3,944 research outputs found
Stabilised finite element methods for ill-posed problems with conditional stability
In this paper we discuss the adjoint stabilised finite element method
introduced in, E. Burman, Stabilized finite element methods for nonsymmetric,
noncoercive and ill-posed problems. Part I: elliptic equations, SIAM Journal on
Scientific Computing, and how it may be used for the computation of solutions
to problems for which the standard stability theory given by the Lax-Milgram
Lemma or the Babuska-Brezzi Theorem fails. We pay particular attention to
ill-posed problems that have some conditional stability property and prove
(conditional) error estimates in an abstract framework. As a model problem we
consider the elliptic Cauchy problem and provide a complete numerical analysis
for this case. Some numerical examples are given to illustrate the theory.Comment: Accepted in the proceedings from the EPSRC Durham Symposium Building
Bridges: Connections and Challenges in Modern Approaches to Numerical Partial
Differential Equation
From the viscous Cahn-Hilliard equation to a regularized forward-backward parabolic equation
A rigorous proof is given for the convergence of the solutions of a viscous
Cahn-Hilliard system to the solution of the regularized version of the
forward-backward parabolic equation, as the coefficient of the diffusive term
goes to 0. Non-homogenous Neumann boundary condition are handled for the
chemical potential and the subdifferential of a possible non-smooth double-well
functional is considered in the equation. An error estimate for the difference
of solutions is also proved in a suitable norm and with a specified rate of
convergence.Comment: Key words and phrases: Cahn-Hilliard system, forward-backward
parabolic equation, viscosity, initial-boundary value problem, asymptotic
analysis, well-posednes
Regularization strategy for inverse problem for 1+1 dimensional wave equation
An inverse boundary value problem for a 1+1 dimensional wave equation with
wave speed is considered. We give a regularisation strategy for
inverting the map where is the
hyperbolic Neumann-to-Dirichlet map corresponding to the wave speed . More
precisely, we consider the case when we are given a perturbation of the
Neumann-to-Dirichlet map , where corresponds to the measurement errors, and reconstruct an approximate wave
speed . We emphasize that may not not be in the
range of the map . We show that the reconstructed wave speed
satisfies . Our
regularization strategy is based on a new formula to compute from
Thermoacoustic tomography with an arbitrary elliptic operator
Thermoacoustic tomography is a term for the inverse problem of determining of
one of initial conditions of a hyperbolic equation from boundary measurements.
In the past publications both stability estimates and convergent numerical
methods for this problem were obtained only under some restrictive conditions
imposed on the principal part of the elliptic operator. In this paper
logarithmic stability estimates are obatined for an arbitrary variable
principal part of that operator. Convergence of the Quasi-Reversibility Method
to the exact solution is also established for this case. Both complete and
incomplete data collection cases are considered.Comment: 16 page
A finite element data assimilation method for the wave equation
We design a primal-dual stabilized finite element method for the numerical
approximation of a data assimilation problem subject to the acoustic wave
equation. For the forward problem, piecewise affine, continuous, finite element
functions are used for the approximation in space and backward differentiation
is used in time. Stabilizing terms are added on the discrete level. The design
of these terms is driven by numerical stability and the stability of the
continuous problem, with the objective of minimizing the computational error.
Error estimates are then derived that are optimal with respect to the
approximation properties of the numerical scheme and the stability properties
of the continuous problem. The effects of discretizing the (smooth) domain
boundary and other perturbations in data are included in the analysis.Comment: 23 page
A variational method for quantitative photoacoustic tomography with piecewise constant coefficients
In this article, we consider the inverse problem of determining spatially
heterogeneous absorption and diffusion coefficients from a single measurement
of the absorbed energy (in the steady-state diffusion approximation of light
transfer). This problem, which is central in quantitative photoacoustic
tomography, is in general ill-posed since it admits an infinite number of
solution pairs. We show that when the coefficients are known to be piecewise
constant functions, a unique solution can be obtained. For the numerical
determination of the coefficients, we suggest a variational method based based
on an Ambrosio-Tortorelli-approximation of a Mumford-Shah-like functional,
which we implemented numerically and tested on simulated two-dimensional data
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