In this talk we propose a probabilistic interpretation of the parameters in the system of differential equations describing a complex cellular brain metabolism model. Uncertainty in the parameter values, variability of the data over a population and errors in the collected data contribute to the variance of the distributions of the parameters. Markov chain Monte Carlo sampling schemes are employed to draw parameter sets which identify models in statistical agreement with the available data and with the a priori belief about the system. The ensemble of solutions of the differential equations corresponding to the different parameter sets provides a measure of how uncertainty in the parameters translates into variability of the predictive output of the model.

We compute the expectation and the two-point correlation of the solution to elliptic boundary value problems with stochastic input data. Besides stochastic loadings, via perturbation theory, our approach covers also elliptic problems on stochastic domains or with stochastic coefficients. The solution's two-point correlation satisfies a deterministic boundary value problem with the two-fold tensor product operator on the two-fold tensor tensor product domain. We discuss the efficient solution of such tensor product problems by either a sparse grid approach based on multilevel frames or by the pivoted Cholesky decomposition. Both approaches involve only standard finite element techniques. Numerical results illustrate the algorithms.

1. Review on convergence of sequences of sets and functions in the deterministic case.
   
   Painleve-Kuratowski's Convergence, epi-convergence, variational properties of epi-convergence.
   Convex Analysis : conjugate of an extended real-valued function, epi-sum (alias inf-convolution)...



2. Convergence of sequences of sets and functions in a stochastic context
   
   Random sets and random functions : denition, notion of equi-distribution and independence, set-valued integral.
   Strong laws of large numbers, Birkhoś Ergodic Theorem.
   Conditional expectation and martingales of random sets and random functions, almost sure convergence.
   Set-valued versions of Fatou's Lemma.



3. Application to the approximation of optimization problems
   
   Convergence of discrete epi-sums to continuous epi-sum.
   Almost sure convergence of estimators.
   Convergence of integral functionals.

• inadequacies in the model,
• measurement errors (which includes the measurements in the quantities used as the model's inputs, and in the quantities it is trying to predict.)

Combining recent results on numerical integration and optimization, we derive a polynomial bound on the worst case complexity of a class of static stochastic optimization problems. We then describe a technique for reducing dynamic problems to static ones. The reduction technique is only a heuristic but it can effectively employ good guesses for good solutions. This is illustrated on an 82-period problem coming from pension insurance industry.