Integrated Applied Mathematics (IAM)

https://ceps.unh.edu/integrated-applied-mathematics

Degrees Offered: Ph.D.

This program is offered in Durham.

Applied mathematics is the study of mathematical and computational methods for solving problems in science and engineering. Over the last several decades, the tools of applied mathematics have dramatically expanded the boundaries of scientific inquiry. It is becoming increasingly important to exploit the full power of mathematical and computational analysis to address the grand challenge problems facing society in areas including energy, climate, environmental sustainability, materials, finance and economics, biophysics and healthcare, and beyond. 

The Integrated Applied Mathematics (IAM) Program at the University of New Hampshire (UNH) is a unique inter-departmental graduate program designed to facilitate interdisciplinary research by preparing highly motivated and capable students to become experts in mathematical and computational problem solving. The IAM Program draws its faculty from the nine departments within the UNH College of Engineering and Physical Sciences (CEPS) and thus is inherently interdisciplinary. The Program gives students the opportunity to explore the frontier where the sciences meet cutting-edge mathematical analysis and high-performance computing. Current areas of mathematical and scientific focus include (but are not limited to): applied nonlinear (asymptotic, variational) analysis of ODEs and PDEs; high-dimensional dynamical systems and bifurcation theory; numerical algorithms for PDEs; machine learning and optimization theory; fluid dynamics; biophysics; materials and solid mechanics; space and plasma physics; and mathematical geo- and environmental science. Graduates of the program acquire the skills and training needed to enter numerous professions, including careers in research, education, and industry.

Curriculum

The IAM Program offers a unique curriculum emphasizing advanced applied mathematical methods and high-performance computing. A Ph.D. candidate in IAM is expected to develop expertise in both applied and computational mathematics as well as one area of specialization (e.g., fluid dynamics, plasma physics and space physics, geoscience, dynamical systems, biophysics, etc.).

Facilities

UNH hosts a state-of-the-art CRAY CS500 supercomputer with over 3000 compute cores and over 12 terabytes of memory. This supercomputer supports multiple projects involving IAM students and faculty.

Admission and Financial Support

Graduate students are admitted to the IAM program with a variety of backgrounds. The expectation is that a student entering the program will have a B.S. or M.S. degree in a technical field such as a mathematics, science, or an engineering discipline. Students with a M.S. degree in a technical field are particularly well suited for admission to the IAM Program. Students have been supported through Teaching Assistantships, CEPS Fellowships, and Research Assistantships funded by federal grants.

Integrated Applied Mathematics (IAM)

IAM 830 - Graduate Ordinary Differential Equations

Credits: 3

Course is a graduate-level course on ordinary differential equations. It is designed to be accessible to first-year graduate students from math, science or engineering backgrounds who have had a first undergraduate course in differential equations, along with a standard calculus sequence. The course is designed to begin with an intensive review of undergraduate differential equations and then will proceed to handle more advanced concepts, starting with multi-dimensional coupled systems of ordinary differential equations, exponential matrix solutions, using coordinate transformations for conversion to standard forms, nonlinear systems and transform-based solutions, using coordinate transformations for conversion to standard forms, nonlinear systems and transform-based techniques. The course will have an interdisciplinary and applied style and will cover the following topics: Intense review of undergraduate differential equations, Power Series and Fourier Series solutions, Multi-dimensional D.E.s, eigenvectors and Jordan forms, Numerical Methods, Nonlinear D.E.s Dynamical Systems and Chaos.

Grade Mode: Letter Grade

IAM 851 - Introduction to High-Performance Computing

Credits: 3

Course gives an introduction to select areas of high-performance computing, providing a basis for writing and working with high-performance simulation codes. The three main topics are: 1) basic software engineering, 2) high-performance and parallel programming, and 3) performance analysis and modeling. Additional topics may include heterogeneous architectures like GPUs and data analysis/visualization. Prereq: Enrollment in a CEPS graduate program, MATH 753, working knowledge of a programming language (C or Fortran), or by permission of instructor.

Grade Mode: Letter Grade

IAM 932 - Graduate Partial Differential Equations

Credits: 3

Graduate level introduction to the analysis of linear and nonlinear partial differential equations. topics include: separation of variables, Fourier series, weak and strong solutions, eigenfunction expansions, the Strum-Liouville problem, Green's functions and fundamental solutions, method of characteristics, and conservation laws. Prereq: Ordinary Differential Equations and Linear Algebra.

Grade Mode: Letter Grade

IAM 933 - Applied Functional Analysis

Credits: 3

Introduction to rigorous mathematical analysis from the perspective of applications. Topics include: metric and normed spaces; convergence; completeness; continuity; Lebesgue measure theory; convergence theorems; Banach, Hilbert, Lp, and Sobolev spaces; orthogonality, bases, and projections; Sturm-Liouville theory; spectral theory; distributions; and weak solutions. Applications including to differential and integral equations, are presented throughout. Prereq: real analysis or graduate introductory courses in mathematical physics or applied mathematics.

Grade Mode: Letter Grade

IAM 940 - Asymptotic and Perturbation Methods

Credits: 3

Introduction to the asymptotic analysis of linear and nonlinear algebraic equations, ODEs, and PDEs and the asymptotic approximation of integrals arising as transform solutions to ODEs/PDEs. Topics include: algebraic equations and dominant balance; asymptotic approximations; complex variable theory and the asymptotic evaluation of integrals via Laplace's method, stationary phase, and steepest descents; the method of matched asymptotic expansions (boundary-layer theory), coordinate straining, multiple scales, averaging, homogenization theory, and WKBJ analysis for singularly perturbed ODEs and PDEs. Prereq: MATH 527, MATH 528, MATH 644 or equivalent. Pre- or Coreq: PHYS 931.

Grade Mode: Letter Grade

IAM 950 - Spatiotemporal and Turbulent Dynamics

Credits: 3

Advanced graduate course on the dynamics of spatiotemporal patterns in nonlinear time-dependent PDEs. Topics include nonlinear pattern formation, bifurcations and symmetry, nonlinear WKB analysis, phase diffusion/amplitude modulation theory, unstable coherent structures in turbulence, and periodic orbit theory. Example systems include 1d and 2d Swift-Hohenberg equation, the 1d Kuramoto-Sivashinsky equation, Rayleigh-Benard convection, and Navier-Stokes in plane Couette and pipe flows. Prereq: MATH 847 and IAM 932, or equivalent; or permission.

Grade Mode: Letter Grade

IAM 961 - Numerical Analysis I: Numerical Linear Algebra

Credits: 3

Introduction to numerical analysis and computational methods for linear systems. Topics include: IEEE floating point arithmetic; vector norms and induced norms; conditioning; projectors; LU decompositions; pivoting; Cholesky factorization; QR decompositions; Gram-Schmidt orthogonalization; Householder triangularization; Singular Value decompositions; least squares problems; stability; eigenvalue problems; power iterations; QR algorithm; Krylov methods; Arnoldi iteration; GMRES; Lanczos iteration; Conjugate gradient algorithms; and Preconditioning. Prereq: scientific programming and linear algebra.

Grade Mode: Letter Grade

IAM 962 - Numerical Partial Differential Equations

Credits: 3

Numerical analysis applied to partial differential equations. Initial topics include the implementation of finite difference and spectral methods applied to the heat equation, wave equation, Burger's equation, and other model equations. The remainder of the course treats numerical analysis, starting with a brief review of function spaces. The primary topics include approximation theory for Sobolov spaces, projection operators, completeness, convergence, and error estimates. Prereq: IAM 961 or permission.

Grade Mode: Letter Grade

IAM 995 - IAM Special Topics

Credits: 1-4

Investigations of graduate-level problems or topics in Integrated Applied Mathematics. May be repeated barring duplication of subject.

Repeat Rule: May be repeated up to unlimited times.

Grade Mode: Letter Grade

IAM 999 - Doctoral Research

Credits: 0

Doctoral Research.

Grade Mode: