950 research outputs found
On a selection principle for multivalued semiclassical flows
We study the semiclassical behaviour of solutions of a Schr ̈odinger equation with a scalar po- tential displaying a conical singularity. When a pure state interacts strongly with the singularity of the flow, there are several possible classical evolutions, and it is not known whether the semiclassical limit cor- responds to one of them. Based on recent results, we propose that one of the classical evolutions captures the semiclassical dynamics; moreover, we propose a selection principle for the straightforward calculation of the regularized semiclassical asymptotics. We proceed to investigate numerically the validity of the proposed scheme, by employing a solver based on a posteriori error control for the Schr ̈odinger equation. Thus, for the problems we study, we generate rigorous upper bounds for the error in our asymptotic approximation. For 1-dimensional problems without interference, we obtain compelling agreement between the regularized asymptotics and the full solution. In problems with interference, there is a quantum effect that seems to survive in the classical limit. We discuss the scope of applicability of the proposed regularization approach, and formulate a precise conjecture
Adaptive Reconstruction for Electrical Impedance Tomography with a Piecewise Constant Conductivity
In this work we propose and analyze a numerical method for electrical
impedance tomography of recovering a piecewise constant conductivity from
boundary voltage measurements. It is based on standard Tikhonov regularization
with a Modica-Mortola penalty functional and adaptive mesh refinement using
suitable a posteriori error estimators of residual type that involve the state,
adjoint and variational inequality in the necessary optimality condition and a
separate marking strategy. We prove the convergence of the adaptive algorithm
in the following sense: the sequence of discrete solutions contains a
subsequence convergent to a solution of the continuous necessary optimality
system. Several numerical examples are presented to illustrate the convergence
behavior of the algorithm.Comment: 26 pages, 12 figure
A Posteriori Error Control for the Binary Mumford-Shah Model
The binary Mumford-Shah model is a widespread tool for image segmentation and
can be considered as a basic model in shape optimization with a broad range of
applications in computer vision, ranging from basic segmentation and labeling
to object reconstruction. This paper presents robust a posteriori error
estimates for a natural error quantity, namely the area of the non properly
segmented region. To this end, a suitable strictly convex and non-constrained
relaxation of the originally non-convex functional is investigated and Repin's
functional approach for a posteriori error estimation is used to control the
numerical error for the relaxed problem in the -norm. In combination with
a suitable cut out argument, a fully practical estimate for the area mismatch
is derived. This estimate is incorporated in an adaptive meshing strategy. Two
different adaptive primal-dual finite element schemes, and the most frequently
used finite difference discretization are investigated and compared. Numerical
experiments show qualitative and quantitative properties of the estimates and
demonstrate their usefulness in practical applications.Comment: 18 pages, 7 figures, 1 tabl
Singular solutions, graded meshes, and adaptivity for total-variation regularized minimization problems
Recent quasi-optimal error estimates for the finite element approximation of
total-variation regularized minimization problems require the existence of a
Lipschitz continuous dual solution. We discuss the validity of this condition
and devise numerical methods using locally refined meshes that lead to improved
convergence rates despite the occurrence of discontinuities. It turns out that
nearly linear convergence is possible on suitably constructed meshes
Unconditional stability of semi-implicit discretizations of singular flows
A popular and efficient discretization of evolutions involving the singular
-Laplace operator is based on a factorization of the differential operator
into a linear part which is treated implicitly and a regularized singular
factor which is treated explicitly. It is shown that an unconditional energy
stability property for this semi-implicit time stepping strategy holds. Related
error estimates depend critically on a required regularization parameter.
Numerical experiments reveal reduced experimental convergence rates for smaller
regularization parameters and thereby confirm that this dependence cannot be
avoided in general.Comment: 21 pages, 8 figure
Regularized semiclassical limits:linear flows with infinite Lyapunov exponents
Semiclassical asymptotics for linear Schr\"odinger equations with non-smooth
potentials give rise to ill-posed formal semiclassical limits. These problems
have attracted a lot of attention in the last few years, as a proxy for the
treatment of eigenvalue crossings, i.e. general systems. It has recently been
shown that the semiclassical limit for conical singularities is in fact
well-posed, as long as the Wigner measure (WM) stays away from singular saddle
points. In this work we develop a family of refined semiclassical estimates,
and use them to derive regularized transport equations for saddle points with
infinite Lyapunov exponents, extending the aforementioned recent results. In
the process we answer a related question posed by P. L. Lions and T. Paul in
1993. If we consider more singular potentials, our rigorous estimates break
down. To investigate whether conical saddle points, such as , admit a
regularized transport asymptotic approximation, we employ a numerical solver
based on posteriori error control. Thus rigorous upper bounds for the
asymptotic error in concrete problems are generated. In particular, specific
phenomena which render invalid any regularized transport for are
identified and quantified. In that sense our rigorous results are sharp.
Finally, we use our findings to formulate a precise conjecture for the
condition under which conical saddle points admit a regularized transport
solution for the WM
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