FAST AUTOMATIC BAYESIAN CUBATURE USING MATCHING KERNELS AND DESIGNS BY

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FAST AUTOMATIC BAYESIAN CUBATURE USING MATCHING KERNELS

AND DESIGNS

JAGADEESWARAN RATHINAVEL

Submitted in partial fulﬁllment of the

requirements for the degree of

Doctor of Philosophy in Applied Mathematics

in the Graduate College of the

Illinois Institute of Technology

Approved

Advisor

Chicago, Illinois

December 2019

arXiv:2210.03253v1 [math.NA] 6 Oct 2022

ACKNOWLEDGMENT

I want to thank my advisor Prof. Fred J Hickernell for his support and guid-

ance in my completion of this thesis and throughout my studies here at IIT. His

support and motivation have given me the conﬁdence to endure through the research.

I would like to also thank the GAIL project collaborators with whom I have

worked to add my new algorithms to the GAIL MATLAB toolbox: Prof. Sou-Cheng

Choi, Yuhan Ding, Lan Jiang, Xin Tong, and Kan Zhang. Especially, Prof. Sou-

Cheng Choi’s support and guidance as the project leader helped me to focus on my

cubature algorithms.

My special gratitude also goes to my thesis committee members, Prof. Jinqiao

Duan, Prof. Fred J Hickernell, Prof. Shuwang Li, and Prof. Geoﬀrey Williamson.

Above all, I want to thank them because they were ﬂexible and willing to dedicate

time to review my work and attend my comprehensive and defense examinations.

I would like to thank Prof. Dirk Nuyens for suggestions, valuable tips and

notes when we were researching higher order nets and kernels.

I would like to thank the organizers of the SAMSI-Lloyds-Turing Workshop

on Probabilistic Numerical Methods, where a part of preliminary version of this work

was discussed. I also thank Prof. Chris Oates and Prof. Sou-Cheng Choi for valuable

comments.

I would like to speciﬁcally thank my friend Samuel Davidson for reviewing and

suggesting the improvements on the text.

Last but not least, I would not be able to make it without the support of my

family. I would like to thank my wife for her continuous support and sacriﬁce. I also

would like to thank my parents for their endless support.

iii

TABLE OF CONTENTS

Page

ACKNOWLEDGEMENT......................... iii

LISTOFTABLES ............................ vi

LISTOFFIGURES............................ viii

ABSTRACT ............................... ix

CHAPTER

1.INTRODUCTION....................... 1

1.1.Cubature......................... 1

1.2. Stopping Criterion . . . . . . . . . . . . . . . . . . . . 2

1.3. Low Discrepancy Points . . . . . . . . . . . . . . . . . . 3

1.4.PriorWork........................ 3

2. BAYESIAN CUBATURE . . . . . . . . . . . . . . . . . . . 6

2.1. Bayesian Posterior Error . . . . . . . . . . . . . . . . . 6

2.2. Hyperparameter Estimation . . . . . . . . . . . . . . . . 8

2.3. Empirical Bayes . . . . . . . . . . . . . . . . . . . . . 9

2.4.FullBayes ........................ 11

2.5. Generalized Cross-Validation . . . . . . . . . . . . . . . 16

2.6. Cone of Functions and the Credible interval . . . . . . . . 19

2.7. The Automatic Bayesian Cubature Algorithm . . . . . . . 22

2.8. Example with the Mat´ern Kernel . . . . . . . . . . . . . 24

3. FAST AUTOMATIC BAYESIAN CUBATURE . . . . . . . . . 28

3.1. Fast Bayesian Transform Kernel . . . . . . . . . . . . . . 28

3.2. Empirical Bayes . . . . . . . . . . . . . . . . . . . . . 32

3.3.FullBayes ........................ 34

3.4. Generalized Cross-Validation . . . . . . . . . . . . . . . 35

3.5. Product Kernels . . . . . . . . . . . . . . . . . . . . . 36

4. INTEGRATION LATTICES AND

SHIFT INVARIANT KERNELS . . . . . . . . . . . . . . . . 40

4.1. Extensible Integration Lattice Node Sets . . . . . . . . . 40

4.2. Shift Invariant Kernels . . . . . . . . . . . . . . . . . . 41

4.3. Continuous Valued Kernel Order . . . . . . . . . . . . . 45

4.4.Summary......................... 50

4.5. Periodizing Variable Transformations . . . . . . . . . . . 51

5. SOBOL’ NETS AND WALSH KERNELS . . . . . . . . . . . 54

5.1.Sobol’Nets........................ 54

5.2. Walsh Kernels . . . . . . . . . . . . . . . . . . . . . . 57

5.3. Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . 61

5.4. Higher Order Nets . . . . . . . . . . . . . . . . . . . . 68

6. NUMERICAL IMPLEMENTATION . . . . . . . . . . . . . . 69

6.1. Overcoming Cancellation Error . . . . . . . . . . . . . . 69

6.2. Kernel Hyperparameters Search . . . . . . . . . . . . . . 72

7. NUMERICAL RESULTS AND OBSERVATIONS . . . . . . . 75

7.1. Testing Methodology . . . . . . . . . . . . . . . . . . . 75

7.2. Multivariate Gaussian Probability . . . . . . . . . . . . . 76

7.3. Keister’s Example . . . . . . . . . . . . . . . . . . . . 78

7.4. Option Pricing . . . . . . . . . . . . . . . . . . . . . . 83

7.5.Discussion ........................ 85

7.6. Comparison with cubMC g,cubLattice g and cubSobol g . 89

7.7. Shape Parameter Fine-tuning . . . . . . . . . . . . . . . 92

8. CONCLUSION AND FUTURE WORK . . . . . . . . . . . . 94

8.1.Conclusion........................ 94

8.2.FutureWork ....................... 95

APPENDIX ............................... 98

BIBLIOGRAPHY............................. 98

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FASTAUTOMATICBAYESIANCUBATUREUSINGMATCHINGKERNELSANDDESIGNSBYJAGADEESWARANRATHINAVELSubmittedinpartialfulllmentoftherequirementsforthedegreeofDoctorofPhilosophyinAppliedMathematicsintheGraduateCollegeoftheIllinoisInstituteofTechnologyApprovedAdvisorChicago,IllinoisDecember2019ACKNOWLEDGMENTIwanttot...

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