Interpolation by piecewise-linear radial basis functions, I

AbstractIn the two-dimensional plane, a set of points x1, x2, …, xn (called “nodes”) is given. It is desired to interpolate arbitrary data given on the nodes by continuous functions having piecewise-linear (“PL”) structure. For this purpose, one can employ the space of all PL-functions on a rectangular grid generated by the nodes. We study this space first. Next, we investigate the special PL-functions that are linear combinations of functions hi(x) = ∥x − xi∥1, in which the l1-norm on R2 is employed. The “dual” case, involving the two-dimensional l∞-norm, is included in our results, as are certain general interpolating functions of the form (s, t) ↦ F(s − si) + G(t − ti)

Nonparametric Markovian Learning of Triggering Kernels for Mutually Exciting and Mutually Inhibiting Multivariate Hawkes Processes

In this paper, we address the problem of fitting multivariate Hawkes processes to potentially large-scale data in a setting where series of events are not only mutually-exciting but can also exhibit inhibitive patterns. We focus on nonparametric learning and propose a novel algorithm called MEMIP (Markovian Estimation of Mutually Interacting Processes) that makes use of polynomial approximation theory and self-concordant analysis in order to learn both triggering kernels and base intensities of events. Moreover, considering that N historical observations are available, the algorithm performs log-likelihood maximization in $O(N)$ operations, while the complexity of non-Markovian methods is in $O(N^{2})$. Numerical experiments on simulated data, as well as real-world data, show that our method enjoys improved prediction performance when compared to state-of-the art methods like MMEL and exponential kernels

Solving variational inequalities defined on a domain with infinitely many linear constraints

We study a variational inequality problem whose domain is defined by infinitely many linear inequalities. A discretization method and an analytic center based inexact cutting plane method are proposed. Under proper assumptions, the convergence results for both methods are given. We also provide numerical examples to illustrate the proposed method