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Author

DeVore, Ronald (4)

Cohen, Albert (2)Bachmayr, Markus (1)Dahmen, Wolfgang (1)Daubechies, Ingrid (1)View MoreJournalConstructive Approximation (3)Foundations of Computational Mathematics (1)KAUST Grant Number
KUS-C1-016-04 (4)

Publisher
Springer Nature (4)

SubjectActive learning (1)Adaptivity (1)Approximation classes (1)Cardinality (1)Compressed sensing (1)View MoreType
Article (4)

Year (Issue Date)2013 (2)2012 (1)2011 (1)Item AvailabilityMetadata Only (4)

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Capturing Ridge Functions in High Dimensions from Point Queries

Cohen, Albert; Daubechies, Ingrid; DeVore, Ronald; Kerkyacharian, Gerard; Picard, Dominique (Constructive Approximation, Springer Nature, 2011-12-21) [Article]

Constructing a good approximation to a function of many variables suffers from the "curse of dimensionality". Namely, functions on ℝ N with smoothness of order s can in general be captured with accuracy at most O(n -s/N) using linear spaces or nonlinear manifolds of dimension n. If N is large and s is not, then n has to be chosen inordinately large for good accuracy. The large value of N often precludes reasonable numerical procedures. On the other hand, there is the common belief that real world problems in high dimensions have as their solution, functions which are more amenable to numerical recovery. This has led to the introduction of models for these functions that do not depend on smoothness alone but also involve some form of variable reduction. In these models it is assumed that, although the function depends on N variables, only a small number of them are significant. Another variant of this principle is that the function lives on a low dimensional manifold. Since the dominant variables (respectively the manifold) are unknown, this leads to new problems of how to organize point queries to capture such functions. The present paper studies where to query the values of a ridge function f(x)=g(a · x) when both a∈ℝ N and g ∈ C[0,1] are unknown. We establish estimates on how well f can be approximated using these point queries under the assumptions that g ∈ C s[0,1]. We also study the role of sparsity or compressibility of a in such query problems. © 2011 Springer Science+Business Media, LLC.

Convergence Rates of AFEM with H −1 Data

Cohen, Albert; DeVore, Ronald; Nochetto, Ricardo H. (Foundations of Computational Mathematics, Springer Nature, 2012-06-29) [Article]

This paper studies adaptive finite element methods (AFEMs), based on piecewise linear elements and newest vertex bisection, for solving second order elliptic equations with piecewise constant coefficients on a polygonal domain Ω⊂ℝ2. The main contribution is to build algorithms that hold for a general right-hand side f∈H-1(Ω). Prior work assumes almost exclusively that f∈L2(Ω). New data indicators based on local H-1 norms are introduced, and then the AFEMs are based on a standard bulk chasing strategy (or Dörfler marking) combined with a procedure that adapts the mesh to reduce these new indicators. An analysis of our AFEM is given which establishes a contraction property and optimal convergence rates N-s with 0<s≤1/2. In contrast to previous work, it is shown that it is not necessary to assume a compatible decay s<1/2 of the data estimator, but rather that this is automatically guaranteed by the approximability assumptions on the solution by adaptive meshes, without further assumptions on f; the borderline case s=1/2 yields an additional factor logN. Computable surrogates for the data indicators are introduced and shown to also yield optimal convergence rates N-s with s≤1/2. © 2012 SFoCM.

Greedy Algorithms for Reduced Bases in Banach Spaces

DeVore, Ronald; Petrova, Guergana; Wojtaszczyk, Przemyslaw (Constructive Approximation, Springer Nature, 2013-02-26) [Article]

Given a Banach space X and one of its compact sets F, we consider the problem of finding a good n-dimensional space X n⊂X which can be used to approximate the elements of F. The best possible error we can achieve for such an approximation is given by the Kolmogorov width dn(F)X. However, finding the space which gives this performance is typically numerically intractable. Recently, a new greedy strategy for obtaining good spaces was given in the context of the reduced basis method for solving a parametric family of PDEs. The performance of this greedy algorithm was initially analyzed in Buffa et al. (Modél. Math. Anal. Numér. 46:595-603, 2012) in the case X=H is a Hilbert space. The results of Buffa et al. (Modél. Math. Anal. Numér. 46:595-603, 2012) were significantly improved upon in Binev et al. (SIAM J. Math. Anal. 43:1457-1472, 2011). The purpose of the present paper is to give a new analysis of the performance of such greedy algorithms. Our analysis not only gives improved results for the Hilbert space case but can also be applied to the same greedy procedure in general Banach spaces. © 2013 Springer Science+Business Media New York.

Approximation of High-Dimensional Rank One Tensors

Bachmayr, Markus; Dahmen, Wolfgang; DeVore, Ronald; Grasedyck, Lars (Constructive Approximation, Springer Nature, 2013-11-12) [Article]

Many real world problems are high-dimensional in that their solution is a function which depends on many variables or parameters. This presents a computational challenge since traditional numerical techniques are built on model classes for functions based solely on smoothness. It is known that the approximation of smoothness classes of functions suffers from the so-called 'curse of dimensionality'. Avoiding this curse requires new model classes for real world functions that match applications. This has led to the introduction of notions such as sparsity, variable reduction, and reduced modeling. One theme that is particularly common is to assume a tensor structure for the target function. This paper investigates how well a rank one function f(x 1,...,x d)=f 1(x 1)⋯f d(x d), defined on Ω=[0,1]d can be captured through point queries. It is shown that such a rank one function with component functions f j in W∞ r([0,1]) can be captured (in L ∞) to accuracy O(C(d,r)N -r) from N well-chosen point evaluations. The constant C(d,r) scales like d dr. The queries in our algorithms have two ingredients, a set of points built on the results from discrepancy theory and a second adaptive set of queries dependent on the information drawn from the first set. Under the assumption that a point z∈Ω with nonvanishing f(z) is known, the accuracy improves to O(dN -r). © 2013 Springer Science+Business Media New York.

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