MAT 280 Laplacian Eigenfunctions Reference
Page (Spring 2007)
Course: MAT 280
CRN: 44770
Title: Laplacian Eigenfunctions: Theory, Applications, and Computations
Class: TTh 4:40pm-6:00pm, 3106 Math. Sci. Bldg.
Instructor: Naoki Saito
Office: 2142 MSB
Email: saito@math dot ucdavis dot edu
Office Hours: By appointment
The general introductory references
- W. A. Strauss: Partial Differential Equations: An Introduction,
Chap. 10 & 11, John Wiley & Sons, 1992.
- R. Courant and D. Hilbert: Methods of Mathematical Physics, Vol. I, Chap. V, VI, & VII, Wiley-Interscience, 1953.
- E. B. Davies: Spectral Theory and Differential Operators,
Cambridge Univ. Press, 1995.
- F. R. K. Chung: Spectral Graph Theory, AMS, 1997.
Lecture 1: Overture: Motivations, Scope and Structure of the Course
Good references on basic Fourier analysis, its relation to PDEs, and some of its history are
- H. Dym and H. McKean: Fourier Series and Integrals, Academic Press, 1972.
- G. B. Folland: Fourier Analysis and Its Applications,
Brooks/Cole, 1992.
- M. A. Pinsky: Introduction to Fourier Analysis and Wavelets,
Brooks/Cole, 2002.
As I mentioned in the class, I will not cover the Laplacian eigenvalue problems
defined on Riemannian manifolds due to the time constraint. If you are interested
in delving into this subject, see -
S. Rosenberg: The Laplacian on a Riemannian Manifold, vol. 31, London Mathematical Society Student Texts, Cambridge Univ. Press, 1997.
Lecture 2: Generalization of Vibrations of 1D String: The Sturm-Liouville
Theory; History of Laplacian Eigenvalue Problems in Rd
Good and elementary references on the Sturm-Liouville theory are
- G. B. Folland: Fourier Analysis and Its Applications,
Sec. 3.5, 3.6, as well as other chapters, Brooks/Cole, 1992.
- K. Yosida: Lectures on Differential and Integral Equations,
Chap. 2, Wiley-Interscience, 1960. Republished by Dover in 1991.
For the derivation of the wave equation (as well as the other PDEs, such as
heat and Poisson's equations), check Appendix 1 of the Folland book above.
For a nice and detailed derivation of the wave equation and the physical meaning
of various boundary conditions can be found in the following wonderful classic:
- P. M. Morse and H. Feshbach: Methods of Theoretical Physics, Part I,
Sec. 2.1, 6.3, McGraw-Hill, 1953. Republished by Feshbach Publishing in 2005.
The following book is an excellent introduction to Hilbert space,
and as an application, the Sturm-Liouville theory is discussed.
You should get one if you have not read this book yet. It is not only
relatively inexpensive ($40) but also includes a lot of interesting comments by
the author throughout the text, in particular, Afterword.
- N. Young: An Introduction to Hilbert Space, Chap. 9, 10, 11,
Cambridge Univ. Press, 1988.
History of Laplacian eigenvalue problems, in particular, prehistory of
spectral geometry can be found in
- M. Kac: "Can one hear the shape of a drum?" Amer. Math. Monthly, vol. 73, no. 4, part 2, pp. 1-23, 1966.
- M. H. Protter:
"Can one hear the shape of a drum? Revisited" SIAM Review, vol. 29, no. 2, pp. 185-197, 1987.
Lecture 3: Problems of Spectral Geometry (Summary)
Also useful is the following book:
- I. Chavel: Eigenvalues in Riemannian Geometry, Academic Press, 1984.
For the derivation of the volume of the unit ball and the surface area of the unit
sphere in Rd, see Sec. 0B of
- G. B. Folland: Introduction to Partial Differential Equations,
2nd Ed., Princeton Univ. Press, 1995.
For the review of the Gamma function, see e.g.,
- G. B. Folland: Fourier Analysis and Its Applications,
Appendix 3, Brooks/Cole, 1992.
Lecture 4: Diffusions on and Vibrations of a Membrane in 2D/3D--I. Basics
The basic reference here is
- W. A. Strauss: Partial Differential Equations: An Introduction,
Sec. 10.1, John Wiley & Sons, 1992.
For the elementary discussion on the completeness of a set of functions
(e.g., a basis) in L2(Ω)
can be found in
- G. B. Folland: Fourier Analysis and Its Applications,
Sec. 3.3-3.4, Brooks/Cole, 1992.
- N. Young: An Introduction to Hilbert Space, Chap. 4,
Cambridge Univ. Press, 1988.
Lecture 5: Diffusions on and Vibrations of a Membrane in 2D/3D--II. 2D Disk
The basic references here are:
- W. A. Strauss: Partial Differential Equations: An Introduction,
Sec. 10.2, John Wiley & Sons, 1992.
- R. Courant and D. Hilbert: Methods of Mathematical Physics, Vol. I,
Sec. V.5.5, Wiley-Interscience, 1953.
For more about the method of series solution for ODEs as well as regular singular
points, check:
- P. M. Morse and H. Feshbach: Methods of Theoretical Physics, Part I,
Sec. 5.2, McGraw-Hill, 1953. Republished by Feshbach Publishing in 2005.
- E. L. Ince: Ordinary Differential Equations, Chap. VII, Dover, 1956.
For more about ODEs of the Euler type, see:
- E. L. Ince: Ordinary Differential Equations, Sec. 6.3, Dover, 1956.
For more about the Bessel functions, the following book contains
everything about them:
- G. N. Watson: A Treatise on the Theory of Bessel Functions, 2nd Ed.,
Cambridge Mathematical Library, Cambridge Univ. Press, 1995 (Originally published
in 1948).
The following books are much shorter than Watson's book but cover the essentials:
- F. Bowman: Introduction to Bessel Functions, Dover, 1958.
- G. B. Folland: Fourier Analysis and Its Applications,
Chap. 5, Brooks/Cole, 1992.
The following are very nice books from Dover and you should definitely
get at least one because they contain not only Bessel functions but also a lot of
other important special functions you should know:
- H. Hochstadt: The Functions of Mathematical Physics, Chap. 8, Dover,
1986 (Originally published by Wiley-Interscience in 1971).
- N. N. Lebedev: Special Functions and Their Applications, Chap. 5, 6, Dover,
1972 (Originally published by Prentice-Hall in 1965).
For the table of zeros of the Bessel functions, see the book of Watson above
or the following must-have book:
- M. Abramowitz and I. A. Stegun: Handbook of Mathematical Functions,
Chap. 9, Dover, 1972.
This book has the following online link:
http://www.math.sfu.ca/~cbm/aands/.
Finally for the science of real drums and percussions, see the following
fascinating book:
- T. D. Rossing: Science of Percussion Instruments, Series in Popular
Science, Vol. 3, World Scientific, 2000.
Lecture 6: Diffusions on and Vibrations of a Membrane in 2D/3D--III. 3D Ball
The basic references here are
- W. A. Strauss: Partial Differential Equations: An Introduction,
Sec. 10.3, John Wiley & Sons, 1992.
- R. Courant and D. Hilbert: Methods of Mathematical Physics, Vol. I,
Sec. V.8, V.9.1, VII.5, Wiley-Interscience, 1953.
- G. B. Folland: Fourier Analysis and Its Applications,
Sec. 6.3, Brooks/Cole, 1992.
Spherical harmonics in Rd with d
≥ 3 are treated in
- G. B. Folland: Introduction to Partial Differential Equations,
2nd Ed., Sec. 2H, Princeton Univ. Press, 1995.
- E. M. Stein and G. Weiss: Introduction to Fourier Analysis on Euclidean Spaces, Sec. IV.2, Princeton Univ. Press, 1971.
Lecture 7: Nodal Sets of Laplacian Eigenfunctions
The basic references here are
- W. A. Strauss: Partial Differential Equations: An Introduction,
Sec. 10.4, John Wiley & Sons, 1992.
- R. Courant and D. Hilbert: Methods of Mathematical Physics, Vol. I,
Sec. V.5, VI.6, Wiley-Interscience, 1953.
Both books contains the famous "Courant Nodal Domain Theorem" claiming
that the kth Laplacian eigenfunction divides the domain Ω (assuming
it is connected) into at most k subdomains.
The following book of Trefethen contains the MATLAB problem to compute
the nodal lines of the Laplacian eigenfunctions for a 2D disk:
- L. N. Trefethen: Spectral Methods in MATLAB, Chap. 11, SIAM, 2000.
This book is really worth reading. You will gain a lot of insights in
programming for solving certain differential equations numerically.
All the MATLAB codes are also available from Prof. Trefethen's website.
Lecture 8: Laplacian Eigenvalue Problems for General Domains--I. Eigenvalues as Minima of the Potential Energy
The basic references here are
- W. A. Strauss: Partial Differential Equations: An Introduction,
Sec. 11.1-11.2, John Wiley & Sons, 1992.
- R. Courant and D. Hilbert: Methods of Mathematical Physics, Vol. I,
Sec. VI.1, Wiley-Interscience, 1953.
Classical but excellent general references on Calculus of Variations are
- R. Courant and D. Hilbert: Methods of Mathematical Physics, Vol. I,
Chap. IV, Wiley-Interscience, 1953.
- S. G. Mikhlin: Mathematical Physics, An Advanced Course, Part II,
North-Holland Publ., 1970.
- V. I. Smirnov: A Course of Higher Mathematics, Vol. IV, Chap. II,
Pergamon Press/Addison-Wesley, 1964.
For more advanced treatment of Calculus of Variations related to PDEs, see
- L. C. Evans: Partial Differential Equations, Chap. 8, AMS, 1998.
Lectures 9, 10: Laplacian Eigenvalue Problems for General Domains--II.
Computation of Eigenvalues
The basic references here are
- W. A. Strauss: Partial Differential Equations: An Introduction,
Sec. 11.3, John Wiley & Sons, 1992.
- R. Courant and D. Hilbert: Methods of Mathematical Physics, Vol. I,
Sec. VI.1-VI.2, Wiley-Interscience, 1953.
For more advanced treatment, see e.g.,
- E. B. Davies: Spectral Theory and Differential Operators, Chap. 4,
Cambridge Univ. Press, 1995.
- A. Henrot: Extremum Problems for Eigenvalues of Elliptic Operators,
Chap.1, Birkhäuser, 2006.
Sec. 1.3 details the Minimax Principle, and also give an example that
the Neumann eigenvalues may not decrease even if the domain volume increases,
which is quite different from the Dirichlet case.
The Rayleigh quotient is the basis of the so-called Power Iterations
for computing eigenvalues of a symmetric real-valued matrix. For the details
of this and other related algorithms, see e.g.,
- L. N. Trefethen and D. Bau, III: Numerical Linear Algebra,
Lecture 27, SIAM, 1997.
- G. H. Golub and C. F. Van Loan: Matrix Computations, 3rd Ed., Sec. 8.2, Johns Hopkins Univ. Press, 1996.
In particular, the "Intermission" in Lecture 10 was based on Lecture 27 of
Trefethen-Bau.
In the beginning of Lecture 9, I briefly mentioned about the Sobolev spaces instead of the space of C20(Ω).
We will not delve into the Sobolev spaces due to the time constraint, but
you should check out the following excellent introductions to them:
- R. A. Adams and J. J. F. Fournier: Sobolev Spaces, 2nd Ed., Elsevier
Science Ltd., Oxford, UK, 2003.
- L. C. Evans: Partial Differential Equations, Chap. 5, AMS, 1998.
- G. B. Folland: Introduction to Partial Differential Equations,
Chap. 6, Princeton Univ. Press, 1995.
- E. H. Loss and M. Loss: Analysis, 2nd Ed., Chap. 7-8, AMS, 2001.
Lecture 11: Laplacian Eigenvalue Problems for General Domains--III.
Completeness of a Set of Eigenfunctions and the Justification of the Separation of
Variables
The basic references are
- W. A. Strauss: Partial Differential Equations: An Introduction,
Sec. 11.3, 11.5, John Wiley & Sons, 1992.
- N. Young: An Introduction to Hilbert Space, Chap. 11,
Cambridge Univ. Press, 1988.
- G. B. Folland: Fourier Analysis and Its Applications,
Sec. 3.3, Brooks/Cole, 1992.
For more advanced treatment, see e.g.,
- L. C. Evans: Partial Differential Equations, Sec. 6.5, AMS, 1998.
In the lecture, I also mentioned the advanced treatment of the heat equation
with the Dirichlet boundary condition using the Sobolev space,
H10(Ω). For the details, see e.g.,
- L. C. Evans: Partial Differential Equations, Chap. 7, AMS, 1998.
Lectures 12,13: Laplacian Eigenvalue Problems for General Domains--IV. Asymptotics of the Eigenvalues
The basic references are
- W. A. Strauss: Partial Differential Equations: An Introduction,
Sec. 11.6, John Wiley & Sons, 1992.
- R. Courant and D. Hilbert: Methods of Mathematical Physics, Vol. I,
Sec. VI.2, Wiley-Interscience, 1953.
- P. R. Garabedian: Partial Differential Equations, Sec. 11.2,
AMS Chelsea Publishing, 1964. This book is another masterpiece on classical
aspects of PDEs; it is also a good buy (inexpensive considering the hardbound and
its size).
For two bounded domains Ω ⊆ Ω', the Dirichlet-Laplacian eigenvalues satisfy the relationship λn ≥ λ'n.
However, this does not hold for the Neumann-Laplacian eigenvalues.
See e.g.,
- A. Henrot: Extremum Problems for Eigenvalues of Elliptic Operators,
Sec. 1.3.2, Birkhäuser, 2006, for a simple example with two rectangles in 2D.
Lecture 14: Shape Recognition using Laplacian Eigenvalues and
Computational Methods of Laplacian Eigenvalues/Eigenfunctions
This lecture is based on the following papers:
- M.A. Khabou, L. Hermi, and M.B.H. Rhouma: "Shape recognition using eigenvalues of the Dirichlet Laplacian," Pattern Recognition, vol. 40, no. 1, pp. 141-153, 2007.
This paper uses the Finite Difference Scheme for eigenvalue computation.
- M. Reuter, F.-E. Wolter, and N. Peinecke: "Laplace-Beltrami spectra as "Shape-DNA" of surfaces and solids," Computer-Aided Design, vol. 38, no. 4, pp. 342-366, 2006. This paper uses a sophisticated version of the Finite Element Method for eigenvalue computation.
-
T. Betcke and L. N. Trefethen:
"Reviving the Method of Particular Solutions," SIAM Review, vol. 47, no. 3,
pp. 469-491, 2005. This is a much improved follow-up on the MPS of
the classic paper:
- L. Fox, P. Henrici, and C. Moler: "Approximations and Bounds for Eigenvalues of Elliptic Operators," SIAM J. Numer. Anal., vol. 4, no. 1, pp. 89-102, 1967.
This is the original paper of the Method of Particular Solutions.
The famous MATLAB logo was derived from this.
All of the above papers rely on the Courant-Hilbert Vol. I, which show
the fundamental nature of this book.
Lecture 15: Computing Laplacian Eigenfunctions via
Diagonalizing the Integral Operator Commuting with Laplacian
And click here for the slides I used in the lecture.
For the background of mouse retinal ganglion cell datasets, see:
- J. Coombs, D. van der List, G.-Y. Wang, and L.M. Chalupa: "Morphological properties of mouse retinal ganglion cells," Neuroscience, vol. 140, no. 1, pp. 123-136, 2006.
- J.-H. Kong, D. R. Fish, R. L. Rockhill, and R. H. Masland: "Diversity of ganglion cells in the mouse retina: Unsupervised morphological classification and its limits,"
vol. 489, no. 3, pp. 293-310, 2005.
Lectures 16, 17: Fast Multipole Methods (FMMs) and
Hierarchically Semi-Separable (HSS) Representations for
Computing Large Scale Laplacian Eigenvalues/Eigenfunctions (by Allen Xue as
an instructor)
These lectures are based on Allen's Ph.D. thesis and the following ground breaking papers:
-
V. Rokhlin: "Rapid solution of integral equations of classical potential theory," J. Comput. Phys., vol. 60, no. 2, pp. 187-207, 1985.
-
L. Greengard and V. Rokhlin: "A fast algorithm for particle simulations," J. Comput. Phys., vol. 73, no. 2, pp. 325-348, 1987.
- S. Chandrasekaran, M. Gu, and T. Pals: "A fast ULV decomposition solver for hierarchically semiseparable representations," SIAM J. Matrix Anal. Appl., vol. 28, no. 3, pp. 603-622, 2006.
Lecture 18: Introduction to Spectral Graph Theory--I. Basics of Graph Theory
This lecture is based on the following material:
- J. H. van Lint and R. M. Wilson: A Course in Combinatorics, 2nd Ed.,
Chap. 1, Cambridge Univ. Press, 2001. This is a fabulous book on combinatorics
and discrete math including many interesting topics and applications. Highly recommended!
- H. Urakawa: "Spectral geometry and graph theory," Ouyou Suuri (Bulletin of the Japan Society of Industrial and Applied Mathematics), vol. 12, no. 1, pp.29-45, 2002. In Japanese. This is one of the best short survey papers on the subject in my opinion, but is written in Japanese.
Lecture 19: Introduction to Spectral Graph Theory--II. Graph Laplacians and Eigenvalues of Adjacency Matrices and Laplacians
This lecture is based on the following material:
- J. H. van Lint and R. M. Wilson: A Course in Combinatorics, 2nd Ed.,
Chap. 31, Cambridge Univ. Press, 2001.
- H. Urakawa: "Spectral geometry and graph theory," Ouyou Suuri (Bulletin of the Japan Society of Industrial and Applied Mathematics), vol. 12, no. 1, pp.29-45, 2002.
- F. R. K. Chung: Spectral Graph Theory, Sec. 1.1-1.3, AMS, 1997.
This is one of the most fundamental and important books in spectral graph theory.
I discussed that the adjacency Laplacian ΔA of a 1D lattice graph is the same
as the matrix used in the Discrete Cosine Transform Type II (DCT-II).
More precisely, the eigenvectors of that adjacency Laplacian are the DCT-II
basis vectors. Subtle changes in the discrete boundary condition generates
matrices whose eigenvectors are DCT-I, III, IV, ... More about this interesting
relationship between the eigenvectors of discrete boundary value problems
and the various types of DCT and Discrete Sine Transform (DST) can be
found in
- G. Strang: "The discrete cosine transform," SIAM Review, vol. 41, no. 1, pp. 135-147, 1999.
The other papers we touched upon in this lecture are
- J. Dodziuk: "Difference equations, isoperimetric inequality and transience of certain random walks," Trans. Amer. Math. Soc., vol. 284, no. 2. pp. 787-794, 1984.
-
H. S. Wilf: "The eigenvalues of a graph and its chromatic number," J. London Math. Soc., vol. s1-42, no.1, pp. 330-332, 1967.
-
J. Tan: "Laplacian spectra and chromatic numbers of graphs," Kyushu J. Math., vol. 54, no. 1, pp. 165-170, 2000.
Lecture 20: Introduction to Spectral Graph Theory--III. Graph Cut &
the Cheeger Constants; Isospectral Graphs; Discrete Laplacian Eigenvalue Problems
This lecture is based on the following material:
- H. Urakawa: "Spectral geometry and graph theory," Ouyou Suuri (Bulletin of the Japan Society of Industrial and Applied Mathematics), vol. 12, no. 1, pp.29-45, 2002.
- F. R. K. Chung: Spectral Graph Theory, Sec. 2.1-2.3, AMS, 1997.
The lower bounds of the second eigenvalue λ2
of graph Laplacians were derived by
- J. Dodziuk and W. S. Kendall: "Combinatorial Laplacians and isoperimetric
inequality," in From Local Times to Global Geometry, Control and Physics (K. D. Elworthy, ed.), Pitman Res. Notes Math. Ser., vol. 150, pp. 68-74,
Longman Sci. Tech., 1986.
-
B. Mohar: "Isoperimetric numbers of graphs,"
J. Comb. Theory, Ser. B, vol. 47, no. 3, pp. 274-291, 1989.
-
J. Tan: "On Cheeger inequalities of a graph," Discrete Math., vol. 269,
no. 1-3, pp. 315-323, 2003.
On cospectrality and isospectrality of graphs, see
- M. E. Fisher: "On hearing the shape of a drum," J. Comb. Theory,
vol. 1, pp. 105-125, 1966.
- G. A. Baker: "Drum shapes and isospectral graphs," J. Math. Phys.,
vol. 7, pp.2238-2242, 1966.
- J. Tan: "On Isospectral Graphs," Interdisciplinary Information Sciences,
vol. 4, no. 2, pp. 117-124, 1998.
-
H. Fujii and A. Katsuda: "Isospectral graphs and isoperimetric constants,"
Discrete Math., vol. 207, no. 1-3, pp. 33-52, 1999.
As for the isoperimetric inequalities of graphs, see e.g.,
- A. Katsuda and H. Urakawa: "The first eigenvalue of the discrete Dirichlet problem for a graph," J. Combin. Math. Combin. Comput. vol. 27, pp. 217-225, 1998.
- A. Katsuda and H. Urakawa: "The Faber-Krahn type isoperimetric inequalities for a graph," Tohoku Math. J. (2) vol. 51, no. 2, pp. 267-281, 1999.
- See also Chap.2 of F. Chung: Spectral Graph Theory.
Applications of spectral graph theory to image segmentation and clustering
have been a hot topic since late 1990s. There are a huge number of
papers written in this area. I would like to list just a few basic papers
in this area.
-
J. Shi and J. Malik: "Normalized cuts and image segmentation,"
IEEE Trans. Pattern Anal. Machine Intell., vol. 22, no. 8, pp. 888-905,
2000.
-
Y. Weiss: "Segmentation using eigenvectors: a unifying view,"
in Proc. the Seventh IEEE International Conference on Computer Vision,
vol. 2, pp. 975-982, 1999.
-
L. Grady and E. L. Schwartz: "Isoperimetric Graph Partitioning for Image Segmentation," IEEE Trans. on Pattern Anal. Machine Intell., vol. 28, no. 3, pp. 469-475, 2006.
- M. Belkin and P. Niyogi: "Laplacian eigenmaps for dimensionality reduction and data representation," Neural Comp., vol. 15, no. 6, pp. 1373-1396, 2003.
-
R. Kannan, S. Vempala, and A. Vetta: "On clusterings: good, bad and spectral,"
J. Assoc. Comput. Mach. vol. 51 no. 3, pp. 497-515, 2004.
Please email
me if you have any comments or questions!
Go
back to MAT 280: Laplacian Eigenfunctions home page