Albert Cohen - Academia.edu (original) (raw)

Papers by Albert Cohen

ESAIM: Mathematical Modelling and Numerical Analysis, 2016

We consider the linear elliptic equation − div(a∇u) = f on some bounded domain D, where a has the... more We consider the linear elliptic equation − div(a∇u) = f on some bounded domain D, where a has the affine form a = a(y) = ā + ∑j≥1yjψj for some parameter vector y = (yj)j ≥ 1 ∈ U = [−1,1]N. We study the summability properties of polynomial expansions of the solution map y → u(y) ∈ V := H01(D) . We consider both Taylor series and Legendre series. Previous results [A. Cohen, R. DeVore and C. Schwab, Anal. Appl. 9 (2011) 11–47] show that, under a uniform ellipticity assuption, for any 0

Orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with ... more Orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide. We show here that under fairly general conditions, exact reconstruction schemes with synthesis filters different from the analysis filters give rise: to two dual Riesz bases of compactly supported wavelets. We give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions, and we present a sufficient condition for the decay of their Fourier transforms. We study the regularity of these biorthogonal bases. We provide several families of examples, all symmetric (corresponding to “linear phase ” filters). In particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity; we also show how to construct symmetric biorthogonal wavelet bases “close ” to a (nonsymmetric) orthonormal basis. 1.

ESAIM: Mathematical Modelling and Numerical Analysis, 2014

Motivated by the numerical treatment of parametric and stochastic PDEs, we analyze the least squa... more Motivated by the numerical treatment of parametric and stochastic PDEs, we analyze the least squares method for polynomial approximation of multivariate functions based on random sampling according to a given probability measure. Recent work has shown that in the univariate case and for the uniform distribution, the least squares method is optimal in expectation in [1] and in probability in [7], under the condition that the number of samples scales quadratically with respect to the dimension of the polynomial space. Here "optimal" means that the accuracy of the least square approximation is comparable with that of the best approximation in the given polynomial space. In this paper, we discuss the optimality of the polynomial least squares method in arbitrary dimension. Our analysis applies to any arbitrary multivariate polynomial space (including tensor product, total degree or hyperbolic crosses), under the minimal requirement that its associated index set is downward closed. The optimality criterion only involves the relation between the number of samples and the dimension of the polynomial space. We extend our results to the approximation of Hilbert space-valued functions in order to apply them to the approximation of parametric and stochastic elliptic PDEs. As a particular case, we discuss "inclusion type" elliptic PDE models, and derive an exponential convergence estimate for the least squares method. Numerical results confirm our estimate, yet pointing out a gap between the condition necessary to achieve optimality in the theory, and the condition that in practice yields the optimal convergence rate.

Sampling Theory in Signal and Image Processing

We consider the inverse problem of reconstructing general solutions to the Helmholtz equation on ... more We consider the inverse problem of reconstructing general solutions to the Helmholtz equation on some domain Ω from their values at scattered points x 1 ,. .. , x n ⊂ Ω. This problem typically arises when sampling acoustic fields with n microphones for the purpose of reconstructing this field over a region of interest Ω contained in a larger domain D in which the acoustic field propagates. In many applied settings, the shape of D and the boundary conditions on its border are unknown. Our reconstruction method is based on the approximation of a general solution u by linear combinations of Fourier-Bessel functions or plane waves. We analyze the convergence of the least-squares estimates to u using these families of functions based on the samples (u(x i)) i=1,...,n. Our analysis describes the amount of regularization needed to guarantee the convergence of the least squares estimate towards u, in terms of a condition that depends on the dimension of the approximation subspace, the sample size n and the distribution of the samples. It reveals the advantage of using non-uniform distributions that have more points on the boundary of Ω. Numerical illustrations show that our approach compares favorably with reconstruction methods using other basis functions, and other types of regularization.

Siam Journal on Scientific Computing, 1997

1995 International Conference on Acoustics, Speech, and Signal Processing, 1995

ABSTRACT We study the non-orthogonality of perfect-reconstruction (PR) biorthogonal filter banks ... more ABSTRACT We study the non-orthogonality of perfect-reconstruction (PR) biorthogonal filter banks by measuring the en-ergy preservation between the spatial and transform domains. The mathematical formulation of that issue leads to the computation of the Riesz constants, ...

Mathematics of Computation, 2000

This paper is concerned with the construction and analysis of wavelet-based adaptive algorithms f... more This paper is concerned with the construction and analysis of wavelet-based adaptive algorithms for the numerical solution of elliptic equations. These algorithms approximate the solution u u of the equation by a linear combination of N N wavelets. Therefore, a benchmark for their performance is provided by the rate of best approximation to u u by an arbitrary linear combination of N N wavelets (so called N N -term approximation), which would be obtained by keeping the N N largest wavelet coefficients of the real solution (which of course is unknown). The main result of the paper is the construction of an adaptive scheme which produces an approximation to u u with error O ( N − s ) O(N^{-s}) in the energy norm, whenever such a rate is possible by N N -term approximation. The range of s > 0 s>0 for which this holds is only limited by the approximation properties of the wavelets together with their ability to compress the elliptic operator. Moreover, it is shown that the number ...

Confluentes Mathematici, 2011

We characterize the lack of compactness in the critical embedding of functions spaces X ⊂ Y havin... more We characterize the lack of compactness in the critical embedding of functions spaces X ⊂ Y having similar scaling properties in the following terms : a sequence (u n) n≥0 bounded in X has a subsequence that can be expressed as a finite sum of translations and dilations of functions (φ l) l>0 such that the remainder converges to zero in Y as the number of functions in the sum and n tend to +∞. Such a decomposition was established by Gérard in [13] for the embedding of the homogeneous Sobolev space X =Ḣ s into the Y = L p in d dimensions with 0 < s = d/2 − d/p, and then generalized by Jaffard in [15] to the case where X is a Riesz potential space, using wavelet expansions. In this paper, we revisit the wavelet-based profile decomposition, in order to treat a larger range of examples of critical embedding in a hopefully simplified way. In particular we identify two generic properties on the spaces X and Y that are of key use in building the profile decomposition. These properties may then easily be checked for typical choices of X and Y satisfying critical embedding properties. These includes Sobolev, Besov, Triebel-Lizorkin, Lorentz, Hölder and BMO spaces. * The third author was supported by the EPSRC Science and Innovation award to the Oxford Centre for Nonlinear PDE (EP/E035027/1). 1 where (φ l) l>0 is a family of functions inḢ s,p (IR d) and where lim L→+∞ lim sup n→+∞ r n,L L q = 0. This decomposition is "asymptotically orthogonal" in the sense that for k = l | log(h l,n /h k,n)| → +∞ or |x l,n − x k,n |/h l,n → +∞, as n → +∞. This type of decomposition was also obtained earlier in [5] for a bounded sequence in H 1 0 (D, R 3) of solutions of an elliptic problem, with D the open unit disk of R 2 and in [27] and [26] for the critical injections of W 1,2 (Ω) in Lebesgue space and of W 1,p (Ω) in Lorentz spaces respectively, with Ω a bounded domain of R d. They were also studied in [25] in an abstract Hilbert space framework and in [4] in the Heisenberg group context.

Lecture Notes in Computer Science, 2012

SIAM Journal on Numerical Analysis, 2004

We introduce and analyze numerical methods for the treatment of inverse problems, based on an ada... more We introduce and analyze numerical methods for the treatment of inverse problems, based on an adaptive wavelet Galerkin discretization. These methods combine the theoretical advantages of the wavelet-vaguelette decomposition (WVD) in terms of optimally adapting to the unknown smoothness of the solution, together with the numerical simplicity of Galerkin methods. In a first step, we simply combine a thresholding algorithm on the data with a Galerkin inversion on a fixed linear space. In a second step, a more elaborate method performs the inversion by an adaptive procedure in which a smaller space adapted to the solution is iteratively constructed; this leads to a significant reduction of the computational cost.

SIAM Journal on Mathematical Analysis, 2011

The reduced basis method was introduced for the accurate online evaluation of solutions to a para... more 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.

Revista Matemática Iberoamericana, 2000

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

Mathematics of Computation, 2011

We study the properties of a simple greedy algorithm introduced in [9] for the generation of data... more We study the properties of a simple greedy algorithm introduced in [9] for the generation of dataadapted anisotropic triangulations. Given a function f , the algorithm produces nested triangulations TN and corresponding piecewise polynomial approximations fN of f. The refinement procedure picks the triangle which maximizes the local L p approximation error, and bisect it in a direction which is chosen so to minimize this error at the next step. We study the approximation error in the L p norm when the algorithm is applied to C 2 functions with piecewise linear approximations. We prove that as the algorithm progresses, the triangles tend to adopt an optimal aspect ratio which is dictated by the local hessian of f. For convex functions, we also prove that the adaptive triangulations satisfy the convergence bound f − fN L p ≤ CN −1 p |det(d 2 f)| L τ with 1 τ := 1 p + 1, which is known to be asymptotically optimal among all possible triangulations.