Adaptive Wavelet Methods for SPDEs

We review a series of results that have been obtained in the context of the DFG-SPP 1324 project “Adaptive wavelet methods for SPDEs”. This project has been concerned with the construction and analysis of adaptive wavelet methods for second order parabolic stochastic partial differential equations on bounded, possibly nonsmooth domains \(\mathcal{O}\subset \mathbb{R}^{d}\). A detailed regularity analysis for the solution process u in the scale of Besov spaces \(B_{\tau,\tau }^{s}(\mathcal{O})\), 1∕τ = s∕d + 1∕p, α > 0, p ≥ 2, is presented. The regularity in this scale is known to determine the order of convergence that can be achieved by adaptive wavelet algorithms and other nonlinear appro…

Spatial Besov regularity for stochastic partial differential equations on Lipschitz domains

We use the scale of Besov spaces B^\alpha_{\tau,\tau}(O), \alpha>0, 1/\tau=\alpha/d+1/p, p fixed, to study the spatial regularity of the solutions of linear parabolic stochastic partial differential equations on bounded Lipschitz domains O\subset R^d. The Besov smoothness determines the order of convergence that can be achieved by nonlinear approximation schemes. The proofs are based on a combination of weighted Sobolev estimates and characterizations of Besov spaces by wavelet expansions.

Multilevel preconditioning and adaptive sparse solution of inverse problems

A note on quarkonial systems and multilevel partition of unity methods

We discuss the connection between the theory of quarkonial decompositions for function spaces developed by Hans Triebel, and the multilevel partition of unity method. The central result is an alternative approach to the stability of quarkonial decompositions in Besov spaces , s > n(1/p − 1)+, which leads to relaxed decay assumptions on the elements of a quarkonial system as the monomial degree grows.

A globally convergent and locally quadratically convergent modified B-semismooth Newton method for $\ell_1$-penalized minimization

We consider the efficient minimization of a nonlinear, strictly convex functional with $\ell_1$-penalty term. Such minimization problems appear in a wide range of applications like Tikhonov regularization of (non)linear inverse problems with sparsity constraints. In (2015 Inverse Problems (31) 025005), a globalized Bouligand-semismooth Newton method was presented for $\ell_1$-Tikhonov regularization of linear inverse problems. Nevertheless, a technical assumption on the accumulation point of the sequence of iterates was necessary to prove global convergence. Here, we generalize this method to general nonlinear problems and present a modified semismooth Newton method for which global converg…

Application of clustering techniques to electron-diffraction data: determination of unit-cell parameters.

A new approach to determining the unit-cell vectors from single-crystal diffraction data based on clustering analysis is proposed. The method uses the density-based clustering algorithm DBSCAN. Unit-cell determination through the clustering procedure is particularly useful for limited tilt sequences and noisy data, and therefore is optimal for single-crystal electron-diffraction automated diffraction tomography (ADT) data. The unit-cell determination of various materials from ADT data as well as single-crystal X-ray data is demonstrated.