SC740 SEMINAR REPORT 11 for Frederick L. Jones

PRESENTER: Michael Eckhoff

OVERVIEW

The central theme of stochastic analysis according to Michael Eckhoff is "white noise", which is defined as the derivative of Brownian motion B, i.e., the motion of a random walk where the root mean square of the distance traveled is proportional to time. The development of Brownian motion in Banach and Hilbert spaces led the way for the development of Stochastic Partial Differential Equations (SPDE’s).

As Michael Eckhoff stated, The traditional engineering Ritz-Galerkin methods have been extended to include "white noise" by introducing "stochastic finite elements" for random media. Michael Eckhoff further argues that the Galerkin approach provides a natural implementation for SPDE’s with "white noise" forcing.

STOCHASTIC ANALYSIS AND THE CONCEPT OF "WHITE NOISE"

As noted by Michael Eckhoff, the central theme of stochastic analysis is "white noise", which is defined as the derivative of Brownian motion B. The source of noise can be either internal or external. Internal noise is caused by internal system dynamics. External noise comes from the system’s environment. If the noise is external, then the mathematical model usually takes the following form:

Where u is the "response" of the system, x is the noise and A is the differential operator.

KEY QUESTIONS ADDRESSED

Some of the key questions that Michael Eckhoff addressed were the following:

1. If "white noise" is defined as the derivative of Brownian motion, then how do you determine its regularity and the function space(s) in which it resides?
2. Given that the function spaces used are either Banach spaces or else Hilbert spaces, how do you extend the differential operator A so that A u = x has a solution?
3. How do you modify traditional Galerkin schemes to handle this problem numerically?
4. Given that the random solution is an ensemble {u(x;w )} w ' W , how do you describe the solution?

STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS (SPDE’s)

As Michael Eckhoff stated, The traditional engineering Ritz-Galerkin methods have been extended to include "white noise" by introducing "stochastic finite elements" for random media. Michael Eckhoff further argues that the Galerkin approach provides a natural implementation for SPDE’s with "white noise" forcing.

According to G. H. Golub and J. M. Ortega in Scientific Computing and Differential Equations, pp.181-186, Galerkin’s method is one of a set of "projection methods" that attempts to approximate the differential equation solution by a finite linear combination of known functions such as trigonometric, spline, etc. , functions. The approximate solution is the "projection" of the solution on the finite-dimensional subspace.

For the selection of the solution technique for the SPDE’s, Michael Eckhoff compared two different variations of Galerkin methods, i.e., (1) the Bubnov-Galerkin method and (2) the Petrov-Galerkin method. Both of these variations are described in Michael Eckhoff’s online copy of his presentation and will not be repeated here. For the specifics of the above two Galerkin method variations as well as a rather extensive list of references on stochastic PDE’s , see Michael Eckhoff’s online copy of his seminar presentation at pax.st.usm.edu/~meckhoff/seminar.

SUMMARY AND CONCLUSIONS

The central theme of stochastic analysis according to Michael Eckhoff is "white noise", which is defined as the derivative of Brownian motion B. The development of Brownian motion in Banach and Hilbert spaces led the way for the development of Stochastic Partial Differential Equations (SPDE’s).

As Michael Eckhoff stated, The traditional engineering Ritz-Galerkin methods have been extended to include "white noise" by introducing "stochastic finite elements" for random media. Michael Eckhoff further argued that the Galerkin approach provides a natural implementation for SPDE’s with "white noise" forcing.

The only question I had concerning the use of Stochastic PDE solutioin methods was whether or not there were ways to reduce "white noise" or else cancel out its effects, rather than to explicitly include it in a mathematical model. Michael Eckhoff did indicate that since some "white noise" is just measurement error, the selection of the "best" incremental value h in the SPDE’s could reduce the "white noise" in the system.

However, even statistical techniques for time series analysis such as Box-Jenkins Autocorrelation have been applied to random signals, i.e., "white noise", in signal processing applications published in the IEEE Transactions on Pattern Analysis and Machine Intelligence to determine if there are any underlying patterns in the signals. Therefore, Stochastic Partial Differential Equations methods definitely can provide very useful information.