Bayesian whole-brain functional magnetic resonance imaging (fMRI) analysis with three-dimensional spatial smoothing priors has been shown to produce state-of-the-art activity maps without pre-smoothing the data. The proposed inference algorithms are computationally demanding however, and used have several less appealing properties, such as being improper having infinite range. We propose a stat...

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In this paper we prove necessary conditions for optimality of a stochastic control problem for a class of stochastic partial differential equations that is controlled through the boundary. This kind of problems can be interpreted as a stochastic control problem for an evolution system in a Hilbert space. The regularity of the solution of the adjoint equation, that is a backward stochastic equat...

Stochastic reduced-order modeling based on time-dependent bases (TDBs) has proven successful for extracting and exploiting low-dimensional manifold from stochastic partial differential equations (SPDEs). The nominal computational cost of solving a rank-$r$ model (ROM) basis, a.k.a. TDB-ROM, is roughly equal to that the full-order $r$ random samples. As now, this performance can only be achieved...

As a model for multiscale systems under random influences on physical boundary, a stochastic partial differential equation under a fast random dynamical boundary condition is investigated. An effective equation is derived and justified by reducing the random dynamical boundary condition to a random static boundary condition. The effective system is still a stochastic partial differential equati...

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 condi...

State-of-the-art computer codes for simulating real physical systems are often characterized by vast number of input parameters. Performing uncertainty quantification (UQ) tasks with Monte Carlo (MC) methods is almost always infeasible because of the need to perform hundreds of thousands or even millions of forward model evaluations in order to obtain convergent statistics. One, thus, tries to ...

In this study, annual runoff is estimated by using a Bayesian geostatistical model for interpolation of hydrological data different spatial support. That is, streamflow observations from catchments (areal data), and precipitation evaporation (point data). The contains one climatic effect that common all years under year specific effect. Hence, the framework enables quantification variability du...

In this paper we discuss the Stochastic Control Problem, which deals with the best way to control a stochastic system and has applications in fields as diverse as robotics, finance, and industry. After quickly reviewing mathematical probability and stochastic integration, we prove the HamiltonJacobi-Bellman equations for stochastic control, which transform the Stochastic Control Problem into a ...