Standard Regularization Techniques Research Articles

Dealing with the center and boundary problems in 1D numerical relativity

Instabilities in finite difference codes due to the singularity of spherical coordinates at the center are studied. In typical Numerical Relativity applications, standard regularization techniques by themselves do not ensure long term stability. A proposal to remedy that problem is presented, which takes advantage of redundant quantities introduced in recent hyperbolic formulations of Einstein's evolution equations. The results are discussed through the example case of a boson star, where a significant improvement in the implementation of boundary conditions is also presented.

Second-Order Approximation of the Pseudoinverse for Operator Deconvolutions and Families of Ill-Posed Problems

We identify a more accurate means of deconvolving compact nondegenerate operators than that provided by independent application of standard regularization techniques to individual members of the implied continuum of first kind integral equations. For the problem defined by $$ h(x,t) = \int_Y f(x,y)g(y,t)dY, $$ where it is required to find g(y,t) given noise-corrupted versions of square-integrable f(x,y) and h(x,t), it is shown that greater accuracy results from optimizing solutions for each member of the sequence of integral equations derived from individual components of the singular value expansion of h(x,t), than from optimizing solutions to each of the equations determined by different fixed values of t. This result has particular implications for treatment of the bioelectric/biomagnetic source imaging problems, where the characteristic need to perform such a deconvolution has been previously handled by producing an individually regularized solution to the above equation for each t in a collection of ...

Fredholm integral equations on the Euclidean motion group

In this work, methods for the solution of Fredholm equations of the first kind with convolution kernel are presented, where all the functions in the integral equation are functions on the Euclidean motion group, and the convolution product is defined relative to the group operation. An application in which such equations arise is examined in detail. The properties of the Fourier transform of scalar-valued functions on the Euclidean motion group are reviewed and applied to an exactly solvable example. Standard regularization techniques are then adapted and illustrated for cases in which exact solutions are not possible.

An Inverse Convolution Method for Regular Parabolic Equations

In this paper, the problem of determining an unknown boundary control of a parabolic distributed parameter system, evolving over finite or infinite time, from incomplete, approximate interior temperature measurements is investigated. First, an exact solution is obtained for square integrable boundary controls associated with a general class of parabolic equations on a finite spatial interval under the assumption that the interior temperature is known for all times. Then, it is shown that this exact solution can be used to develop a stable algorithm for the numerical solution of this problem, without the introduction of standard regularization techniques. This algorithm assumes only knowledge of a finite, discrete set of approximate temperature readings. One advantage of this inversion process is the availability of a priori error bounds based on the measurement errors and frequency of sampling that are obtained in this paper. Another useful feature is that it encompasses boundary controls which are arbitrary linear combinations of surface temperature and flux. Numerical results are presented.

Biased anisotropic diffusion: a unified regularization and diffusion approach to edge detection

A global edge detection algorithm based on variational regularization is presented and analysed. The algorithm can also be viewed as an anisotropic diffusion method. These two quite different methods are thereby unified from the original outlook. This puts anisotropic diffusion, as a method in early vision, on more solid grounds; it is just as well founded as the well-accepted standard regularization techniques. The unification also brings the anisotropic diffusion method an appealing sense of optimality, thereby intuitively explaining its extraordinary performance. The algorithm to be presented, moreover, has the following attractive properties:o1. It only requires the solution of a single boundary value problem over the entire image domain — almost always a very simple (rectangular) region.2. It converges to the solution of interest. 1. It only requires the solution of a single boundary value problem over the entire image domain — almost always a very simple (rectangular) region. 2. It converges to the solution of interest. The first of these properties implies very significant advantages over other existing regularization methods; the computation cost is typically cut by an order of magnitude or more. The second property represents considerable advantages over the existing diffusion methods; it removes the problem of deciding when to stop, as well as that of actually stopping the diffusion process.

A Bidiagonalization-Regularization Procedure for Large Scale Discretizations of Ill-Posed Problems

In this paper, we consider ill-posed problems which discretize to linear least squares problems with matrices K of high dimensions. The algorithm proposed uses K only as an operator and does not need to explicitly store or modify it. A method related to one of Lanczos is used to project the problem onto a subspace for which K is bidiagonal. It is then an easy matter to solve the projected problem by standard regularization techniques. These ideas are illustrated with some integral equations of the first kind with convolution kernels, and sample numerical results are given.