One of the most significant challenges faced by hydrogeologic modelers is the disparity
between the spatial and temporal scales at which fundamental flow, transport, and reaction …

In his forward-looking paper [374] at the conference “Mathematics Towards the Third
Millennium,” our esteemed colleague at Brown University Prof. David Mumford argued that …

The present paper proposes a Galerkin finite element projection scheme for the solution of
the partial differential equations (pde's) involved in the probability density evolution method …

Mathematical models of advection–reaction phenomena rely on advective flow velocity and
(bio) chemical reaction rates that are notoriously random. By using functional integral …

Quantitative analysis of the risk for reservoir real-time operation is a hard task owing to the
difficulty of accurate description of inflow uncertainties. The ensemble-based hydrologic …

The Buckley--Leverett (nonlinear advection) equation is often used to describe two-phase
flow in porous media. We develop a new probabilistic method to quantify parametric …

Nonlinear hyperbolic balance laws with uncertain (random) initial data are ubiquitous in a
plethora of transport phenomena that often exhibit shocks. We develop the method of …

We derive deterministic cumulative distribution function (CDF) equations that govern the
evolution of CDFs of state variables whose dynamics are described by the first-order …

Deterministic Eulerian–Lagrangian models represent the interaction between particles and
carrier flow through the drag force. Its analytical descriptions are only feasible in special …

Hyperbolic balance laws with uncertain (random) parameters and inputs are ubiquitous in
science and engineering. Quantification of uncertainty in predictions derived from such laws …