# MATHS 787 : Special Topic: Inverse Problems and Stochastic Differential Equations

## Science

### Course Prescription

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

### 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 769, 766

### Capabilities Developed in this Course

 Capability 1: Disciplinary Knowledge and Practice Capability 2: Critical Thinking Capability 3: Solution Seeking Capability 4: Communication and Engagement Capability 5: Independence and Integrity

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

### Assessments

Assessment Type Percentage Classification
Four Assignments (equally weighted) 40% Individual Coursework
Final Exam (2 hours) 60% Individual Examination
1 2 3 4 5 6
Four Assignments (equally weighted)
Final Exam (2 hours)

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

### 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

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

### Key Topics

• Stochastic Differential Equations: Probability, random variables, Random walks and Wiener processes, Itô calculus, Applications of SDE's.
• Inverse Problems: Hilbert spaces and linear operator theory, Singular value decomposition and pseudoinverses, Tikhonov regularisation, Nonlinear problems and iterative methods.

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

• Lectures will be available as recordings, if classrooms are equipped with 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.

The University of Auckland will not tolerate cheating, or assisting others to cheat, and views cheating in coursework 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. This requirement also applies to sources on the internet. A student's assessed work may be reviewed against online source material using computerised detection mechanisms.

### Class Representatives

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

### 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 .

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 course assessment continues to meet the principles of the University’s assessment policy. Some adjustments may need to be made in emergencies. You will be kept fully informed by your course co-ordinator/director, 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 .

### 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 students may be asked to submit 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. In exceptional circumstances changes to elements of this course may be necessary at short notice. Students enrolled in this course will be informed of any such changes and the reasons for them, as soon as possible, through Canvas.

Published on 28/10/2022 11:28 a.m.