10,996 research outputs found
Boosting the Maxwell double layer potential using a right spin factor
We construct new spin singular integral equations for solving scattering
problems for Maxwell's equations, both against perfect conductors and in media
with piecewise constant permittivity, permeability and conductivity, improving
and extending earlier formulations by the author. These differ in a fundamental
way from classical integral equations, which use double layer potential
operators, and have the advantage of having a better condition number, in
particular in Fredholm sense and on Lipschitz regular interfaces, and do not
suffer from spurious resonances. The construction of the integral equations
builds on the observation that the double layer potential factorises into a
boundary value problem and an ansatz. We modify the ansatz, inspired by a
non-selfadjoint local elliptic boundary condition for Dirac equations
A fast and well-conditioned spectral method
A novel spectral method is developed for the direct solution of linear ordinary differential equations with variable coefficients. The method leads to matrices which are almost banded, and a numerical solver is presented that takes operations, where is the number of Chebyshev points needed to resolve the coefficients of the differential operator and is the number of Chebyshev points needed to resolve the solution to the differential equation. We prove stability of the method by relating it to a diagonally preconditioned system which has a bounded condition number, in a suitable norm. For Dirichlet boundary conditions, this reduces to stability in the standard 2-norm
Nonlinear morphoelastic plates II: exodus to buckled states
Morphoelasticity is the theory of growing elastic materials. This theory is based on the multiple decomposition of the deformation gradient and provides a formulation of the deformation and stresses induced by growth. Following a companion paper, a general theory of growing nonlinear elastic Kirchhoff plate is described. First, a complete geometric description of incompatibility with simple examples is given. Second, the stability of growing Kirchhoff plates is analyzed
The exponentially convergent trapezoidal rule
It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators
Thermoacoustic tomography arising in brain imaging
We study the mathematical model of thermoacoustic and photoacoustic
tomography when the sound speed has a jump across a smooth surface. This models
the change of the sound speed in the skull when trying to image the human
brain. We derive an explicit inversion formula in the form of a convergent
Neumann series under the assumptions that all singularities from the support of
the source reach the boundary
Fast integral equation methods for the Laplace-Beltrami equation on the sphere
Integral equation methods for solving the Laplace-Beltrami equation on the
unit sphere in the presence of multiple "islands" are presented. The surface of
the sphere is first mapped to a multiply-connected region in the complex plane
via a stereographic projection. After discretizing the integral equation, the
resulting dense linear system is solved iteratively using the fast multipole
method for the 2D Coulomb potential in order to calculate the matrix-vector
products. This numerical scheme requires only O(N) operations, where is the
number of nodes in the discretization of the boundary. The performance of the
method is demonstrated on several examples
- …