A Nonlocal Helmholtz Decomposition

Andrew Haar Author

04/05/2021 Added

47 Plays

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A discussion of a new nonlocal framework which admits a Helmholtz decomposition. Further we discuss parallels between the local and nonlocal frameworks.

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[00:00:00.350]My name is Andrew Haar.
[00:00:01.480]I'm a senior, studying mathematics here at UNL,
[00:00:04.650]and I've been doing research for the past two years
[00:00:06.800]on nonlocal vector calculus,
[00:00:08.970]with my advisor, Dr. Petronela Radu.
[00:00:11.750]And so I'm here to talk to you about, more specifically,
[00:00:15.960]my research on nonlocal Helmholtz decompositions.
[00:00:21.670]There we go.
[00:00:22.503]So, local Helmholtz decompositions, just to start, are,
[00:00:26.860]the local Helmholtz decomposition says
[00:00:28.740]that we can decompose any vector function
[00:00:30.990]by any three-dimensional vector function
[00:00:32.760]that's sufficiently nice.
[00:00:34.710]That means it decays fast enough in infinity,
[00:00:38.020]into a solenoidal and an irrotational component,
[00:00:41.890]i.e. a curl-free and a divergence-free component,
[00:00:46.690]just said differently.
[00:00:47.680]And this has been really nice
[00:00:49.200]in a great deal of scientific disciplines,
[00:00:54.700]and partially because it allows us
[00:00:56.430]to prescribe a scalar field
[00:00:58.770]and a divergence-free vector field
[00:01:01.250]of another vector field.
[00:01:02.760]But for many other reasons, for example,
[00:01:04.310]in solving numerically incompressible
[00:01:06.800]Navier-Stokes equations.
[00:01:09.490]But anyway, moving on to nonlocal vector calculus.
[00:01:13.190]In mathematical modeling, we have a bit of a problem.
[00:01:16.900]And that problem is that the world is not smooth.
[00:01:21.460]From the middle of the 1600s until these days,
[00:01:23.670]we have been using derivatives in many models
[00:01:25.970]to approximate real-world phenomena,
[00:01:28.800]but derivatives do tend to break,
[00:01:31.290]well they do break down
[00:01:35.335]whenever there are discontinuities.
[00:01:39.050]So we, if phenomena are nice enough,
[00:01:42.200]such as in classical diffusion, wave propagation,
[00:01:45.460]and so forth,
[00:01:47.380]they can still be modeled using these derivatives.
[00:01:52.565]But these models break down
[00:01:55.520]when their material discontinuities.
[00:01:58.310]For example, in crack formulation,
[00:02:01.600]or in image processing, is another example.
[00:02:03.950]And weak derivatives have sort of solved this,
[00:02:05.680]but not really.
[00:02:07.350]There still are problems with it.
[00:02:08.540]So in comes peridynamics or nonlocal calculus,
[00:02:13.010]peri meaning near, dynamics meaning force.
[00:02:15.820]So in nonlocal, really all that,
[00:02:18.130]what that means is that objects interact with other objects
[00:02:22.740]that are immediately next to them.
[00:02:27.050]And I have a picture for that on this slide,
[00:02:29.360]is that in this local view we're looking at depth.
[00:02:32.230]We're really looking at only points
[00:02:33.970]that are infinitesimally close to this value X.
[00:02:36.950]But in the nonlocal view,
[00:02:38.210]we're thinking that all of these points around X,
[00:02:40.980]interact with each other.
[00:02:42.320]So it's more of a breadth kind of a view.
[00:02:45.870]And in order to do that,
[00:02:47.360]we replace our differential operators
[00:02:49.010]with weakly singular integral operators.
[00:02:52.820]And we have this horizon of interaction.
[00:02:54.890]All of these points in this circle,
[00:02:57.320]in this horizon around X
[00:02:58.910]are interacting in this circle as a radius.
[00:03:01.260]Generally we say delta.
[00:03:03.450]So getting into my research, specifically,
[00:03:06.870]we have to define two preliminary types of convolutions.
[00:03:11.070]We have the regular convolution, of course,
[00:03:13.840]but we can also define this dot convolution
[00:03:16.370]which is exactly like a regular convolution,
[00:03:18.370]except you take, it's between two vectors,
[00:03:20.390]and you take the dot product instead of a regular product.
[00:03:24.350]And then the cross-convolution
[00:03:25.850]is also exactly what you'd expect it to be.
[00:03:29.520]It's between two vectors in R3.
[00:03:31.790]And when you cross-convolve them
[00:03:33.270]you take the cross-product,
[00:03:34.980]but instead of multiplying,
[00:03:36.050]each of these become convolutions.
[00:03:38.846]So those definitions allow us
[00:03:41.010]to define our nonlocal divergence gradient,
[00:03:45.540]and curl operators.
[00:03:46.930]And these are actually new operators, I should say,
[00:03:50.330]new operators that my research advisor and I
[00:03:52.580]have come up with in order to,
[00:03:54.600]we've created this new nonlocal framework
[00:03:56.620]that admits a Helmholtz decomposition,
[00:03:58.080]as what we'll see later.
[00:03:59.720]So the nonlocal gradient, we need a kernel alpha,
[00:04:02.730]this in a sense defines our interaction.
[00:04:05.510]This is how things interact,
[00:04:07.490]and it's just gonna be myself convolved with you.
[00:04:09.570]This is point-wise convolution.
[00:04:11.830]And the divergence is what you would expect.
[00:04:13.720]It's a dot convolution between alpha and a vector.
[00:04:17.450]The curl is a cross-convolution
[00:04:19.110]and this minus sign comes from convergence issues
[00:04:22.030]that we won't get into in this.
[00:04:23.120]And the Laplacian is also defined as in the local case,
[00:04:25.610]that divergence of the gradient,
[00:04:28.730]and this gives us our Helmholtz decomposition.
[00:04:31.830]So if we have a vector function in L1,
[00:04:37.980]and our vector kernel is alpha,
[00:04:39.670]then we can decompose it into minus a gradient plus a curl,
[00:04:44.480]a gradient of a function plus a curl of another function,
[00:04:47.190]which is exactly the Helmholtz decomposition
[00:04:49.750]as in the local case.
[00:04:51.960]And these operators, I should say,
[00:04:53.380]were defined specifically
[00:04:54.700]to make this Helmholtz decomposition work out
[00:04:57.830]because a Helmholtz decomposition,
[00:04:59.040]as I said, is very desirable,
[00:05:00.850]in studying the theory,
[00:05:04.720]for example, Poisson problems.
[00:05:09.290]So just an overview of the other results.
[00:05:11.510]We defined the operators
[00:05:12.740]to make the Helmholtz decomposition work,
[00:05:14.850]but it turns out
[00:05:16.300]that all of these other really nice results hold.
[00:05:19.520]So these new operators
[00:05:21.540]introduce a new nonlocal calculus framework,
[00:05:24.030]which includes the classical setting,
[00:05:25.740]mostly as a limiting case
[00:05:27.010]because we've shown convergence.
[00:05:28.530]But also if we're willing to consider functions in this,
[00:05:32.150]well not functions,
[00:05:32.983]but things in the space of the tempered distributions,
[00:05:36.750]we can actually take our kernel
[00:05:39.110]to be minus the derivative of the Dirac mass,
[00:05:40.980]and not actually just directly,
[00:05:42.820]you can recover the local operator,
[00:05:46.020]the local gradient divergence and curl in this,
[00:05:48.630]by letting alpha equal minus the derivative
[00:05:51.120]of the Dirac mass.
[00:05:53.600]So, and in this framework,
[00:05:55.940]we have a bunch of other identities that,
[00:05:59.440]they're specifically important
[00:06:01.090]in studying the Poisson equation,
[00:06:02.800]but also just important identities
[00:06:04.550]that highlight how similar
[00:06:07.010]the theory is between the local and the nonlocal case,
[00:06:10.010]at least in this setting.
[00:06:11.180]So for example, we have the kernel of the operators.
[00:06:13.300]This is for the first time.
[00:06:14.330]No other people have identified kernels for operators
[00:06:17.740]in their nonlocal setting.
[00:06:19.560]And by kernel here, I mean,
[00:06:20.895]where they're zero, not alpha.
[00:06:25.120]We've also identified an integration by parts,
[00:06:29.590]which of course is incredibly useful
[00:06:32.340]when studying any PDE,
[00:06:33.700]especially because it allows us to do multiplier methods.
[00:06:37.464]And we've also discovered a lot of identities
[00:06:40.380]that parallel the local setting.
[00:06:41.500]For example, the curl of the curl identity, which give,
[00:06:44.170]the curl of the curl
[00:06:45.050]is gradient of the divergence minus the Poisson,
[00:06:47.860]just as in the local case,
[00:06:49.900]and that's incredibly important
[00:06:51.480]in studying the Poisson equation,
[00:06:53.500]the curl of the curl identity.
[00:06:56.440]Furthermore, we've obtained a delta squared convergence
[00:07:00.170]of our operators, which is very nice,
[00:07:01.900]and we can extend them to other domains,
[00:07:05.230]essentially all that last point is saying.
[00:07:07.220]So possible areas of further investigation,
[00:07:09.870]we'd like to strengthen our results a little bit,
[00:07:13.100]and that's all this one is.
[00:07:15.970]The biggest thing we would like to do
[00:07:17.100]is look at how they act on bounded domains
[00:07:19.780]which would be nontrivial
[00:07:20.700]because we rely heavily on the Fourier transform for these.
[00:07:24.650]And we'd like to see
[00:07:25.483]if there are any other parallels to local calculus,
[00:07:27.950]such as, we could see these through numerical analysis.
[00:07:31.380]We could see these parallels through studying classical PDEs
[00:07:34.330]but in this nonlocal setting.
[00:07:36.010]We've already obtained some well-posedness results
[00:07:38.130]for the Poisson problem.
[00:07:39.750]And we've, of course,
[00:07:40.583]like to use these to model physical phenomenon.
[00:07:43.670]So with that, thank you.
[00:07:44.790]Thank you to UCARE for supporting me.
[00:07:46.570]Thank you to my amazing research advisor, also,
[00:07:49.710]for supporting me.

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A Nonlocal Helmholtz Decomposition

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