WEBVTT
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My name is Andrew Haar.
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I'm a senior, studying mathematics here at UNL,
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and I've been doing research for the past two years
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on nonlocal vector calculus,
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with my advisor, Dr. Petronela Radu.
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And so I'm here to talk to you about, more specifically,
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my research on nonlocal Helmholtz decompositions.
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There we go.
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So, local Helmholtz decompositions, just to start, are,
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the local Helmholtz decomposition says
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that we can decompose any vector function
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by any three-dimensional vector function
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that's sufficiently nice.
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That means it decays fast enough in infinity,
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into a solenoidal and an irrotational component,
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i.e. a curl-free and a divergence-free component,
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just said differently.
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And this has been really nice
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in a great deal of scientific disciplines,
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and partially because it allows us
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to prescribe a scalar field
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and a divergence-free vector field
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of another vector field.
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But for many other reasons, for example,
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in solving numerically incompressible
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Navier-Stokes equations.
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But anyway, moving on to nonlocal vector calculus.
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In mathematical modeling, we have a bit of a problem.
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And that problem is that the world is not smooth.
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From the middle of the 1600s until these days,
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we have been using derivatives in many models
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to approximate real-world phenomena,
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but derivatives do tend to break,
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well they do break down
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whenever there are discontinuities.
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So we, if phenomena are nice enough,
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such as in classical diffusion, wave propagation,
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and so forth,
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they can still be modeled using these derivatives.
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But these models break down
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when their material discontinuities.
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For example, in crack formulation,
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or in image processing, is another example.
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And weak derivatives have sort of solved this,
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but not really.
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There still are problems with it.
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So in comes peridynamics or nonlocal calculus,
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peri meaning near, dynamics meaning force.
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So in nonlocal, really all that,
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what that means is that objects interact with other objects
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that are immediately next to them.
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And I have a picture for that on this slide,
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is that in this local view we're looking at depth.
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We're really looking at only points
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that are infinitesimally close to this value X.
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But in the nonlocal view,
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we're thinking that all of these points around X,
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interact with each other.
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So it's more of a breadth kind of a view.
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And in order to do that,
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we replace our differential operators
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with weakly singular integral operators.
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And we have this horizon of interaction.
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All of these points in this circle,
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in this horizon around X
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are interacting in this circle as a radius.
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Generally we say delta.
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So getting into my research, specifically,
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we have to define two preliminary types of convolutions.
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We have the regular convolution, of course,
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but we can also define this dot convolution
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which is exactly like a regular convolution,
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except you take, it's between two vectors,
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and you take the dot product instead of a regular product.
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And then the cross-convolution
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is also exactly what you'd expect it to be.
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It's between two vectors in R3.
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And when you cross-convolve them
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you take the cross-product,
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but instead of multiplying,
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each of these become convolutions.
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So those definitions allow us
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to define our nonlocal divergence gradient,
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and curl operators.
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And these are actually new operators, I should say,
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new operators that my research advisor and I
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have come up with in order to,
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we've created this new nonlocal framework
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that admits a Helmholtz decomposition,
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as what we'll see later.
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So the nonlocal gradient, we need a kernel alpha,
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this in a sense defines our interaction.
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This is how things interact,
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and it's just gonna be myself convolved with you.
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This is point-wise convolution.
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And the divergence is what you would expect.
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It's a dot convolution between alpha and a vector.
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The curl is a cross-convolution
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and this minus sign comes from convergence issues
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that we won't get into in this.
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And the Laplacian is also defined as in the local case,
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that divergence of the gradient,
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and this gives us our Helmholtz decomposition.
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So if we have a vector function in L1,
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and our vector kernel is alpha,
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then we can decompose it into minus a gradient plus a curl,
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a gradient of a function plus a curl of another function,
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which is exactly the Helmholtz decomposition
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as in the local case.
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And these operators, I should say,
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were defined specifically
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to make this Helmholtz decomposition work out
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because a Helmholtz decomposition,
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as I said, is very desirable,
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in studying the theory,
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for example, Poisson problems.
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So just an overview of the other results.
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We defined the operators
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to make the Helmholtz decomposition work,
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but it turns out
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that all of these other really nice results hold.
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So these new operators
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introduce a new nonlocal calculus framework,
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which includes the classical setting,
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mostly as a limiting case
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because we've shown convergence.
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But also if we're willing to consider functions in this,
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well not functions,
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but things in the space of the tempered distributions,
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we can actually take our kernel
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to be minus the derivative of the Dirac mass,
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and not actually just directly,
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you can recover the local operator,
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the local gradient divergence and curl in this,
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by letting alpha equal minus the derivative
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of the Dirac mass.
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So, and in this framework,
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we have a bunch of other identities that,
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they're specifically important
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in studying the Poisson equation,
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but also just important identities
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that highlight how similar
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the theory is between the local and the nonlocal case,
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at least in this setting.
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So for example, we have the kernel of the operators.
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This is for the first time.
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No other people have identified kernels for operators
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in their nonlocal setting.
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And by kernel here, I mean,
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where they're zero, not alpha.
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We've also identified an integration by parts,
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which of course is incredibly useful
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when studying any PDE,
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especially because it allows us to do multiplier methods.
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And we've also discovered a lot of identities
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that parallel the local setting.
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For example, the curl of the curl identity, which give,
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the curl of the curl
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is gradient of the divergence minus the Poisson,
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just as in the local case,
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and that's incredibly important
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in studying the Poisson equation,
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the curl of the curl identity.
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Furthermore, we've obtained a delta squared convergence
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of our operators, which is very nice,
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and we can extend them to other domains,
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essentially all that last point is saying.
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So possible areas of further investigation,
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we'd like to strengthen our results a little bit,
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and that's all this one is.
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The biggest thing we would like to do
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is look at how they act on bounded domains
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which would be nontrivial
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because we rely heavily on the Fourier transform for these.
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And we'd like to see
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if there are any other parallels to local calculus,
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such as, we could see these through numerical analysis.
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We could see these parallels through studying classical PDEs
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but in this nonlocal setting.
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We've already obtained some well-posedness results
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for the Poisson problem.
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And we've, of course,
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like to use these to model physical phenomenon.
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So with that, thank you.
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Thank you to UCARE for supporting me.
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Thank you to my amazing research advisor, also,
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for supporting me.