993 research outputs found
Numerical identification of a variable parameter in 2d elliptic boundary value problem by extragradient methods
This work focuses on the inverse problem of identifying a variable parameter in a 2-D scalar elliptic boundary value problem. It is well-known that this inverse problem is highly ill-posed and regularization is necessary for its stable solution. The inverse problem is studied in an optimization framework, which is the most suitable framework for incorporating regularization. This optimization problem is a constrained optimization problem where the constraint set is a closed and convex set of the admissible coefficients. As an objective functional, we use both the output least squares and modified output least squares functionals. It is known that the most commonly used iterative schemes for such problems require strong monotonicity of the objective functionals derivative. In the context of the considered inverse problem, this is a very stringent requirement and is achieved through a careful selection of the regularization parameter. In contrast, extragradient type methods only require the derivative of the objective functional to be monotone and this allows a greater flexibility for the selection of the regularization parameter. In this work, we use the finite element method for the discretization of the inverse problem and apply the most commonly studied extragradient methods
Iterative Methods for the Elasticity Imaging Inverse Problem
Cancers of the soft tissue reign among the deadliest diseases throughout the world and effective treatments for such cancers rely on early and accurate detection of tumors within the interior of the body. One such diagnostic tool, known as elasticity imaging or elastography, uses measurements of tissue displacement to reconstruct the variable elasticity between healthy and unhealthy tissue inside the body. This gives rise to a challenging parameter identification inverse problem, that of identifying the LamĂ© parameter ÎŒ in a system of partial differential equations in linear elasticity. Due to the near incompressibility of human tissue, however, common techniques for solving the direct and inverse problems are rendered ineffective due to a phenomenon known as the âlocking effectâ. Alternative methods, such as mixed finite element methods, must be applied to overcome this complication. Using these methods, this work reposes the problem as a generalized saddle point problem along with a presentation of several optimization formulations, including the modified output least squares (MOLS), energy output least squares (EOLS), and equation error (EE) frameworks, for solving the elasticity imaging inverse problem. Subsequently, numerous iterative optimization methods, including gradient, extragradient, and proximal point methods, are explored and applied to solve the related optimization problem. Implementations of all of the iterative techniques under consideration are applied to all of the developed optimization frameworks using a representative numerical example in elasticity imaging. A thorough analysis and comparison of the methods is subsequently presented
Quantum Aspects of GMS Solutions of Noncommutative Field Theory and Large N Limit of Matrix Models
We investigate quantum aspects of Gopakumar-Minwalla-Strominger (GMS)
solutions of noncommutative field theory (NCFT) at large noncommutativity
limit, . Building upon a quantitative map between operator
formulation of 2-(respectively, (2+1))-dimensional NCFTs and large matrix
models of (respectively, ) noncritical strings, we show that GMS
solutions are quantum mechanically sensible only if we make appropriate joint
scaling of and . For 't Hooft's planar scaling, GMS solutions are
replaced by large saddle-point solutions. GMS solutions are recovered from
saddle-point solutions at small 't Hooft coupling regime, but are destabilized
at large 'tHooft coupling regime by quantum effects. We make comparisons
between these large effects and recently studied infrared effects in NCFTs.
We estimate U(N) symmetry breaking gradient effects and argue that they are
suppressed only at small 't Hooft coupling regime.Comment: 39 pages, Latex, JHEP3.cls, 7 .eps figures v2. typos corrected,
references adde
First order algorithms in variational image processing
Variational methods in imaging are nowadays developing towards a quite
universal and flexible tool, allowing for highly successful approaches on tasks
like denoising, deblurring, inpainting, segmentation, super-resolution,
disparity, and optical flow estimation. The overall structure of such
approaches is of the form ; where the functional is a data fidelity term also
depending on some input data and measuring the deviation of from such
and is a regularization functional. Moreover is a (often linear)
forward operator modeling the dependence of data on an underlying image, and
is a positive regularization parameter. While is often
smooth and (strictly) convex, the current practice almost exclusively uses
nonsmooth regularization functionals. The majority of successful techniques is
using nonsmooth and convex functionals like the total variation and
generalizations thereof or -norms of coefficients arising from scalar
products with some frame system. The efficient solution of such variational
problems in imaging demands for appropriate algorithms. Taking into account the
specific structure as a sum of two very different terms to be minimized,
splitting algorithms are a quite canonical choice. Consequently this field has
revived the interest in techniques like operator splittings or augmented
Lagrangians. Here we shall provide an overview of methods currently developed
and recent results as well as some computational studies providing a comparison
of different methods and also illustrating their success in applications.Comment: 60 pages, 33 figure
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