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Asymptotically improved solvers for the variable coefficient Helmholtz equation
Abstract:
Existing numerical methods for the variable coefficient Helmholtz equation are prohibitively expensive in the high-frequency regime --- that is, when the wavenumber $k$ of the problem is large. A fundamental obstacle is that in $d$ dimensions, standard discretization methods (e.g., finite element bases, orthogonal polynomials, collocation methods) require $\mathcal{O}\left(k^d\right)$ points to represent it solutions. The computational cost to solve the Helmholtz equation must then be at least $\mathcal{O}\left(k^d\right)$, and is, in fact, often far larger due to the difficulties which arise when solving the linear systems which arise in the highly oscillatory regime.
We will discuss a new class of solvers which are able overcome the difficulties inherent in standard techniques for representing oscillatory functions, at least in certain special cases. They operate by constructing a basis in the space of solutions of the variable coefficient Helmholtz equation whose logarithms are nonoscillatory functions.
First we will show that this approach can be used to solve one-dimensional variable coefficient with near optimal accuracy in time independent of the wavenumber $k$. Standard methods which achieve similar levels of accuracy require $\mathcal{O}\left(k\right)$ time.
Then, we will discuss a method for the numerical simulation of scattering from a radially symmetric potential in two spatial dimensions. Existing methods for solving this class of problems have running times which grow superlinearly in $N=\mathcal{O}\left(k^2\right)$, while the asymptotic running time of our algorithm is $\mathcal{O}\left(k \log(k)\right)$.
We will conclude by briefly discuss the prospects for extending these methods to more general problems.
If you would like to attend, please email crampersad@flatironinstitute.org for the Zoom details.