computational method based on goal-oriented error estimation for heterogeneous geological materials

Computational modeling of geological materials is challenging. Firstly, they are heterogeneous with numerous uncertainties in the input parameters and secondly, the computational cost of modeling geological structures is time consuming due to the large and different length scales involved. In this article, we propose an efficient computational method for heterogeneous geological materials based on goal oriented error estimation and adaptive mesh refinement. Instead of estimating the error in a specific norm, the proposed novel error estimation approach which is called dual-weighted residual error estimation, approximates the error with respect to the quantity of interest. The dual-weighted residual error estimation is a dual-based scheme which requires an adjoint problem. The adjoint or dual problem is described by defining the quantity of interest in a functional form. Then by solving the primal and dual problems, errors in terms of the specified quantities are calculated. In many applications in engineering geology, the material is heterogeneous. In such cases, the material properties can be regarded as random fields. This variety of material properties leads to non-uniform distributions of the solution gradients, e.g. stresses. Therefore, it is vital to apply a reliable error estimation approach to be able to do efficiently the mesh-adaptivity procedure with regard to varying material parameters with pre-defined correlation lengths. Hence, the proposed error estimator is extended by accounting for a random field model to describe the material properties. Local estimated errors are exploited in order to accomplish the mesh adaptivity procedure. The goal-oriented mesh adaptivity controls the local errors in terms of the prescribed quantities. Both refinement and coarsening processes are applied to raise the efficiency. The performance of the proposed computational approach is demonstrated for several examples.

Goal-oriented error estimation and adaptive mesh refinement in dynamic coupled thermoelasticity

We present an adaptive method for dynamic coupled thermoelasticity problems based on goal-oriented error estimation. We use dimensionless variables in the finite element formulation and for the time integration we employ the acceleration-based Newmark method. Different recovery methods such as superconvergent patch recovery (SPR), L2-projection patch recovery (L2-PR) and weighted superconvergent patch recovery (WSPR) are used to estimate the error in the quantity of interest (QoI). By using adaptive refinement in space, the error in the quantity of interest is minimized. Therefore, the discretization is refined such that the error is equally distributed on the refined mesh. We demonstrate the efficiency of this method by numerous numerical examples.

Goal-oriented error estimation and mesh adaptivity in 3d elastoplasticity problems

We propose a 3D adaptive method for plasticity problems based on goal-oriented error estimation, which computes the error with respect to a prescribed quantity of interest. It is a dual-based scheme which requires an adjoint problem. The computed element-wise errors at each load/displacement increment are utilized for the mesh adaptivity purpose. Mesh adaptivity procedure is performed based on refinement and coarsening by introducing hanging nodes in quadrilateral and hexahedral elements in 2d and 3d, respectively. Several numerical simulations are investigated and the results are compared with available analytical solutions, existing experimental data and results of mesh adaptivity based on other conventional error estimation methods.

A goal-oriented adaptive finite element method for 3D resistivity modeling using dual-error weighting approach

A goal-oriented adaptive finite element (FE) method for solving 3D direct current (DC) resistivity modeling problem is presented. The model domain is subdivided into unstructured tetrahedral elements that allow for efficient local mesh refinement and flexible description of complex models. The elements that affect the solution at each receiver location are adaptively refined according to a goal-oriented posteriori error estimator using dual-error weighting approach. The FE method with adapting mesh can easily handle such structures at almost any level of complexity. The method is demonstrated on two synthetic resistivity models with analytical solutions and available results from integral equation method, so the errors can be quantified. The applicability of the numerical method is illustrated on a resistivity model with a topographic ridge. Numerical examples show that this method is flexible and accurate for geometrically complex situations.

Goal-Oriented Error Estimation for the Reduced Basis Method, with Application to Sensitivity Analysis

The reduced basis method is a powerful model reduction technique designed to speed up the computation of multiple numerical solutions of parametrized partial differential equations. We consider a quantity of interest, which is a linear functional of the PDE solution. A new probabilistic error bound for the reduced model is proposed. It is efficiently and explicitly computable, and we show on different examples that this error bound is sharper than existing ones. We include application of our work to sensitivity analysis studies.

Strict upper and lower bounds of quantities for beams on elastic foundation by dual analysis

Purpose – The purpose of this paper is to acquire strict upper and lower bounds on quantities of slender beams on Winkler foundation in finite element analysis. Design/methodology/approach – It leans on the dual analysis wherein the constitutive relation error (CRE) is used to perform goal-oriented error estimation. Due to the coupling of the displacement field and the stress field in the equilibrium equations of the beam, the prolongation condition for the stress field which is the key ingredient of CRE estimation is not directly applicable. To circumvent this difficulty, an approximate problem and the solution thereof are introduced, enabling the CRE estimation to proceed. It is shown that the strict bounding property for CRE estimation is preserved and strict bounds of quantities of the beam are obtainable thereafter. Findings – Numerical examples are presented to validate the strict upper and lower bounds for quantities of beams on elastic foundation by dual analysis. Research limitations/implications – This paper deals with one-dimensional (1D) beams on Winkler foundation. Nevertheless, the present work can be naturally extended to analysis of shells and 2D and 3D reaction-diffusion problems for future research. Originality/value – CRE estimation is extended to analysis of beams on elastic foundation by a decoupling strategy; strict upper bounds of global energy norm error for beams on elastic foundation are obtained; strict bounds of quantities for beams on elastic foundation are also obtained; unified representation and corresponding dual analysis of various quantities of the beam are presented; rigorous derivation of admissible stresses for beams is given.

The damped Crank–Nicolson time-marching scheme for the adaptive solution of the Black–Scholes equation

This paper is concerned with the derivation of a residual-based a posteriori error estimator and mesh-adaptation strategies for the space-time finite element approximation of parabolic problems with irregular data. Typical applications arise in the field of mathematical finance, where the Black–Scholes equation is used for modeling the pricing of European options. A conforming finite element discretization in space is combined with second-order time discretization by a damped Crank–Nicolson scheme for coping with data irregularities in the model. The a posteriori error analysis is developed within the general framework of the dual weighted residual method for sensitivity-based, goal-oriented error estimation and mesh optimization. In particular, the correct form of the dual problem with damping is considered.

Goal-oriented error estimation for beams on elastic foundation with double shear effect

In this paper, goal-oriented error estimation for Timoshenko beams on Pasternak foundation, which involves double shear effect, is performed. The constitutive relation error (CRE) estimation is used in finite element analysis (FEA) to acquire strict bounds on quantities of interest. Due to the coupling of the displacement field and the internal force field in the equilibrium equations of the beam, the prolongation condition for construction of the admissible internal force field, a pillar of the CRE estimation, is not directly applicable. To overcome this difficulty, an auxiliary approximate problem whose stress solution enables the CRE estimation to proceed is introduced. Thereafter, strict bounds of outputs for the beam are obtained by dual analysis to which a significant adjunct is the circumvention of shear locking in low-order finite element analysis. Numerical results are presented to validate the strict bounding properties for quantities of beams on elastic foundation and accurate estimation of displacement quantities that is impervious to shear locking.

A Goal-Oriented Error Estimator for Reduced Basis Method Modeling of Microwave Devices

This letter proposes a novel a-posteriori error estimator suitable for the reduced order modeling of microwave circuits. Unlike the existing error estimators based on impedance function residuals, the new one exploits the residual error associated with the computation of the scattering matrix. The estimator can be effectively used in the Reduced Basis Method (RBM) to automatically generate reduced-order models. The results of numerical experiments show that the new error estimator predicts the errors in the computed S-parameters more accurately than the other existing approaches, and that it can be used to automate the generation of wideband reduced models.

Adaptive thermo-mechanical finite element formulation based on goal-oriented error estimation

Abstract In this paper, a rh-adaptive thermo-mechanical formulation based on goal-oriented error estimation is proposed. The goal-oriented error estimation relies on different recovery-based error estimators, i.e. the superconvergent patch recovery (SPR), L 2 -projection patch recovery ( L 2 -PR) and weighted superconvergent patch recovery (WSPR). A new adaptive refinement strategy (ARS) is presented that minimizes the error in a quantity of interest and refines the discretization such that the error is equally distributed on the refined mesh. The method is validated by numerous numerical examples where an analytical solution or reference solution is available.

Locally equilibrated stress recovery for goal oriented error estimation in the extended finite element method

Goal oriented error estimation and adaptive procedures are essential for the accurate and efficient evaluation of finite element numerical simulations that involve complex domains. By locally improving the approximation quality, for example, by using the extended finite element method (XFEM), we can solve expensive problems which could result intractable otherwise. Here, we present an error estimation technique for enriched finite element approximations that is based on an equilibrated recovery technique, which considers the stress intensity factor as the quantity of interest. The locally equilibrated superconvergent patch recovery is used to obtain enhanced stress fields for the primal and dual problems defined to evaluate the error estimate.