Optimization under adaptive error control for finite element based simulations

The Poisson-Boltzmann equation is a widely used model to study electrostatics in molecular solvation. Its numerical solution using a boundary integral formulation requires a mesh on the molecular surface only, yielding accurate representations of the solute, which is usually a complicated geometry. Here, we utilize adjoint-based analyses to form two goal-oriented error estimates that allow us to determine the contribution of each discretization element (panel) to the numerical error in the solvation free energy. This information is useful to identify high-error panels to then refine them adaptively to find optimal surface meshes. We present results for spheres and real molecular geometries, and see that elements with large error tend to be in regions where there is a high electrostatic potential. We also find that even though both estimates predict different total errors, they have similar performance as part of an adaptive mesh refinement scheme. Our test cases suggest that the adaptive mesh refinement scheme is very effective, as we are able to reduce the error one order of magnitude by increasing the mesh size less than 20% and come out to be more efficient than uniform refinement when computing error estimations. This result sets the basis toward efficient automatic mesh refinement schemes that produce optimal meshes for solvation energy calculations.

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

A goal-oriented a-posteriori error estimator is developed for transient coupled thermo-hydro-mechanical (THM) parametric problems solved with a reduced basis approximation. The estimator assesses the error in some specific Quantity of Interest (QoI). The goal-oriented error estimate is derived based on explicitly-computed weak residual of the primal problem and implicitly-computed adjoint solution associated with the QoI. The time-dependence of the coupled THM system poses an additional complexity as the auxiliary adjoint problem evolves backwards in time. The error estimator guides a greedy adaptive procedure that constructs progressively an optimal reduced basis by smartly selecting snapshot points over a given parametric training sample. The reduced basis obtained is used to drastically reduce the coupled system spatial degrees of freedom by several orders of magnitude. The computational gain obtained from the developed methodology is demonstrated through applications in 2D and 3D parametrized problems simulating the evolution of coupled THM processes in rock masses.

This paper studies the coupling between anisotropic mesh adaptation and goal-oriented error estimate. The former is very well suited to the control of the interpolation error. It is generally interpreted as a local geometric error estimate. On the contrary, the latter is preferred when studying approximation errors for PDEs. It generally involves non local error contributions. Consequently, a full and strong coupling between both is hard to achieve due to this apparent incompatibility. This paper shows how to achieve this coupling. This is done in three steps. First, a new ap riorierror estimate is proved in a formal framework adapted to goal-oriented mesh adaptation for output functionals. Second, the error estimate is applied to the set of steady compressible Euler equations which are solved by a stabilized Galerkin finite element discretization. A goal-oriented error estimation is derived. Third, rewritten in the continuous mesh framework, the previous estimate is minimized on the set of continuous meshes thanks to a calculus of variations. The optimal mesh is then derive. 3D examples of steady flows around supersonic and transsonic aircrafts are presented to validate the current approach and to demonstrate its efficiency.

In this paper, nonconforming finite element methods (FEMs) are proposed for the constrained optimal control problems (OCPs) governed by the nonsmooth elliptic equations, in which the popular $$EQ_1^{rot}$$ element is employed to approximate the state and adjoint state, and the piecewise constant element is used to approximate the control. Firstly, the convergence and superconvergence properties for the nonsmooth elliptic equation are obtained by introducing an auxiliary problem. Secondly, the goal-oriented error estimates are obtained for the objective function through establishing the negative norm error estimate. Lastly, the methods are extended to some other well-known nonconforming elements.

In this paper we review the basic concepts to obtain a posteriori error estimates for the finite element solution of an elliptic linear model problem. We give the basic ideas to establish global error estimates for the energy norm as well as goal-oriented error estimates. While we show how these error estimation techniques are employed for our simple model problem, the emphasis of the paper is on assessing whether these procedures are ready for use in practical linear finite element analysis. We conclude that the actually practical error estimation techniques do not provide mathematically proven bounds on the error and need to be used with care. The more accurate estimation procedures also do not provide proven bounds that, in general, can be computed efficiently. We also briefly comment upon the state of error estimations in nonlinear and transient analyses and when mixed methods are used.

Goal-oriented error estimation and adaptivity for the finite element method

We introduce a goal-oriented strategy for multiscale computations performed using the Multiscale Finite Element Method (MsFEM). In a previous work, we have shown how to use, in the MsFEM framework, the concept of Constitutive Relation Error (CRE) to obtain a guaranteed and fully computable a posteriori error estimate in the energy norm (as well as error indicators on various error sources). Here, the CRE concept is coupled with the solution of an adjoint problem to control the error and drive an adaptive procedure with respect to a given output of interest. Furthermore, a local and non-intrusive enrichment technique is proposed to enhance the accuracy of error bounds. The overall strategy, which is fully automatic and robust, enables to reach an appropriate trade-off between certified reliability and computational cost in the MsFEM context. The performances of the proposed method are investigated on several illustrative numerical test cases. In particular, the error estimation is observed to be very accurate, yielding a very efficient adaptive procedure.

The negative impacts of microgrids (µGs) on the load frequency highlight the importance of implementing a robust, efficient, and adaptable controller to ensure stability. This work introduces an adaptive load frequency control (LFC) for an isolated µG that includes a PV system and electric vehicles (EVs), which have a significant impact on frequency. This control utilizes a combination of sine cosine optimization (SCO) and balloon effect identifier (BEI) algorithms. The controller presented in this work transforms the LFC process into an optimization problem that is highly compatible with various random situations encountered in the control process. The suggested control method is a novel approach by utilizing SCO+BEI for adaptive LFC application, resulting in a highly efficient response. The effectiveness of the proposed adaptive controller is assessed under the conditions of 17 MW variable load, system parameters uncertainties, and installed PV systems of 6 MW. MATLAB / Simulink package is rummage-sale as a digital test environment. According to simulation results, the proposed adaptive controller succeeds in regulating the frequency and power of an islanded µG. To measure the efficiency of the proposed control scheme, a comparison between other control techniques (such as adaptive controller using Jaya+BEI and classical integral controller) is done. The findings of the studied scenarios assured that the not compulsory control method using (SCO+BEI) has an obvious superiority over other control methods in terms of frequency solidity in case of random load instabilities and parameter uncertainties. Finally, it can be said that the proposed controller can better ensure the safe operation of the µGs.

BackgroundDynamic mathematical models in the form of systems of ordinary differential equations (ODEs) play an important role in systems biology. For any sufficiently complex model, the speed and accuracy of solving the ODEs by numerical integration is critical. This applies especially to systems identification problems where the parameter sensitivities must be integrated alongside the system variables. Although several very good general purpose ODE solvers exist, few of them compute the parameter sensitivities automatically.ResultsWe present a novel integration algorithm that is based on second derivatives and contains other unique features such as improved error estimates. These features allow the integrator to take larger time steps than other methods. In practical applications, i.e. systems biology models of different sizes and behaviors, the method competes well with established integrators in solving the system equations, and it outperforms them significantly when local parameter sensitivities are evaluated. For ease-of-use, the solver is embedded in a framework that automatically generates the integrator input from an SBML description of the system of interest.ConclusionsFor future applications, comparatively ‘cheap’ parameter sensitivities will enable advances in solving large, otherwise computationally expensive parameter estimation and optimization problems. More generally, we argue that substantially better computational performance can be achieved by exploiting characteristics specific to the problem domain; elements of our methods such as the error estimation could find broader use in other, more general numerical algorithms.

In this article, we set up a non-intrusive procedure that yields for strict and highquality error bounds of quantities of interest in linear viscoelasticity problems solved by means of the Finite Element Method. The non-intrusive feature is achieved by introducing, via a partition of unity, enrichment functions in the solution of the adjoint problem (handbook techniques). The resulting goal-oriented error estimation method is thus easy to implement in a FE code and enables to consider trully pointwise quantities of interest.

In this article, we set up a non-intrusive procedure that yields for strict and highquality error bounds of quantities of interest in linear viscoelasticity problems solved by means of the Finite Element Method. The non-intrusive feature is achieved by introducing, via a partition of unity, enrichment functions in the solution of the adjoint problem (handbook techniques). The resulting goal-oriented error estimation method is thus easy to implement in a FE code and enables to consider trully pointwise quantities of interest.

Scattering parameters computed by the finite-element method (FEM) are necessarily subject to discretisation error. It has been shown how this error can be estimated by a quite inexpensive postprocessing operation. An estimate of the error in the field is first obtained by increasing the polynomial order of all the elements and solving the resulting system approximately. Then a carefully chosen linear functional is applied to obtain the error in the scattering parameter from the error in the field. In the paper, the method is extended to obtain estimates over a frequency band, thus providing information about the frequency error of key points, such as transmission peaks or nulls. Moreover, it is shown that the error estimates can be used to correct the computed solution, offering significant improvements in accuracy. Numerical results are obtained with hierarchal, tetrahedral finite elements, applied to an E-plane T-junction, a metallic post in a waveguide, and a waveguide bend.

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

Goal-oriented error estimation and h-adaptivity for Maxwell’s equations