Numerical Optimization of Low Eigenvalues of the Dirichlet and Neumann Laplacians

We perform a numerical optimization of the first ten nontrivial eigenvalues of the Neumann Laplacian for planar Euclidean domains. The optimization procedure is done via a gradient method, while the computation of the eigenvalues themselves is done by means of an efficient meshless numerical method which allows for the computation of the eigenvalues for large numbers of domains within a reasonable time frame. The Dirichlet problem, previously studied by Oudet using a different numerical method, is also studied and we obtain similar (but improved) results for a larger number of eigenvalues. These results reveal an underlying structure to the optimizers regarding symmetry and connectedness, for instance, but also show that there are exceptions to these preventing general results from holding.

[1]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[2]  Dorin Bucur,et al.  Minimization of the third eigenvalue of the Dirichlet Laplacian , 2000, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[3]  Pedro R. S. Antunes.  Numerical calculation of eigensolutions of 3D shapes using the method of fundamental solutions , 2011 .

[4]  Edouard Oudet,et al.  Numerical minimization of eigenmodes of a membrane with respect to the domain , 2004 .

[5]  Antoine Henrot,et al.  Extremum Problems for Eigenvalues of Elliptic Operators , 2006 .

[6]  Hans F. Weinberger,et al.  An Isoperimetric Inequality for the N-Dimensional Free Membrane Problem , 1956 .

[7]  George Polya,et al.  On the Eigenvalues of Vibrating Membranes(In Memoriam Hermann Weyl) , 1961 .

[8]  Pedro R. S. Antunes,et al.  A numerical study of the spectral gap , 2008 .

[9]  Giuseppe Buttazzo,et al.  An existence result for a class of shape optimization problems , 1993 .

[10]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .

[11]  Dan Boneh,et al.  On genetic algorithms , 1995, COLT '95.

[12]  Jan Sokolowski,et al.  Introduction to shape optimization , 1992 .

[13]  Timo Betcke,et al.  Stability and convergence of the method of fundamental solutions for Helmholtz problems on analytic domains , 2007, J. Comput. Phys..

[14]  Iosif Polterovich,et al.  Maximization of the second positive Neumann eigenvalue for planar domains , 2008, 0801.2142.

[15]  C. B. Moler,et al.  Bounds for Eigenvalues and Eigenvectors of Symmetric Operators , 1968 .

[16]  G. Poliquin,et al.  Wolf-Keller theorem for Neumann eigenvalues , 2010, 1007.4771.

[17]  Carlos J. S. Alves,et al.  The Method of Fundamental Solutions Applied to the Calculation of Eigenfrequencies and Eigenmodes of 2D Simply Connected Shapes , 2005 .

[18]  J. Keller,et al.  Range of the first two eigenvalues of the laplacian , 1994, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[19]  Giovanni Landi,et al.  Chern–Simons forms on principal superfiber bundles , 1990 .

[20]  G. Szegő,et al.  Inequalities for Certain Eigenvalues of a Membrane of Given Area , 1954 .

[21]  Jan A Snyman,et al.  Practical Mathematical Optimization: An Introduction to Basic Optimization Theory and Classical and New Gradient-Based Algorithms , 2005 .

[22]  S. Cox.  EXTREMAL EIGENVALUE PROBLEMS FOR STARLIKE PLANAR DOMAINS , 1995 .

[23]  Jeng-Tzong Chen,et al.  Eigensolutions of multiply connected membranes using the method of fundamental solutions , 2005 .

[24]  A. Bogomolny.  Fundamental Solutions Method for Elliptic Boundary Value Problems , 1985 .

[25]  E. Krahn,et al.  Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises , 1925 .

[26]  Pedro Freitas,et al.  New Bounds for the Principal Dirichlet Eigenvalue of Planar Regions , 2006, Exp. Math..

[27]  George Polya,et al.  On the characteristic frequencies of a symmetric membrane , 1955 .

[28]  I. Stakgold,et al.  A Variational Theorem for ∇2u+ λu= 0 and its Application , 1952 .

[29]  Edouard Oudet,et al.  Minimizing the Second Eigenvalue of the Laplace Operator with Dirichlet Boundary Conditions , 2003 .