Math 273C (Advanced Techniques in Computational Mathematics III)

Instructor: Prof. Michael Holst (5739 AP&M, mholst@math.ucsd.edu)
Term: Spring 2010
Lecture: 9:30a-10:45a TuThu, APM 2402
Class webpage: http://ccom.ucsd.edu/~mholst/teaching/ucsd/273c_s10/

Specific Course Topics for Spring 2010:
  • As part of a book project with Ivar Stakgold, we will give a series of lectures in the Winter and Spring 2010 quarters on the theory and numerical treatment of nonlinear elliptic equations. The focus of the Spring 2010 quarter is on mathematical tools in Applied Harmonic Analysis and Approximation Theory, as needed for the numerical treatment of nonlinear elliptic equations.
Textbooks/References:
  • AdFo03: R. A. Adams and J. F. Fornier. Sobolev Spaces. Academic Press, San Diego, CA, second edition, 2003.
  • StHo10: I. Stakgold and M. Holst. Green's Functions and Boundary Value Problems. John Wiley & Sons, Inc., NY, NY, third edition, 2010.
  • DeLo93: R. A. DeVore and G. G. Lorentz. Constructive Approximation. Springer-Verlag, New York, NY, 1993.
  • Trie78: H. Triebel. Interpolation Theory, Function Spaces, and Differential Operators. North-Holland, Amsterdam, 1978.
  • Daub92: I. Daubechies. Ten Lectures on Wavelets. SIAM, Philadelphia, PA, 1992.
  • Urba09: Karstan Urban. Wavelet Methods for Elliptic Partial Differential Equations. Oxford University Press, New York, 2009.
  • Brae80: D. Braess. Nonlinear Approximation Theory. Springer-Verlag, Berlin, Germany, 1980.
  • Oswa94: P. Oswald. Multilevel Approximation. B. G. Teubner, Stuttgart, Germany, 1994.

SYLLABUS: We will cover the five major topics listed below, roughly every two weeks of the quarter. The individual topics listed within each of the five major topics will be covered in varying degree, depending on the available lecture time.
  1. Sobolev spaces (AdFo03,StHo10):
    Classification of open bounded domains in Rn. Banach spaces Ck and Lp. Weak derivatives, W[k,p] norm, Lp-based Sobolev spaces W[k,p]. H[k,p] via completion of Coo. Meyers-Serin Theorem (W=H). Embedding, compactness, density, extension, trace theorems. Poincare inequality. Hilbert spaces H[k] = W[k,2]. Fractional spaces W[s,p] via fractional norm or Fourier transform.
  2. Besov spaces (DeLo93,Trie78):
    Holder spaces C[k,alpha]. Modulus of continuity, Lp modulus of smoothness. The B[s;p,q] norm and Besov space B[s;p,q]. K-functionals. Norm, operator, and space interpolation. Relation to Sobolev and Potential spaces, third distinct construction of fractional order Sobolev spaces. The DeVore diagram. Approximation spaces. Interpolation and quasi-interpolation.
  3. Wavelet Decomposition (Daub92,Urba09):
    Haar system, piecewise linear systems, detail spaces. B-splines. Refinable functions, scaling function, multiresolution analysis. Orthogonal and biorthogonal wavelets. Jackson and Bernstein inequalities, Riesz stable bases. Interpolatory wavelets, semi-orthogonal wavelets, curvelets, ridglets, Fast wavelet transform. Vanishing moments and compression.
  4. Approximation Theory (DeLo93,Brae80,StHo10):
    Bernstein polynomials, Weierstrass Theorem. Best linear approximation in Banach Spaces, the Hilbert Space Projection Theorem. Petrov-Galerkin methods, inf-sup conditions, a priori and a posteriori error estimates. Nonlinear approximation and N-widths. Convergence and optimality of adaptive methods. Applications to linear and nonlinear PDE.
  5. Applications to Operator Equations (Urba09,Oswa94,StHo10):
    Iterative methods for linear operator equations. Integral equations, boundary integral methods. Sparse representation of integral operators. Optimal preconditioners for elliptic operators. Littlewood-Paley decomposition. Compression, compressed sensing, l1-approximation.