337 research outputs found
Discretization of variational regularization in Banach spaces
Consider a nonlinear ill-posed operator equation where is
defined on a Banach space . In general, for solving this equation
numerically, a finite dimensional approximation of and an approximation of
are required. Moreover, in general the given data \yd of are noisy.
In this paper we analyze finite dimensional variational regularization, which
takes into account operator approximations and noisy data: We show
(semi-)convergence of the regularized solution of the finite dimensional
problems and establish convergence rates in terms of Bregman distances under
appropriate sourcewise representation of a solution of the equation. The more
involved case of regularization in nonseparable Banach spaces is discussed in
detail. In particular we consider the space of finite total variation
functions, the space of functions of finite bounded deformation, and the
--space
An Optimal Interpolation Set for Model-Based Derivative-Free Optimization Methods
This paper demonstrates the optimality of an interpolation set employed in
derivative-free trust-region methods. This set is optimal in the sense that it
minimizes the constant of well-poisedness in a ball centred at the starting
point. It is chosen as the default initial interpolation set by many
derivative-free trust-region methods based on underdetermined quadratic
interpolation, including NEWUOA, BOBYQA, LINCOA, and COBYQA. Our analysis
provides a theoretical justification for this choice
Local times for typical price paths and pathwise Tanaka formulas
Following a hedging based approach to model free financial mathematics, we
prove that it should be possible to make an arbitrarily large profit by
investing in those one-dimensional paths which do not possess local times. The
local time is constructed from discrete approximations, and it is shown that it
is -H\"older continuous for all . Additionally, we provide
various generalizations of F\"ollmer's pathwise It\^o formula
A piecewise linear FEM for an optimal control problem of fractional operators: error analysis on curved domains
We propose and analyze a new discretization technique for a linear-quadratic
optimal control problem involving the fractional powers of a symmetric and
uniformly elliptic second oder operator; control constraints are considered.
Since these fractional operators can be realized as the Dirichlet-to-Neumann
map for a nonuniformly elliptic equation, we recast our problem as a
nonuniformly elliptic optimal control problem. The rapid decay of the solution
to this problem suggests a truncation that is suitable for numerical
approximation. We propose a fully discrete scheme that is based on piecewise
linear functions on quasi-uniform meshes to approximate the optimal control and
first-degree tensor product functions on anisotropic meshes for the optimal
state variable. We provide an a priori error analysis that relies on derived
Holder and Sobolev regularity estimates for the optimal variables and error
estimates for an scheme that approximates fractional diffusion on curved
domains; the latter being an extension of previous available results. The
analysis is valid in any dimension. We conclude by presenting some numerical
experiments that validate the derived error estimates
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