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We introduce and analyze nonsmooth Schur-Newton methods for a class of nonsmooth saddle point problems. The method is able to solve problems where the primal energy decomposes into a convex smooth part and a convex separable but nonsmooth part. The method is based on nonsmooth Newton techniques for an equivalent unconstrained dual problem. Using this we show that it is globally convergent even for inexact evaluation of the linear subproblems.
A theorem on error estimates for smooth nonlinear programming
problems in Banach spaces is proved that can be used to derive
optimal error estimates for optimal control problems. This theorem is applied
to a class of optimal control problems for quasilinear elliptic equations.
The state equation is approximated by a finite element scheme, while different
discretization methods are used for the control functions. The distance of
locally optimal controls to their discrete approximations is estimated.
Explicit Error Estimates for Courant, Crouzeix-Raviart and Raviart-Thomas Finite Element Methods
(2011)
The elementary analysis of this paper presents explicit expressions of the constants in the a priori error estimates for the lowest-order Courant, Crouzeix-Raviart nonconforming and Raviart-Thomas mixed finite element methods in the Poisson model problem. The three constants and their dependences on some maximal angle in the triangulation are indeed all comparable and allow accurate a priori error control.
This paper establishes the equivalence of conforming Courant finite element method and nonconforming Crouzeix-Raviart finite element method in the sense that the respective energy error norms are equivalent up to generic constants and higher-order data oscillations in a Poisson model problem. The Raviart-Thomas mixed finite element method is better than the previous two whereas the conjecture of the converse relation is proved to be false.
This paper completes the analysis of comparison initiated by Braess in Calcolo (2010). Two numerical benchmarks illustrate the comparison theorems and the possible strict superiority of the Raviart-Thomas mixed finite element method. Applications include least-squares finite element methods and equality of approximation classes for concepts of optimality for adaptive finite element methods.
Regularization and Numerical Solution of the Inverse Scattering Problem using Shearlet Frames
(2014)
Regularization techniques for the numerical solution of nonlinear inverse scattering
problems in two space dimensions are discussed. Assuming that the boundary of a scatterer is its most prominent feature, we exploit as model the class of cartoon-like functions.
Since functions in this class are asymptotically optimally sparsely approximated by shearlet frames, we consider shearlets as a means for the regularization in a Tikhonov method.
We examine both directly the nonlinear problem and a linearized problem obtained by
the Born approximation technique. As problem classes we study the acoustic inverse
scattering problem and the electromagnetic inverse scattering problem. We show that
this approach introduces a sparse regularization for the nonlinear setting and we present
a result describing the behavior of the local regularity of a scatterer under linearization,
which shows that the linearization does not affect the sparsity of the problem. The analytical results are illustrated by numerical examples for the acoustic inverse scattering problem that highlight the effectiveness of this approach.