Heterogenous Diffusion Operator in Nonlocal Framework

Anjaneshwar Ganesan Author

08/03/2021 Added

7 Plays

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Exploration of a subfield of Mathematics called Nonlocal Operators, specifically the Heterogenous Diffusion Operator.

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[00:00:01.200]Hey all I'm Anjan and today my presentation is going to be about heterogeneous
[00:00:05.160]diffusion operator in the non-local frame work.
[00:00:10.490]So, uh,
[00:00:13.010]first let's talk about the mathematical modeling using non-local methods and
[00:00:17.420]sort
[00:00:17.660]Of what it means. Uh, we have been using classical derivatives
[00:00:20.870]to model our real world phenomena. However, there are a few exceptions that,
[00:00:25.490]you know,
[00:00:25.780]we can think about one is that not all phenomenon are small enough to be modeled
[00:00:30.020]by the classical derivative, and that's a real problem because for example,
[00:00:34.700]if you take examples like breaking clay,
[00:00:36.680]and if you take examples like shattering glass, uh,
[00:00:40.490]those are not phenomenon that you can easily model with the derivative,
[00:00:45.530]if not impossible.
[00:00:47.180]So essentially non-local calculus provides us a simple
[00:00:52.010]solution well actually not simple, but it provides us a solution,
[00:00:56.990]uh, to sort of work around that issue, which is, uh,
[00:01:00.860]we use integral sort of convert derivative to intergral.
[00:01:05.750]If, you know, that's a simple way to think about it and I can show you sort of,
[00:01:10.760]you know, uh, an example using this, this surface, if you could call, right.
[00:01:15.140]You know,
[00:01:15.440]think about some squishy clay being here and when you're breaking apart,
[00:01:19.550]the clay, basically we are analyzing all the forces between,
[00:01:24.320]between the points in this fault. So, you know,
[00:01:26.690]if this is the point you want to find out what forces are acting on it,
[00:01:30.170]you can essentially find out the force from this point acting on this point.
[00:01:34.040]And from this point acting on center point on this ball,
[00:01:37.400]and this ball has a radius of Delta. And, um, you know,
[00:01:42.380]if you do that to all the points, I guess, in this, in this, uh, the surface,
[00:01:47.060]you, you can figure out, you know, how,
[00:01:49.080]how the forces are acting on the entire
[00:01:53.990]system essentially. Uh,
[00:01:57.620]so the method that I, uh, I sort of, I'm concentrating on, uh,
[00:02:02.430]it's called, uh,
[00:02:04.010]the non-local heterogeneous laplacian rather though the subsection of
[00:02:08.900]non-local calculus. And I'll, I'll sort of, uh,
[00:02:12.980]have an sort of create an analogy of passing notes in class.
[00:02:17.660]Uh,
[00:02:18.410]you sort of an analogy of how the non-local heterogeneous supply chain or I'll,
[00:02:23.120]I'll call it the NHL and how it works essentially.
[00:02:27.650]So, uh, let you have.
[00:02:31.130]XD with a concentration, uh, you know, think about division if you want it.
[00:02:35.720]That also helps out of a quantity under observation at X at point X
[00:02:40.730]and comma T I'll.
[00:02:41.780]Use T uh, you know, I I'll use time.
[00:02:45.830]As a lot of a variable in, in, in this, uh, presentation,
[00:02:50.480]but right now I'm concentrating on space. So X,
[00:02:54.380]Y um, variables,
[00:02:57.050]not T yet I'll do it sometime the future, but not now, then, you know,
[00:03:01.480]UT, uh,
[00:03:03.130]you subscript T X of D is the rate of change of the concentration with respect
[00:03:07.480]of time. And so think about this.
[00:03:09.930]Let's say a class was.
[00:03:11.910]Given an assignment right now,
[00:03:13.420]a classroom was given an assignment and teacher said,
[00:03:17.340]the only way you can communicate with each other in the room was by passing
[00:03:21.990]notes, right? Um,
[00:03:24.750]so all the students have to work on this group assignment,
[00:03:27.030]and the only way they can transfer information is by passing notes. Uh,
[00:03:31.020]so they would have to start off, um.
[00:03:34.430]Pass notes in, in.
[00:03:36.570]It depends, right? Because some students may pass a lot of notes. Some people,
[00:03:39.960]some students may pass less notes and some people may have more notes to pass
[00:03:43.890]and some people have lessons to pass.
[00:03:45.870]So the number of notes that a person has,
[00:03:49.080]you could think of it as a concentration sort of thing, right?
[00:03:52.140]And the mew of X, Y is the probability that.
[00:03:56.490]Massive Bush and X moves to do.
[00:03:59.670]Push. And why. So, for example, if I'm one of the students,
[00:04:02.220]and if you're a student that's sitting in front of me,
[00:04:04.320]then there's a probability that I'll give you a piece of paper, because,
[00:04:09.300]you know, in a, in a classroom idea classroom,
[00:04:11.730]one person may like the other person,
[00:04:13.380]or may not like the other person there such things to think about,
[00:04:16.500]but I guess this is a much more complex, like much more general example because,
[00:04:21.420]uh, you know, such calculus can be applied to more scenarios. Right.
[00:04:26.100]Um, so yeah, X like for example, concert diffusion process,
[00:04:30.360]which has a preferred direction like uphill or downhill, uh,
[00:04:33.600]similarly you have X,
[00:04:35.010]Y is the probability of max at Bush and white to push an X.
[00:04:38.700]So just like how may I may have like a probability to give you a note,
[00:04:43.830]uh, you may have probability to give me a note.
[00:04:46.350]So I might maybe give you a note, like 50% chance of giving you a certain note.
[00:04:51.000]You have 50% chance of giving me, uh,
[00:04:53.730]or maybe 0.6 or 0.7% chance of, uh, a percent sorry,
[00:04:58.110]but 70% chance of giving, uh, note, right?
[00:05:03.660]So this can be represented by the, by an integral sort of.
[00:05:07.920]So this interview, well is the net outgoing at location X.
[00:05:12.630]So for example, this, this integral represents how many,
[00:05:17.250]uh,
[00:05:18.480]like all the outgoing notes that I give out to everyone around me.
[00:05:22.950]So this is just me giving up notes to everybody around me.
[00:05:26.430]And this integral represents how many notes I give to everyone and
[00:05:31.050]this, and this integral represents the net flow incoming.
[00:05:34.620]So it tells you, uh,
[00:05:37.650]the people are on me and what they give me the what,
[00:05:40.830]what are the notes that they essentially, um,
[00:05:43.560]the number of notes that I get from everyone around me and this sort of makes
[00:05:48.120]sense because, you know, when I'm giving out a note to someone,
[00:05:51.330]there's a chance that, you know, they have, they have a certain, um,
[00:05:55.800]I have probability to give, so this is the probability,
[00:05:58.850]and this is the note itself. So, and then you're summing up, uh,
[00:06:03.590]the central symbol as busy something.
[00:06:05.360]So you're summing up everything that I'm giving to people.
[00:06:09.470]And this is summing everything that I received from people. And, you know,
[00:06:14.090]when there's no variation in time, we get such, you know,
[00:06:16.880]such a quantity and you subtract those both.
[00:06:19.190]And this is like the net concentration at, at, at a moment in time,
[00:06:23.780]essentially. And we did not, you know, an integral,
[00:06:27.230]I called it a passion of you at you visited is given by this integral
[00:06:32.150]and I call it the non-local heterogeneous supplication for non symmetric
[00:06:36.360]Cardinals. So the deficient law is given by, uh, such
[00:06:41.990]and essentially let's,
[00:06:43.130]let's break it down even more to get under the interesting result.
[00:06:46.010]Let us try decomposing that nutshell into its symmetric and anti-Semitic Cornel
[00:06:50.200]parts for any kernel is symmetric part is given by this.
[00:06:53.420]And its antisemetic part is given by this expression. Uh, this,
[00:06:57.950]this is pretty important because the next slide explains why they're important.
[00:07:01.700]Um, if you some, you know, the symmetric antisemetic, I mean, yeah,
[00:07:06.230]the address and metric parts, uh, you get me off X, Y.
[00:07:09.830]So the probability of me, me passing a note to you,
[00:07:13.850]and if you subtract the symmetric with the ADESA metric part,
[00:07:18.470]you get X call my wife. So the probability of you giving to me,
[00:07:21.770]and we can use these facts to sort of, uh, substitute in,
[00:07:26.600]uh, for this part of the interval essentially,
[00:07:31.310]right? Because you can substitute the quantities, uh, with, with these,
[00:07:35.420]with these meals here, right? And once we do that, we derive this expression.
[00:07:40.190]And essentially what that means is, uh, the [inaudible] of the function.
[00:07:44.600]You essentially.
[00:07:46.520]The, uh, the [inaudible].
[00:07:48.980]Of the symmetry Cornel minus the application of the antique symmetric Cornel
[00:07:53.810]minus the two times minus a constant term,
[00:07:58.430]essentially. Right.
[00:08:02.590]And we can also derive another interest in result from the non-local
[00:08:06.880]heterogeneous whiplash. And for example, if the function you, uh,
[00:08:11.230]is this constant, what would the plush in, uh,
[00:08:15.340]or for constant yield? So we can take our original equation again,
[00:08:18.820]which is equation, uh, pause 0.1.
[00:08:22.240]We can take that and then sort of substitute, and you can see that, you know,
[00:08:26.440]we get this result and essentially, and this, the integral of, uh,
[00:08:31.150]asymmetric, uh, anti-Semitic sorry, antisemetic Cornel. Uh,
[00:08:35.830]the integral of that may not necessarily be zero.
[00:08:38.050]So this is interesting because this implies the existence of skilling factor as
[00:08:42.880]the classical application, constant is zero.
[00:08:46.120]So there might be a scaling factor involved.
[00:08:48.310]And this also shows that it does go here into our calculation or decomposition
[00:08:53.320]we did above which, you know, uh, if, if you as a constant,
[00:08:58.320]then you get zero here, zero here, and also, uh,
[00:09:02.610]you symmetric there. Okay. So let's jump to an example real quick.
[00:09:06.240]So for example, uh, take you affects is equal to one, and this is the,
[00:09:11.040]you know,
[00:09:11.340]the Colonel and we can use the equation to determine the antisemetic part.
[00:09:16.680]And again, since this is a constant, uh,
[00:09:19.320]we can take the integral and we get this result and let us take X is equal to
[00:09:24.150]one and graph this result.
[00:09:25.470]I graph to see if it actually gets us close to the classical flashing that we
[00:09:30.150]expect and wallah, it does, it's close to zero. And she,
[00:09:34.350]the main concept is as the Delta approaches,
[00:09:36.540]zero should approach the classical of plush and basically the classical during a
[00:09:40.410]bit of, uh, concepts that we know of. Right? So as Delta approaches,
[00:09:45.150]zero, this example converges to the classical oppression of your effects,
[00:09:49.380]which is equal to zero. So that, that's pretty interesting to think about, um,
[00:09:54.240]from previous papers to return on non-local applications,
[00:09:57.660]the symmetric Cardinal flashing has been shown to converge the classical
[00:10:01.890][inaudible].
[00:10:03.180]Hence I'll be my future step is to explore whether the classical counterpart,
[00:10:07.920]uh, of, uh, of the asymmetric Cornel,
[00:10:11.790]and as a metric Cornel convergence to anything I'll be doing numerical
[00:10:15.720]simulation analysis, uh, to do that. And yeah, thank you for listening.
[00:10:19.800]It has been my pleasure to presenting the research to you all. Thank you.

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Heterogenous Diffusion Operator in Nonlocal Framework

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