Stochastic Partial Differential Equations      

Stochastic problems arise for a variety of reasons, e.g., when system parameters are fluctuating, when fluctuating external forces exist, and when a small set of variables are extracted from a larger set of variables, the rest being treated as a "heat bath." Stochastic modeling is often successful even in cases where it is not obvious that a simple systematic/stochastic split exists. On the other hand, if too naive stochastic models are used, the results can be dangerously misleading. Modeling of this kind often involves a delicate balance between mathematical rigor and physical intuition. Computer simulation operating at this interface is often a crucial component of the analysis. My interest in this field dates from graduate school days learning the ABCs at the feet of Bob Dorfman and Robert Zwanzig.

The theory of stochastic ODEs is by now more or less understood. However, in complex system modeling and field theoretic dynamical problems the equations one needs to solve are most often stochastic partial differential equations (SPDEs). While formal works on SPDEs abound in the mathematical literature, there is much less information on practical computational approaches to solving these equations.

Aside from considerations regarding the physical basis of SPDEs, I am interested in various other aspects, including: (1) new algorithms, (2) error control theory, (3) implementation of global and/or gauge constraints, (4) multiplicative noise, and (5) nonlinear stochastic Master equations, a relatively new class of SPDEs. Most of my work on "conventional" SPDEs has been carried out with Ji Qiang (LBNL) and with Luis Bettencourt (CCS-3) and Grant Lythe (Leeds) (all originally LANL post-docs). Grant and I once started writing a long review article on SPDEs (but will it ever be finished?).

Papers

  1. Stochastic PDEs: convergence to the continuum?, Grant Lythe and Salman Habib, Comp. Phys. Comm. 142, 29 (2001)
  2. A Second-Order Stochastic Leap-Frog Algorithm for Multiplicative Noise Brownian Motion, Ji Qiang and Salman Habib, Phys. Rev. E 62, 7430 (2000) physics/9912055
  3. Controlling One-Dimensional Langevin Dynamics on the Lattice, Luis M.A. Bettencourt, Salman Habib, and Grant Lythe, Phys. Rev. D. 60, 105039 (1999) hep-lat/9903007
  4. Multiplicative Noise: Applications in Cosmology and Field Theory, Salman Habib, in Stochastic Processes in Astrophysics, Annals of the New York Academy of Sciences, Vol. 706, Proceedings of the Eighth Annual Workshop in Nonlinear Astronomy, Gainesville, Florida, 4 - 6 Feb., 1993, edited by R. Buchler and H.E. Kandrup (New York Academy of Sciences, 1993)
  5. Nonlinear Noise in Cosmology, Salman Habib and Henry E. Kandrup, Phys. Rev. D 46, 5303 (1992) gr-qc/9208005
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Salman Habib / LANL / revised February 2005
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