Ostoja-Starzewski gives talk at Los Alamos

Professor Martin Ostoja-Starzewski recently presented a talk, “Stochastic Partial Differential Equations Why? How? What?” in the Computational Physics Division at Los Alamos National Laboratory. The focus of his lecture was on PDEs with random field (RF) coefficients, source (forcing) term, and/or boundary/initial conditions.  

The RF coefficients are introduced to describe the medium’s non-deterministic spatial variability. Thus, one arrives at stochastic PDEs (or SPDEs). Already the simplest case of steady-state conductivity gives rise to serious physical modeling and mathematical challenges. In general, there are two research communities studying SPDEs: mathematical and applied (engineering science). 

The mathematical community deals primarily with idealized random functions of space and time—for example, white noise and Brownian sheet. The studied SPDEs are generalizations of elliptic, parabolic, hyperbolic, and then nonlinear deterministic PDEs.

In the applied research community, the random field coefficients are intended to reflect some real material (concrete, polycrystal, granular or fibrous medium, etc.) under practically relevant loading conditions. These problems lead to numerical methods – typically the stochastic finite elements (SFEs) – which in turn involve various approximate solution techniques. Alas, most SFEs and Uncertainty Quantification (UQ) models typically miss a link with the micromechanics and homogenization theories of random media. 

Since all the continuum theories involve tensors, major advances in tensor-valued random field (TRF) models are needed. Recently, explicit representations (and spectra) have been given for the most general, statistically homogeneous and isotropic TRFs of ranks 1…4, only subject to restrictions imposed by physical laws. These representations account for arbitrary local anisotropies and spatial correlation structures, where the case of rank 1 is known in the statistical turbulence. Ostoja-Starzewski’s talk ended with a methodology for SPDEs (e.g. modeling impact waves) on scalar RFs with spatial fractal-and-Hurst effects.

Ostoja-Starzewski joined MechSE in 2006. He earned a PhD in 1983 in mechanical engineering from McGill University in Canada.