Algorithms For Such Equations Research Articles

Classification of variational multiplicative fourth-order difference equations

We use a recently found method to characterize all multiplicative fourth-order difference equations admitting a variational structure. We use this result to give a simplified classification algorithm for such equations and a characterization of the associated symplectic structures. Finally, we discuss several examples of equations of this family from the literature and propose new ones.

Finite difference schemes for second order systems describing black holes

In the harmonic description of general relativity, the principal part of Einstein's equations reduces to 10 curved space wave equations for the components of the space-time metric. We present theorems regarding the stability of several evolution-boundary algorithms for such equations when treated in second order differential form. The theorems apply to a model black hole space-time consisting of a spacelike inner boundary excising the singularity, a timelike outer boundary and a horizon in between. These algorithms are implemented as stable, convergent numerical codes and their performance is compared in a 2-dimensional excision problem.

Some relations among RKHS norms, Fredholm equations, and innovations representations

We first show how reproducing kernel Hilbert space (RKHS) norms can be determined for a large class of covariance functions by methods based on the solution of a Riccati differential equation or a Wiener-Hopf integral equation. Efficient numerical algorithms for such equations have been extensively studied, especially in the control literature. The innovations representations enter in that it is they that suggest the form of the RKHS norms. From the RKHS norms, we show how recursive solutions can be obtained for certain Fredholm equations of the first kind that are widely used in certain approaches to detection theory. Our approach specifies a unique solution: moreover, the algorithms used are well suited to the treatment of increasing observation intervals.