||Rose-Hulman Institute of Technology
||sneedbj (at) rose-hulman (dot) edu
||Perfectly Matched Layers and the Art of Cloaking
Given a wave-like partial differential equation on a domain, Perfectly Matched Layers (PML) is a numerical technique used to simulate an open boundary.
This is accomplished by introducing a small layer just inside the boundary of the domain.
This method has two strengths.
Firstly, the layer absorbs incoming disturbances with drastically reduced reflection.
Secondly, outside the layer the solution behaves exactly as if it were computed on an infinite domain.
This is the reason for identifying the layer as "perfectly matched".
The link between this method and cloaking is because the boundary of the domain is undetectable outside the layer.
Cloaking (in a weak sense) refers to a method in which an object is not noticeably perceivable by an observer.
In this setting, PML can be used to motivate a framework for objects which are not perceived.
(The analogy is to the layer disguising the boundary of the computation domain.)
- Week 1: (Week of June 2nd)
Read several papers motivating PML.
Wrote several Finite Difference Methods (FDM) in Matlab attempting to implement PML for the 1D and 2D Cartesian wave equation.
All methods were numerically unstable for choices of parameters well below those which cause absorption at boundary.
Gave talk listed below as "First Presentation".
- Week 2: (Week of June 9th)
Implemented several FDMs demonstrating how PML can be applied to the coupled first order wave equation.
Motivated by Steven G. Johnson "Notes on Perfectly Matched Layers (PMLs)" available as part of his course on Nanophotonics.
These methods achieved short term numerical stability -- long enough to see one wave be absorbed by the layer;
however, I was unable to achieve long term stability with these methods.
- Week 3: (Week of June 16th)
Confirmed notions of systematic error as the source of the instability from the first week.
Took a step back and looked at more of the theory behind solutions to the wave equation without PML.
Visited High Line Park in NYC.
- Week 4: (Week of June 23rd)
Implemented methods which keep the wave equation in its second order form.
These methods were long term numerical stabile, generalized very easily to higher dimensions, and cheap in their use of auxiliary functions.
Motivated by Grote and Sim's "Efficient PML for the wave equation" available at arXiv:1001.0319 [math.NA].
- Week 5: (Week of June 30th)
Did more simulations confirming the error produced from PML in the 1D and 2D cases.
Revised FDM schemes to be more in line with the ideas shown in the "Efficient PML" paper above.
Finalized the gif image used above to use as a visualization of PML.
I currently have over 7000 lines of Matlab code saved between in 46 important scripts at this point.
- Week 6: (Week of July 7th)
As an REU group, we visited the AT&T Lab in Florham Park and heard several talks about the research being done there.
The talks varied from using differential privacy for protecting individuals in publicly released data sets to optimizing phone applications for efficient use of battery and data.
Did some algebra showing the coordinate stretching nature of PML in each of the different frameworks.
Did some work with the wave equation in polar coordinates.
Visited "the rest of NYC".
- Week 7: (Week of July 14th)
This was by far my most productive week.
I was very close to getting my answers previously, and this week I just flushed out the rest of the steps.
Completed FDMs for the wave equation with PML in polar coordinates.
Did some trend fitting to show the error produced by PML was exponentially small.
Finalized my stability argument which now explains all the stability issues I have had.
Solved the 1D wave equation with PML via looking for soliton solutions.
Began looking at the Helmholtz equation and wrote FDMs for the 1D and 2D cases.
Gave talk listed below as "Second Presentation".
- Week 8: (Week of July 21th)
Proved a bound for the decay of the 1D PML Wave Equation.
Began working with energy arguments to determine an error bound in a more general setting.
Began writing the final report.