Discrete Regularization Research Articles

Relations between SVD and GSVD of Discrete regularization problems in standard and general form

We consider an algorithm, due to Eldén, that transforms general discrete regularization problems into standard form. We derive a simple relationship between the SVD associated with the standard-form problem and the GSVD associated with the general problem, and we discuss the accuracy of the GSVD computed via this transformation.

Particle scattering at the Planck scale and the Aichelburg-Sexl geometry

At energies of the order or larger than the Planck mass, the curved space-time geometry created by the particles dominates their collision process. The so-called Aichelburg-Sexl (AS) metric is relevant in this problem. We study the exact quantum particle scattering by this geometry in D space-time dimensions. The Klein-Gordon equation is ill defined and a regularization procedure is needed to give a sense to it (this does not happen for string equations). We find the troublesome result that the exact solution depends on the regularization chosen. Continuous regularization yields a scattering phase shift [ ϕ cont] that agrees with that found by 't Hooft in D=4. Discrete (lattice) regularization yields in the continuum limit a phase-shift ϕ = 2arctang[ ϕ cont]. (Actually, the same lattice results hold for the one-dimensional Dirac equation with a Dirac-δ potential and for fermionic two-dimensional field-theoretical models.) In the AS metric, we find that the scattering amplitude [ f lattice], obtained from lattice regularization, exhibits cuts in both s and t variables, and can be expressed as a coherent superposition of relativistic coulombian amplitudes [ f cont obtained from continuous regularization exhibits poles]. For small s, both f cont and f lattice= iGs( πt) −1 (i.e. the one-graviton exchange amplitude) but for small t, f lattice =i[Gsπt ln 2 (− 1 4 t)]−1 (f cont =iGs(πt) −1 for both small s and t). We compute and analyze the partial-wave amplitudes. We also compare both amplitudes in the intermediate t region and discuss their connection with the eikonal (and improved eikonal) approximation.

One-dimensional regularization with discontinuities

Regularization is equivalent to fitting smoothing splines to the data so that efficient and reliable numerical algorithms exist for finding solutions. however, the results exhibit poor performance along edges and boundaries. To cope with such anomalies, a more general class of smoothing splines that preserve corners and discontinuities is studied. Cubic splines are investigated in detail, since they are easy to implement and produce smooth curves near all data points except those marked as discontinuities or creases. A discrete regularization method is introduced to locate corners and discontinuities in the data points before the continuous regularization is applied.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Identification of spatially varying parameters in distributed parameter systems by discrete regularization

Identification of spatially varying parameters in distributed parameter systems from noisy data is an ill-posed problem. The regularization identification approach, developed by C. Kravaris and J. H. Seinfeld [ SIAM J. Control Optim. 23 (1985), 217–241] provides stable approximate solutions to that problem. In this work, a discretized minimization of the smoothing functional is proposed by using finite-dimensional convergent approximations in Sobolev spaces. A convergence theorem for the discretized minimization of the smoothing functional is established. The performance of this discrete regularization approach is evaluated by numerical experiments on the identification of spatially varying diffusivity in the two-dimensional diffusion equation.

A three-dimensional quadratic nonconforming element

We define a second-degree nonconforming element on tetrahedra. We build a basis for the opproximation space derived from this element. We prove a discrete regularity property similar to the one that holds for the corresponding two-dimensional element.