Affine Decomposition Research Articles

A Space-Time Petrov--Galerkin Certified Reduced Basis Method: Application to the Boussinesq Equations

We present a space-time certified reduced basis method for long-time integration of parametrized parabolic equations with quadratic nonlinearity which admit an affine decomposition in parameter but with no restriction on coercivity of the linearized operator. We first consider a finite element discretization based on discontinuous Galerkin time integration and introduce associated Petrov--Galerkin space-time trial- and test-space norms that yield optimal and asymptotically mesh independent stability constants. We then employ an $hp$ Petrov--Galerkin (or minimum residual) space-time reduced basis approximation. We provide the Brezzi--Rappaz--Raviart a posteriori error bounds which admit efficient offline-online computational procedures for the three key ingredients: the dual norm of the residual, an inf-sup lower bound, and the Sobolev embedding constant. The latter are based, respectively, on a more round-off resistant residual norm evaluation procedure, a variant of the successive constraint method, and a time-marching implementation of a fixed-point iteration of the embedding constant for the discontinuous Galerkin norm. Finally, we apply the method to a natural convection problem governed by the Boussinesq equations. The result indicates that the space-time formulation enables rapid and certified characterization of moderate-Grashof-number flows exhibiting steady periodic responses. However, the space-time reduced basis convergence is slow, and the Brezzi--Rappaz--Raviart threshold condition is rather restrictive, such that offline effort will be acceptable only for very few parameters.

On the approximation of stability factors for general parametrized partial differential equations with a two-level affine decomposition

A new approach for computationally efficient estimation of stability factors for parametric partial differential equations is presented. The general parametric bilinear form of the problem is approximated by two affinely parametrized bilinear forms at different levels of accuracy (after an empirical interpolation procedure). The successive constraint method is applied on the coarse level to obtain a lower bound for the stability factors, and this bound is extended to the fine level by adding a proper correction term. Because the approximate problems are affine, an efficient offline/online computational scheme can be developed for the certified solution (error bounds and stability factors) of the parametric equations considered. We experiment with different correction terms suited for a posteriori error estimation of the reduced basis solution of elliptic coercive and noncoercive problems.

Affine Decompositions of Parametric Stochastic Processes for Application within Reduced Basis Methods

AbstractWe consider parameter dependent spatial stochastic processes in the context of partial differential equations (PDEs) and model order reduction. For a given parameter, a random sample of such a process specifies a sample coefficient function of a PDE, e.g. characteristics of porous media such as Li-ion batteries or random influences in biomechanical systems. To apply the Reduced Basis Method (RBM) to parametrized systems (with stochastic or deterministic parameter dependencies), it is necessary to obtain affine decompositions of the systems in parameter and space (cf. Patera and Rozza (2006); Wieland (2010)). For deterministic problems, it is common to use the Empirical Interpolation Method (EIM) (cf. Barrault et al. (2004)). For stochastic coefficients one can apply the Karhunen-Loève (KL) expansion (cf. Karhunen (1947)) where the terms with stochastic dependencies are assumed to satisfy certain distributions and are modeled using polynomial chaos (PC) expansions (cf. Ghanem and Spanos (1991)).In this paper, we extend the EIM to parametrized spatial stochastic processes. The goal is to develop efficiently computable affine decompositions of not only parameter dependent but also stochastic systems that separate spatial dependencies from parametric and probabilistic influences without any assumptions on the distribution of non-spatial terms. We will use the basic concept of the EIM together with ideas from Proper Orthogonal Decomposition (POD) as well as from the Discrete Empirical Interpolation Method (DEIM) (cf. Chaturantabut and Sorensen (2010)), and we use least-squares approxmations.

Approximation and Spanning in the Hardy Space, by Affine Systems

We show that every function in the Hardy space can be approximated by linear combinations of translates and dilates of a synthesizer \(\psi \in L^1({\bf R}^d)\), provided only that \(\widehat{\psi}(0)=1\) and \(\psi\) satisfies a mild regularity condition. Explicitly, we prove scale averaged approximation for each \(f \in H^1({\bf R}^d)\), $$f(x) = \lim_{J \to \infty} \frac{1}{J} \sum_{j=1}^J \sum_{k \in {\bf Z}^d} c_{j,k} \psi(a_j x - k),$$ where \(a_j\) is an arbitrary lacunary sequence (such as \(a_j=2^j\)) and the coefficients \(c_{j,k}\) are local averages of f. This formula holds in particular if the synthesizer \(\psi\) is in the Schwartz class, or if it has compact support and belongs to \(L^p\) for some \(1<p<\infty\). A corollary is a new affine decomposition of \(H^1\) in terms of differences of \(\psi\).

Affine frame decompositions and shift-invariant spaces

In this paper, we show that the property of tight affine frame decomposition of functions in L 2 can be extended in a stable way to functions in Sobolev spaces when the generators of the tight affine frames satisfy certain mild regularity and vanishing moment conditions. Applying the affine frame operators Q j on jth levels to any function f in a Sobolev space reveals the detailed information Q j f of f in such tight affine decompositions. We also study certain basic properties of the range of the affine frame operators Q j such as the topological property of closedness and the notion of angles between the ranges for different levels, and thus establishing some interesting connection to (tight) frames of shift-invariant spaces.

Dyadic Affine Decompositions and Functional Wavelet Transforms

Decomposition of continuous functions can be accomplished by considering the difference of consecutive interpolation operators. When such a difference is expressed as an infinite series of some “wavelets” basis, the coefficient sequence becomes Donoho’s “interpolating wavelet transform.” Here, in contrast to the usual $L^2 $-setting, no analyzing wavelet is used to describe the wavelet transform. The objective of this paper is to study the structure of such decomposition spaces, including the formulation of bases and their duals, which leads to the notion of functional wavelet transforms (FnWT) using the duals as analyzing wavelets. Such a transform retains some of the most important properties of the integral wavelet transform of Grossmann and Morlet, such as the property of vanishing moments, which has significant applications to engineering problems.

Perihelion precession

By means of affine decomposition of oblique frame, a simple general formula is established which is convenient for calculating the precession of the perihelion point of a Keplerian ellipse under perturbation.