Designing Optical Fibers: Fitting the Derivatives of a Nonlinear Pde-Eigenvalue Problem

When trying to fit data to functions of the eigensystem of a pde-eigenvalue problem, such as Maxwell’s equation, numerical differentiation is ineffective and analytic gradients must be supplied. In our motivating example of trying to determine the chemical composition of the layers of specialty optical fibers, the function involved fitting the higher order derivatives with respect to frequency of the positive eigenvalues. The computation of the gradient was the most time consuming part of the minimization problem. It was realized that if one interchanged the order of differentiation, and differentiated first with respect to the design parameters, fewer derivatives of the eigenvectors would be required and one could take full advantage that each grid point was affected by only a few variables. As the model was expanded to cover a fiber wrapped around a spool, the bandwidth of the linearized symmetric eigenvalue problem increased. At the heart of each of the iterative methods used to find the few positive eigenvalues was a symmetric, banded, indefinite matrix. Here we present an algorithm for this problem which reduces a symmetric banded matrix to a block diagonal matrix of 1 x 1 and 2 x 2 blocks. Fillin outside the band because of pivoting for stability is prevented by a sequence of planar transformations. Computationally the algorithm is compared to the block unsymmetric banded solver and the block positive definite symmetric band solver in LAPACK.