| Abstract | 第1-8页 |
| 摘要 | 第8-44页 |
| 1 Introduction | 第44-54页 |
| ·Uncertainty quantification | 第44-46页 |
| ·Verification and validation | 第44页 |
| ·Types of uncertainty | 第44-46页 |
| ·On stochastic discretization | 第46-47页 |
| ·General discretization in random space | 第46-47页 |
| ·General comments on aforementioned stochastic discretization | 第47页 |
| ·Representation of random processes | 第47-50页 |
| ·Karhunen–Loe`ve expansion | 第47-48页 |
| ·Wiener polynomial chaos | 第48-49页 |
| ·Representation of white noise | 第49-50页 |
| ·Elliptic equation: an illustration solving SPDEs | 第50-52页 |
| ·Stochastic Galerkin method with model reduction | 第52页 |
| ·Discretizations in spatial and temporal spaces | 第52-53页 |
| ·Main contribution of this work | 第53-54页 |
| 2 Error estimates of numerical solutions to di?usion equations with multiplicative white noise in space | 第54-68页 |
| ·Elliptic equation with purely multiplicative spatial white noise | 第55-63页 |
| ·Notations | 第55-56页 |
| ·Regularity verification | 第56-57页 |
| ·Total error behavior | 第57-58页 |
| ·Error estimates for each equation in the propagator | 第58-61页 |
| ·Error estimates for truncating white noise | 第61-63页 |
| ·Parabolic equation with purely multiplicative spatial white noise | 第63-68页 |
| ·Conclusions | 第64-66页 |
| ·Proof of Theorem 2.2.1 | 第66-68页 |
| 3 Spectral separating scheme for passive scalar equation in Gaussian field | 第68-96页 |
| ·Assumptions | 第69页 |
| ·Propagator using cosine basis in time | 第69-74页 |
| ·Existence and uniqueness of the solution | 第70-71页 |
| ·Error estimates for the direct method | 第71-72页 |
| ·Proof of Theorem 3.2.2 | 第72-74页 |
| ·Propagator using multilevel basis | 第74-77页 |
| ·Error estimates for multilevel basis case | 第74-77页 |
| ·Proof of Lemma 3.2.4 | 第77-79页 |
| ·Proof of (3.2.13) | 第78-79页 |
| ·Numerical results | 第79-96页 |
| ·Propagator and its deterministic solver | 第80-82页 |
| ·Computing the moments | 第82页 |
| ·Performance comparison between online and o?ine computation | 第82-84页 |
| ·Long-term integration | 第84-91页 |
| ·Testing a non-degenerate parabolic equation | 第91-92页 |
| ·Testing a di?usion equation | 第92页 |
| ·Concluding remark | 第92-96页 |
| 4 Long-term integration for a stochastic ordinary di?erential equation | 第96-106页 |
| ·Problem setting | 第96-100页 |
| ·Dynamics of numerical solution | 第97-98页 |
| ·Error estimate of numerical solution | 第98-99页 |
| ·Error estimates for behavior at large t | 第99-100页 |
| ·Behavior of error estimates for small t | 第100页 |
| ·Error estimate of weak approximation: long time inte-gration | 第100-101页 |
| ·Estimates of moments of the numerical solution | 第100-101页 |
| ·Numerical results | 第101-102页 |
| ·Discussion | 第102-103页 |
| ·Appendix | 第103-106页 |
| ·Roots of Legendre polynomials | 第103-104页 |
| ·Fourier expansion of exp(?xt) | 第104-106页 |
| 5 Comparison between di?erent stochastic advection with multiplicative noise | 第106-112页 |
| ·Exact Solutions | 第107-110页 |
| ·The solution to stochastic advection equation (5.0.2) | 第107-108页 |
| ·The solution to (5.0.1) | 第108-109页 |
| ·The eigenvalues of exponential kernel | 第109-110页 |
| ·A di?erent approach to represent the process | 第110-111页 |
| ·Discussions | 第111-112页 |
| 6 Error estimate for functional ANOVA method with interpolation | 第112-142页 |
| ·Introduction | 第113-115页 |
| ·Further description of ANOVA: decompose the underlying space | 第113-114页 |
| ·The application of ANOVA to solving SPDEs | 第114-115页 |
| ·Weights and e?ective dimension for ANOVA | 第115-121页 |
| ·Calculating e?ective dimension: practical consideration | 第116-117页 |
| ·How are the weights chosen for a particular use? | 第117-118页 |
| ·E?ective dimension and weights | 第118-121页 |
| ·Weights and e?ective dimension for anchored ANOVA | 第121-128页 |
| ·Comparison of di?erent choices of anchor points | 第126-128页 |
| ·Error estimates of anchored ANOVA for continuous func- tions | 第128-133页 |
| ·Truncation Error | 第129-131页 |
| ·Interpolation error | 第131-133页 |
| ·Error estimates of (Lebesgue) ANOVA for continuous functions | 第133-134页 |
| ·Numerical Results | 第134-138页 |
| ·Verification of the error estimates | 第134页 |
| ·Genz function f4 | 第134-136页 |
| ·Genz function f5 | 第136-138页 |
| ·summary | 第138-139页 |
| ·Appendix: Detailed Proof of Lemma 6.4.2 | 第139-142页 |
| 7 Conclusion | 第142-144页 |
| Bibliography | 第144-152页 |
| List of Papers | 第152-154页 |
| Acknowledgements | 第154-155页 |
