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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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As you may know, waves are all all around us.

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They are really important in our daily lives,

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for the fact that we're able to

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actually conduct this interview remotely or even in person.

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If you can hear me then, that is because of waves,

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because of our ability to harness waves,

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there are certain waves that can be destructive.

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For example, waves coming from earthquakes.

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Um, we need to understand waves

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and harness them for, our use.

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So, but the, the problem is

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that the mathematical equation describing

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waves are complicated.

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We cannot solve them by hand.

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We have to solve them using computers.

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Supercomputers actually, then the problem is waves

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do not decay, or they decay rather slowly in, in time

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and space because that is why they are useful.

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Um, that's why we can use them to probe, nature.

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We can use them to determine where minerals

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or resources are located.

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This property of wave that is the radiation of wave

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to far field is also what makes them to be difficult

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to simulate on computers.

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In particular, if we're computing on a

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finite computational domain.

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So take for example, I know London is not an quake count,

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quake country.

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Take for example, an quake that happened in London,

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or if London was an quake country.

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We need to do seismic hazard assessment.

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So we have to, place our resources,

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our computational resources around the, the region.

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So although ways that are emitted by earthquakes travel far,

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so if we truncate our domain so that we focus our resources

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where these, where earthquake is harmful,

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we place artificial boundaries

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and ways leaving the domain will come back

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and corrupt the solution to our problem.

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So this is really difficult.

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Um, but this has been

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with us since we started using computers to simulate waves.

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And the, the idea

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or the question of how do we efficiently

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truncate our domain has been around

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for more than 40 years now.

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So one of the techniques that mathematicians

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and scientists have developed is the,

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so-called perfectly match layer.

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So this is an absorbing layer that we put

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around the competitional domain, such that ways

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that are living the domain leave would've been

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reflected back.

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So that if we place these absorbing layers around

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our competition domains, we can compute

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accurately and efficiently.

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So, but when we derive these equations,

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we modify the equations that are given to us by nature

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and introduce new terms.

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We may not know

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What the properties of these equations are, so we need

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to understand if they are well POed,

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which is really important from a mathematical point of view,

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and if they are stable,

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which is also important from mathematical

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and numerical point of view,

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because we want to keep our solutions stable.

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We don't want things to blurb,

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but in many cases, the solution to blow up.

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So my work has been to mathematically address this

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question of stability or instability of the PML.

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When will this, when would the

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PML be unstable?

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PML here means perfectly much layer.

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So the, the acronym PML is perfectly much layer.

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So, in particular, if you have boundary condition,

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for example, PML may interact with the boundary condition

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and lead to, to some instability.

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So my, the question has been,

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or my question has been, what so types

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of boundary condition will allow us have a stable PML

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and how can we implement those boundary conditions

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in a numerical framework?

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This idea of PML was first introduced in the nineties by

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a French mathematician

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or engineer called them JP .

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So, but, mathematicians have started analyzing this

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also for the past 20 years.

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So the, the leading theory was for, I mean,

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when we do mathematical analysis,

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we try to simplify the problem.

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So the leading theory has been

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a simple constant coefficient problem on an infinite domain.

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So from that theory one can understand,

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when would the PML be unstable or stable?

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But when we have boundary conditions

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or when we introduce heterogeneities, which is often appear

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in nature, this type

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of the theory governing these problems have alluded us

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until the recent times.

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Yeah, it's contributing to the literature

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and trying to establish mathematically, the theory

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of web ness

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and stability of the perfectly match layers so

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that we can use them or,

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or we can understand when we can use them

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or when they are, when they can grow up.

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When we do, numerical competitions,

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my methods is really rooted in, deep mathematical,

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theoretical framework.

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So I start from the continuous model

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because these equations are asymmetric.

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What this means is that, the equations

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that we get from physics are often symmetric systems.

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It means that there is some energy that is being conserved.

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I mean, nature tries to conserve energy or minimize energy.

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Um, but when we derive these PML equations,

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we have as symmetric systems,

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We may not, at least in the physical space,

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what we call physical space, time

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and space, we may not derive estimate

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or energy conservation law,

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conservation of these equations.

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So my idea is to go to transform spaces like lap plus

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and full spaces where we can do,

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rigorous mathematical analysis.

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So these tools come from pure mathematics, actually,

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although I'm an applied mathematician,

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but these tools come from pure mathematics.

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These methods apply to different wave equation,

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particularly say acoustic wave equation

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or elastic wave equation governing seismic waves.

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So my theory

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or the theory I developed with colleagues,

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have been implemented in large scale,

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computational framework to simulate, waves,

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and it saves as much as 96% of computational cost

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in standard seismic benchmark problems.

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So they can be applied to earthquake,

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simulation or when we are doing simulation

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for seismic software.

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The conclusion so far is that

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the PM is good for is actually

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can be a stable system for different scenarios

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for computing waves.

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Where it has not succeeded is in the, so-called,

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systems that have backward propagating modes.

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So this means, this means that the, the, the,

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the, the, the direction

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or the flow of information is

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opposite to the flow of energy.

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So there are certain MA materials particularly

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and also tropical, elastic materials that do

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that support this kind of, wave motion.

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So the PML has, been known to be unstable for these,

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type of systems.

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So we may, we need to develop perfectly match layers

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for these, systems.

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Another problem is for nonlinear problems,

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the perfectly match layers basically a linear theory,

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but we often have, we have non-linear problems, for example,

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coming from, climate modeling

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or solving, oil equations.

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So we need to extend the theory to, to be able

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to handle non-linear problem.

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But a good understanding

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or mathematical understanding

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of the linear theory will help us to extend the res,

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this, or derive good absorbing layers

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or perfectly much layer for nonlinear problems.

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