Capítulos (Ecuaciones Diferenciales y Análisis Numérico)
https://hdl.handle.net/11441/10835
Tue, 17 May 2022 09:56:55 GMT2022-05-17T09:56:55ZLow rank approximation of multidimensional data
https://hdl.handle.net/11441/89456
Low rank approximation of multidimensional data
In the last decades, numerical simulation has experienced tremendous improvements driven by massive growth of computing power. Exascale computing has been achieved this year and will allow solving ever more complex problems. But such large systems produce colossal amounts of data which leads to its own difficulties. Moreover, many engineering problems such as multiphysics or optimisation and control, require far more power that any computer architecture could achieve within the current scientific computing paradigm. In this chapter, we propose to shift the paradigm in order to break the curse of dimensionality by introducing decomposition to reduced data. We present an extended review of data reduction techniques and intends to bridge between applied mathematics
community and the computational mechanics one. The chapter is organized into two parts. In the first one bivariate separation is studied, including discussions on the equivalence of proper orthogonal decomposition (POD, continuous framework) and singular value decomposition (SVD, discrete matrices). Then, in the second part, a wide review of tensor formats and their approximation is proposed. Such work has already been provided in
the literature but either on separate papers or into a pure applied
mathematics framework. Here, we offer to the data enthusiast scientist a description of Canonical, Tucker, Hierarchical and Tensor train formats including their approximation algorithms. When it is possible, a careful analysis of the link between continuous and discrete methods will be performed.
Tue, 01 Jan 2019 00:00:00 GMThttps://hdl.handle.net/11441/894562019-01-01T00:00:00ZA random model for immune response to virus in fluctuating environments
https://hdl.handle.net/11441/58710
A random model for immune response to virus in fluctuating environments
Sadovnichiy, Victor A.; Zgurovsky, Mikhail Z.
In this work we study a model for virus dynamics with a random immune
response and a random production rate of susceptible cells from cell proliferation. In traditional models for virus dynamics, the rate at which the viruses are cleared by the immune system is constant, and the rate at which susceptible cells are provided is constant or a function depending on the population of all cells. However, the human body in general is never stationary, and thus these rates can barely be constant. Here we assume that the human body is a random environment and models the rates by random processes, which result in a system of random differential equations. We then analyze the long term behavior of the random system, in particular the existence and geometric structure of the random attractor, by using the theory of random dynamical systems. Numerical simulations are provided to illustrate the theoretical result.
Fri, 01 Jan 2016 00:00:00 GMThttps://hdl.handle.net/11441/587102016-01-01T00:00:00ZSome aspects concerning the dynamics of stochastic chemostats
https://hdl.handle.net/11441/58688
Some aspects concerning the dynamics of stochastic chemostats
Sadovnichiy, Victor A.; Zgurovsky, Mikhail Z.
In this paper we study a simple chemostat model influenced by white
noise which makes this kind of models more realistic. We use the theory of random attractors and, to that end, we first perform a change of variable using the OrnsteinUhlenbeck process, transforming our stochastic model into a system of differential equations with random coefficients. After proving that this random system possesses a unique solution for any initial value, we analyze the existence of random attractors. Finally we illustrate our results with some numerical simulations.
Fri, 01 Jan 2016 00:00:00 GMThttps://hdl.handle.net/11441/586882016-01-01T00:00:00ZControl of weakly blowing up semilinear heat equations
https://hdl.handle.net/11441/55140
Control of weakly blowing up semilinear heat equations
Berestycki, Henri; Pomeau, Yves
In these notes we consider a semilinear heat equation in a bounded domain of IRd
, with
control on a subdomain and homogeneous Dirichlet boundary conditions. We consider nonlinearities
for which, in the absence of control, blow up arises.
We prove that when the nonlinearity grows at infinity fast enough, due to the local (in
space) nature of the blow up phenomena, the control may not avoid the blow up to occur
for suitable initial data. This is done by means of localized energy estimates.
However, we also show that when the nonlinearity is weak enough, and provided the
system admits a globally defined solution (for some initial data and control), the choice of a
suitable control guarantees the global existence of solutions and moreover that the solution
may be driven in any finite time to the globally defined solution. In order for this to be true
we require the nonlinearity f to satisfy at infinity the growth condition
f(s)
|s| log3/2
(1 + |s|)
→ 0 as |s| → ∞.
This is done by means of a fixed point argument and a careful analysis of the control of
linearized heat equations relying on global Carleman estimates. The problem of controlling the blow up in this sense remains open for nonlinearities growing at infinity like f(s) ∼
|s|logp
(1 + |s|) with 3/2 ≤ p ≤ 2.
Tue, 01 Jan 2002 00:00:00 GMThttps://hdl.handle.net/11441/551402002-01-01T00:00:00Z