88,221 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
Compatible finite element spaces for geophysical fluid dynamics
Compatible finite elements provide a framework for preserving important structures in equations of geophysical uid dynamics, and are becoming important in their use for building atmosphere and ocean models. We survey the application of compatible finite element spaces to geophysical uid dynamics, including the application to the nonlinear rotating shallow water equations, and the three-dimensional compressible Euler equations. We summarise analytic results about dispersion relations and conservation properties, and present new results on approximation properties in three dimensions on the sphere, and on hydrostatic balance properties
Neural and spectral operator surrogates: unified construction and expression rate bounds
Approximation rates are analyzed for deep surrogates of maps between
infinite-dimensional function spaces, arising e.g. as data-to-solution maps of
linear and nonlinear partial differential equations. Specifically, we study
approximation rates for Deep Neural Operator and Generalized Polynomial Chaos
(gpc) Operator surrogates for nonlinear, holomorphic maps between
infinite-dimensional, separable Hilbert spaces. Operator in- and outputs from
function spaces are assumed to be parametrized by stable, affine representation
systems. Admissible representation systems comprise orthonormal bases, Riesz
bases or suitable tight frames of the spaces under consideration. Algebraic
expression rate bounds are established for both, deep neural and spectral
operator surrogates acting in scales of separable Hilbert spaces containing
domain and range of the map to be expressed, with finite Sobolev or Besov
regularity. We illustrate the abstract concepts by expression rate bounds for
the coefficient-to-solution map for a linear elliptic PDE on the torus
A Mean-field statistical theory for the nonlinear Schrodinger equation
A statistical model of self-organization in a generic class of
one-dimensional nonlinear Schrodinger (NLS) equations on a bounded interval is
developed. The main prediction of this model is that the statistically
preferred state for such equations consists of a deterministic coherent
structure coupled with fine-scale, random fluctuations, or radiation. The model
is derived from equilibrium statistical mechanics by using a mean-field
approximation of the conserved Hamiltonian and particle number for
finite-dimensional spectral truncations of the NLS dynamics. The continuum
limits of these approximated statistical equilibrium ensembles on
finite-dimensional phase spaces are analyzed, holding the energy and particle
number at fixed, finite values. The analysis shows that the coherent structure
minimizes total energy for a given value of particle number and hence is a
solution to the NLS ground state equation, and that the remaining energy
resides in Gaussian fluctuations equipartitioned over wavenumbers. Some results
of direct numerical integration of the NLS equation are included to validate
empirically these properties of the most probable states for the statistical
model. Moreover, a theoretical justification of the mean-field approximation is
given, in which the approximate ensembles are shown to concentrate on the
associated microcanonical ensemble in the continuum limit.Comment: 24 pages, 2 figure
Finite dimensional approximation of a class of constrained nonlinear optimal control problems
An abstract framework for the analysis and approximation of a class of nonlinear optimal control and optimization problems is constructed. Nonlinearities occur in both the objective functional and in the constraints. The framework includes an abstract nonlinear optimization problem posed on infinite dimensional spaces, and approximate problem posed on finite dimensional spaces, together with a number of hypotheses concerning the two problems. The framework is used to show that optimal solutions exist, to show that Lagrange multipliers may be used to enforce the constraints, to derive an optimality system from which optimal states and controls may be deduced, and to derive existence results and error estimates for solutions of the approximate problem. The abstract framework and the results derived from that framework are then applied to three concrete control or optimization problems and their approximation by finite element methods. The first involves the von Karman plate equations of nonlinear elasticity, the second, the Ginzburg-Landau equations of superconductivity, and the third, the Navier-Stokes equations for incompressible, viscous flows
Identification of nonlinear coefficient in a transport equation
Considered a problem of identification a nonlinear coefficient in a first order PDE via final observation. The problem is stated as an optimal control problem and solved numerically. Implicit finite difference scheme is used for the approximation of the state equation. A space of control variables is approximated by a sequence of finite-dimensional spaces with increaing dimensions. Finite dimensional problems are solved by a gradient method and numerical results are presented
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