this is my dissertation outline

1. Outline of dissertation project
Mengdi Zheng
Title of dissertation: Numerical methods for stochastic systems subject to generalized
Levy noise.
1 Introduction
1.1 Motivation
1.2 Introduction of Levy processes
1.3 History of simulation of stochastic systems with Levy noise
1.4 Organization of dissertation
2 Simulation of Levy jump processes
2.1 Random walk approximation to Poisson processes
2.2 Karhunen-Loeve expansion for Poisson processes
2.3 Compound Poisson approximation to Levy jump processes
2.4 Series representation to Levy jump processes
Simulation of Gamma processes, tempered stable processes, and inverse Gaussian subordinators as
examples.
3 Adaptive multi-element polynomial chaos with discrete
measure: Algorithms and application to SPDEs
The past work has been done for continuous measures. Therefore, we do discrete measure here.
1

2. 3.1 Generation of orthogonal polynomials for discrete measures
3.1.1 Nowak method
3.1.2 Stieltjes method
3.1.3 Fischer method
3.1.4 Modi

3. ed Chebyshev method
3.1.5 Lanczos method
3.1.6 Gaussian quadrature rule associated with a discrete measure
This allows us to do Polynomial Chaos for SPDEs excited by discrete RVs.
3.1.7 Orthogonality tests of numerically generated polynomials
3.2 Discussion about the error of numerical integration
3.2.1 Theorem of numerical integration on discrete measure
3.2.2 Testing numerical integration with one RV
3.2.3 Testing numerical integration with multiple RVs on sparse grids
We did the numerical test with GENZ functions.
3.3 Application to stochastic reaction equation and KdV equation
3.3.1 Reaction equation with discrete random coecients
3.3.2 KdV equation with random forcing
We simulated the KdV equation with Stochastic excitation given by two discrete RVs, by a discrete
RV and a continuous RV, and by eight discrete RVs.
3.4 Conclusion
4 Adaptive Wick-Malliavin (WM) approximation to non-
linear SPDEs with discrete random variables
4.1 WM approximation
4.1.1 WM series expansion
4.1.2 WM propagators
We derive the WM propagators for a stochastic reaction equation and a stochastic Burgers equation.
2

4. 4.2 Moment statistics byWMapproximation of stochastic reaction equa-
tions
4.2.1 Reaction equation with one RV
4.2.2 Reaction equation with multiple RVs
We simulate the reaction equation with

5. ve Poisson RVs, and with a Poisson RV and a Binomial RV.
4.3 Moment statistics byWMapproximation of stochastic Burgers equa-
tions
4.3.1 Burgers equation with one RV
4.3.2 Burgers equation with multiple RVs
We simulate the Burgers equation with three Poisson RVs.
4.4 Adaptive WM method
We introduce P-Q re

6. nements to keep the error under a satisfactory level. Q is the WM approxima-
tion order. Both reaction equations and Burgers equations are simulated.
4.5 Computational complexity
4.5.1 Burgers equation with one RV
4.5.2 Burgers equation with d RVs
We compare the cost from WM approximation and gPC for Burgers equation with d RVs with the
same accuracy.
3

7. 5 Numerical methods for SPDEs with 1D tempered -stable
(TS) processes
5.1 Background of TS processes
5.2 Numerical simulation of 1D TS processes
5.2.1 Simulation of 1D TS processes by compound Poisson (CP) approximation
5.2.2 Simulation of 1D TS processes by series representation
5.2.3 Example: simulation of inverse Gaussian subordinators by CP approximation
and series representation
5.3 Stochastic models driven by TS white noises
5.3.1 Stochastic reaction-diusion model driven by TS white noises
5.3.2 1D stochastic overdamped Langevin equation driven by TS white noises
5.4 Simulation of stochastic reaction-diusion model driven by TS white
noises
5.4.1 Moment statistics by Monte Carlo (MC) with CP approximation and series rep-
resentation
5.4.2 Moment statistics by PCM with CP approximation and series representation
5.4.3 Comparing MC and PCM in CP approximation or series representation
5.5 Simulation of 1D stochastic overdamped Langevin equation driven
by TS white noises
5.5.1 Generalized Fokker-Planck (FP) equations for overdamped Langevin equations
with TS white noises
5.5.2 Simulating density by CP approximation
5.5.3 Simulating density by TFPDEs
We simulate the moment statistics of the over damped Langevin equation here by the deterministic
TFPDEs and PCM/CP. Also, we simulate the density of the solution by both the deterministic
TFPDEs and MC/CP.
4

8. 6 Numerical methods for SPDEs with additive multi-dimensional
Levy jump processes
6.1 Diusion model driven by multi-dimensional Levy jump process
6.2 Simulating multi-dimensional Levy pure jump processes
6.2.1 LePage's series representation with radial decomposition of Levy measure
6.2.2 Series representation with Levy copula
6.3 Generalize FP equation for joint PDF of SODEs with correlated
Levy jump processes
6.3.1 Generalized FP equation for SODEs with correlated Levy jump processes
6.3.2 Generalized FP equation for SODEs with multi-dimensional TS processes
6.4 Simulation of heat equation driven by bivariate Levy jump process
in LePage's representation
6.4.1 Simulating the moment statistics by PCM with series representation
6.4.2 Simulating the joint PDF of spatial modes by the generalized FP equation
6.4.3 Simulating moment statistics by TFPDE and PCM with series representation
6.5 Simulation of heat equation driven by bivariate tempered stable
Clayton Levy process
6.5.1 Simulating the moment statistics by PCM with series representation
6.5.2 Simulating the joint PDF of spatial modes by the generalized FP equation
6.5.3 Simulating moment statistics by TFPDE and PCM with series representation
6.6 Simulation of heat equation driven by correlated multi-component
Levy process
(i'm still working on this. i need to do 4 dimensions.)
7 Application of fractional dynamics on networks: proba-
bilistic and deterministic approaches
5

9. 8 Summary
A Simulation of 2D stochastic Navier-Stokes (NS)
ow past
a cylinder
A.1 2D NS
ow past a cylinder with Gaussian stochastic boundary con-
ditions (SBC)
We simulated the errors of the mean and variance of the velocity and the pressure of the
ow by
gPC, MEgPC, PCM, and MEPCM.
A.1.1 2D NS
ow past a cylinder with SBC by general Polynomial Chaos (gPC)
A.1.2 2D NS
ow past a cylinder with SBC by Multi-Element general Polynomial
Chaos (MEgPC)
A.1.3 2D NS
ow past a cylinder with SBC by probability collocation method (PCM)
A.1.4 2D NS
ow past a cylinder with SBC by Multi-Element probability collocation
method (MEPCM)
A.1.5 Convergence comparison study of gPC, MEgPC, PCM, and MEPCM
A.2 2D NS
ow past a cylinder with Poisson SBC by MEPCM
We simulated the mean and variance of the lift and drag forces on the cylinder by MEPCM.
A.3 2D NS
ow past a cylinder with hybrid type SBC (Gaussian and
Poisson) by MEPCM
6