3 research outputs found
E2N: Error Estimation Networks for Goal-Oriented Mesh Adaptation
Given a partial differential equation (PDE), goal-oriented error estimation
allows us to understand how errors in a diagnostic quantity of interest (QoI),
or goal, occur and accumulate in a numerical approximation, for example using
the finite element method. By decomposing the error estimates into
contributions from individual elements, it is possible to formulate adaptation
methods, which modify the mesh with the objective of minimising the resulting
QoI error. However, the standard error estimate formulation involves the true
adjoint solution, which is unknown in practice. As such, it is common practice
to approximate it with an 'enriched' approximation (e.g. in a higher order
space or on a refined mesh). Doing so generally results in a significant
increase in computational cost, which can be a bottleneck compromising the
competitiveness of (goal-oriented) adaptive simulations. The central idea of
this paper is to develop a "data-driven" goal-oriented mesh adaptation approach
through the selective replacement of the expensive error estimation step with
an appropriately configured and trained neural network. In doing so, the error
estimator may be obtained without even constructing the enriched spaces. An
element-by-element construction is employed here, whereby local values of
various parameters related to the mesh geometry and underlying problem physics
are taken as inputs, and the corresponding contribution to the error estimator
is taken as output. We demonstrate that this approach is able to obtain the
same accuracy with a reduced computational cost, for adaptive mesh test cases
related to flow around tidal turbines, which interact via their downstream
wakes, and where the overall power output of the farm is taken as the QoI.
Moreover, we demonstrate that the element-by-element approach implies
reasonably low training costs.Comment: 27 pages, 14 figure
Goal-oriented a posteriori error estimates and adaptivity for the numerical solution of partial differential equations
Aposteriorní odhady chyby jsou nedílnou součástí každé spolehlivé numerické metody pro řešení parciálních diferenciálních rovnic. Účelem odhadů chyby cílové veličiny je kontrolovat výpočetní chyby předem dané veličiny. Díky tomu je tato metoda velmi vhodná pro řadu praktických aplikací. Výsledné odhady chyby mohou být rovněž využity k adaptaci výpočetní sítě. To umožňuje nalézt numerickou aproximaci cílové veličiny velmi efektivním způsobem. V této práci jsou odhady chyby cílové veličiny odvozeny pro nespojitou Galerkinovu metodu použitou pro numerické řešení lineární skalární úlohy a pro nelineární Eulerovy rovnice popisující proudění nevazké stlačitelné kapaliny. Dále se práce zaměřuje na několik aspektů metody odhadů cílové veličiny, konkrétně na: rekonstrukci diskrétního řešení, adjungovanou konzistenci diskretizace, kontrolu algebraických chyb vznikajících při řešení algebraických problémů pro primární i adjungovaný problém a propojení odhadů s hp-anizotropní adaptací sítě. Vlastnosti a chování metody jsou ověřeny numerickými experimenty.A posteriori error estimation is an inseparable component of any reliable numerical method for solving partial differential equations. The aim of the goal-oriented a posteriori error estimates is to control the computational error directly with respect to some quantity of interest, which makes the method very convenient for many engineering applications. The resulting error estimates may be employed for mesh adaptation which enables to find a numerical approximation of the quantity of interest under some given tolerance in a very efficient manner. In this thesis, the goal-oriented error estimates are derived for discontinuous Galerkin discretizations of the linear scalar model problems, as well as of the Euler equations describing inviscid compressible flows. It focuses on several aspects of the goal-oriented error estimation method, in particular, higher order reconstructions, adjoint consistency of the discretizations, control of the algebraic errors arising from iterative solutions of both algebraic systems, and linking the estimates with the hp-anisotropic mesh adaptation. The computational performance is demonstrated by numerical experiments.Katedra numerické matematikyDepartment of Numerical MathematicsMatematicko-fyzikální fakultaFaculty of Mathematics and Physic