As my research area is in Applied Mathematics and Computational Science, this advice is for students who might be interested working in this area. This may also be useful for engineers who are looking for descirptions of our applied mathematics offerings.
I have listed most of the applied math courses offered in our department below. While every student's plan of study will vary depending on interests, I have starred (*) those courses that are typically taken by graduate students who work with me. Students may also take a few courses outside of the mathematics department if it suites their interest/research project.
Marty Day also has a website with general advice for VT math graduate students, links to interesting courses offered in other departments as well as a number of good links to generic graduate school advice.
Bill Layton at the University of Pittsburgh has posted a manifesto on how to give a 20 minute talk.
I have compiled a list of job opportunities here. These don't include opportunities for M.S. and Ph.D. positions, which usually follow from finding a good place to study and a good advisor to work with. We can discuss those opportunities personally.
Virginia Tech has a very active initiative in computational science. This is supported by a number of faculty members in the Department of Mathematics and the Interdisciplinary Center for Applied Mathematics. There is also a new undergraduate program in Computational Modeling and Data Analytics (CMDA) offered in the College of Science which has a lot of interesting appled mathematics courses. Our faculty are heavily engaged in research areas that can be classified as applied and computational math, numerical analysis, computational science and/or scientific computing. There have been a number of recent faculty hires across the Colleges of Engineering and Science that naturally interface with our research interests. Although a formal program does not yet exist, we can put together a plan of study that emphasizes computational science for students with a good background in math and computer programming (e.g., C, Julia, or Matlab).
Below is a listing of courses of interest to graduate students interested in applied mathematics, numerical analysis, and scientific computing/computational science courses offered in our program. Graduate level courses (5000+) are described in more detail here. Unless otherwise stated, these courses run every year and two semester sequences run fall then spring. Although each student has different objectives, the courses that I usually recommend for my students are marked with a (*).
This is essential background for understanding material in later courses such as Differential Equations, Numerical Analysis, etc. It is also a required course for graduate work in the Mathematics department. This is often taken by first year graduate students depending on your preparation.
This one semester course varies quite a bit depending on who teaches it. It is usually project oriented and requires students to teach themselves how to program in Matlab. It would be very rare for a graduate student to take this course, though it satisfies the computationally intensive course requirement for our graduate program.
Most graduate students will skip this course and go directly into the 5000 level Numerical Analysis sequence after taking 4225-6. This depends on background. Although I haven't yet recommended this course to one of my students, it offers a good introduction to numerical analysis to graduate students who may not know much about applied math. It also checks the computationally intensive course requirement in our program.
Basic set theoretic and topological notions, fundamental theorems of measure and integration, differentiation, applications to linear analysis.
This provides good mathematical background and would be more important if stochastic variables are of interest. This sequence is primarily important since it provides preparation for the Functional Analysis sequence.
Existence theorems, linear theory, stability theory, periodic solutions, Poincare-Bendixson theory, boundary-value problems, functional differential equations.
This course is generally a classical treatment of differential equations. A general description of topics is given and the emphasis and order of topics may change depending on the instructor.
The first semester focuses on theorems of existence, uniqueness, continuous dependence and differentiability of solutions on parameters. It does touch on solutions of linear systems, periodic solutions, as well as linear stability analysis of nonlinear systems. The second semester continues with stability analysis, Floquet theory, the Poincare-Bendixson theorem as well as some topics from dynamical systems theory.
Various graduate level topics in applied mathematics for graduate students in mathematics and qualified students in other areas. May be taken for credit more than once with department consent. Consent required.
The topics offered in these courses change every year. General comments on special topics will be given at the end.
Partial differential equations of first and second order, hyperbolic equations, elliptic equations and Green's functions, parabolic equations, canonic forms, application to physics and engineering.
This course varies a lot depending on the instructor and the appropriateness of this course depends on student background.
Computational procedures for ordinary differential equations including Runge-Kutta methods, variable-step Runge-Kutta methods, predictor-corrector methods, applications to two-point boundary-value problems and parameter estimation. Error control, relative and absolute stability, methods for stiff equations; with computer assignments. Senior standing in engineering, science, or mathematics, and some programming ability required.
This course runs in the fall, though not every year.
A survey of the construction, analysis, and implementation of numerical algorithms in linear algebra, nonlinear equations and optimization, approximation by polynomials, quadrature, and ordinary differential equations. High-level programming language required.
THE sequence for numerical analysts. The first semester primarily consists of numerical linear algebra including forward and backward error analysis (stability and conditioning), numerous matrix factorizations including the singular value decomposition, QR, LU, Cholesky, etc. As well as iterative algorithms for linear systems and eigenvalue problems.
The second semester is contains approximation theory. This includes polynomial approximation and interpolation, quadrature, nonlinear equations (Newton's method, continuation, etc.), and optimization (secant methods, globalization methods such as trust-region methods).
Finite difference methods for initial and boundary value problems for partial differential equations. Consistency, stability, convergence, dispersion, and dissipation. Methods for linear and nonlinear elliptic and parabolic equations, first- and second-order hyperbolic equations, and nonlinear conservation laws. High-level programming language required.
Finite difference and finite volume methods for approximation of partial differential equations. Frequently taught from the excellent book by Strikwerda with additional notes.
Weak formulations of boundary-value problems for elliptic partial differential equations. Finite element spaces. Approximation theory for finite element spaces. Error estimates. Effects of numerical integration and curved boundaries. Nonconforming methods. Concrete examples of the application of the finite element method. Efficient implementation strategies. Time dependent problems. High-level programming language required.
This is the course we all fight over to teach. The course description covers the standard topics. Other topics are highlighted in the last few weeks depending on the research emphasis of the instructor.
Presentation and analysis of numerical methods for solving common mathematical and physical problems. Methods of solving large sparse linear systems of equations, algebraic eigenvalue problems, and linear least squares problems. Numerical algorithms for solving constrained and unconstrained optimization problems. Numerical solutions of nonlinear algebraic systems. Convergence, error analysis. Hardware and software influences. Efficiency, accuracy, and reliability of software. Robust computer codes.
This course is taught in Math every other year and could be labeled Scientific Computing 101. It would be a foundational course in a computational science / scientific computing program. However, there is too much overlap between this sequence and MATH 5465-6 to take them both. This sequence specifically addresses high performance computing issues that computational scientists need to know.
This course is offered in the fall semester. The graduate catalog states
Introduction to mathematical techniques for modeling and simulation, parameter identification and analysis of biological systems. Emphasis on both theoretical and practical issues and methods of computation, with concrete applications. Suitable for students from the mathematical and life sciences who have a basic foundation in multivariate calculus and ordinary differential equations. 5515: Continuous models and methods. 5516: Discrete models and methods.
The follow-up course 5516 rarely runs, so don't expect to take 5515-6 as a sequence.
This course is offered in the spring semester. The graduate catalog states
Determinants, rank, linear systems, eigenvalues, diagonalization, Gram-Schmidt process, Hermitian and unitary matrices, Jordan canonical form, variational principles, perturbation theory, Courant minimax theorem, Weyl's inequality, numerical methods for solving linear systems and for determining eigenvalues. science or engineering.
A lot of overlap with MATH 5465 but is better described as an advanced linear algebra course. Often taken by students outside of math to fulfill background or course requirements.
Unified course in the calculus of variations and control theory including multiple integral problems and distributed parameter control systems. Necessary conditions, sufficient conditions, nonclassical problems, optimal control, distributed parameter control, computational methods.
A classical course extending concepts from calculus to function spaces. It has important applications to control theory and optimization.
Introduction to stochastic models used in financial market analysis and associated computational methods. 5725: Brownian motion, stochastic integration, Ito calculus, martingales, no-arbitrage pricing, Black-Scholes formula, basic term-structure models. 5726: PDE characterizations for American, Asian and various other path-dependent options, development and application of numerical methods for computation. Must meet pre-requisites or have instructor's consent.
This sequence is offered every other year and alternates with Stochastic Processes.
Banach spaces, Hilbert spaces, linear operators on Banach and Hilbert spaces, Riesz Representation Theorems, spectral theory, topological vector spaces, other topics in functional analysis.
The sequence has foundational results that are important in control and optimization of distributed parameter systems (i.e. PDEs). It does vary quite a bit depending on the instructor, but the essentials are always covered.
Advanced topics in applied mathematics for graduate students in mathematics, science, and engineering. May be taken for credit more than once with department consent. Consent required.
Topics courses are discussed below.
Our department offers multiple special topics courses each year that are either developed by faculty or are in response to an interest of a group of graduate students (frequently a group of motivated students have proposed courses in topics that are of interest to them leading to new special topics courses, the minimum number of registered students must be seven).
Historically, the courses of interest to applied math students fall under the Topics in Analysis or Topics in Applied Math numbers. An incomplete list includes:
- Advanced Finite Element Methods (has been offered several times and included special studies in e.g., mixed finite elements, error estimation, and discontinuous Galerkin finite elements)
- Advanced Topics in Functional Analysis
- Approximation of Large-Scale Dynamical Systems
- Discrete Dynamical Systems
- Distributed Parameter Systems
- Finite Element Methods for Quantum Nanoscience
- Mathematical Modeling Of Atmospheric And Oceanic Flows
- Partial Differential Equations
- Robust Control
- Sensitivity Analysis
- Stochastic Processes (offered almost every other year)
