Faculty of Science

Department of Mathematics

Land Acknowledgement

Toronto is in the 'Dish With One Spoon Territory’. The Dish With One Spoon is a treaty between the Anishinaabe, Mississaugas and Haudenosaunee that bound them to share the territory and protect the land. Subsequent Indigenous Nations and peoples, Europeans and all newcomers have been invited into this treaty in the spirit of peace, friendship and respect.

Course Outline (Winter 2021)

MTH600 – Computational Methods in Mathematics

Instructor

Wei Xu, Dept. of Mathematics, Ryerson University

Office: EPH 420

Tel: (416) 979-5000x544325

Email: [email protected]

Office Hours: Only by appointment

Prerequisites

Please consult the University Calendar for these.

Calendar Description

Monte Carlo simulation: Introduction. Generation of random variables. Monte Carlo integration. variance reduction techniques. Monte Carlo techniques for stochastic processes: Markov Chains, numerical solutions of stochastic differential equations. Numerical methods for PDEs: Introduction, finite-differences methods for PDEs, Crank-Nicolson method, boundary conditions, error estimates. Optimization Methods: Descent direction method, solving nonlinear least squares problems, Gauss-Newton method, Levenberg-Marquardt method, trust region method, solving optimization problems with linear constraints.

Lecture: 3 hours (online live teaching)

Labs: 1 hour (online lab)

Compulsory Textbook

Most of the material will be laid out in lecture notes. Additional reading maybe assigned as we proceed in the course.

Reference Textbooks

Monte Carlo Simulation

1. Glasserman, P: Monte Carlo methods in financial engineering.

2. R. Rubsinstein and D.P. Kroses, Simulation and Monte Carlo method.

Numerical methods for PDE

1.L. Edsberg. Introduction to computation to computation and model for Differential equations.

Linear/Nonlinear optimization:

1. Bertsimas, D: Introduction to Linear Optimization

2. Bertsekas, D: Nonlinear Programming

Course Organization

Lectures: Thursday 3pm- 6 pm (online live teaching on Zoom)

Labs: Friday 11am -12 pm (online live lab sessions on Zoom)

Learning Objectives

At the end of this course, the successful student will be able to:

1. Simulate common random variables and stochastic processes.

2. Implement numerical methods to solve Partial Differential Equations(PDEs)

3. Find optimum values for linear and non-linear multivariate functions with constraints.

4. Apply simulations, numerical methods for PDEs, and optimization to real world problems.

Evaluation & Feedback

Assignment 1	16%
Assignment 2	16%
Assignment 3	16%
Quiz (5 out of 6)	15%
Project	37%

● Results for assignments will be returned within 2 weeks of due date.

● Some problems in Assignments may require coding in MATLAB.

Grading requirements

To pass the course you have to get at least 50% of the total course marks.

Labs

Labs are in the form of computer labs and will provide hands-on instruction and practice using MATLAB for course-related problems. Examples and Exercise questions will be made available prior to the Lab, and Lab solutions will be posted after the Lab has been completed by all sections for the course.

Information Technology and electronic devices

The course will extensively use MATLAB.

Evaluation Guidelines

● Assignments will contain problem questions.

● Sharing of calculators, pencils, pens or erasers is not permitted during any course examination.

Missed Evaluations

Students are required to inform their instructors of any situation which arises during the semester which may have an adverse effect upon their academic performance, and must request any considerations and accommodations according to the relevant policies and well in advance. Failure to do so will jeopardize any academic appeals.

● Medical certificates – If a student misses the deadline for submitting an assignment, or the date of an exam or other evaluation component because of illness, he or she must submit a Ryerson Student Medical Certificate AND an Academic Consideration form within 3 working days of the missed date. Both documents are available at www.ryerson.ca/senate/forms/medical.pdf. If you are a full-time or part-time degree student, then you submit your forms to your own program department or school. If you are a certificate or non-certificate student, then you submit your forms to the staff at the front desk of the Chang School.

● Religious observance – If a student needs accommodation because of religious observance, he or she must submit a Request for Accommodation of Student Religious, Aboriginal and Spiritual Observance AND an Academic Consideration form within the first 2 weeks of the class or, for a final examination, within 2 weeks of the posting of the examination schedule. If the required absence occurs within the first 2 weeks of classes, or the dates are not known well in advance as they are linked to other conditions, these forms should be submitted with as much lead time as possible in advance of the required absence. Both documents are available at http://www.ryerson.ca/senate/forms/relobservforminstr.pdf. If you are a fulltime or part-time degree student, then you submit the forms to your own program department or school. If you are a certificate or non-certificate student, then you submit the forms to the staff at the front desk of the Chang School.

● Students who need academic accommodation support should register with the Academic Accommodation Support office (formerly called the Access Centre). Before the first graded work is due, registered students should inform their instructors through an “Accommodation Form for Professors” that they are registered with Academic Accommodation Support and what accommodations are required.

Communication with Students

Ryerson’s email policy http://www.ryerson.ca/content/dam/senate/policies/pol157.pdf states that only Ryerson e-mail accounts are to be used for communication with students. All students, including continuing education students, have access to Ryerson email through their my.ryerson.ca site, and this is the official way in which they receive communication. All students are required to register for and maintain this account.

Emails sent from other accounts may not be answered!

Course Content

Week	Chapters	Evaluation
Week 1	Introduction to option pricing
Week 2	Introduction to Brownian motion and stochastic process
Week 3	Introduction to Stochastic differential equations and Ito’s Lemma
Week 4	Introduction to Monte Carlo simulation	Assignment 1
Week 5	Application of Monte Carlo simulation and variance reduction technique
Week 6	Simulation of stochastics processes and error analysis
Week 7	Introduction to PDE	Assignment 2
Week 8	Finite difference method(Explicit and implicit method)
Week 9	Crank-Nicolson method and convergence analysis
Week 10	Introduction to Optimization and Dynamics Programming	Assignment 3
Week 11	Numerical methods for nonlinear least squares problems	Project
Week 12	Numerical methods for nonlinear least squares problems with linear constraints

This outline is tentative. You are expected to regularly check the course website for updates, announcements as well as dates for term tests (in the event they are moved).

Academic Policies

a. Ryerson Policies of Interest

Ryerson Senate Policies - http://www.ryerson.ca/senate/policies/

Ryerson Academic Integrity - http://www.ryerson.ca/academicintegrity/

Policy 46 - Undergraduate Grading, Promotion and Academic Standing

Policy 60 - Student Code of Academic Conduct

Policy 61 - Student Code of Non-academic Conduct

Policy 134 - Undergraduate Academic Consideration and Appeals

Policy 135 - Examination Policy

Policy 150 - Accommodation of Student Religious Observance Obligations

Policy 157 - Student Email Accounts for Official University Communication

b. Obligations – Students need to inform faculty of any situation arising during the semester which may have an adverse effect upon their academic performance; they must request any necessary considerations (e.g. medical or compassionate), or accommodations [e.g. religious observance, disability (should be registered with the Access Centre), etc.] according to policies and well in advance. Failure to do so will jeopardize any academic appeals.

c. Re-grading and Re-calculation – Must be requested within 10 working days of the return of the graded assignment to the class.

Academic Conduct

http://www.ryerson.ca/academicintegrity/

In order to create an environment conducive to learning and respectful of others’ rights, phones and pagers must be silenced during lectures, lab sessions and evaluations.

Students should refrain from disrupting the lectures by arriving late and/or leaving the classroom before the lecture is finished.

Academic Misconduct

According to the Ryerson policy 60

(http://www.ryerson.ca/content/dam/senate/policies/pol60.pdf), academic misconduct includes, but is not limited to:

▪ Plagiarism which is the claiming of words, ideas, artistry, drawings or data of another person. This also includes submitting your own work in whole or in part for credit in two or more courses.

▪ Cheating

▪ Misrepresentation of personal identity or performance

▪ Submission of false information

▪ Contributing to academic misconduct

▪ Damaging, tampering, or interfering with the scholarly environment

▪ Unauthorized copying or use of copyrighted materials

▪ Violations of departmental policies or professional behavior

▪ Violations of specific departmental or course requirements

Committing academic misconduct will trigger academic penalties, including:

● a course-grade reduction that may be more severe than assigning a grade of “zero” (0) on course work; failing grades, suspension and possibly expulsion from the University. As a Ryerson student, you are responsible for familiarizing yourself with Ryerson conduct policies.

Non-Academic Conduct

Ryerson's Student Code of Non-academic Conduct is described in Senate Policy 61:

http://www.ryerson.ca/content/dam/senate/policies/pol61.pdf

Among many other infractions, the code specifically refers to the following as a violation: “Disruption of Learning and Teaching - Students shall not behave in disruptive ways that obstruct the learning and teaching environment”.

Diversity and Inclusion Statement

In this course I would like to create a learning environment that supports a diversity of thoughts, perspectives and experiences, and honors your identities (including race, gender, class, sexuality, religion, ability, etc.) For more information about our University’s resources and services on Equity, Diversity, and Inclusion please visit https://www.ryerson.ca/equity/