Games for eigenvalues of the Hessian and concave/convex envelopes

We study the PDE $\lambda_j(D^2 u) = 0$, in $\Omega$, with $u=g$, on $\partial \Omega$. Here $\lambda_1(D^2 u) \leq ... \leq \lambda_N (D^2 u)$ are the ordered eigenvalues of the Hessian $D^2 u$. First, we show a geometric interpretation of the viscosity solutions to the problem in terms of convex/concave envelopes over affine spaces of dimension $j$. In one of our main results, we give necessary and sufficient conditions on the domain so that the problem has a continuous solution for every continuous datum $g$. Next, we introduce a two-player zero-sum game whose values approximate solutions to this PDE problem. In addition, we show an asymptotic mean value characterization for the solution the the PDE

On Markov Chains with Uncertain Data

In this paper, a general method is described to determine uncertainty intervals for performance measures of Markov chains given an uncertainty region for the parameters of the Markov chains. We investigate the effects of uncertainties in the transition probabilities on the limiting distributions, on the state probabilities after n steps, on mean sojourn times in transient states, and on absorption probabilities for absorbing states. We show that the uncertainty effects can be calculated by solving linear programming problems in the case of interval uncertainty for the transition probabilities, and by second order cone optimization in the case of ellipsoidal uncertainty. Many examples are given, especially Markovian queueing examples, to illustrate the theory.Markov chain;Interval uncertainty;Ellipsoidal uncertainty;Linear Programming;Second Order Cone Optimization

The evolution problem associated with eigenvalues of the Hessian

In this paper we study the evolution problem $$\left\lbrace\begin{array}{ll} u_t (x,t)- \lambda_j(D^2 u(x,t)) = 0, & \text{in } \Omega\times (0,+\infty), \\ u(x,t) = g(x,t), & \text{on } \partial \Omega \times (0,+\infty), \\ u(x,0) = u_0(x), & \text{in } \Omega, \end{array}\right.$$ where $\Omega$ is a bounded domain in $\mathbb{R}^N$ (that verifies a suitable geometric condition on its boundary) and $\lambda_j(D^2 u)$ stands for the $j-$st eigenvalue of the Hessian matrix $D^2u$. We assume that $u_0$ and $g$ are continuous functions with the compatibility condition $u_0(x) = g(x,0)$, $x\in \partial \Omega$. We show that the (unique) solution to this problem exists in the viscosity sense and can be approximated by the value function of a two-player zero-sum game as the parameter measuring the size of the step that we move in each round of the game goes to zero. In addition, when the boundary datum is independent of time, $g(x,t) =g(x)$, we show that viscosity solutions to this evolution problem stabilize and converge exponentially fast to the unique stationary solution as $t\to \infty$. For $j=1$ the limit profile is just the convex envelope inside $\Omega$ of the boundary datum $g$, while for $j=N$ it is the concave envelope. We obtain this result with two different techniques: with PDE tools and and with game theoretical arguments. Moreover, in some special cases (for affine boundary data) we can show that solutions coincide with the stationary solution in finite time (that depends only on $\Omega$ and not on the initial condition $u_0$)

Fuzzy Logic Application for Optimization of the Cooling Towers Control System

The control system for the SPS-BA6 cooling towers station is considered in order to introduce the concept of a multivariable process. Multivariable control means the maintenace of several controlled variables at independent set points. In a single-variable system, to keep the single process variables within their critical values is considered a rather simple operation. In a complex multivariable system, the determination of the optimal operation point results in a combination of all set values of the variables. Control of a multivariable system requires therefore a more complex analysis. As the solution based on a mathematical model of the process is far beyond acceptable complexity, most mathematical models involve extensive simplifications and linearizations to optimize the resulting controllers. In this report the author will demonstrate how fuzzy logic might provide elegant and efficient solutions in the design of multivariable control based on experimental results rather than on mathematical models

Compressive Pattern Matching on Multispectral Data

We introduce a new constrained minimization problem that performs template and pattern detection on a multispectral image in a compressive sensing context. We use an original minimization problem from Guo and Osher that uses $L_1$ minimization techniques to perform template detection in a multispectral image. We first adapt this minimization problem to work with compressive sensing data. Then we extend it to perform pattern detection using a formal transform called the spectralization along a pattern. That extension brings out the problem of measurement reconstruction. We introduce shifted measurements that allow us to reconstruct all the measurement with a small overhead and we give an optimality constraint for simple patterns. We present numerical results showing the performances of the original minimization problem and the compressed ones with different measurement rates and applied on remotely sensed data.Comment: Published in IEEE Transactions on Geoscience and Remote Sensin

BaBar simulation production - A millennium of work in under a year

The BaBar experiment requires simulated events beyond the ability of a single computing site to provide. This paper describes the evolution of simulation and job management methods to meet the physics community requirements and how production became distributed to use resources beyond any one computing center. The evolution of BaBar simulation along with the development of the distribution of the computing effort is described. As the computing effort is distributed to more sites there is a need to simplify production so the effort does not multiply with number of production centers. Tools are created to be flexible in handling errors and failures that happen in the system and respond accordingly, this reduces failure rates and production effort. This paper will focus on one cycle of simulation production within BaBar as a description of a large scale computing effort which was fully performed, and provided new simulation data to the users on time