Convergence Rates for Greedy Algorithms in Reduced Basis Methods

Abstract

The reduced basis method was introduced for the accurate online evaluation of solutions to a parameter dependent family of elliptic partial differential equations. Abstractly, it can be viewed as determining a "good" n dimensional space H n to be used in approximating the elements of a compact set F in a Hilbert space H. One, by now popular, computational approach is to find H n through a greedy strategy. It is natural to compare the approximation performance of the H n generated by this strategy with that of the Kolmogorov widths d n (F) since the latter gives the smallest error that can be achieved by subspaces of fixed dimension n. The first such comparisons, given in [1], show that the approximation error, σ n (F) := dist(F, H n ), obtained by the greedy strategy satisfies σ n (F) ≤ Cn2 n d n (F). In this paper, various improvements of this result will be given. Among these, it is shown that whenever d n (F) ≤ M n −α , for all n > 0, and some M, α > 0, we also have σ n (F) ≤ C α M n −α for all n > 0, where C α depends only on α. Similar results are derived for generalized exponential rates of the form M e −an α . The exact greedy algorithm is not always computationally feasible and a commonly used computationally friendly variant can be formulated as a "weak greedy algorithm". The results of this paper are established for this version as well.

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

“…Chapter 3 guides the reader through different sampling strategies, providing a comparison between classic techniques based on Singular Value Decomposition (SVD), Proper Orthogonal Decomposition (POD) and greedy algorithms. In this context it also discusses recent results on a priori convergence in the context of the concept of the Kolmogorov N-width [10]. Chapter 4 contains a thorough discussion of the computation of lower bounds for stability factors lower bounds and a comparative discussion of the various techniques.…”

“…Our vector field variable u(µ) = (u 1 (µ), u 2 (µ)) is the displacement of the elastic block under the applied load: the displacement satisfies the plane-strain linear elasticity equations in Ω in combination with the following boundary conditions: homogeneous Neumann (load-free) conditions are imposed on the top and bottom boundaries Γ top and Γ base of the block; homogeneous Dirichlet n · u = µ [9] on Γ 1 , n · u = µ [10] on Γ 2 , n · u = µ [11] on Γ 3 , representing traction loads. The output of interest s(µ) is the integrated horizontal (traction/compression) displacement over the full loaded boundary Γ right , given by…”

“…As already mentioned, details are postponed to Chapter 4. During this iterative basis selection process and if at the n-th step a n-dimensional reduced basis space V rb is given, the next basis function is the one that maximizes the estimated model order reduction error given the n-dimensional space V rb over P. That is, we select 10) and compute u δ (µ n+1 ) to enrich the reduced basis space as V rb = span{u δ (µ 1 ), . .…”

“…We define an original problem, denoted by subscript o, posed over the parameter-dependent physical domain Ω o = Ω o (µ). We denote by V o (µ) a suitable Hilbert space defined on Ω o (µ) and consider an elliptic problem of the following form: 10) where…”

“…. , f n }, we have the following convergence result for the basic greedy approximation [10] (see also [36] for a generalization to Banach spaces): Theorem 3.2. Assume that F has an exponentially small Kolmogorov N -width, d N (F ) ≤ ce −aN with a > log 2.…”