Past

Self-interacting diffusions are solutions to SDEs with a drift term depending

on the process and its normalized occupation measure $\mu_t$ (via an interaction

potential and a confinement potential): $$\mathrm{d}X_t = \mathrm{d}B_t -\left(

\nabla V(X_t)+ \nabla W*{\mu_t}(X_t) \right) \mathrm{d}t ; \mathrm{d}\mu_t = (\delta_{X_t}

- \mu_t)\frac{\mathrm{d}t}{r+t}; X_0 = x,\,\ \mu_0=\mu$$ where $(\mu_t)$ is the

process defined by $$\mu_t := \frac{r\mu + \int_0^t \delta_{X_s}\mathrm{d}s}{r+t}.$$

We establish a relation between the asymptotic behaviour of $\mu_t$ and the

asymptotic behaviour of a deterministic dynamical flow (defined on the space of

the Borel probability measures). We will also give some sufficient conditions

for the convergence of $\mu_t$. Finally, we will illustrate our study with an

example in the case $d=2$.

A wide variety of problems arising in applications require the sampling of a

probability measure on the space of functions. Examples from econometrics,

signal processing, molecular dynamics and data assimilation will be given.

In this situation it is of interest to understand the computational

complexity of MCMC methods for sampling the desired probability measure. We

overview recent results of this type, highlighting the importance of measures

which are absolutely continuous with respect to a Guassian measure.

We consider two-stage inflationary models in which a superheavy scale F-term hybrid inflation is followed by an intermediate scale modular inflation. We confront these models with the restrictions on the power spectrum of density perturbations P_R and the spectral index n_s from the recent data within the power-law cosmological model with cold dark matter and a cosmological constant. We show that these restrictions can be met provided that the number of e-foldings N_HI* of the pivot scale k*=0.002/Mpc during hybrid inflation is appropriately restricted. The additional e-foldings required for solving the horizon and flatness problems can be naturally generated by the subsequent modular inflation realized by a string axion.
   

Many recent algorithmic approaches involve the construction of a differential equation model for computational purposes, typically by introducing an artificial time variable. The actual computational model involves a discretization of the now time-dependent differential system, usually employing forward Euler. The resulting dynamics of such an algorithm is then a discrete dynamics, and it is expected to be ''close enough'' to the dynamics of the continuous system (which is typically easier to analyze) provided that small -- hence many -- time steps, or iterations, are taken. Indeed, recent papers in inverse problems and image processing routinely report results requiring thousands of iterations to converge. This makes one wonder if and how the computational modeling process can be improved to better reflect the actual properties sought.

In this talk we elaborate on several problem instances that illustrate the above observations. Algorithms may often lend themselves to a dual interpretation, in terms of a simply discretized differential equation with artificial time and in terms of a simple optimization algorithm; such a dual interpretation can be advantageous. We show how a broader computational modeling approach may possibly lead to algorithms with improved efficiency.