Welcome to the class website for the course Finite Difference Methods for Partial Differential Equations. The class will focus on numerical approximation of partial differential equations using finite difference approaches. The class will cover both mathematical foundations and practical aspects of effective implementation of such numerical methods. Many examples will also be discussed drawn from problems arising in the sciences, engineering, and finance. For more details see the syllabus and the topics listed below.
Please be sure to read the prerequisites and grading policies for the class.
Selection of Topics
- Continuous-time Stochastic Processes
- Stochastic Differential Equations
- Ito's Stochastic Calculus
- Ito Integral, Ito Lemma, Ito Isometry
- Girsanov Theorem
- Forward and Backward Kolomogorov PDEs
- First Passage Time Problems and Elliptic PDEs
- Numerical Methods for SDEs and SPDEs
- Weak vs Strong Numerical Approximation
- Introduction to Statistical Mechanics
- Langevin Equations, Fluctuation-Dissipation Principle, Boltzmann H-Theorem, Green-Kubo Relations
- Adiabatic elimination of fast degrees of freedom
- Analysis of metastability and rare events
- Applications in physics, biology, and engineering.
For more information see the syllabus Δ.
Grading:
The grade for the class will be based on the homework assignments (see policy below), a midterm exam, and take-home final exam as follows:
Homework Assignments 30%
Midterm Exam 30%
Final Exam 40%
Homework Policy:
Assignments will be made weekly and posted on the class website. While it is permissible for you to discuss materials with classmates, the submitted homework must be your own work.
Exams:
The midterm exam will be given in the class on Thursday, October 27th.
The take-home final exam will be given toward the end of the quarter.
Supplemental Class Notes:
Class Annoucements:
Homework Assignments:
Turn all homeworks into my mailbox in South Hall 6th floor by 4:30pm on the due date. Homeworks will be returned in class.
HW1: (Due Thursday, ??)
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