Outline of dissertation project

Mengdi Zheng

Title of dissertation: Numerical methods for stochastic systems subject to generalizedLevy 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 asexamples.

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.

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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 Modified 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 coefficients

3.3.2 KdV equation with random forcing

We simulated the KdV equation with Stochastic excitation given by two discrete RVs, by a discreteRV 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.

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4.2 Moment statistics by WM approximation 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 five Poisson RVs, and with a Poisson RV and a Binomial RV.

4.3 Moment statistics by WM approximation 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 refinements 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 thesame accuracy.

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5 Numerical methods for SPDEs with 1D tempered α-stable

(TαS) processes

5.1 Background of TαS processes

5.2 Numerical simulation of 1D TαS processes

5.2.1 Simulation of 1D TαS processes by compound Poisson (CP) approximation

5.2.2 Simulation of 1D TαS processes by series representation

5.2.3 Example: simulation of inverse Gaussian subordinators by CP approximationand series representation

5.3 Stochastic models driven by TαS white noises

5.3.1 Stochastic reaction-diffusion model driven by TαS white noises

5.3.2 1D stochastic overdamped Langevin equation driven by TαS white noises

5.4 Simulation of stochastic reaction-diffusion model driven by TαS whitenoises

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 drivenby TαS white noises

5.5.1 Generalized Fokker-Planck (FP) equations for overdamped Langevin equationswith TαS 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 deterministicTFPDEs and PCM/CP. Also, we simulate the density of the solution by both the deterministicTFPDEs and MC/CP.

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6 Numerical methods for SPDEs with additive multi-dimensional

Levy jump processes

6.1 Diffusion 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 correlatedLevy 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 TαS processes

6.4 Simulation of heat equation driven by bivariate Levy jump processin 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 stableClayton 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-componentLevy 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

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8 Summary

A Simulation of 2D stochastic Navier-Stokes (NS) flow past

a cylinder

A.1 2D NS flow 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 flow bygPC, MEgPC, PCM, and MEPCM.

A.1.1 2D NS flow past a cylinder with SBC by general Polynomial Chaos (gPC)

A.1.2 2D NS flow past a cylinder with SBC by Multi-Element general PolynomialChaos (MEgPC)

A.1.3 2D NS flow past a cylinder with SBC by probability collocation method (PCM)

A.1.4 2D NS flow past a cylinder with SBC by Multi-Element probability collocationmethod (MEPCM)

A.1.5 Convergence comparison study of gPC, MEgPC, PCM, and MEPCM

A.2 2D NS flow 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 flow past a cylinder with hybrid type SBC (Gaussian andPoisson) by MEPCM

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