An Advanced (Graduate) Certificate in Computational Science

April 26, 2000

The State Education Department and the Chancellor of the State University of New York have approved an Advanced (Graduate) Certificate in Computational Science at the University at Buffalo. Computational Science is an emerging discipline, uniting ideas in mathematics and computer science together with applications arising in science and engineering. Because it complements theoretical and experimental investigations, Computational Science is often referred to as "the third science". The Certificate recognizes a student's special training in scientific computing and applications. This Certificate is a cooperative program involving the Center for Computational Research and the Departments of Mathematics, Physics, Chemical Engineering and Mechanical and Aerospace Engineering; other departments have expressed an interest in joining the program.

The Certificate requires a total of 15 hours of graduate coursework (usually 5 courses). A two-semester course in High Performance Computing is common for all students in the program; the remaining nine hours consists of options chosen from computing and applications courses specific to each participating department. By exposing students to some of the fundamental methods in computational science, while also providing them with an understanding of the application of computing within their chosen discipline, the requirements highlight the interdisciplinary nature of computational science.

The University at Buffalo has made a significant investment in creating the Center for Computational Research, a facility that provides leading edge computing and visualization resources to the University and Western New York communities. The Certificate Program and the new courses in High Performance Computing were designed to help train UB students in the methods of scientific computing on modern computing hardware available at CCR.

View-Download PDF Version

Rationale

This Advanced (Graduate) Certificate in Computational Science is designed to provide students at the University at Buffalo with training in advanced scientific computing in combination with specialized education in traditional disciplines of science and engineering. Computational Science is an emerging discipline, uniting ideas of Mathematics and Computer Science together with applications arising in science and engineering. Computational Science must be distinguished from Computer Science. Computer Science concerns the design of hardware and software for the computer systems of the future. Computational Science concerns the exploitation of current hardware and software to address large-scale computational problems that arise in fields of engineering and science. As computers become more integrated into the working environment of business and industry, students will require thorough and systematic training in computational science in order to secure employment in advanced technology fields. This Advanced Certificate program is designed to address this need. The Program provides for coursework in current techniques of high performance computing, and for advanced study in a chosen discipline.

More specifically, Computational Science applies techniques from Mathematics and Computer Science to problems of applied interest. Active areas of research in Computational Science include the design of fast algorithms for the solution of linear algebra and differential equations, optimization, database management, and visualization. In addition, solving cutting-edge problems in any of these fields necessitates exploiting advanced computer architectures. Thus working in computational science requires a background in high performance computing (the knowledge of computational methods on advanced architectures) together with training in an application area (differential equations or linear algebra or optimization or database or visualization or management of advanced computing systems, among others).

Because of the inherently interdisciplinary nature of Computational Science, the program of study for this Advanced Certificate is designed as a cooperative venture among several departments and organized research units. Part of the program (high performance computing) is common to all students, and part is specific to participating departments. This arrangement promotes the study of the methods of scientific computing while simultaneously advancing the study of disciplinary science and engineering. Study in computational science complements study in traditional departmental sub-specialties, and a Certificate Program thus broadens the educational background of students.

This Certificate Program will be comprised of several options. The Certificate Program requires a total of 15 hours of graduate coursework, 6 hours (40%) of this coursework is common to all students in the Program and consists of a two-semester course in High Performance Computing, while the remaining 9 hours (60%) consists of options chosen from approved disciplinary courses related to computing, computational simulations, and numerical modeling. This arrangement exposes students to some of the fundamental methods of computational science, while also providing them with an understanding of the application of computing within their chosen discipline.

Students wishing to earn this Advanced Certificate must be admitted into a department participating in the Certificate Program, either in the graduate program (for a degree or for the Certificate) or in an approved 5-Year Combined BA-MA degree program. All coursework, including the course hours in High Performance Computing, must be approved by the participating departments for graduate credit within their programs and must be applicable toward a registered graduate degree. At present, there are four departments participating in this Certificate Program, each of which has registered graduate programs leading to Masters and Ph.D. degrees. The participating departments are:

• Chemical Engineering
• Mathematics
• Mechanical and Aerospace Engineering
• Physics

Thus, an Advanced (Graduate) Certificate in Computational Science provides unique training for students in science and engineering. Approval of the Certificate by the State Education Department will provide appropriate recognition of the special education earned by these students through their work in Computational Science.

Curriculum

Requirements for the Advanced Certificate consist of:

1. Acceptance into the graduate or combined BA-MA degree program of a participating department
2. Completion of 15 hours of coursework, including:
   • 6 credit hours consisting of the required courses High Performance Computing I and High Performance Computing II;
   • 9 credit hours of elective coursework consisting of departmentally approved courses as detailed below.

Students must maintain a B average in all Certificate courses and must be in good academic standing in their home department.
The courses High Performance Computing I and II (HPC I and II) were created in Spring, 1999, through the efforts of the University at Buffalo's Center for Computational Research (CCR), and serve to educate students in the methods of scientific computing on modern hardware architectures. HPC I and II are cross-listed among CCR and three departments participating in the Certificate Program, ensuring departmental acceptance of this coursework toward their degree programs. [ Note. Physics has agreed to cross-list the HPC course, and is preparing the necessary paperwork. In the interim, the Physics Department will accept the currently listed HPC courses as applying towards a degree in Physics. ] The participating departments and CCR will coordinate the instruction assignments for HPC I and II.

In addition to the required HPC courses, students must complete elective courses in their home departments, chosen from a program designed by the individual departments. This additional coursework falls into four options, one from each of the departments:

1. Option in Chemical Engineering. In addition to High Performance Computing I and II, students must take CE 531 Chemical Engineering Analysis I. Students must also choose two additional courses from among: CE 532, Chemical Engineering Analysis II, CE 526 Statistical Mechanics, CE 533 Introduction to Finite Elements, or CE 580 Nonlinear Analysis. Students may request a waiver to allow a substitution for these requirements, but any substitution requires prior approval of the Director of Graduate Studies in consultation with the Director of CCR.
2. Option in Mathematics. In addition to High Performance Computing I and II, students must take MTH 537, Introduction to Numerical Analysis I and MTH 538, Introduction to Numerical Analysis II, and one course from among MTH 545 Introduction to Ordinary Differential Equations, MTH 549, Introduction to Partial Differential Equations, MTH 645, Advanced Ordinary Differential Equations, MTH 649, Partial Differential Equations, MTH 543, Fundamentals of Applied Mathematics, MTH 539, Methods of Applied Mathematics. Students may request a waiver to allow a substitution for these requirements, but any substitution requires prior approval of the Director of Graduate Studies in consultation with the Director of CCR.
3. Option in Mechanical and Aerospace Engineering. In addition to High Performance Computing I and II, students must take three courses from among: MAE 510, Virtual Reality, MAE 529, Finite Element Structural Analysis, MAE 541, Topics in Finite Element Analysis, MAE 542, Engineering Applications of Computational Fluid Dynamics, MAE 550, Optimization in Engineering Design, MAE 564, Manufacturing Automation, Sys 500, 506, 507, 508, 509 Special Topics in System Identification, Sys 571, System Analysis, Sys 581, Optimal Estimation Methods. Students may request a waiver to allow a substitution for these requirements, but any substitution requires prior approval of the Director of Graduate Studies in consultation with the Director of CCR.
4. Option in Physics. In addition to High Performance Computing I and II, students must take three courses from among PHY 501, Mathematical Physics, PHY 505, Computational Physics I, PHY 506, Computational Physics II, PHY 507, Quantum Mechanics I, PHY 508, Quantum Mechanics II, PHY 509, Classical Dynamics, PHY 513 Electrodynamics I, PHY 514, Electrodynamics II, and PHY 519 Statistical Mechanics I, PHY 520, Statistical Mechanics II. Students may request a waiver to allow a substitution for these requirements, but any substitution requires prior approval of the Director of Graduate Studies in consultation with the Director of CCR.

Participating departments, in consultation with the Director of CCR, will approve the awarding of the Advanced Certificate for students registered in that department.

Catalog description of courses

Cross-listed HPC courses

COR 501/CE 620/MTH 667/MAE609/PHY 515 High Performance Computing I: The first semester of a two semester course sequence that will introduce students to the fundamental concepts of scientific computing, with particular attention given to algorithms that are well-suited to high performance computer architectures. The first semester will concentrate on computational linear algebra, including iterative and direct methods for solving linear systems and for eigenvalue problems, and the use of BLAS and other public domain libraries.

COR 502/CE 621/MTH 668/MAE 610/PHY 516 High Performance Computing II: The second semester of a two semester course sequence that will introduce students to the fundamental concepts of scientific computing, with particular attention given to algorithms that are well-suited to high performance computer architectures. The second semester will concentrate on scientific computing in applications, including stochastic methods, FFTs, and finite element and finite difference methods.

Note: PHY 515 and 516 are the designated course numbers for the pending cross-listing of HPC I and II by the Physics Department.

Chemical Engineering

CE 526 Statistical Mechanics: A first approach to statistical mechanics and its methods. Ensembles and the statistical formulation of the laws of thermodynamics. Mono-and poly-atomic ideal gases. Imperfect gases and graph theory. Dense liquids, distribution functions, computer simulation techniques. Lattice models, renormalization group methods. Microscopic dynamics and transport properties. Inhomogeneous fluids.

CE 531 Chemical Engineering Analysis I: Development and application of mathematical techniques of particular interest to chemical engineers. Process of formulating mathematical models for simple chemical processes. Differential equations, ordinary and partial. Analytical (exact and approximate) methods of solving equations.

CE 532 Chemical Engineering Analysis II: Computational methods for solving differential equations that model physical phenomena in chemical engineering.

CE 533 Introduction to Finite Element Methods: Finite element methods will be developed in the general framework of the weighted residual methods. Basis function (Lagrange, Hermite, Spline) will be developed in one, two, and three dimensions. Programming with FEM will be discussed for linear and nonlinear problems as well as for moving boundary problems. Iterative solution schemes will be compared and employed. Physical problems will be solved from the areas of fluid flow, transport phenomena, and reaction engineering.

CE 580 Nonlinear Analysis: Autonomous and nonautonomous systems; nonlinear ODEs; phase plane analysis; linear stability theory; Lyapunovs direct method; bifurcation theory; cusps, isolas, and limit cycles; periodic solutions and Hopf bifurcation and stability analysis; nonlinear PDEs; pattern formation in chemical systems; transition to chaos; hydrodynamic stability. Examples focus on reaction and transport processes.

Mathematics

MTH 537 Introduction to Numerical Analysis I: Lagrangian interpolation. Newton-Cotes quadrature formulas, Gaussian quadrature and orthogonal polynomials. Romberg quadrature, difference equations, numerical solution of ordinary differential equations, predictor-corrector methods, Runge-Kutta methods. Additional reading on selected topics.

MTH 538 Introduction to Numerical Analysis II: Solution of nonlinear equations and simultaneous linear equations, linear least-square approximations. Chebyshev polynomials, minimax approximations, calculation of eigenvalues and eigenvectors. Additional reading on selected topics.

MTH 539 Methods of Applied Mathematics I: Matrices, equivalence, quadratic and hermitian forms, eigenvalues, invariants, function spaces and Sturm-Liouville problems. Calculus of variations, Euler-Lagrange equations, constraints, variable endpoints, Sturm-Liouville theory, Rayleigh-Ritz method. Integral equations. Green's functions, Hilbert-Schmidt theory, Fredholm theory, singular integral equations.

MTH 543 Fundamentals of Applied Mathematics I: Mathematical formulation and analysis of models for phenomena in the natural sciences. Includes derivation of relevant differential equations from conservation laws and constitutive relations. Potential topics include diffusion, stationary solutions, traveling waves, linear stability analysis, scaling and dimensional analysis, perturbation methods, variational and phase space methods, kinematics and laws of motion for continuous media. Examples from areas might include, but are not confined to, biology, fluid dynamics, elasticity, chemistry, astrophysics, geophysics.

MTH 545 Introduction to Ordinary Differential Equations: Existence and uniqueness of solutions, continuation of solutions, dependence on initial conditions and parameters; linear systems of equations with constant and variable coefficients; autonomous systems, phase space and stability. Additional reading on selected topics.

MTH 549 Introduction to Partial Differential Equations: A rigorous study of the wave, heat, and potential equations in two dimensions, focusing on fundamental concepts, methods and properties of solutions. General properties of second order linear equations in two dimensions, classification, characteristics, well-posed problems and approximation. Solution of the three types of equations by the method of separation of variables and Fourier series. Poisson representation formulas. Nonhomogeneous problems and Green's function. Formulation and properties of the Tricomi problem. Discussion of a simplified problem in fluid dynamics. Additional reading on selected topics.

MTH 645 Advanced Ordinary Differential Equations: Existence theorems, linear and nonlinear differential equations, regular and singular boundary value problems, stability theory of linear and nonlinear systems. Liapunov's second method. Geometric theory of differential equations in the plane.

MTH 649 Partial Differential Equations: The Cauchy problem for partial differential equations, classification of second order linear partial differential equations, properties of solutions for elliptic, parabolic and hyperbolic equations, existence of solutions for elliptic partial differential equations. Topics from Fourier and Laplace transforms, potential theory, Green's functions, integral equations, Sobolev spaces, and Schwartz distributions.

Mechanical and Aerospace Engineering

MAE 510 Special Topics'Virtual Reality: Virtual Reality applications and research Virtual Reality refers to techniques used to synthesize real world by recreating its physical, visual and audio models and representations (there are other forms of reflexes and cognition methods also). In this course we will study this field and review various applications and research issues extensively. We will also learn interactive computer graphics programming techniques using OpenGL which is a preferred graphics programming API environment. The following broadly describes the topics
Concepts in Interactive Computer Graphics

Programming in OpenGL/GLUT

Introduction to World Tool Kit Libraries

Virtual Reality Hardware and Software

Application and research in Virtual Reality
The course will be structured into three parts; OpenGL Programming, Weekly VR Seminars and Projects. Your performance in each of the three components will be weighted to determine your grade at the end of the semester. Graphics programming will consist of lecture classes and programming assignments. In the seminar series each student will present a 30-40 minute in-class seminar on a VR topic.

Note: The Department has initiated a petition for new course approval that will formalize this special topics course as MAE 574.

MAE 529 Finite Element Structural Analysis: This course is intended to bridge the gap between the theory and application of finite element modeling. At the end of the course the student will be able to judge if a problem is appropriate for finite element analysis, will know how to determine the model type, will be able to determine the type of elements to use and how many, and will have the background to judge the accuracy of the results obtained. These practical skills are difficult to acquire in a strictly theoretical course. Specific topics will include: fundamentals of FEM, basic input data required, definition of the problem, the finite element model, debugging the model, element performance and distortion, use of refined mesh modeling and substructuring, dynamic FEM, application for thermal analysis, and calibrating the accuracy of finite element models.

MAE 541 Topics in Finite Element Analysis: Advanced topics in finite element analysis including but not necessarily limited to mathematical foundations, linear transient analysis, basic non-linear analysis - material properties and geometric non-linearities, error estimation and adaptivity, advanced solution techniques, and special problems e.g. incompressible materials, thin structures. Course will involve extensive project work on realistic applications.

MAE 542 Engineering Applications of Computational Fluid Dynamics: This course is intended for seniors and beginning graduate students interested in computer based analysis of engineering problems in fluid mechanics and heat transfer. Application of computer analysis to engineering design of fluid/thermal systems will be emphasized. Students need not have a graduate level background in fluid mechanics or heat transfer, however, an undergraduate level is necessary. The general governing equations and methods to solve them, including; finite-difference, finite-volume, panel methods, and finite element methods, will be surveyed. Introduction to the use of state-of-the-art computer tools for analysis and graphical representation of results will give the student a broad view of computational fluid mechanics for engineering applications in the fluid/thermal sciences. This course is particularly suited for Masters of Engineering students.

MAE 550 Optimization in Engineering Design: Optimization techniques with applications in various aspects of engineering design. Concepts of design variables, constraints, objective functions, penalty functions, Lagrange multipliers. Techniques for solving constrained and unconstrained optimization problems: classical approaches, steepest descent, conjugate directions, conjugate gradient, controlled random searches, etc. Discussion of generalized reduced gradient, sequential linear programming, and recursive quadratic programming strategies. Computer implementation of optimization schemes. Applications and examples in the design of engineering components and systems.

MAE 564 Manufacturing Automation: Rapid growth of automation has been a strong motivation for engineers to acquire skills in the area of Computer Aided Manufacturing and Design. This course will serve as an introduction to the theory of automation as related to manufacturing and design integration. We will look at various hardware, software and algorithm issues involved in fast and flexible product development cycles. Following strategies of automated manufacturing systems will be covered: CAD-CAM: integration, programming and simulation; Robotics: applications in welding, material handling and human intensive processes; Reverse Engineering: modeling product from laser and CMM data of parts; Virtual Environments: industrial applications of virtual reality and prototyping; Intelligent Diagnostics: sensor fusion for machine tool monitoring; Automated Inspection: computer vision and methods of automated quality control; Design for Manufacturing: issues involved in concurrent product development

SYS 500, 506, 507, 508, 509 Special Topics/System Identification. This course covers fundamental systems identification techniques. Topics covered include: introduction to the identification process; brief review of mathematical topics necessary for the course; time-domain approach for identification of linear time-invariant systems. This will include continuous and discrete time models, Markov parameters, ARX, ARMA, ARMAX and other models; frequency domain method; Eigensystem Realization Algorithm; and Kalman filters.

Note: The Department has initiated a petition for new course approval that will formalize this special topics course as SYS 584.

SYS 571 Systems Analysis 1: Development of mathematical techniques for the analysis of systems in the time domain. Introduction to state space concepts. Review of matrices and vectors. Vector spaces. Coordinate transformation. Jordan canonical form. State-space representation of control systems. Solutions of state space equations. Controllability and observability. Feedback control structures.

SYS 581 Optimal Estimation Methods: Introduction to linear and nonlinear estimation methods with emphasis on both theory and implementation. Batch and sequential strategies, real-time and post-experiment estimation are covered. Includes both parameter estimation and state estimation.

Physics

PHY 501 Introduction to Mathematical Physics I: Complex variables; ordinary and partial differential equations; special functions; Fourier series and transforms; Laplace transforms; boundary value problems; Green's functions; matrices; function spaces; group theory; tensors.

PHY 505 Computational Physics I: This course integrates the elements of numerical analysis and computer programming to study a variety of problems in classical, quantum ans statistical physics. Basic numerical operations : root finding, interpolation, matrix inversion, numerical differentiation and quadrature. Structured programming of basic operations in a high level language such as FORTRAN, Pascal, C. Numerical solution of ordinary differential equations of classical and quantum physics. Matrix operations and solutions of linear systems. Statistical analysis of data, curve fitting, the Fast Fourier Transform. Computer graphics.

PHY 506 Computational Physics II: Advanced numerical techniques applied to time-independent and time-dependent problems that lead to linear and non-linear partial differential equations, and to the physics of systems with many degrees of freedom. Large linear systems of equations; non-linear systems; boundary value problems; importance sampling and Monte Carlo methods; use of mathematical library routines in numerical programs; introduction to computer algebra and symbolic manipulation programs.

PHY 507-508 : Quantum Mechanics I and II: First course deals with basic principles and formulations of quantum mechanics, classical limit and WKB method, representations and transformations, pure and mixed states, conservation laws, symmetries, central force problems and solutions; approximation methods for stationary states. Second course concentrates on coupling of angular momenta, rotation matrix, time-independent perturbation theory and quantum transitions, many electron systems ans scattering theory.

PHY 509 Classical Dynamics: D'Alembert's principle of virtual work; Lagrange's equations and application to the dynamics of particles, rigid bodies, rotating systems, small oscillations, non-holonomic systems; principle of least action; Hamilton's principle; canonical transformations; Poisson brackets; integral invariants; Hamilton-Jacobi theory.

PHY 513-514 Electrodynamics I and II: First course deals with treatment of boundary-value problems in electrostatics and magnetostatics by method of images, orthogonal function and expansions, Green's functions and conformal mapping; multipoles, dielectrics, and magnetic materials. Second course concentrates on Maxwell's equations, propagation of plane waves in various media, radiating systems, special theory of relativity, radiation my moving charges, radiation damping, scattering and absorption of radiation by a bound system.

PHY 519 Statistical Mechanics I: Properties of physical systems explained in terms of microscopic constituents. Lioville's theorem, distribution functions, and Boltzmann equation; the ergodic theory; the H theorem, Gibbs ensemble theory; classical Bose-Einstein and Fermi-Dirac statistics; discussion of black body radiation, Bose-Einstein gas, specific heat of metals, white dwarfs, viscosity, heat conductivity, and other equilibrium and nonequilibrium phenomena. LEC

PHY 520 Statistical Mechanics II: Determination and use of the distribution functions and partition functions defined in canonical and grand ensembles; fluids in equilibrium and non equilibrium, phase transitions and scaling theory, theory of condensation; electric and magnetic susceptibilities, hard sphere gases; electron gas, Ising lattice theory and density matrix.

Faculty

Faculty participating in the initiative in high performance computing are drawn from the several participating departments at the University at Buffalo.

Resources and Support

The coursework required for this Advanced Certificate consists of classes drawn from regularly scheduled departmental offerings, and the specialized, required courses High Performance Computing I and II are cross-listed by several participating departments. Responsibility for staffing departmental courses resides with those departments. UB's Center for Computational Research (CCR) will help coordinate the program and course offerings. The CCR is not an academic department and provides no credit-bearing course instruction. The CCR does provide hardware, software and consulting support for those faculty and students using high performance computing facilities. CCR thus provides access to the tools required for applying the methods and techniques learned as part of the Certificate coursework to current research problems in science and engineering. This combination of specialized training in computational science, advanced education in a field of science and engineering, and access to a leading center of academic computing, offers students the opportunity for a program of advanced study unmatched in the State of New York.