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Course Criteria
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1.00 Credits
Course description unavailable
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1.00 Credits
This course consists of two parts: a general introduction to variational methods in PDE, and a more focused foray into some special topics. For the former we will cover the direct method in the calculus of variations, various notions of convexity, Noether's theorem, minimax methods, index theory, and gamma-convergence. For the latter we will focus on several specific problems of recent interest, with emphasis on the Ginzburg-Landau energy functional.
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1.00 Credits
The course will cover basic and advanced algorithms for solution of linear and nonlinear systems as well as eigenvalue problems.
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1.00 Credits
Addresses strategies and algorithms in devising efficient discontinuous Galerkin solvers for fluid flow equations such as Euler and Navier-Stokes. The course starts with an introduction to discontinuous Galerkin methods for elliptic and hyperbolic equations and then focuses on the following topics: 1) Serial and parallel implementations of various discontinuous Galerkin operators for curvilinear ele- ments in multiple space dimensions. 2) Explicit, semi-explicit and implicit time discretizations. 3) Multigrid (multi-level) solvers and preconditioners for systems arising from discontinuous Galerkin approximations of the partial differential equations.
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1.00 Credits
Covers fundamental topics in High Performance Computing, with emphasis on Parallel Computing. We will start from basics in programming with C++, creating and compiling sequential codes and employing high performance scientific numerical libraries such as LAPACK and BLAS. Next, we will learn MPI programming and employ parallel linear algebra libraries such as ScaLapack for dense and SuperLU for sparse matrix operations. The focus will be on load balancing, efficient design of communication and multilevel parallelism required for heterogeneous computing. We will continue with advanced topics such as: parallel I/O and GPU-CPU computing. We will use computer laboratory for hands-on tutorials. Grade will be based on 5-6 home assignments/mini-projects and a final project with presentation. Students are encouraged to suggest projects related to the area of their expertise.
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1.00 Credits
Presents the formulation and approximation by the Finite Element Method (FEM) of linear and non-linear problems of Solid Mechanics. The formulation of problems is based on the Virtual Work Principle (VWP). Increasing complexity problems will be considered such as simple bar under traction, beams, plates, plane problems and solids with linear and hyperelastic materials. All problems are formulated using the same sequence of presentation which includes kinematics, strain measure, rigid body deformation, internal work, external work, VWP and constitutive equations. The approximation of the given problems is based on the High-order FEM. Examples will be presented using a Matlab code.
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1.00 Credits
In this advanced topic course, we will go over several aspects of recent mathematical work on kinetic theory. Graduate level PDE is required.
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1.00 Credits
Many modern data sets involve observations about a network of interacting components. Probabilistic and statistical models for graphs and networks play a central role in understanding these data sets. This is an area of active research across many disciplines. Students will read and discuss primary research papers and complete a final project.
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1.00 Credits
The Malliavin calculus is a stochastic calculus for random variables on Gaussian probability spaces, in particular the classical Wiener space. It was originally introduced in the 1970s by the French mathematician Paul Malliavin as a probabilistic approach to the regularity theory of second-order deterministic partial differential equations. Since its introduction, Malliavin's calculus has been extended beyond its original scope and has found applications in many branches of stochastic analysis; e.g. filtering and optimal control, mathematical finance, numerical methods for stochastic differential equations. This course will introduce, starting in a simple setting, the basic concepts and operations of the Malliavin calculus, which will then be applied to the study of regularity of stochastic differential equations and their associated partial differential equations. In addition, applications from optimal control and finance, including the Clark-Ocone foruma and its connection with hedging, will be presented.
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1.00 Credits
We will study a variety of mathematical models for problems in materials science. Mainly we will consider models of phase transformation, static and dynamic. Some of the topics to be treated are: (1) models of phase transformation; (2) gradient flows; (3) kinetic theories of domain growth; (4) stochastic models; (5) free boundary problems. A working familiarity with partial differential equations is required.
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