Pablo Groisman

A symptotic behaviour for a numerical approximation of a parabolic problem with blowing up solutions

Abstract In this paper, we study the asymptotic behaviour of a semidiscrete numerical approximation for u t = u xx + u p in a bounded interval, (0,1), with Dirichlet boundary conditions. We focus in the behaviour of blowing up solutions. We find that the blow-up rate for the numerical scheme is the same as for the continuous problem. Also we find the blow-up set for the numerical approximations and prove that it is contained in a neighbourhood of the blow-up set of the continuous problem when the mesh parameter is small enough.

NUMERICAL BLOW-UP FOR A NONLINEAR PROBLEM WITH A NONLINEAR BOUNDARY CONDITION

In this paper we study numerical approximations for positive solutions of a nonlinear heat equation with a nonlinear boundary condition. We describe in terms of the nonlinearities when solutions of a semidiscretization in space exist globally in time and when they blow up in finite time. We also find the blow-up rates and the blow-up sets. In particular we prove that regional blow-up is not reproduced by the numerical scheme. However, in the appropriate variables we can reproduce the correct blow-up set when the mesh parameter goes to zero.

Totally Discrete Explicit and Semi-implicit Euler Methods for a Blow-up Problem in Several Space Dimensions

The equation ut=Δu+up with homogeneous Dirichlet boundary conditions has solutions with blow-up if p>1. An adaptive time-step procedure is given to reproduce the asymptotic behavior of the solutions in the numerical approximations. We prove that the numerical methods reproduce the blow-up cases, the blow-up rate and the blow-up time. We also localize the numerical blow-up set.

FULLY DISCRETE ADAPTIVE METHODS FOR A BLOW-UP PROBLEM

We present adaptive procedures in space and time for the numerical study of positive solutions to the following problem: with p,m>0. We describe how to perform adaptive methods in order to reproduce the exact asymptotic behavior (the blow-up rate and the blow-up set) of the continuous problem.

Fleming–Viot selects the minimal quasi-stationary distribution: The Galton–Watson case

Consider N particles moving independently, each one according to a subcritical continuous-time Galton-Watson process unless it hits 0, at which time it jumps instantaneously to the position of one of the other particles chosen uniformly at random. The resulting dynamics is called Fleming-Viot process. We show that for each N there exists a unique invariant measure for the Fleming-Viot process, and that its stationary empirical distribution converges, as N goes to infinity, to the minimal quasi-stationary distribution of the Galton-Watson process conditioned on non-extinction.

A robust clustering method for detection of abnormal situations in a process with multiple steady-state operation modes

Abstract Many classical multivariate statistical process monitoring (MSPM) techniques assume normal distribution of the data and independence of the samples. Very often, these assumptions do not hold for real industrial chemical processes, where multiple plant operating modes lead to multiple nominal operation regions. MSPM techniques that do not take account of this fact show increased false alarm and missing alarm rates. In this work, a simple fault detection tool based on a robust clustering technique is implemented to detect abnormal situations in an industrial installation with multiple operation modes. The tool is applied to three case studies: (i) a two-dimensional toy example, (ii) a realistic simulation usually used as a benchmark example, known as the Tennessee–Eastman Process, and (iii) real data from a methanol plant. The clustering technique on which the tool relies assumes that the observations come from multiple populations with a common covariance matrix (i.e., the same underlying physical relations). The clustering technique is also capable of coping with a certain percentage of outliers, thus avoiding the need of extensive preprocessing of the data. Moreover, improvements in detection capacity are found when comparing the results to those obtained with standard methodologies. Hence, the feasibility of implementing fault detection tools based on this technique in the field of chemical industrial processes is discussed.

Numerical Analysis of Stochastic Differential Equations with Explosions

Abstract Stochastic ordinary differential equations may have solutions that explode in finite or infinite time. In this article we design an adaptive numerical scheme that reproduces the explosive behavior. The time step is adapted according to the size of the computed solution in such a way that, under adequate hypotheses, the explosion of the solutions is reproduced.

On the Dependence of the Blow-Up Time with Respect to the Initial Data in a Semilinear Parabolic Problem

Abstract We find a bound for the modulus of continuity of the blow-up time for the semilinear parabolic problem , with respect to the initial data.

Time-space white noise eliminates global solutions in reaction-diffusion equations

Abstract We prove that perturbing the reaction–diffusion equation u t = u x x + ( u + ) p ( p > 1 ), with time–space white noise produces that solutions explodes with probability one for every initial datum, opposite to the deterministic model where a positive stationary solution exists.

Nonparametric likelihood based estimation for a multivariate Lipschitz density

We consider a problem of nonparametric density estimation under shape restrictions. We deal with the case where the density belongs to a class of Lipschitz functions. Devroye [L. Devroye, A Course in Density Estimation, in: Progress in Probability and Statistics, vol. 14, Birkhauser Boston Inc., Boston, MA, 1987] considered these classes of estimates as tailor-made estimates, in contrast in some way to universally consistent estimates. In our framework we get the existence and uniqueness of the maximum likelihood estimate as well as strong consistency. This NPMLE can be easily characterized but it is not easy to compute. Some simpler approximations are also considered.