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Topics in Applied Mathematics

by: Dr. Filomena Hegmann

Topics in Applied Mathematics APPM 7400

Dr. Filomena Hegmann

GPA 3.76


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Class Notes
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This 2 page Class Notes was uploaded by Dr. Filomena Hegmann on Thursday October 29, 2015. The Class Notes belongs to APPM 7400 at University of Colorado at Boulder taught by Staff in Fall. Since its upload, it has received 72 views. For similar materials see /class/231872/appm-7400-university-of-colorado-at-boulder in Applied Math at University of Colorado at Boulder.


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Date Created: 10/29/15
APPM 7400 PSEUDOSPECTRAL METHODS AND RADIAL BASIS FUNCTIONS Pseudospectral PS methods Approx 20 lectures Introduction to Finite Differences FD Lagrange s amp Newton39s interpolation formulas Elementary error estimates and illustrations of the Runge phenomenon Elementary ways to generate nite difference FD stencils Pade approximations Pade method for FD formulas Nonequispaced FD weights algorithm FD stencils in 2D FD limits Differentiation matrices DMs Fourier transform series FFT Relations between different types of Fourier transforms series Gibbs39 phenomenon Convolution theorem Applications of the DFT FFT algorithm via sparse matrix factorization 2D Fourier transform rotation property PS in periodic cases PS DM equivalence to FD limit of increasing order PS in non periodic cases Runge phenomenon theory Lebesque constants Jacobi and Chebyshev polynomials Key properties of PS methods Convergence in nonperiodic periodic and nonsmooth cases DM eigenvalues Time stepping and stability considerations ODE methods and stability concepts root condition stability domain etc von Neumann analysis PS variations and enhancements Staggered grids preconditioning Polar and spherical coordinates double Fourier method spherical harmonics PS methods and timespace domain comer singularities Fictitious points for ODE boundary value problems and for highorder IBV problems Some applications Nonlinear waves numerical uid mechanics turbulence etc Radial basis functions RBFs Approx 23 lectures Introduction to RBFs Cubic splines end conditions Introduction to RBFs Via cubic splines Introduction Via failure of xed basis function sets Nonsingularity results for different types of radial functions Micchelli s theorem Different types of radial functions nonsmooth VS smooth global vs compact Matlab RBF demo code PC7MODEL Shape parameter issues Determinant formula for the REF interpolant Mathematica codes Various shape parameter 8 gt 0 results ill conditioning ContourFade algorithm RBFs on spheres RBFQR algorithm Besseltype radial functions Runge phenomenon issues Boundary effects boundary conditions for RBF approximations Runge phenomenon for RBF interpolants Gibbs phenomenon for RBF interpolants Spatially variable 8 opportunities RBFs for PDEs Kansa s pioneering work elliptic equations 1D equispaced case infinite interval and periodic results some closedform expressions Periodic implementations circlesphere time stepping stability theorem RBFs for wavetype equations RBFs together with least squares RBF with domain decomposition RBF implementation of special operators Laplacian gradient diV etc Some additional topics Fast numerical algorithms eg BFGP Krylov fast multipole methods


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