Ian Tobasco

A dynamic programming approach to the Parisi functional

G.Parisi predicted an important variational formula for the thermodynamic limit of the intensive free energy for a class of mean field spin glasses. In this paper, we present an elementary approach to the study of the Parisi functional using stochastic dynamic programing and semi-linear PDE. We give a derivation of important properties of the Parisi PDE avoiding the use of Ruelle Probability Cascades and Cole-Hopf transformations. As an application, we give a simple proof of the strict convexity of the Parisi functional, which was recently proved by Auffinger and Chen in [2].

Some properties of the phase diagram for mixed p-spin glasses

In this paper we study the Parisi variational problem for mixed p-spin glasses with Ising spins. Our starting point is a characterization of Parisi measures whose origin lies in the first order optimality conditions for the Parisi functional, which is known to be strictly convex. Using this characterization, we study the phase diagram in the temperature-external field plane. We begin by deriving self-consistency conditions for Parisi measures that generalize those of de Almeida and Thouless to all levels of Replica Symmetry Breaking (RSB) and all models. As a consequence, we conjecture that for all models the Replica Symmetric phase is the region determined by the natural analogue of the de Almeida–Thouless condition. We show that for all models, the complement of this region is in the RSB phase. Furthermore, we show that the conjectured phase boundary is exactly the phase boundary in the plane less a bounded set. In the case of the Sherrington–Kirkpatrick model, we extend this last result to show that this bounded set does not contain the critical point at zero external field.

Optimal bounds and extremal trajectories for time averages in nonlinear dynamical systems

Abstract For any quantity of interest in a system governed by ordinary differential equations, it is natural to seek the largest (or smallest) long-time average among solution trajectories, as well as the extremal trajectories themselves. Upper bounds on time averages can be proved a priori using auxiliary functions, the optimal choice of which is a convex optimization problem. We prove that the problems of finding maximal trajectories and minimal auxiliary functions are strongly dual. Thus, auxiliary functions provide arbitrarily sharp upper bounds on time averages. Moreover, any nearly minimal auxiliary function provides phase space volumes in which all nearly maximal trajectories are guaranteed to lie. For polynomial equations, auxiliary functions can be constructed by semidefinite programming, which we illustrate using the Lorenz system.

Navier-Stokes solver using Green's functions I: Channel flow and plane Couette flow

Numerical solvers of the incompressible Navier-Stokes equations have reproduced turbulence phenomena such as the law of the wall, the dependence of turbulence intensities on the Reynolds number, and experimentally observed properties of turbulence energy production. In this article, we begin a sequence of investigations whose eventual aim is to derive and implement numerical solvers that can reach higher Reynolds numbers than is currently possible. Every time step of a Navier-Stokes solver in effect solves a linear boundary value problem. The use of Greens functions leads to numerical solvers which are highly accurate in resolving the boundary layer, which is a source of delicate but exceedingly important physical effects at high Reynolds numbers. The use of Greens functions brings with it a need for careful quadrature rules and a reconsideration of time steppers. We derive and implement Greens function based solvers for the channel flow and plane Couette flow geometries. The solvers are validated by reproducing turbulence phenomena in good agreement with earlier simulations and experiment.

A Probabilistic Approach to Inverse Convection-Diffusion

Initial condition inverse problems are ill-posed and computationally expensive to solve. We present a computational approach for solving inverse problems in the realm of onedimensional contaminant transport. The approach employs finite differencing as a forward solver and probabilistic methods for inversion. Markov Chain Monte Carlo sampling is used to efficiently recover posterior probabilities. The results show that the Bayesian framework is a robust approach for initial condition inversion. I. Introduction Real-time modeling of accidental or intentional contaminant spread in urban settings is critical to homeland security and environmental safety. Knowing with good confidence where and when to evacuate prior to loss of health or even loss of life is of the utmost importance to decision making organizations. However, before such organizations can take action, knowledge must be obtained concerning both the origin and the evolution of the pollutants. The associated computational problem is inherently ill-posed, as many different inputs can give the same or very similar outputs. This problem is also computationally expensive: just modeling a forward dispersion problem in a realistic urban environment can take several hours using stateof-the-art computational fluid dynamics, and the proposed inverse problem is significantly more expensive. This paper presents a computational framework for probabilistic inversion, in which solutions are given as probability distributions for initial conditions. This paper first presents the theory behind our approach. One-dimensional contaminant transport physics is discussed briefly and finite difference schemes used to numerically solve the linear convection-diffusion equation are detailed. The corresponding inverse problem is then posed in a Bayesian framework, and a Monte Carlo Markov Chain (MCMC) algorithm for performing efficient inversions is discussed. Results follow, in an effort to demonstrate the applicability of a probabilistic approach for inverting convectivediffusive processes.