8,731 research outputs found
An L1 Penalty Method for General Obstacle Problems
We construct an efficient numerical scheme for solving obstacle problems in
divergence form. The numerical method is based on a reformulation of the
obstacle in terms of an L1-like penalty on the variational problem. The
reformulation is an exact regularizer in the sense that for large (but finite)
penalty parameter, we recover the exact solution. Our formulation is applied to
classical elliptic obstacle problems as well as some related free boundary
problems, for example the two-phase membrane problem and the Hele-Shaw model.
One advantage of the proposed method is that the free boundary inherent in the
obstacle problem arises naturally in our energy minimization without any need
for problem specific or complicated discretization. In addition, our scheme
also works for nonlinear variational inequalities arising from convex
minimization problems.Comment: 20 pages, 18 figure
Projection method for droplet dynamics on groove-textured surface with merging and splitting
The geometric motion of small droplets placed on an impermeable textured
substrate is mainly driven by the capillary effect, the competition among
surface tensions of three phases at the moving contact lines, and the
impermeable substrate obstacle. After introducing an infinite dimensional
manifold with an admissible tangent space on the boundary of the manifold, by
Onsager's principle for an obstacle problem, we derive the associated parabolic
variational inequalities. These variational inequalities can be used to
simulate the contact line dynamics with unavoidable merging and splitting of
droplets due to the impermeable obstacle. To efficiently solve the parabolic
variational inequality, we propose an unconditional stable explicit boundary
updating scheme coupled with a projection method. The explicit boundary
updating efficiently decouples the computation of the motion by mean curvature
of the capillary surface and the moving contact lines. Meanwhile, the
projection step efficiently splits the difficulties brought by the obstacle and
the motion by mean curvature of the capillary surface. Furthermore, we prove
the unconditional stability of the scheme and present an accuracy check. The
convergence of the proposed scheme is also proved using a nonlinear
Trotter-Kato's product formula under the pinning contact line assumption. After
incorporating the phase transition information at splitting points, several
challenging examples including splitting and merging of droplets are
demonstrated.Comment: 26 page
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
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
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