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Stability of the perfectly matched layer (PML) for waves

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Kenneth Duru discusses the stability analysis of the perfectly matched layer (PML) in two-space dimensional layered elastic media.

Image courtesy of interviewee. May 30, 2024

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The transcript discusses the importance of waves in daily life, their applications, and challenges in simulating them numerically. Waves are crucial for communication and can also be destructive, such as those caused by earthquakes. The mathematical equations describing waves are complex and often require supercomputers for resolution.

A significant issue is truncating computational domains; waves naturally radiate and can reflect back, causing inaccuracies in simulations. To tackle this, the concept of the Perfectly Matched Layer (PML) was developed, which serves as an absorbing boundary to prevent reflections from corrupting calculations. However, the stability of the PML can be affected by boundary conditions, which has been a research focus.

The speaker's research aims to analyze the stability of PMLs, particularly under various conditions, addressing problems that arise from non-standard scenarios like backward propagating modes and nonlinear systems. They emphasize the need for a robust mathematical foundation to ensure the effective application of PMLs in simulations, particularly for seismic events, suggesting that proper implementation can significantly reduce computational costs (up to 96% in some cases). The conclusion highlights the effectiveness of PMLs while acknowledging the need for further development to address nonlinear problems and specific material behaviors that present stability challenges.

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00:00.185 --> 00:03.445
As you may know, waves are all all around us.

00:03.445 --> 00:07.455
They are really important in our daily lives,

00:07.755 --> 00:09.055
for the fact that we're able to

00:09.975 --> 00:13.675
actually conduct this interview remotely or even in person.

00:13.695 --> 00:17.035
If you can hear me then, that is because of waves,

00:17.035 --> 00:19.115
because of our ability to harness waves,

00:19.405 --> 00:21.835
there are certain waves that can be destructive.

00:22.095 --> 00:23.955
For example, waves coming from earthquakes.

00:24.655 --> 00:26.315
Um, we need to understand waves

00:26.615 --> 00:29.785
and harness them for, our use.

00:31.575 --> 00:32.995
So, but the, the problem is

00:32.995 --> 00:35.475
that the mathematical equation describing

00:35.565 --> 00:37.315
waves are complicated.

00:37.375 --> 00:38.635
We cannot solve them by hand.

00:39.595 --> 00:42.375
We have to solve them using computers.

00:42.375 --> 00:46.955
Supercomputers actually, then the problem is waves

00:47.805 --> 00:51.145
do not decay, or they decay rather slowly in, in time

00:51.145 --> 00:53.705
and space because that is why they are useful.

00:54.205 --> 00:57.545
Um, that's why we can use them to probe, nature.

00:57.805 --> 01:00.585
We can use them to determine where minerals

01:00.805 --> 01:02.985
or resources are located.

01:03.575 --> 01:06.145
This property of wave that is the radiation of wave

01:06.145 --> 01:09.505
to far field is also what makes them to be difficult

01:09.965 --> 01:11.505
to simulate on computers.

01:12.205 --> 01:15.145
In particular, if we're computing on a

01:15.465 --> 01:16.625
finite computational domain.

01:17.675 --> 01:21.735
So take for example, I know London is not an quake count,

01:21.795 --> 01:23.175
quake country.

01:23.405 --> 01:25.455
Take for example, an quake that happened in London,

01:25.795 --> 01:27.775
or if London was an quake country.

01:28.235 --> 01:30.735
We need to do seismic hazard assessment.

01:30.995 --> 01:34.495
So we have to, place our resources,

01:34.495 --> 01:38.095
our computational resources around the, the region.

01:38.835 --> 01:42.255
So although ways that are emitted by earthquakes travel far,

01:42.755 --> 01:46.055
so if we truncate our domain so that we focus our resources

01:46.055 --> 01:50.095
where these, where earthquake is harmful,

01:52.325 --> 01:54.265
we place artificial boundaries

01:54.525 --> 01:56.825
and ways leaving the domain will come back

01:56.925 --> 01:59.065
and corrupt the solution to our problem.

01:59.845 --> 02:01.105
So this is really difficult.

02:02.205 --> 02:03.265
Um, but this has been

02:03.265 --> 02:06.545
with us since we started using computers to simulate waves.

02:07.335 --> 02:09.915
And the, the idea

02:10.295 --> 02:13.835
or the question of how do we efficiently

02:14.435 --> 02:16.995
truncate our domain has been around

02:17.015 --> 02:18.315
for more than 40 years now.

02:19.345 --> 02:22.845
So one of the techniques that mathematicians

02:22.865 --> 02:24.365
and scientists have developed is the,

02:24.365 --> 02:25.765
so-called perfectly match layer.

02:26.515 --> 02:28.855
So this is an absorbing layer that we put

02:28.855 --> 02:32.455
around the competitional domain, such that ways

02:32.455 --> 02:34.935
that are living the domain leave would've been

02:35.045 --> 02:36.615
reflected back.

02:37.395 --> 02:40.575
So that if we place these absorbing layers around

02:41.235 --> 02:43.535
our competition domains, we can compute

02:44.575 --> 02:46.055
accurately and efficiently.

02:48.155 --> 02:50.695
So, but when we derive these equations,

02:51.355 --> 02:54.135
we modify the equations that are given to us by nature

02:54.675 --> 02:55.935
and introduce new terms.

02:57.605 --> 02:58.825
We may not know

02:58.825 --> 03:01.375
What the properties of these equations are, so we need

03:01.375 --> 03:03.095
to understand if they are well POed,

03:03.095 --> 03:06.475
which is really important from a mathematical point of view,

03:07.255 --> 03:08.275
and if they are stable,

03:08.365 --> 03:10.355
which is also important from mathematical

03:10.775 --> 03:11.955
and numerical point of view,

03:11.955 --> 03:13.835
because we want to keep our solutions stable.

03:13.835 --> 03:15.115
We don't want things to blurb,

03:15.935 --> 03:18.795
but in many cases, the solution to blow up.

03:19.685 --> 03:24.045
So my work has been to mathematically address this

03:25.395 --> 03:28.075
question of stability or instability of the PML.

03:28.185 --> 03:32.145
When will this, when would the

03:32.225 --> 03:33.665
PML be unstable?

03:33.825 --> 03:35.305
PML here means perfectly much layer.

03:35.405 --> 03:38.665
So the, the acronym PML is perfectly much layer.

03:41.235 --> 03:45.845
So, in particular, if you have boundary condition,

03:45.845 --> 03:48.245
for example, PML may interact with the boundary condition

03:48.245 --> 03:49.765
and lead to, to some instability.

03:50.875 --> 03:52.535
So my, the question has been,

03:53.075 --> 03:56.695
or my question has been, what so types

03:56.695 --> 04:01.315
of boundary condition will allow us have a stable PML

04:01.775 --> 04:03.955
and how can we implement those boundary conditions

04:05.235 --> 04:07.255
in a numerical framework?

04:07.605 --> 04:11.935
This idea of PML was first introduced in the nineties by

04:12.535 --> 04:14.335
a French mathematician

04:14.355 --> 04:17.775
or engineer called them JP .

04:18.715 --> 04:22.505
So, but, mathematicians have started analyzing this

04:23.775 --> 04:25.225
also for the past 20 years.

04:26.075 --> 04:29.015
So the, the leading theory was for, I mean,

04:29.045 --> 04:30.855
when we do mathematical analysis,

04:30.855 --> 04:32.095
we try to simplify the problem.

04:32.835 --> 04:34.305
So the leading theory has been

04:34.945 --> 04:39.225
a simple constant coefficient problem on an infinite domain.

04:40.125 --> 04:44.505
So from that theory one can understand,

04:45.015 --> 04:48.215
when would the PML be unstable or stable?

04:49.115 --> 04:50.895
But when we have boundary conditions

04:50.895 --> 04:54.495
or when we introduce heterogeneities, which is often appear

04:55.515 --> 04:58.475
in nature, this type

04:58.475 --> 05:02.475
of the theory governing these problems have alluded us

05:02.765 --> 05:03.795
until the recent times.

05:04.305 --> 05:06.155
Yeah, it's contributing to the literature

05:06.575 --> 05:10.595
and trying to establish mathematically, the theory

05:10.735 --> 05:11.755
of web ness

05:11.755 --> 05:14.755
and stability of the perfectly match layers so

05:14.875 --> 05:16.835
that we can use them or,

05:16.975 --> 05:19.755
or we can understand when we can use them

05:19.855 --> 05:22.555
or when they are, when they can grow up.

05:22.905 --> 05:24.875
When we do, numerical competitions,

05:25.255 --> 05:29.435
my methods is really rooted in, deep mathematical,

05:29.455 --> 05:30.995
theoretical framework.

05:31.995 --> 05:33.975
So I start from the continuous model

05:33.975 --> 05:37.355
because these equations are asymmetric.

05:37.385 --> 05:41.585
What this means is that, the equations

05:41.585 --> 05:45.025
that we get from physics are often symmetric systems.

05:45.685 --> 05:48.665
It means that there is some energy that is being conserved.

05:48.825 --> 05:52.305
I mean, nature tries to conserve energy or minimize energy.

05:53.235 --> 05:55.535
Um, but when we derive these PML equations,

05:55.535 --> 05:56.775
we have as symmetric systems,

05:57.025 --> 05:59.205
We may not, at least in the physical space,

05:59.205 --> 06:00.765
what we call physical space, time

06:00.865 --> 06:04.405
and space, we may not derive estimate

06:04.545 --> 06:06.645
or energy conservation law,

06:06.645 --> 06:08.325
conservation of these equations.

06:09.025 --> 06:12.965
So my idea is to go to transform spaces like lap plus

06:12.985 --> 06:17.205
and full spaces where we can do,

06:18.045 --> 06:19.325
rigorous mathematical analysis.

06:19.945 --> 06:22.965
So these tools come from pure mathematics, actually,

06:23.285 --> 06:24.565
although I'm an applied mathematician,

06:24.625 --> 06:26.085
but these tools come from pure mathematics.

06:26.775 --> 06:30.885
These methods apply to different wave equation,

06:30.885 --> 06:32.805
particularly say acoustic wave equation

06:32.985 --> 06:35.965
or elastic wave equation governing seismic waves.

06:36.885 --> 06:38.265
So my theory

06:38.925 --> 06:41.625
or the theory I developed with colleagues,

06:41.815 --> 06:44.785
have been implemented in large scale,

06:45.095 --> 06:48.745
computational framework to simulate, waves,

06:49.005 --> 06:52.625
and it saves as much as 96% of computational cost

06:53.445 --> 06:55.585
in standard seismic benchmark problems.

06:56.895 --> 07:01.625
So they can be applied to earthquake,

07:01.625 --> 07:04.625
simulation or when we are doing simulation

07:04.765 --> 07:06.505
for seismic software.

07:06.965 --> 07:09.615
The conclusion so far is that

07:11.265 --> 07:15.725
the PM is good for is actually

07:16.305 --> 07:19.565
can be a stable system for different scenarios

07:19.585 --> 07:20.725
for computing waves.

07:21.375 --> 07:25.765
Where it has not succeeded is in the, so-called,

07:26.915 --> 07:30.045
systems that have backward propagating modes.

07:30.045 --> 07:34.845
So this means, this means that the, the, the,

07:34.905 --> 07:37.565
the, the, the direction

07:37.585 --> 07:40.405
or the flow of information is

07:40.885 --> 07:42.045
opposite to the flow of energy.

07:42.225 --> 07:44.125
So there are certain MA materials particularly

07:44.625 --> 07:46.925
and also tropical, elastic materials that do

07:46.925 --> 07:49.605
that support this kind of, wave motion.

07:50.305 --> 07:54.085
So the PML has, been known to be unstable for these,

07:54.185 --> 07:55.285
type of systems.

07:55.545 --> 07:58.605
So we may, we need to develop perfectly match layers

07:58.905 --> 08:00.085
for these, systems.

08:00.965 --> 08:05.055
Another problem is for nonlinear problems,

08:05.235 --> 08:08.215
the perfectly match layers basically a linear theory,

08:09.515 --> 08:14.265
but we often have, we have non-linear problems, for example,

08:14.265 --> 08:16.745
coming from, climate modeling

08:17.085 --> 08:19.945
or solving, oil equations.

08:20.125 --> 08:24.025
So we need to extend the theory to, to be able

08:24.025 --> 08:26.105
to handle non-linear problem.

08:27.315 --> 08:29.965
But a good understanding

08:30.425 --> 08:31.645
or mathematical understanding

08:31.665 --> 08:34.925
of the linear theory will help us to extend the res,

08:35.035 --> 08:38.285
this, or derive good absorbing layers

08:38.505 --> 08:40.565
or perfectly much layer for nonlinear problems.

00:00 - As you may know, waves are all all around us. They are really important in our daily lives, for the fact that we're able to actually conduct this interview remotely or even in person. If you can hear me then, that is because of waves, because of our ability to harness waves,

00:19 - there are certain waves that can be destructive. For example, waves coming from earthquakes. we need to understand waves and harness them for, our use. So, but the, the problem is that the mathematical equation describing waves are complicated. We cannot solve them by hand. We have to solve them using computers. Supercomputers actually, then the problem is waves

00:47 - do not decay, or they decay rather slowly in, in time and space because that is why they are useful. that's why we can use them to probe, nature. We can use them to determine where minerals or resources are located. This property of wave that is the radiation of wave to far field is also what makes them to be difficult to simulate on computers. In particular, if we're computing on a finite computational domain.

01:17 - So take for example, I know London is not an quake count, quake country. Take for example, an quake that happened in London, or if London was an quake country. We need to do seismic hazard assessment. So we have to, place our resources, our computational resources around the, the region. So although ways that are emitted by earthquakes travel far, so if we truncate our domain so that we focus our resources where these, where earthquake is harmful,

01:52 - we place artificial boundaries and ways leaving the domain will come back and corrupt the solution to our problem. So this is really difficult. but this has been with us since we started using computers to simulate waves. And the, the idea or the question of how do we efficiently truncate our domain has been around for more than 40 years now.

02:19 - So one of the techniques that mathematicians and scientists have developed is the, so-called perfectly match layer. So this is an absorbing layer that we put around the competitional domain, such that ways that are living the domain leave would've been reflected back. So that if we place these absorbing layers around our competition domains, we can compute accurately and efficiently.

02:48 - So, but when we derive these equations, we modify the equations that are given to us by nature and introduce new terms. We may not know What the properties of these equations are, so we need to understand if they are well POed, which is really important from a mathematical point of view, and if they are stable, which is also important from mathematical and numerical point of view,

03:11 - because we want to keep our solutions stable. We don't want things to blurb, but in many cases, the solution to blow up. So my work has been to mathematically address this question of stability or instability of the PML. When will this, when would the PML be unstable? PML here means perfectly much layer. So the, the acronym PML is perfectly much layer. So, in particular, if you have boundary condition,

03:45 - for example, PML may interact with the boundary condition and lead to, to some instability. So my, the question has been, or my question has been, what so types of boundary condition will allow us have a stable PML and how can we implement those boundary conditions in a numerical framework? This idea of PML was first introduced in the nineties by a French mathematician or engineer called them JP .

04:18 - So, but, mathematicians have started analyzing this also for the past 20 years. So the, the leading theory was for, I mean, when we do mathematical analysis, we try to simplify the problem. So the leading theory has been a simple constant coefficient problem on an infinite domain. So from that theory one can understand, when would the PML be unstable or stable? But when we have boundary conditions

04:50 - or when we introduce heterogeneities, which is often appear in nature, this type of the theory governing these problems have alluded us until the recent times. Yeah, it's contributing to the literature and trying to establish mathematically, the theory of web ness and stability of the perfectly match layers so that we can use them or, or we can understand when we can use them

05:19 - or when they are, when they can grow up. When we do, numerical competitions, my methods is really rooted in, deep mathematical, theoretical framework. So I start from the continuous model because these equations are asymmetric. What this means is that, the equations that we get from physics are often symmetric systems. It means that there is some energy that is being conserved. I mean, nature tries to conserve energy or minimize energy.

05:53 - but when we derive these PML equations, we have as symmetric systems, We may not, at least in the physical space, what we call physical space, time and space, we may not derive estimate or energy conservation law, conservation of these equations. So my idea is to go to transform spaces like lap plus and full spaces where we can do, rigorous mathematical analysis.

06:19 - So these tools come from pure mathematics, actually, although I'm an applied mathematician, but these tools come from pure mathematics. These methods apply to different wave equation, particularly say acoustic wave equation or elastic wave equation governing seismic waves. So my theory or the theory I developed with colleagues, have been implemented in large scale, computational framework to simulate, waves,

06:49 - and it saves as much as 96% of computational cost in standard seismic benchmark problems. So they can be applied to earthquake, simulation or when we are doing simulation for seismic software. The conclusion so far is that the PM is good for is actually can be a stable system for different scenarios for computing waves. Where it has not succeeded is in the, so-called,

07:26 - systems that have backward propagating modes. So this means, this means that the, the, the, the, the, the direction or the flow of information is opposite to the flow of energy. So there are certain MA materials particularly and also tropical, elastic materials that do that support this kind of, wave motion. So the PML has, been known to be unstable for these, type of systems.

07:55 - So we may, we need to develop perfectly match layers for these, systems. Another problem is for nonlinear problems, the perfectly match layers basically a linear theory, but we often have, we have non-linear problems, for example, coming from, climate modeling or solving, oil equations. So we need to extend the theory to, to be able to handle non-linear problem. But a good understanding

08:30 - or mathematical understanding of the linear theory will help us to extend the res, this, or derive good absorbing layers or perfectly much layer for nonlinear problems.

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