RT Journal Article
ID 1e3602e830bed37d
A1 Venturi, Daniele
A1 Karniadakis, George Em
T1 DIFFERENTIAL CONSTRAINTS FOR THE PROBABILITY DENSITY FUNCTION OF STOCHASTIC SOLUTIONS TO THE WAVE EQUATION
JF International Journal for Uncertainty Quantification
JO IJUQ
YR 2012
FD 2012-03-29
VO 2
IS 3
SP 195
OP 213
K1 stochastic partial differential equations
K1 high-dimensional methods
K1 random fields
AB By using functional integral methods we determine new types of differential constraints satisfied by the joint probability density function of stochastic solutions to the wave equation subject to uncertain boundary and initial conditions. These differential constraints involve unusual limit partial differential operators and, in general, they can be grouped into two main classes: the first one depends on the specific field equation under consideration (i.e., on the stochastic wave equation), the second class includes a set of intrinsic relations determined by the structure of the joint probability density function of the wave and its derivatives. Preliminary results we have obtained for stochastic dynamical systems and first-order nonlinear stochastic particle differential equations (PDEs) suggest that the set of differential constraints is complete and, therefore, it allows determining uniquely the probability density function of the solution to the stochastic problem. The proposed new approach can be extended to arbitrary nonlinear stochastic PDEs and it could be an effective way to overcome the curse of dimensionality for random boundary and initial conditions. An application of the theory developed is presented and discussed for a simple random wave in one spatial dimension.
PB Begell House
LK http://dl.begellhouse.com/journals/52034eb04b657aea,3495dd1f6253a653,1e3602e830bed37d.html