1,164 research outputs found
Can coercive formulations lead to fast and accurate solution of the Helmholtz equation?
A new, coercive formulation of the Helmholtz equation was introduced in
[Moiola, Spence, SIAM Rev. 2014]. In this paper we investigate -version
Galerkin discretisations of this formulation, and the iterative solution of the
resulting linear systems. We find that the coercive formulation behaves
similarly to the standard formulation in terms of the pollution effect (i.e. to
maintain accuracy as , must decrease with at the same rate
as for the standard formulation). We prove -explicit bounds on the number of
GMRES iterations required to solve the linear system of the new formulation
when it is preconditioned with a prescribed symmetric positive-definite matrix.
Even though the number of iterations grows with , these are the first such
rigorous bounds on the number of GMRES iterations for a preconditioned
formulation of the Helmholtz equation, where the preconditioner is a symmetric
positive-definite matrix.Comment: 27 pages, 7 figure
Iteratively regularized Newton-type methods for general data misfit functionals and applications to Poisson data
We study Newton type methods for inverse problems described by nonlinear
operator equations in Banach spaces where the Newton equations
are regularized variationally using a general
data misfit functional and a convex regularization term. This generalizes the
well-known iteratively regularized Gauss-Newton method (IRGNM). We prove
convergence and convergence rates as the noise level tends to 0 both for an a
priori stopping rule and for a Lepski{\u\i}-type a posteriori stopping rule.
Our analysis includes previous order optimal convergence rate results for the
IRGNM as special cases. The main focus of this paper is on inverse problems
with Poisson data where the natural data misfit functional is given by the
Kullback-Leibler divergence. Two examples of such problems are discussed in
detail: an inverse obstacle scattering problem with amplitude data of the
far-field pattern and a phase retrieval problem. The performence of the
proposed method for these problems is illustrated in numerical examples
A Quasi-Variational Inequality Problem Arising in the Modeling of Growing Sandpiles
Existence of a solution to the quasi-variational inequality problem arising
in a model for sand surface evolution has been an open problem for a long time.
Another long-standing open problem concerns determining the dual variable, the
flux of sand pouring down the evolving sand surface, which is also of practical
interest in a variety of applications of this model. Previously, these problems
were solved for the special case in which the inequality is simply variational.
Here, we introduce a regularized mixed formulation involving both the primal
(sand surface) and dual (sand flux) variables. We derive, analyse and compare
two methods for the approximation, and numerical solution, of this mixed
problem. We prove subsequence convergence of both approximations, as the mesh
discretization parameters tend to zero; and hence prove existence of a solution
to this mixed model and the associated regularized quasi-variational inequality
problem. One of these numerical approximations, in which the flux is
approximated by the divergence-conforming lowest order Raviart-Thomas element,
leads to an efficient algorithm to compute not only the evolving pile surface,
but also the flux of pouring sand. Results of our numerical experiments confirm
the validity of the regularization employed.Comment: 51 p., low resolution fig
Optimal design of plane elastic membranes using the convexified F\"{o}ppl's model
This work puts forth a new optimal design formulation for planar elastic
membranes. The goal is to minimize the membrane's compliance through choosing
the material distribution described by a positive Radon measure. The
deformation of the membrane itself is governed by the convexified F\"{o}ppl's
model. The uniqueness of this model lies in the convexity of its variational
formulation despite the inherent nonlinearity of the strain-displacement
relation. It makes it possible to rewrite the optimization problem as a pair of
mutually dual convex variational problems. In the primal problem a linear
functional is maximized with respect to displacement functions while enforcing
that point-wisely the strain lies in an unbounded closed convex set. The dual
problem consists in finding equilibrated stresses that are to minimize a convex
integral functional of linear growth defined on the space of Radon measures.
The pair of problems is analysed: existence and regularity results are
provided, together with the system of optimality criteria. To demonstrate the
computational potential of the pair, a finite element scheme is developed
around it. Upon reformulation to a conic-quadratic & semi-definite programming
problem, the method is employed to produce numerical simulations for several
load case scenarios.Comment: 55 page
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