CCM Colloquium: James Bremer (U of Toronto)

3rd Floor Classroom/3-Flatiron Institute (162 5th Avenue)

3rd Floor Classroom/3-Flatiron Institute

162 5th Avenue


Asymptotically improved solvers for the variable coefficient Helmholtz equation


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.




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