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Course Criteria
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3.00 Credits
Partial differential equations, including classification, method of characteristics, separation of variables, boundary value and initial value problems, Green's functions, maximum principle, distributions, and weak solutions. Students must have already taken or be concurrently enrolled in Differential Equations 1.
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3.00 Credits
Convergence sequences and series of functions; metric space topology, compactness, completeness and the Ascoli-Arzela Theorem; Contraction Mapping Principle, Implicit Function Theorem; intro to Lebesgue integration leading to $L^p$-spaces; properties of Hilbert spaces, Fourier transform. Students should have knowledge of linear algebra, multivariable calculus, and analysis at the undergraduate level.
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3.00 Credits
Markov chains, Martingales, probability measures, SDE, Brownian motion, Monte Carlo methods. Students should be an APM graduate student.
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3.00 Credits
Fundamentals of linear algebra and numerical linear algebra, including decompositions (LU, QR, SVD), Eigen values, spectral theory, least squares problems. Programming with MATLAB. Students should be an APM graduate student.
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3.00 Credits
Covers interpolation, solution of nonlinear equations and systems, numerical differentiation, numerical integration, numerical solution of ordinary and partial differential equations. Students should be an APM graduate student.
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3.00 Credits
Extends topics of APM 505. Introduces essential iterative methods, Gauss-Seidel, conjugate gradients. Methods for SVD, total least squares and root-finding applications in image analysis. Students should have basic knowledge of numerical linear algebra and a programming language.
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3.00 Credits
Stability, accuracy, consistency, RK methods, finite differences, linear partial differential equations, dispersion, dissipation, method of lines. Applications and programming. Students should have had a previous graduate course in ODE or applied analysis.
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3.00 Credits
Numerical methods for parabolic, elliptic, and hyperbolic partial differential equations, including finite difference/volume, finite element, and spectral methods. Mathematical concepts of stability, consistency, and convergence. Applications to scientific, biomedical, and industrial problems. Students should have a basic knowledge of PDE's and a programming language.
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3.00 Credits
Linear programming, unconstrained nonlinear minimization, line search algorithms, conjugate gradients, quasi-Newton methods, constrained nonlinear optimization, gradient projection, and penalty methods. Completion of courses in Applied Linear Algebra and Computational Methods is strongly recommended prior to enrollment in this course.
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3.00 Credits
Extends approximation theory to global methods, including Fourier and orthogonal polynomial expansions. Applications to imaging and hyperbolic, parabolic, and elliptic partial differential equations. Students should have previous graduate courses in ordinary and partial differential equations, basic programming skills.
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