Overview
My research interests lie mainly in the area of applied and computational mathematics. In particular, I am interested in optimal design of materials using methods derived from the Calculus of Variations, the use of variational methods in inverse problems, and the numerical solution of the equations involved in modeling of materials and their electromagnetic properties.
Recent Work
Often it is convenient to be able to formulate the solution of a linear PDE such as the conductivity equation −∇ · σ∇u = f as the minimizer of a certain functional over a suitable class of admissible functions. Unfortunately, when the parameters in the equation of interest are complex, multiplying by the complex conjugate of a test function and integrating by parts results in a stationary variational principle rather than a minimization principle.
My first result was to find a numerical method to solve the Helmholtz equation that takes advantage of the minimizing nature of the variational principles derived by Graeme Milton and his collaborators. As should be the case, the finite element matrix for the method is positive definite, which allows for the use of Krylov subspace methods that pertain only to such matrices. In my work I employed the preconditioned conjugate gradient method with a block Jacobi preconditioner, which gave good results in that the number of conjugate gradient iterations required grew slowly with respect to the number of points in the numerical grid. Also, I was able to derive an error bound for the method, which gives a convergence estimate based on the spacing in the numerical grid and the smoothness of the solution. To my knowledge, this is the only method for the Helmholtz equation that results in a positive definite finite element matrix, and therefore it may be favored over methods based on stationarity.
For details, see
A. V. Cherkaev and L. V. Gibiansky, Variational principles for complex conductivity, viscoelasticity, and similar problems in media with complex moduli, Journal of Mathematical Physics 35 (1994), 127–145.
G. W. Milton, P. Seppecher, and G. Bouchitt ́e, Minimization variational principles for acoustics, elastodynamics, and electromagnetism in lossy inhomogeneous bodies at fixed frequency, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 465 (2009), 367–396.
Russell B. Richins and David C. Dobson, A numerical minimization scheme for the complex helmholtz equation, ESAIM: Mathematical Modelling and Numerical Analysis 46 (2012), 39–57.
I am currently working on several extensions of these ideas. Specifically, I am attempting to apply the minimization method developed earlier to more complicated problems in wave propagation. The first of these is the problem of electromagnetic scattering off a cavity embedded in a conducting half plane. The cavity is assumed to be filled with a lossy material. Above the half plane, there is only vacuum. The difficulty arises on the boundary between the material and the vacuum. On this boundary, one must enforce a transparent boundary condition, which involves non-local operators. The challenge is to properly deal with these operators while maintaining the positive definiteness of at least a large part of the finite element matrix.
Another application which I am investigating is the use of the minimization variational principles in the inverse problem of determining the properties of a material from boundary measurements of the fields. One advantage of having these variational principles is that they can be used to give constraints on the admissible values for the material parameters. These constraints can then be used to accelerate the convergence of standard methods for this problem.