MATHS 767 : Inverse Problems and Stochastic Differential Equations

Science

2025 Semester One (1253) (15 POINTS)

Course Prescription

Covers stochastic differential equations and inverse problems, including: continuous time processes, random walks and Wiener processes, Itō calculus, and applications of SDEs, Hilbert spaces and linear operator theory, singular value decomposition and pseudoinverses, Tikhonov regularisation, nonlinear problems and iterative methods.

Course Overview

This course will cover two important topics in applied mathematics:
  • Stochastic differential equations arise in science, technology, engineering, business, ecology and several other fields of research. Most often, these models treat phenomena that evolve randomly with time. One half of the course focuses on stochastic differential equations (SDEs) and the mathematical tools to solve them.
  • Inverse problems are unstable problems that are encountered in all fields of science and engineering. Inverse problems can be characterized as problems that tolerate measurement and modelling errors poorly. The other half of the course focuses on the theory of deterministic inverse problems.
This is an advanced applied mathematics course for the BSc(Hons), BAdvSci and PGDipSci. In particular, this course is a major component for the Master of Mathematical Modelling. It is also of interest to students majoring in Pure Mathematics, Physics, Engineering Science, Computer Science or Statistics.

Course Requirements

Prerequisite: B- or higher in MATHS 340 and 361 Restriction: MATHS 766, 769, 787

Capabilities Developed in this Course

Capability 3: Knowledge and Practice
Capability 4: Critical Thinking
Capability 5: Solution Seeking
Capability 6: Communication
Capability 8: Ethics and Professionalism
Graduate Profile: Master of Science

Learning Outcomes

By the end of this course, students will be able to:
  1. Identify analytic and numerical methods suitable for the analysis of SDEs (Capability 3, 4, 5 and 8)
  2. Use and apply analytic and numerical methods, as appropriate, for the analysis of SDEs (Capability 3, 4 and 5)
  3. Write appropriately detailed and structured mathematical arguments in simple contexts (Capability 3, 4, 5 and 6)
  4. Gain and use the necessary notions and results of the linear vector spaces and Hilbert spaces (Capability 3, 4, 5 and 8)
  5. Apply singular value decomposition to minimum norm solutions and describe their relationship to the instability of compact operators (Capability 3, 4, 5, 6 and 8)
  6. Identify and use the four main categories of regularisation approaches: generic projections, truncated expansions, Tikhonov-type methods and truncated iterations. (Capability 3, 4, 5 and 8)

Assessments

Assessment Type Percentage Classification
Assignments 45% Individual Coursework
Final Exam (2 hours) 50% Individual Examination
Quizzes 5% Individual Coursework
Assessment Type Learning Outcome Addressed
1 2 3 4 5 6
Assignments
Final Exam (2 hours)
Quizzes

A minimum of 35% in the exam is required to pass the course.

Key Topics

►  Stochastic Differential Equations:
  • Preliminary on probability: What is a stochastic process?
    Expectation, mean, independent random variables, Central Limit Theorem
  • Discrete stochastic processes: Random walks, Markov chains,
    Transition matrices, Martingales, Optional stopping theorem (OST), hitting times
  • Continuous stochastic processes: Wiener process/Brownian motion,
    Itō calculus, Euler-Maruyama method, Fokker-Plank equation, OST and hitting times
  • Higher dimensions (optional): system of SDEs
►  Inverse Problems:
  • Introduction and motivation: forward vs inverse problems, ill-posedness
    Preliminary tools: norms and Banach spaces, inner products and Hilbert spaces
  • Existence and Uniqueness for linear inverse problems: Pseudoinverse theory
    Classical vs least-squares solutions, normal equations, minimum-norm solution, singular value decomposition (SVD) and pseudoinverse
  • Stability for linear inverse problems: Regularisation theory
    Methods: Truncated SVD, Tikhonov regularisation, Landweber iteration
    Choice of the regularisation parameter (L-curve, Discrepancy Principle of Morozov),
    Variational formulation, Fix-point theorem
  • Nonlinear inverse problems: Nonlinear least-squares, Gauss-Newton algorithm

Special Requirements

No special requirement.

Tuākana

Tuākana Science is a multi-faceted programme for Māori and Pacific students providing topic specific tutorials, one-on-one sessions, test and exam preparation and more. Explore your options at
https://www.auckland.ac.nz/en/science/study-with-us/pacific-in-our-faculty.html
https://www.auckland.ac.nz/en/science/study-with-us/maori-in-our-faculty.html

Whanaungatanga and manaakitanga are fundamental principles of our Tuākana Mathematics programme which provides support for Māori and Pasifika students who are taking mathematics courses. The Tuākana Maths programme consists of workshops and drop-in times, and provides a space where Māori and Pasifika students are able to work alongside our Tuākana tutors and other Māori and Pasifika students who are studying mathematics.

Workload Expectations

This course is a standard 15 point course and students are expected to spend 10 hours per week involved in each 15 point course that they are enrolled in.

For this course, you can expect 3 hours of lectures, no tutorial, 3 hours of reading and thinking about the content and 4 hours of work on assignments and/or test preparation.

Delivery Mode

Campus Experience

  • Recordings will be available if classrooms are equipped with working recording facilities.
  • The course will not include live online events.
  • The activities for the course are scheduled as a standard weekly timetable delivery.

Learning Resources

Course materials are made available in a learning and collaboration tool called Canvas which also includes reading lists and lecture recordings (where available).

Please remember that the recording of any class on a personal device requires the permission of the instructor.

All lecture notes and supporting reading material, as well as some Matlab codes, will be available on CANVAS.

Student Feedback

During the course Class Representatives in each class can take feedback to the staff responsible for the course and staff-student consultative committees.

At the end of the course students will be invited to give feedback on the course and teaching through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback.

Your feedback helps to improve the course and its delivery for all students.

 

Academic Integrity

The University of Auckland will not tolerate cheating, or assisting others to cheat, and views cheating in coursework, tests and examinations as a serious academic offence. The work that a student submits for grading must be the student's own work, reflecting their learning. Where work from other sources is used, it must be properly acknowledged and referenced. A student's assessed work may be reviewed against electronic source material using computerised detection mechanisms. Upon reasonable request, students may be required to provide an electronic version of their work for computerised review.

Class Representatives

Class representatives are students tasked with representing student issues to departments, faculties, and the wider university. If you have a complaint about this course, please contact your class rep who will know how to raise it in the right channels. See your departmental noticeboard for contact details for your class reps.

Copyright

The content and delivery of content in this course are protected by copyright. Material belonging to others may have been used in this course and copied by and solely for the educational purposes of the University under license.

You may copy the course content for the purposes of private study or research, but you may not upload onto any third party site, make a further copy or sell, alter or further reproduce or distribute any part of the course content to another person.

Inclusive Learning

All students are asked to discuss any impairment related requirements privately, face to face and/or in written form with the course coordinator, lecturer or tutor.

Student Disability Services also provides support for students with a wide range of impairments, both visible and invisible, to succeed and excel at the University. For more information and contact details, please visit the Student Disability Services’ website http://disability.auckland.ac.nz

Special Circumstances

If your ability to complete assessed coursework is affected by illness or other personal circumstances outside of your control, contact a member of teaching staff as soon as possible before the assessment is due.

If your personal circumstances significantly affect your performance, or preparation, for an exam or eligible written test, refer to the University’s aegrotat or compassionate consideration page https://www.auckland.ac.nz/en/students/academic-information/exams-and-final-results/during-exams/aegrotat-and-compassionate-consideration.html.

This should be done as soon as possible and no later than seven days after the affected test or exam date.

Learning Continuity

In the event of an unexpected disruption we undertake to maintain the continuity and standard of teaching and learning in all your courses throughout the year. If there are unexpected disruptions the University has contingency plans to ensure that access to your course continues and your assessment is fair, and not compromised. Some adjustments may need to be made in emergencies. You will be kept fully informed by your course co-ordinator, and if disruption occurs you should refer to the University Website for information about how to proceed.

The delivery mode may change depending on COVID restrictions. Any changes will be communicated through Canvas.

Student Charter and Responsibilities

The Student Charter assumes and acknowledges that students are active participants in the learning process and that they have responsibilities to the institution and the international community of scholars. The University expects that students will act at all times in a way that demonstrates respect for the rights of other students and staff so that the learning environment is both safe and productive. For further information visit Student Charter https://www.auckland.ac.nz/en/students/forms-policies-and-guidelines/student-policies-and-guidelines/student-charter.html.

Disclaimer

Elements of this outline may be subject to change. The latest information about the course will be available for enrolled students in Canvas.

In this course you may be asked to submit your coursework assessments digitally. The University reserves the right to conduct scheduled tests and examinations for this course online or through the use of computers or other electronic devices. Where tests or examinations are conducted online remote invigilation arrangements may be used. The final decision on the completion mode for a test or examination, and remote invigilation arrangements where applicable, will be advised to students at least 10 days prior to the scheduled date of the assessment, or in the case of an examination when the examination timetable is published.

Published on 19/10/2024 12:45 p.m.