Sequential design of computer experiments for the estimation of a probability of failure

This paper deals with the problem of estimating the volume of the excursion set of a function $f:\mathbb{R}^d \to \mathbb{R}$ above a given threshold, under a probability measure on $\mathbb{R}^d$ that is assumed to be known. In the industrial world, this corresponds to the problem of estimating a probability of failure of a system. When only an expensive-to-simulate model of the system is available, the budget for simulations is usually severely limited and therefore classical Monte Carlo methods ought to be avoided. One of the main contributions of this article is to derive SUR (stepwise uncertainty reduction) strategies from a Bayesian-theoretic formulation of the problem of estimating a probability of failure. These sequential strategies use a Gaussian process model of $f$ and aim at performing evaluations of $f$ as efficiently as possible to infer the value of the probability of failure. We compare these strategies to other strategies also based on a Gaussian process model for estimating a probability of failure.Comment: This is an author-generated postprint version. The published version is available at http://www.springerlink.co

Homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions

A microscopic heterogeneous system under random influence is considered. The randomness enters the system at physical boundary of small scale obstacles as well as at the interior of the physical medium. This system is modeled by a stochastic partial differential equation defined on a domain perforated with small holes (obstacles or heterogeneities), together with random dynamical boundary conditions on the boundaries of these small holes. A homogenized macroscopic model for this microscopic heterogeneous stochastic system is derived. This homogenized effective model is a new stochastic partial differential equation defined on a unified domain without small holes, with static boundary condition only. In fact, the random dynamical boundary conditions are homogenized out, but the impact of random forces on the small holes' boundaries is quantified as an extra stochastic term in the homogenized stochastic partial differential equation. Moreover, the validity of the homogenized model is justified by showing that the solutions of the microscopic model converge to those of the effective macroscopic model in probability distribution, as the size of small holes diminishes to zero.Comment: Communications in Mathematical Physics, to appear, 200

Stochastic Approximation to Understand Simple Simulation Models

This paper illustrates how a deterministic approximation of a stochastic process can be usefully applied to analyse the dynamics of many simple simulation models. To demonstrate the type of results that can be obtained using this approximation, we present two illustrative examples which are meant to serve as methodological references for researchers exploring this area. Finally, we prove some convergence results for simulations of a family of evolutionary games, namely, intra-population imitation models in n-player games with arbitrary payoffs.Ministerio de Educación (JC2009- 00263), Ministerio de Ciencia e Innovación (CONSOLIDER-INGENIO 2010: CSD2010-00034, DPI2010-16920

Neuroanatomical phenotypes in a mouse model of the 22q11.2 microdeletion

Recurrent deletions at the 22q11.2 locus have been established as a strong genetic risk factor for the development of schizophrenia and cognitive dysfunction. Individuals with 22q11.2 deletions have a range of well-defined volumetric abnormalities in a number of critical brain structures. A mouse model of the 22q11.2 deletion (Df(16)A+/-) has previously been utilized to characterize disease-associated abnormalities on synaptic, cellular, neurocircuitry, and behavioral levels. We performed a high-resolution MRI analysis of mutant mice compared with wild-type littermates. Our analysis revealed a striking similarity in the specific volumetric changes of Df(16)A+/-mice compared with human 22q11.2 deletion carriers, including in cortico-cerebellar, cortico-striatal and cortico-limbic circuits. In addition, higher resolution magnetic resonance imaging compared with neuroimaging in human subjects allowed the detection of previously unknown subtle local differences. The cerebellar findings in Df(16)A+/-mice are particularly instructive as they are localized to specific areas within both the deep cerebellar nuclei and the cerebellar cortex. Our study indicates that the Df(16)A+/-mouse model recapitulates most of the hallmark neuroanatomical changes observed in 22q11.2 deletion carriers. Our findings will help guide the design and interpretation of additional complementary studies and thereby advance our understanding of the abnormal brain development underlying the emergence of 22q11.2 deletion-associated psychiatric and cognitive symptoms.Molecular Psychiatry advance online publication, 3 September 2013; doi:10.1038/mp.2013.112