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Publication# Heuristic strategies for the approximation of stability factors in quadratically nonlinear parametrized PDEs

Abstract

In this paper we present some heuristic strategies to compute rapid and reliable approximations to stability factors in nonlinear, inf-sup stable parametrized PDEs. The efficient evaluation of these quantities is crucial for the rapid construction of a posteriori error estimates to reduced basis approximations. In this context, stability factors depend on the problem’s solution, and in particular on its reduced basis approximation. Their evaluation becomes therefore very expensive and cannot be performed prior to (and independently of) the construction of the reduced space. As a remedy, we first propose a linearized, heuristic version of the Successive Constraint Method (SCM), providing a suitable estimate – rather than a rigorous lower bound as in the original SCM – of the stability factor. Moreover, for the sake of computational efficiency, we develop an alternative heuristic strategy, which combines a radial basis interpolant, suitable criteria to ensure its positiveness, and an adaptive choice of interpolation points through a greedy procedure. We provide some theoretical results to support the proposed strategies, which are then applied to a set of test cases dealing with parametrized Navier-Stokes equations. Finally, we show that the interpolation strategy is inexpensive to apply and robust even in the proximity of bifurcation points, where the estimate of stability factors is particularly critical.

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Heuristics is the process by which humans use mental short cuts to arrive at decisions. Heuristics are simple strategies that humans, animals, organizations, and even machines use to quickly form judgments, make decisions, and find solutions to complex problems. Often this involves focusing on the most relevant aspects of a problem or situation to formulate a solution. While heuristic processes are used to find the answers and solutions that are most likely to work or be correct, they are not always right or the most accurate.

In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has a number of data points, obtained by sampling or experimentation, which represent the values of a function for a limited number of values of the independent variable. It is often required to interpolate; that is, estimate the value of that function for an intermediate value of the independent variable.

In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. What is meant by best and simpler will depend on the application. A closely related topic is the approximation of functions by generalized Fourier series, that is, approximations based upon summation of a series of terms based upon orthogonal polynomials.

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