Compressive sensing Petrov-Galerkin approximation of high-dimensional parametric operator equations

We analyze the convergence of compressive sensing based sampling techniques for the efficient evaluation of functionals of solutions for a class of high-dimensional, affine-parametric, linear operator equations which depend on possibly infinitely many parameters. The proposed algorithms are based on so-called "non-intrusive" sampling of the high-dimensional parameter space, reminiscent of Monte-Carlo sampling. In contrast to Monte-Carlo, however, a functional of the parametric solution is then computed via compressive sensing methods from samples of functionals of the solution. A key ingredient in our analysis of independent interest consists in a generalization of recent results on the approximate sparsity of generalized polynomial chaos representations (gpc) of the parametric solution families, in terms of the gpc series with respect to tensorized Chebyshev polynomials. In particular, we establish sufficient conditions on the parametric inputs to the parametric operator equation such that the Chebyshev coefficients of the gpc expansion are contained in certain weighted $\ell_p$-spaces for $0<p\leq 1$. Based on this we show that reconstructions of the parametric solutions computed from the sampled problems converge, with high probability, at the $L_2$, resp. $L_\infty$ convergence rates afforded by best $s$-term approximations of the parametric solution up to logarithmic factors.Comment: revised version, 27 page

Accelerated Projected Gradient Method for Linear Inverse Problems with Sparsity Constraints

Regularization of ill-posed linear inverse problems via $\ell_1$ penalization has been proposed for cases where the solution is known to be (almost) sparse. One way to obtain the minimizer of such an $\ell_1$ penalized functional is via an iterative soft-thresholding algorithm. We propose an alternative implementation to $\ell_1$-constraints, using a gradient method, with projection on $\ell_1$-balls. The corresponding algorithm uses again iterative soft-thresholding, now with a variable thresholding parameter. We also propose accelerated versions of this iterative method, using ingredients of the (linear) steepest descent method. We prove convergence in norm for one of these projected gradient methods, without and with acceleration.Comment: 24 pages, 5 figures. v2: added reference, some amendments, 27 page

Wavelet and Multiscale Methods

Various scientific models demand finer and finer resolutions of relevant features. Paradoxically, increasing computational power serves to even heighten this demand. Namely, the wealth of available data itself becomes a major obstruction. Extracting essential information from complex structures and developing rigorous models to quantify the quality of information leads to tasks that are not tractable by standard numerical techniques. The last decade has seen the emergence of several new computational methodologies to address this situation. Their common features are the nonlinearity of the solution methods as well as the ability of separating solution characteristics living on different length scales. Perhaps the most prominent examples lie in multigrid methods and adaptive grid solvers for partial differential equations. These have substantially advanced the frontiers of computability for certain problem classes in numerical analysis. Other highly visible examples are: regression techniques in nonparametric statistical estimation, the design of universal estimators in the context of mathematical learning theory and machine learning; the investigation of greedy algorithms in complexity theory, compression techniques and encoding in signal and image processing; the solution of global operator equations through the compression of fully populated matrices arising from boundary integral equations with the aid of multipole expansions and hierarchical matrices; attacking problems in high spatial dimensions by sparse grid or hyperbolic wavelet concepts. This workshop proposed to deepen the understanding of the underlying mathematical concepts that drive this new evolution of computation and to promote the exchange of ideas emerging in various disciplines

Wavelet and Multiscale Methods

[no abstract available

An optimal adaptive wavelet method for First Order System Least Squares

In this paper, it is shown that any well-posed 2nd order PDE can be reformulated as a well-posed first order least squares system. This system will be solved by an adaptive wavelet solver in optimal computational complexity. The applications that are considered are second order elliptic PDEs with general inhomogeneous boundary conditions, and the stationary Navier-Stokes equations.Comment: 40 page