Compressed sensing is a new concept in signal processing where one seeks to minimize the number of measurements be taken from signals while still retaining information necessary approximate them well. The ideas have their origins certain abstract results functional analysis and approximation theory by Kashin but were recently brought into forefront work Candès, Romberg, Tao Donoho who constructed concrete algorithms showed promise application. There remain several fundamental questions on...

This paper is concerned with the construction and analysis of wavelet-based adaptive algorithms for numerical solution elliptic equations. These approximate solution<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u"><mml:semantics><mml:mi>u</mml:mi><mml:annotation encoding="application/x-tex">u</mml:annotation></mml:semantics></mml:math></inline-formula>of equation by a linear combination of<inline-formula alttext="upper...

Parametric partial differential equations are commonly used to model physical systems. They also arise when Wiener chaos expansions as an alternative Monte Carlo solving stochastic elliptic problems. This paper considers a class of second order, linear, parametric, PDE's in bounded domain D with coefficients depending on possibly countably many parameters. It shows that the dependence solution parameters diffusion coefficient is analytically smooth. analyticity then exploited prove under...

Parametrized families of PDEs arise in various contexts such as inverse problems, control and optimization, risk assessment, uncertainty quantification. In most these applications, the number parameters is large or perhaps even infinite. Thus, development numerical methods for parametric problems faced with possible curse dimensionality. This article directed at (i) identifying understanding which properties equations allow one to avoid this (ii) developing analysing effective fully exploit...

We consider the problem of reconstructing an unknown bounded function u defined on a domain X⊂ℝ d from noiseless or noisy samples at n points (x i ) i=1,⋯,n . measure reconstruction error in norm L 2 (X,dρ) for some given probability dρ. Given linear space V m with dim (V )=m≤n, we study general terms weighted least-squares approximations spaces based independent random samples. It is well known that can be inaccurate and unstable when too close to n, even case. Recent results [6, 7] have...

We consider the problem of approximating a given element f from Hilbert space $\mathcal{H}$ by means greedy algorithms and application such procedures to regression in statistical learning theory. improve on existing theory convergence rates for both orthogonal algorithm relaxed algorithm, as well forward stepwise projection algorithm. For all these algorithms, we prove results variety function classes not simply those that are related convex hull dictionary. then show how bounds lead new...

Given a function f 2 L (Q), Q := [0, 1) and real number t 0, let U( , t) inf g2BV (Q) kf gk (I) + V ( g), where the infimum is taken over all functions g BV of bounded variation on I.This related extremal problems arise in several areas mathematics such as interpolation operators statistical estimation, well digital image processing.Techniques for finding minimizers based variational calculus nonlinear partial differential equations have been put forward by authorsThe main disadvantage these...

The use of multiresolution decompositions in the context finite volume schemes for conservation laws was first proposed by A. Harten purpose accelerating evaluation numerical fluxes through an adaptive computation. In this approach solution is still represented at each time step on finest grid, resulting inherent limitation potential gain memory space and computational time. present paper concerned with development analysis fully schemes, which computed a dynamically evolved grid. A crucial...

Motivated by the numerical treatment of parametric and stochastic PDEs, we analyze least-squares method for polynomial approximation multivariate functions based on random sampling according to a given probability measure. Recent work has shown that in univariate case, is quasi-optimal expectation [A. Cohen, M A. Davenport D. Leviatan. Found. Comput. Math. 13 (2013) 819–834] [G. Migliorati, F. Nobile, E. von Schwerin, R. Tempone, 14 (2014) 419–456], under suitable conditions relate number...

The numerical approximation of parametric partial differential equations is a computational challenge, in particular when the number involved parameter large. This paper considers model class second order, linear, parametric, elliptic PDEs on bounded domain D with diffusion coefficients depending parameters an affine manner. For such models, it was shown [9, 10] that under very weak assumptions coefficients, entire family solutions to can be simultaneously approximated Hilbert space V =...

In this paper we propose and analyze stable variational formulations for convection diffusion problems starting from concepts introduced by Sangalli.We derive efficient reliable a posteriori error estimators that are based on these formulations.The analysis of resulting adaptive solution concepts, when specialized to the setting suggested Sangalli's work, reveals partly unexpected phenomena related specific nature norms induced formulation.Several remedies, other specifications, explored...

This paper considers the problem of optimal recovery an element $u$ a Hilbert space ${\cal H}$ from measurements form $\ell_j(u)$, $j=1,\dots,m$, where $\ell_j$ are known linear functionals on H}$. Problems this type well studied [C. A. Micchelli, T. J. Rivlin, and S. Winograd, Numer. Math., 26 (1976), pp. 191--200] usually carried out under assumption that belongs to prescribed model class, typically compact subset Motivated by reduced modeling for solving parametric partial differential...

We establish new results on the space BV of functions with bounded variation. While it is well known that this admits no unconditional basis, we show "almost" characterized by wavelet expansions in following sense: if a function f BV, its coefficient sequence normalized basis satisfies class weak- \ell^1 type estimates. These weak estimates can be employed to prove many interesting results. use them identify interpolation spaces between and Sobolev or Besov spaces, derive...

If |$L$| is a bounded linear operator mapping the Banach space |$X$| into |$Y$| and |$K$| compact set in , then Kolmogorov widths of image |$L(K)$| do not exceed those multiplied by norm . We extend this result from maps to holomorphic mappings |$u$| following sense: when |$n$| -widths are |$\mathcal {O}(n^{-r})$| for some |$r>1$| |$u(K)$| {O}(n^{-s})$| any |$s\lt r-1$| use these results prove various theorems about manifolds consisting solutions certain parametrized partial differential...

We consider the linear elliptic equation − div( a ∇ u ) = f on some bounded domain D , where has form exp( b with random function defined as ( y ∑ j ≥ 1 ψ ∈ ℝ N are i.i.d. standard scalar Gaussian variables and is given sequence of functions in L ∞ ). study summability properties Hermite-type expansions solution map → V := H 0 that is, ν ℱ ), ∏ ≥1 tensorized Hermite polynomials indexed by set finitely supported sequences nonnegative integers. Previous results [V.H. Hoang C. Schwab, M3AS 24...

We consider the linear elliptic equation − div( a ∇ u ) = f on some bounded domain D , where has affine form a(y) ā + ∑ j≥1 y j ψ for parameter vector ( ≥ 1 ∈ U [−1,1] N . study summability properties of polynomial expansions solution map → V := H 0 both Taylor series and Legendre series. Previous results [A. Cohen, R. DeVore C. Schwab, Anal. Appl. 9 (2011) 11–47] show that, under uniform ellipticity assuption, any &lt; p 1, ℓ (∥ ∥ L ∞ implies -norms or coefficients. Such ensure convergence...

Kolmogorov $n$-widths and low-rank approximations are studied for families of elliptic diffusion PDEs parametrized by the coefficients. The decay can be controlled that error achieved best $n$-term using polynomials in parametric variable. However, we prove certain relevant instances where coefficients piecewise constant over a partition physical domain, exhibit significantly faster decay. This, turn, yields theoretical justification fast convergence reduced basis or POD methods when...