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Directors:
Randolph E. Bank
Philip E. Gill
Michael Holst

Administrative Contact:
Terry Le

Office: AP&M 7431
Phone: (858)534-9813
Fax: (858)534-5273
E-mail: tele@ucsd.edu
Computational and Applied Mathematics Courses
The following courses in computational and applied mathematics are offered by the mathematics department in support of the computational science-oriented undergraduate and graduate education and training programs across the campus. Some are cross-listed with our departmental partners in the CSME program.
170ABC. Introduction to Numerical Analysis (4-4-4)
170A. Introduction to Numerical Analsysis: Linear Algebra (4)
Analysis of numerical methods for linear algebraic systems and least squares problems. Orthogonalization methods. Ill conditioned problems. Eigenvalue and singular value computations. Three lectures, one recitation. Knowledge of programming recommended. Prerequisite: Math. 20F. (F,S)

170B. Introduction to Numerical Analysis: Approximation and Nonlinear Equations (4)
Rounding and discretization errors. Calculation of roots of polynomials and nonlinear equations. Interpolation. Approximation of functions. Three lectures, one recitation. Knowledge of programming recommended. Prerequisite: Math. 170A. (W)

170C. Introduction to Numerical Analysis: Ordinary Differential Equations (4)
Numerical differentiation and integration. Ordinary differential equations and their numerical solution. Basic existence and stability theory. Difference equations. Boundary value problems. Three lectures, one recitation. Prerequisite: Math. 170B or consent of instructor. (S)
171AB. Introduction to Numerical Optimization (4-4)
171A. Introduction to Numerical Optimization: Linear Problems (4)
Linear optimization and applications. Linear programming, the simplex method, duality. Selected topics from integer programming, network flows, transportation problems, inventory problems, and other applications. Three lectures, one recitation. Knowledge of programming recommended. Credit not allowed for both Math. 171A and Econ. 172A. Prerequisite: Math. 20F.

171B. Introduction to Numerical Optimization: Nonlinear Problems (4)
Convergence of sequences in Rn, multivariate Taylor series. Bisection and related methods for nonlinear equations in one variable. Newton’s methods for nonlinear equations in one and many variables. Unconstrained optimization and Newton’s method. Equality-constrained optimization, Kuhn-Tucker theorem. Inequality-constrained optimization. Three lectures, one recitation. Knowledge of programming recommended. Credit not allowed for both Math. 171B and Econ. 172B. Prerequisite: Math. 171A.
173. Mathematical Software Scientific Programming (4)
Development of high quality mathematical software for the computer solution of mathematical problems. Three lectures, one recitation. Prerequisites: Math. 170A or Math. 174 and knowledge of FORTRAN. (W)
174/274. Numerical Methods for Physical Modeling (4)
(Math 174 and 274 are Conjoined.) Floating point arithmetic, direct and iterative solution of linear equations, iterative solution of nonlinear equations, optimization, approximation theory, interpolation, quadrature, numerical methods for initial and boundary value problems in ordinary differential equations. Students may not receive credit for both Math. 174 and PHYS 105, AMES 153 or 154. Students may not receive credit for Math. 174 if Math. 170A, B, or C has already been taken. Graduate students will do an extra assignment/exam. Prerequisites: Math. 20D with a grade of C or better and Math. 20F with a grade of C or better, or consent of instructor.
175/275. Numerical Methods for Partial Differential Equations (4)
(Math 174 and 274 are Conjoined.) Mathematical background for working with partial differential equations. Survey of finite difference, finite element, and other numerical methods for the solution of elliptic, parabolic, and hyperbolic partial differential equations. Formerly Math. 172; students may not receive credit for Math. 175/275 and Math. 172. Graduate students do an extra paper, project, or presentation, per instructor. Prerequisite: Math. 174 or Math. 274, or consent of instructor.
176. Advanced Data Structures (4)
Descriptive and analytical presentation of data structures and algorithms. Lists, tables, priority queues, disjoint subsets, and dictionaries data types. Data structuring techniques include linked lists, arrays, hashing, and trees. Performance evaluation involving worst case, average and expected case, and amortized analysis. Crecit not offered for both Math. 176 and CSE 100. Equivalent to CSE 100. Prerequisites: CSE 12, CSE 21, or Math. 15B, and CSE 30, or consent of instructor.
179/279. Projects in Computational and Applied Mathematics (4)
(Math 179 and Math 279 are Conjoined.) Mathematical models of physical systems arising in science and engineering, good models and well-posedness, numerical and other approximation techniques, solution algorithms for linear and nonlinear approximation problems, scientific visualizations, scientific software design and engineering, project-oriented. Graduate students will do an extra paper, project, or presentation per instructor. Prerequisite: Math. 174 or Math. 274 or consent of instructor.
270ABC. Numerical Analysis (4-4-4)
270A. Numerical Linear Algebra (4)
Error analysis of the numerical solution of linear equations and least squares problems for the full rank and rank deficient cases. Error analysis of numerical methods for eigenvalue problems and singular value problems. Iterative methods for large sparse systems of linear equations. Prerequisites: graduate standing or consent of instructor.

270B. Numerical Approximation and Nonlinear Equations (4)
Iterative methods for nonlinear systems of equations, Newton’s method. Unconstrained and constrained optimization. The Weierstrass theorem, best uniform approximation, least-squares approximation, orthogonal polynomials. Polynomial interpolation, piecewise polynomial interpolation, piecewise uniform approximation. Numerical differentiation: divided differences, degree of precision. Numerical quadrature: interpolature quadrature, Richardson extrapolation, Romberg Integration, Gaussian quadrature, singular integrals, adaptive quadrature. Prerequisites: Math. 270A or consent of instructor.

270C. Numerical Ordinary Differential Equations (4)
Initial value problems (IVP) and boundary value problems (BVP) in ordinary differential equations. Linear methods for IVP: one and multistep methods, local truncation error, stability, convergence, global error accumulation. Runge-Kutta (RK) Methods for IVP: RK methods, predictor-corrector methods, stiff systems, error indicators, adaptive time-stepping. Finite difference, finite volume, collocation, spectral, and finite element methods for BVP; a priori and a posteriori error analysis, stability, convergence, adaptivity. Prerequisites: Math. 270B or consent of instructor.
271ABC. Numerical Optimization (4-4-4)
Formulation and analysis of algorithms for constrained optimization. Optimality conditions; linear and quadratic programming; interior methods; penalty and barrier function methods; sequential quadratic programming methods. Prerequisite: consent of instructor. (F,W,S)
272ABC. Numerical Partial Differential Equations (4-4-4)
272A. Numerical Partial Differential Equations I (4)
Survey of discretization techniques for elliptic partial differential equations, including finite difference, finite element and finite volume methods. Lax-Milgram Theorem and LBB stability. A priori error estimates. Mixed methods. Convection-diffusion equations. Systems of elliptic PDEs. Prerequisites: graduate standing or consent of instructor.

272B. Numerical Partial Differential Equations II (4)
Survey of solution techniques for partial differential equations. Basic iterative methods. Preconditioned conjugate gradients. Multigrid methods. Hierarchical basis methods. Domain decomposition. Nonlinear PDEs. Sparse direct methods. Prerequisites: Math. 272A or consent of instructor.

272C. Numerical Partial Differential Equations III (4)
Time dependent (parabolic and hyperbolic) PDEs. Method of lines. Stiff systems of ODEs. Space-time finite element methods. Adaptive meshing algorithms. A posteriori error estimates. Prerequisites: Math. 272B or consent of instructor.
273ABC. Advanced Techniques in Computational Mathematics (4-4-4)
273A. Advanced Techniques in Computational Mathematics I (4)
Models of physical systems, calculus of variations, principle of least action. Discretization techniques for variational problems, geometric integrators, advanced techniques in numerical discretization. Project-oriented; projects designed around problems of current interest in science, mathematics, and engineering. Prerequisite: graduate standing or consent of instructor.

273B. Advanced Techniques in Computational Mathematics II (4)
Nonlinear functional analysis for numerical treatment of nonlinear PDE. Numerical continuation methods, pseudo-arclength continuation, gradient flow techniques, and other advanced techniques in computational nonlinear PDE. Project-oriented; projects designed around problems of current interest in science, mathematics, and engineering. Prerequisite: Math. 273A or consent of instructor.

273C. Advanced Techniques in Computational Mathematics III (4)
Adaptive numerical methods for capturing all scales in one model, multiscale and multiphysics modeling frameworks, and other advanced techniques in computational multiscale/multiphysics modeling. Project-oriented; projects designed around problems of current interest in science, mathematics, and engineering. Prerequisite: Math. 273B or consent of instructor.
276. Numerical Analysis in Multi-Scale Biology (4)
(Cross-listed with BENG 276/CHEM 276) Introduces mathematical tools to simulate biological processes at multiple scales. Numerical methods for ordinary and partial differential equations (deterministic and stochastic), and methods for parallel computing and visualization. Hands-on use of computers emphasized, students will apply numerical methods in individual projects. Prerequisite: consent of instructor.
277AB. Topics in Computational Mathematics (4-4)
277A. Topics in Computational and Applied Mathematics (4)
Topics vary from year to year. May be repeated for credit with consent of advisor. Prerequisite: graduate standing or consent of instructor.

277B. Topics in Numerical Mathematics (4)
Topics vary from year to year. May be repeated for credit with consent of advisor. Prerequisite: consent of instructor.
278AB. Seminar in Computational Mathematics (1-1)
278A. Seminar in Computational Mathematics (1)
Various topics in computational mathematics. Prerequisite: graduate standing or consent of instructor. (S/U grade only.)

278B. Seminar in Mathematical Physics/PDE (1)
Various topics in mathematical physics and partial differential equations. Prerequisite: graduate standing or consent of instructor. (S/U grade only.)
The following is the new syllabus for the Numerical Analysis Qualifying Exam based on Math 270ABC; it was revised in Fall 2009.
270ABC Qualifying Exam Syllabus

Numerical Analysis Qualifying Exam

The Numerical Analysis Qualifying Exam is based in the syllabus for Mathematics 270ABC. The course is divided in to six segments, each approximately five weeks in length, that survey major topics in Numerical Analysis and Scientific Computation. These basic topics are:

  • Numerical Linear Algebra for Dense Matrices (270A)
  • Numerical Linear Algebra for Sparse Matrices (270A)
  • Nonlinear Equations and Optimization (270B)
  • Approximation Theory (270B)
  • Initial Value Problems for Ordinary Differential Equations (270C)
  • Two Point Boundary Value Problems for Ordinary Differential Equations (270C)

Because of the timing of the qualifying exam, questions on the last topic (2-point BVP) are not included in the exam.

In five weeks time, none of these topics can be addressed very deeply. However, those choosing to specialize in some area of numerical analysis or scientific computation need to have some understanding of the issues, both theoretical and algorithmic, in all of these areas. There exist more specialized and advanced courses (e.g., Mathematics 271ABC and Mathematics 272ABC) that address some of these topics in a more comprehensive fashion.

The textbook for Mathematics 270ABC, Numerical Mathematics, by Quarteroni, Sacco, and Saleri ([1] below) has chapters that cover most of the material in the syllabus. This textbook is also available online through SpringerLink at http://www.springerlink.com/content/q67046/. Other texts that provide a survey of many of the topics covered in this course include Dalquist and Björck [2,3], Atkinson [4], and Isaacson and Keller [5].

General References for Mathematics 270ABC

  1. Alfio Quarteroni, Riccardo Sacco, and Fausto Saleri. Numerical mathematics, volume 37 of Texts in Applied Mathematics. Springer-Verlag, Berlin, second edition, 2007.
  2. Ake Björck and Germund Dahlquist. Numerical methods. Prentice-Hall Inc., Englewood Cliffs, N.J., 1974. Translated from the Swedish by Ned Anderson, Prentice-Hall Series in Automatic Computation.
  3. Germund Dahlquist and Åke Björck. Numerical methods. Dover Publications Inc., Mineola, NY, 2003. Translated from the Swedish by Ned Anderson, Reprint of the 1974 English translation.
  4. Kendall E. Atkinson. An introduction to numerical analysis. John Wiley & Sons Inc., New York, second edition, 1989.
  5. Eugene Isaacson and Herbert Bishop Keller. Analysis of numerical methods. Dover Publications Inc., New York, 1994. Corrected reprint of the 1966 original [Wiley, New York; MR0201039 (34 #924)].


270A: Numerical Linear Algebra

Numerical Linear Algebra for Dense Matrices

  1. Gaussian Elimination: Condition number and stability of linear systems of equations, LU factorization and its variants, pivoting, iterative refinement, roundoff error analysis, complexity.
  2. Linear Least Squares Problems: basic theory (projection, best approximation), over and under-determined systems, Gram-Schmidt and its variants, Householder and Givens transformations, QR factorization, singular value decomposition (SVD), stability and conditioning of least squares problems.
  3. Algebraic Eigenvalue Problem: Schur Decomposition, power method, inverse iteration, Rayleigh Quotient iteration, Hessenberg form, QR method for eigenvalues, Sturm sequences, Gerschgorin estimates, conditioning of eigenvalues and eigenvectors.

Numerical Linear Algebra for Sparse Matrices

  1. Sparse Gaussian Elimination: sparse matrix data structures, graph model for Gaussian Elimination, computing the fill-in, ordering (minimum degree, nested dissection), complexity, band matrices.
  2. Iterative Methods for Linear Equations: convergence theory, basic iterative methods (Jacobi, Gauss-Seidel, SOR, SSOR, ILU), complexity, orderings, convergence criteria.
  3. Krylov Subspace Methods: Conjugate gradients, preconditioning, generalized condition number, Lanczos method, GMRES, Biconjugate gradients.

Specialized References for Mathematics 270A

  1. Gene H. Golub and Charles F. Van Loan. Matrix computations. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, MD, third edition, 1996.
  2. G. W. Stewart. Matrix algorithms. Vol. I. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1998. Basic decompositions.
  3. G. W. Stewart. Matrix algorithms. Vol. II. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. Eigensystems.
  4. J. H. Wilkinson. The algebraic eigenvalue problem. Monographs on Numerical Analysis. The Clarendon Press Oxford University Press, New York, 1988. Oxford Science Publications.
  5. Richard S. Varga. Matrix iterative analysis, volume 27 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, expanded edition, 2000.
  6. Alan George and Joseph W. H. Liu. Computer solution of large sparse positive definite systems. Prentice-Hall Inc., Englewood Cliffs, N.J., 1981. Prentice-Hall Series in Computational Mathematics.


270B: Numerical Approximation and Nonlinear Equations

Nonlinear Equations and Optimization

  1. Scalar Equations: Fixed point iteration, bisection, Newton's method, secant method, order of convergence.
  2. Nonlinear Systems: Steepest descent, Approximate Newton Methods, Kantorovich Theorem, Quasi Newton Methods, line search, trust region.
  3. Optimization: unconstrained, equality and inequality constraints, necessary and sufficient conditions, penalty and barrier methods, active set methods, interior point methods, KKT systems, linear and quadratic programming.

Approximation Theory

  1. Polynomial Interpolation: Lagrange Interpolation, divided differences, Aiken-Neville and Barycentric formulas, error estimates, Runge's example, Chebyshev approximation, orthogonal polynomials, least squares approximation.
  2. Piecewise Polynomial Approximation: C^0 approximation spaces, error estimates for interpolation and least squares, Peano Kernel Theorem, Hermite and spline approximation.
  3. Numerical Quadrature: Newton-Cotes and Gaussian Quadrature, basic and composite formulas, error estimates, singular integrals, Richardson extrapolation, Euler-Maclaurin Formula, Romberg integration, adaptive quadrature.

Specialized References for Mathematics 270B

  1. Philip E. Gill, Walter Murray, and Margaret H. Wright. Practical optimization. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1981.
  2. J. M. Ortega and W. C. Rheinboldt. Iterative solution of nonlinear equations in several variables, volume 30 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. Reprint of the 1970 original.
  3. Philip J. Davis. Interpolation and approximation. Dover Publications Inc., New York, 1975. Republication, with minor corrections, of the 1963 original, with a new preface and bibliography.
  4. Carl de Boor. A practical guide to splines, volume 27 of Applied Mathematical Sciences. Springer-Verlag, New York, revised edition, 2001.
  5. Kendall Atkinson and Weimin Han. Theoretical numerical analysis, volume 39 of Texts in Applied Mathematics. Springer, Dordrecht, third edition, 2009. A functional analysis framework.


270C: Numerical Ordinary Differential Equations

Initial Value Problems for Ordinary Differential Equations

  1. Theoretical Background: existence, uniqueness and stability for first order IVP systems, Lipschitz continuity, Gronwall's Lemma, converting IVP's to standard form (first order systems).
  2. Single-Step Methods: explicit and diagonally implicit Runge-Kutta and Runge-Kutta-Fehlberg methods, error estimates and order of convergence, absolute stability, A and L stability, stiff equations, adaptive time stepping.
  3. Multi-Step Methods: solution of linear difference equations, Adams methods, predictor corrector, backward difference formulas, weak and strong stability, error analysis and orders of convergence.

Two Point Boundary Value Problems for Ordinary Differential Equations

  1. Theoretical Background: existence, uniqueness and stability for the 2-point BVP, Ritz and Galerkin variational formulations, Lax-Milgram Theorem.
  2. Finite Difference Methods: derivation of finite difference equations, consistency, stability and error estimates, convection dominated problems.
  3. Finite Element Methods: basic formulation, stability and error estimates, duality and Nitsche's trick, reference elements, streamline diffusion Petrov-Galerkin method for convection dominated problems.

Specialized References for Mathematics 270C

  1. K. Eriksson, D. Estep, P. Hansbo, and C. Johnson. Computational differential equations. Cambridge University Press, Cambridge, 1996.
  2. C. William Gear. Numerical initial value problems in ordinary differential equations. Prentice-Hall Inc., Englewood Cliffs, N.J., 1971.
  3. E. Hairer, S. P. Nørsett, and G. Wanner. Solving ordinary differential equations. I, volume 8 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, second edition, 1993. Nonstiff problems.
  4. E. Hairer and G. Wanner. Solving ordinary differential equations. II, volume 14 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 1991. Stiff and differential-algebraic problems.
  5. Kendall Atkinson and Weimin Han. Theoretical numerical analysis, volume 39 of Texts in Applied Mathematics. Springer, Dordrecht, third edition, 2009. A functional analysis framework.
  6. Gilbert Strang and George J. Fix. An analysis of the finite element method. Prentice-Hall Inc., Englewood Cliffs, N. J., 1973. Prentice-Hall Series in Automatic Computation.

During the 2009-2010 academic year, a special version of Math 273ABC was run, with lectures based in part on the upcoming new third edition of ``Green's Functions and Boundary Value Problems'' by Stakgold and Holst (John-Wiley, 2010, 888 pages). The syllabus and outline of the lectures for this special version of 273ABC is below.
273ABC Syllabus (Used in 2009-2010, lectures based in part on Stakgold-Holst Book)

Math 273ABC Syllabus (2009-2010 version)

Math 273ABC was designed as a project-oriented course in computational mathematics, created as the educational component of Michael Holst's NSF CAREER Award in 1999. It has run every other year (in Fall of odd years) since 1999, and became one of the core math contributions to the campus-wide CSME doctoral program when it was launched in 2007. The structure of the course has been based on a sequence of quarter-length projects, whereby the student designs a mathematical model using partial differential equations, analyzes the model using techniques of applied analysis, analyzes the quality of discretizations of the model using modern approximation theory, and then develops an efficient computational solution strategy. (See the 273ABC syllabi from the UCSD catalog as listed above.) During the 2009-2010 academic year, a more traditionally structured version of Math 273ABC was run, with lectures based in part on the upcoming third edition of the book ``Green's Functions and Boundary Value Problems'', by Stakgold and Holst. This new third edition of Stakgold's classic graduate text in applied analysis will appear in Fall 2010 (John-Wiley, 888 pages), and includes about 275 pages of new material on nonlinear functional analysis, nonlinear PDE, approximation theory, and modern computational methods for nonlinear elliptic equations. The lectures for this special version of 273ABC were based in part on the new material written for the book; a syllabus and outline of the lectures is given below. Depending on whether the material is covered in other graduate courses a particular year, and depending on the interests of the CSME students, this more traditionally structured version of 273ABC may be run again in some future year (the 273ABC course usually runs every other year, alternating with Math 272).

273A: Ordinary Differential Equations and Dynamical Systems

  1. Linear operator theory [1,2]: Linear systems and exponentials of operators. Canonical forms of operators (nilpotent, Jordan, and real canonical forms).
  2. Existence and Uniqueness [3,4]: Existence and uniqueness of solutions to initial value problems. Continuous dependence on initial conditions. Gronwall's inequality, Cauchy-Peano theorem, Picard-Lindelöf theorem.
  3. Flow Maps [5,6]: Time-t maps, phase space and flow maps, first integrals, phase volume and the connection to the divergence of a vector field.
  4. Dynamical Systems [7,2]: Qualitative analysis for nonlinear systems, phase portraits, classification of equilibria, Hartman-Grobman Theorem, Lyapunov stability theorem, Poincaré-Bendixson theorem. Brouwer fixed point theorem and its proof via the no-retraction theorem and the determinant lemma [8].
  5. Classical mechanics [6,9]: Lagrangian and Hamiltonian systems with one degree of freedom. Hamilton's and Hamilton's phase space principle, Euler-Lagrange and Hamilton's equations. Generating functions and Hamilton-Jacobi theory.

References for Mathematics 273A

  1. Morris W. Hirsch and Stephen Smale. Differential equations, dynamical systems, and linear algebra. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Pure and Applied Mathematics, Vol. 60.
  2. Lawrence Perko. Differential equations and dynamical systems, volume 7 of Texts in Applied Mathematics. Springer-Verlag, New York, third edition, 2001.
  3. Jack K. Hale. Ordinary differential equations. Robert E. Krieger Publishing Co. Inc., Huntington, N.Y., second edition, 1980.
  4. Earl A. Coddington and Norman Levinson. Theory of ordinary differential equations. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.
  5. Vladimir I. Arnol'd. Ordinary differential equations. Springer Textbook. Springer-Verlag, Berlin, 1992. Translated from the third Russian edition by Roger Cooke.
  6. Ralph Abraham and Jerrold E. Marsden. Foundations of mechanics. Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, Reading, Mass., 1978. Second edition, revised and enlarged, With the assistance of Tudor Ratiu and Richard Cushman.
  7. John Guckenheimer and Philip Holmes. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, volume 42 of Applied Mathematical Sciences. Springer-Verlag, New York, 1990. Revised and corrected reprint of the 1983 original.
  8. Yakar Kannai. An elementary proof of the no-retraction theorem. Amer. Math. Monthly, 88(4):264-268, 1981.
  9. V. I. Arnol'd. Mathematical methods of classical mechanics, volume 60 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1989. Translated from the Russian by K. Vogtmann and A. Weinstein.


273B: Partial Differential Equations and Nonlinear Analysis

  1. Linear Analysis [1,2]: Banach and Hilbert spaces. Linear operators, adjoints, four fundamental subspaces. Operator norms. Bounded, unbounded, closed, compact, projection operators. Fredholm alternative. Semi-groups. Linear functionals, dual space, reflexive and separable spaces. Strong, weak, weak-* topologies. Riesz, Hahn-Banach, Banach-Schauder, and Banach-Steinhaus Theorems.
  2. Hyperbolic Partial Differential Equations [3,4]: First order systems and characteristics. Classification and well-posedness. Cauchy problem for hyperbolic equations, wave equation, conservation laws. Weak solutions, jump conditions, shocks. Weak, strong, symmetric hyperbolicity. Local and global existence.
  3. Elliptic and Parabolic Partial Differential Equations [2,4]: Dirichlet and Neumann problems for elliptic and parabolic equations. Classical, strong, weak solutions. Distributions and fundamental solutions. Definition of Sobolev spaces. Ellipticity, inf-sup conditions, energy estimates, Lax-Milgram Theorem, existence, uniqueness. Maximum principles, a priori L^oo estimates, Harnack inequality.
  4. Nonlinear Analysis [2,5,6]: Nonlinear maps in Banach spaces, variations, Gateaux and Frechet derivatives. Taylor expansion, Newton's Method. Inverse and Implicit Function Theorems, Bifurcation. Stationarity of functionals. Weakly closed sets, weak-lower-semi-continuity, reflexive Banach spaces, weak convergence. Existence of minimizers. Applications to nonlinear PDE.
  5. Fixed-Point Techniques [2,5,7]: Fixed-point iteration, rates of convergence. Banach Fixed-Point Theorem. Ordered Banach spaces, maximum principle, monotone increasing maps. Method of sub- and super-solutions, compatible barriers. Extension of the Brouwer Theorem to the Schauder Fixed-Point Theorem. Homotopy and continuation methods. Applications to nonlinear PDE.

References for Mathematics 273B

  1. E. Kreyszig. Introductory Functional Analysis with Applications. John Wiley & Sons, Inc., New York, NY, 1990.
  2. I. Stakgold and M. Holst. Green's Functions and Boundary Value Problems. John Wiley & Sons, Inc., New York, NY, third edition, 2010.
  3. M. Renardy and R. C. Rogers. An Introduction to Partial Differential Equations. Springer-Verlag, New York, NY, 1993.
  4. L. C. Evans. Partial Differential Equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1998.
  5. E. Zeidler. Nonlinear Functional Analysis and its Applications, volume I: Fixed Point Theorems. Springer-Verlag, New York, NY, 1991.
  6. S. Fucik and A. Kufner. Nonlinear Differential Equations. Elsevier Scientific Publishing Company, New York, NY, 1980.
  7. S. Kesavan. Topics in Functional Analysis and Applications. John Wiley & Sons, Inc., New York, NY, 1989.


273C: Applied Harmonic Analysis and Approximation Theory

  1. Sobolev spaces [1,2] Classification of open bounded domains in R^n. Banach spaces C_k and L_p. Weak derivatives, W[k,p] norm, L_p-based Sobolev spaces W[k,p]. H[k,p] via completion of C_oo. 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 [3,4]: Holder spaces C[k,alpha]. Modulus of continuity, L_p 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 [5,6]: 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 [2,3,7]: 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 [2,6,8]: 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, L_1-approximation.

References for Mathematics 273C

  1. R. A. Adams and J. F. Fornier. Sobolev Spaces. Academic Press, San Diego, CA, second edition, 2003.
  2. I. Stakgold and M. Holst. Green's Functions and Boundary Value Problems. John Wiley & Sons, Inc., New York, NY, third edition, 2010.
  3. R. A. DeVore and G. G. Lorentz. Constructive Approximation. Springer-Verlag, New York, NY, 1993.
  4. H. Triebel. Interpolation Theory, Function Spaces, and Differential Operators. North-Holland, Amsterdam, 1978.
  5. I. Daubechies. Ten Lectures on Wavelets. SIAM, Philadelphia, PA, 1992.
  6. Karstan Urban. Wavelet Methods for Elliptic Partial Differential Equations. Oxford University Press, New York, 2009.
  7. D. Braess. Nonlinear Approximation Theory. Springer-Verlag, Berlin, Germany, 1980.
  8. P. Oswald. Multilevel Approximation. B. G. Teubner, Stuttgart, Germany, 1994.