 Ok, so, the first talk in the afternoon is going to be Michela Jakobeli from the University of Cambridge. She will speak about quantization of measures in a great approach. So, thank you very much. It's a great pleasure to be here. Thank you to the organizers for inviting me. So today I'm going to talk about quantization of measures. That in this contest means approximation. So, let me start. The outline of the talk is the following. I will start with a brief introduction of the problem of quantization of measures. Then I will talk a bit about a variational approach to this problem. And I will explain the heuristic of the results obtained in one dimension and dimension two. And then finally I will tell something about some new perspective on the subject. So, let me start with a simple question that will motivate the problem. So, the question is the following. Suppose that we are in an urban region and we have a density of population rule. I mean, we would like to find the best way to locate n-clinics in our urban region in order to meet the demand of the population. So, clearly we want to put clinics when we have more people. And we also would like the clinic to be big proportional to the density of population, so in order to meet the demand. So, to answer to this question, so to find the optimal location, we have first to find a notion of optimality. So we want to understand in which sense we want to optimize. But also we need to decide where to locate the clinics. And so, we have to deal with position XI. And we have to choose how big each clinic should be, and so at each location XI we will assign a mass MI. So, just a bit of history. This problem of quantization of measure has been introduced in the late 40s in the context of information theory for the task of finding a way to transform an analogic signal to a digital one in such a way to lose as little information as possible. But also, so this was started by Shannon in the 50s, but then it had application in numerical integration because when you have multidimensional integrals with respect to some measures and you want to find some quadrature formulas to compute this integral with respect to simpler measures. But also, it has application in crystallography and in economics. For example, the question that I stated before of the location of the clinics is a problem called optimal location of service centers. And this clearly arises in the context of economics. And much other things going on. I just discovered that there is a subject that is called filtering of measure that is linked to stochastic differential equation and is actually the same thing, but I have no idea. So I am continuing discovering new application of this subject. So let me go to mathematics. So this problem can be stated, the quantization problem can be stated at follow. So we fix n, and we want to find the best approximation of rho by an atomic measure sum over i and my delta xi. So I will just write at the blackboard what I would like to remain. So we have rho dy dy and my delta xi. So I just goes from one to n. To do that, we have to find the notion of optimization, as I told before, so what is optimal. And this comes via the notion of buses and distance. So here, since the distance that we want to approximate is absolutely continuum, we can give an easier definition of buses and distance. So we fix r better equal than 1. So the buses and distance of order r between rho and our discrete measure is the infimum over omega of 1 minus dy to the power r, rho dy, where t varies among all transport maps. So we want to transport the map rho onto our discrete measure, the measure rho onto our discrete measure. And the property that t has to have in order to be a transport map is that if we look at the set, ai, that is the counter image of xi via the map t, we want that rho integrated on this region, ai on this Borel set will be exactly mi. So this is a mass condition. Some comments about why using these distances for approaching this problem. So a very easy comment is the following. When we want to find approximation of probability measures, there are several distances that are used to do that. For example, the total variation distance or procur of distance. But if you just think about the total variation distance, if you have the total variation distance between delta zero and delta epsilon, it's always two, it's always a constant. So it's not sensitive to when you want to deal with atomic measures. While the vasterstane distance between two deltas, delta zero and delta epsilon is exactly epsilon. So it's sensitive to when one of the two measures of both are atomic. But another reason to use that is that these kind of distances that demetrize weak convergence of measures. And this will be very useful later on. And a posteriori, another reason, is that when you want to find, when you want to look for a variational approach to solve a problem like that, you want to end up with a functional to study. And this distance will lead to a good functional for studying this problem. So, I mean, at the beginning I told you that we want to both find the optimal location of the points and also the optimal masses. We can start minimizing on the masses. So we fix some point, x1, xn in red omega and we want to minimize with respect to mi, mn with the constraint of being probability measure. So the answer to this question, so the optimal masses mi will be the integral of the Voronoi cell of the point xi rho dy dy. So what is a Voronoi diagram? A Voronoi diagram is a way of partitioning the plane with respect of closeness of points. So if you have n points in the plane that we will call seeds x1, xn, then you will have a partition of the plane so that the Voronoi cell of the point xi will be the set of all points that are closer to xi to respect to all the other points in the plane. So this is an answer at least to the problem of the choice of the right masses. So once we have this information, we plug this information into our vasterstane distance and it can be proved that this minimization problem translates into a minimization problem only with respect to position. So now what we would like to do is to study the functional F and R. I will refer to this functional as discrete function, just to differentiate it from the continuous one. So F and R of x1, xn is the integral over omega, minimus over i that goes from one to n, xi minus y to the power r, rho dy dy. So we started from a problem with parameters that masses and position and now we just have to deal with the optimal positions. So the goal from now on will be minimizing this functional. But before finding the minimizer, we can ask ourselves, if we find the minimizer, what is going to do asymptotically? How these points are going to distribute with respect to rho? And an answer to this question has been given from the point of view of calculus of variation by Buckley and Weiss in the 80s and later on improved by Graf and Luzki. So if we take a probability density with finite r plus delta moment and if x1, xn is a global minimizer of the functional F and R, then we have that the empirical measure centered in this minimizer, we click on verge as n goes to infinity to a certain power of rho, a power that depends by the dimension D and the power of the cost R. So at the beginning I told you that I would like to find an approximation of rho while here we are asymptotically approximating a power of rho, but this is why, the reason is that in this moment, we are giving the same weight to all the masses. So all the masses weight the same, one over n, the weight is one over n for all the masses. We are not considering the optimal choice of the masses. So if here then we plug in the mi defined as before via the Voronoi diagrams, then we approximate exactly rho. Yes, but in general, the problem is we use this information to find the formula for the function that we want to study, but then we first would like to locate the points, the optimal points, minimizing this function, then once you locate this problem, these points, you compute who are the Voronoi cell and then you compute the masses. So this should be the algorithm. But at this point, this give us the good information to study another function that is just dependent on the position and not on both position and masses. And this does not depend on the power R. So the Voronoi diagram is always the same. So at least this give us an answer about the asymptotic distribution of these points. So now we would like to find a more deterministic approach to find in actually these points and this is where our gradient flow approach comes into play. So the heuristic is very easy, the following. So imagine we take an initial configuration of point, x1, not xn0, then we would like this point, x0 to evolve under the direction of minus gradient of Fnr. So we would like to solve the gradient flow of Fnr. And if we are lucky, so if Fnr has good properties, we expect that as time goes to infinity, we will convert to global minimizer. So once we find this global minimizer, at this point we will perform the limit in the number of particles and we will hope to approximate our measure. But you see that here we have two different limits in t and in n that in principle are not commuting. And also the functional, the discrete functional Fnr is non convex. And there's many local minimizer. It's a functional that you can imagine a bit like that. So what happened is that you could just get stuck in a local minimizer and never reach the global minimizer. So this is the first and deeper difficulty in this problem because we would like to understand both the behavior in time, in the long time behavior and the limiters number of particles goes to infinity because clearly coming back to the question that I stated at the beginning, if you want to locate clinics and you just want to locate five, there are efficient methods to find where to locate these clinics. But this problem become mathematically more challenging and interesting in the limiters, the number of points goes to infinity. So what's a way to approach this problem? So since we want to pass to the limit in the system of ODEs, we want to embed our Iran in a bigger space that we choose to be L2. And to do that, we consider a set of reference point, x1 at xn up, and we parameterize the general family of point xi using a smooth map capital X. Because the idea is to translate this discrete problem into a continuous problem and hopefully use the good properties of the continuous version of this problem to deduce some information on the discrete one. So, because once we can reparametize all the points using this map capital X, that's in L2, we can rewrite our discrete function with respect to capital X. At this point, we show that a suitable normalization of F and R converges to a continuum functional. So we will have that N to the power R, F and R will converge to another functional, a calligraphic F of X with X in L2. Or, see, infinity. So the idea would be the following. We would like to study the gradient flow of the continuous functional, the gradient flow of the discrete one, and see whether these two gradient flow remain uniformly close for all time. Because if the answer is yes, if these two gradient flow are close uniformly in time, we can say that the dynamic induced by the continuous gradient flow approach very well, describes very well the dynamic induced by the discrete one. So we can deduce what we need on the original function. And this was a bit the problem. The program that we wanted to approach in dimension one and dimension two, clearly the real difficulty is that this is a non-convex problem. But as you can imagine from the description of Voronoi tassillation, this problem degenerates very easily with the dimension because finding the Voronoi cell in an interval is very easy, but finding Voronoi tassillation in i-th dimension is much more complicated. So let me start with the one-dimensional case. Okay, now our omega is zero one, and in this situation, okay we order the point, and in this situation it's very easy to understand where the Voronoi cell is because we just consider the midpoints. And when we have an explicit description of the Voronoi cell at this point, we can find a more manageable formula for the discrete functional because now our f and r is exactly this, okay, apart the boundary terms, but it's this function, and then we just differentiate it and we find the discrete gradient flow, but apart that we expand here in rho, and we find the continuous function that we wanted to choose, so just to visualize it, to visualize this parametrization that I mentioned before, imagine that here we have zero, one, one, here I have the general, I have the grid i over n, so the point xi will be given by, so calligraphic x is going from zero one to zero one, and it's increasing, so we will have that xi, the general point xi is capital X of i over n, or okay, i minus one half to be precise, but this is the idea. So at this point, we can find the continuous version of our functional, theta will be in this, so at this function theta will be in zero one, and from here we do a first variation, and we find the parabolic equation that will govern the dynamic of the continuous problem. So just few comments of this equation, let's see now excess Dirichlet boundary condition, because we want zero to be in zero and zero to be in one, and just to get a flavor of what is like this equation, we put rho equal to one, and we find an equation of pilaplacian time. Yes. No, yes, but I will come back also on this, because it's the main equation. So you see that this is a second order parabolic equation that will possibly degenerate, and one of the main tasks will be to avoid the generality issue on this equation. That is a priori not easy, because this equation, you see that constants are not solution, and it depends by x d theta of x and d theta of x, so it doesn't have a straightforward maximum principle, because now all the point will be to see that if we impose a time zero, that the theta of x is positive, it will remain positive for all time, and the priori is not clear on that equation that we have comparison principle. We have comparison principle for the pilaplacian equation, but when rho is not one, the equation is much more complicated. So a way to prove, I mean maybe there are several, but a comparison principle of our original equation is to pass through a change of variable that we can inherit from the kinetic theory perspective. So when you have a many particle system, you can choose to use a Lagrangian approach that means that you follow the path of each particle, so you are sitting on the particle and you see each particle which position and velocity it has, or you can also use an Eulerian point of view that means that you are sitting on and you are looking your particle flows and looking what are doing the particles that are passing through the point you are sitting on. So this is just to explain a bit. In mathematical term, we will define f of t of x as the push forward of the Lebesgue measure, the theta, via x and the push forward condition gives exactly this condition. So we will have that f of t, capital X, is exactly one of the theta x. So, I mean, just a bit of computation and we can rewrite our equation for x in an equation for f. And we see that here, I mean, for who is expert of gradient flows, it is immediate to understand that this is a gradient flow in Busterstein 2 of a certain functional that can be thought as a Lyapunov functional in the spirit of the Loran talk of this morning. But in this moment we will just focus on the structure of this theory in B, because we would like to find a comparison principle for this equation that is if flow is equal to one in equation of very fast diffusion type. Just a remark, in this situation I have periodic boundary condition that comes from the fact that for x we have Dirichlet boundary condition because we have that x of 0 is 0 and x of 1 is 1, but the very fast diffusion equation if set it on the whole r and Dirichlet boundary condition have no solution because the mass immediately disappears. So in this situation at least we know that we have solution because the mass remains trapped. And so also for this equation we don't have really a comparison principle so we have to do another smart change of variable that comes using what would be the limiting and what would be our u infinity with respect to the talk of this morning. So the asymptotic equilibrium that is rho at the power 1 over 1 per ser because in the spirit of the theorem of Buckley and Weiss I told at the beginning that if we find the minimizer we will have that 1 over n and some over i delta xi bar where xi bar is the minimizer will converge to rho to a constant rho over d d plus r in this situation d is equal to 1. So at least for you now constants are solution and we can prove a comparison principle. So once we have a comparison principle for you we can unwrap our change of variables and so we will recover information for x. So in this way we can prove that if we impose a time zero that the delta of x is positive it will remain positive for all times. Now the problem here is that we want to use this condition to hold both at the continuous and at the discrete level because we want to compare a discrete gradient flow with a continuous one and also when you have to compare the continuous one you have to evaluate the continuous gradient flow on a grid and then at the point to compare the two. So we also need this condition to hold at a discrete level. So let me just state the result that has been obtained in collaboration with Emanuele Cagliotti from Ron Sapienza and Francois Gauls from Ecole Polytechnique. So now for stating the theorem r will be equal to 2 and we assume that rho is close to 1 in c2 and this is very important that we will come back to this hypothesis commenting on this hypothesis. So we assume that x1, xn is the gradient flow of the discrete functional starting from x1, 0, xn, 0 so under some regularity assumption of rho, so rho is c3, alpha and on the initial datum because we want that at time 0 the two starting point of the discrete gradient flow in the continuous one are close of an order 1 over n square so this is a closeness condition at time 0 that is necessary when rho is different from 1 otherwise we don't need it. At this point we can say that the continuous and discrete gradient flow remain quantitatively close for all time. So you see that here we have this estimate with a time shift that is n cubed and now I will explain why. This is because this n cubed comes from the fact that I told that the discrete gradient flow will convert to a continuous one if scaled with n to the powers r that in this case will be n square but since we are comparing gradients and we are and we embedded rn to in l2 we have another power of n that pops out and so we have this time shift of order n cubed and in some sense is saying us that these two limits are not exactly commuting but they are going in a diagonal way and once we have these estimates where we estimate on the continuous function by triangular inequality we can prove that we have a quantitative version of the theorem of buckling wise this means that provided that we wait enough time the empirical measure centered in xi will converge asymptotically to rho to the power 1 over 3 that is the one dictated by buckling wise exactly what we was expecting so I mean, is it clear up to now? Yes I think so I don't remember I mean, I assumed a priori that that rho is very small in c2r, is very close to 1 because if it is not close to 1 the continuous functional is not convex so we have a counter example in case it's not close to 1 no, no, no it's not degenerated yes, clearly it's not degenerating in n, no it doesn't depend on n no, no, it doesn't depend on n, it doesn't depend on n it depends on the c3r normal bar of rho and the initial bound that we put on rho that is between lambda and capital lambda but it's not depending on the number of points so just some comments on the strategy of the proof so if rho is equal to 1 it will be much easier because we have comparison principle on the continuous equation and also at the discrete we can prove comparison principle by using maximum principles and we will have that this bound that preserved in time so that we can perform grombala argument of that but if if rho is different from 1 then we have to be more careful because first we had to prove comparison principle on the continuous function then we have to show that the continuous gradient flow evaluated on a grid solves the equation for the discrete one up to an error over the 1 over n square this means that the continuous gradient flow so if we have xi minus s cubed df and 2 so yes gradient f and 2 plus ri equal to 0 and ri is bounded by c over n square so we want the scheme to be good up to an order and to square at this point we can we can prove that the discrete and the continuous gradient flow are 1 over n square close on a finite interval of time and we can use this information to make pass the information that we recovered on the continuous functional using maximum principle to xi because in this small interval of time we can perform the L2-grombal estimates that will tell us that actually we are going closer exponentially faster and since the scheme is very good so it's converging up to an order 1 over n square we can bootstrap this estimate and show that actually it holds for all time so in this sense it was really crucial that the model was a priori good because otherwise there is no way the scheme not to be degenerate in the number of particles so the comment on the hypothesis on the floor is that actually our discrete functional is highly non-convex so the idea would be to find a continuous functional calligraphic path that has some convexity properties and then use the closeness of the two problems but if also the continuous functional is non-convex then we cannot perform any kind of gradient flow arguments so in case rho is not close to 1 we have a counter example for this so just disclosed the one-dimensional case so I would like to say something on a more challenging situation so where we are in dimension 2 so now the difficulties are many because at least in dimension 1 since the Voronoi cell were very easy we could find a very practical formula for the discrete gradient flow doing calculations but in dimension 2 we don't have the definition of the functional and also so this is a spoiler because this say that we find it once we find the discrete functional and a continuous version of it we will see that the continuous functional has not the good properties that have the continuous functional in dimension 1 but depends on the determinant of grad x and this will make this functional island of convex and unfortunately there is still no theory for gradient flow for island of convex function so this was the bad news but the good news is that for enlarge we know who are the asymptotically best configuration so if we want to approximate the constant then we know that in dimension 2 if we choose the regular hexagonal pattern this is a configuration of minimal energy so in this situation when you know at least who is the configuration of minimal energy we can think to do an analysis close to that minimal energy state and this is exactly what we do but before let me show you some pictures from a simulation a pattern these are all polygons the gray ones are the polygons that are less hexagonal it means that they have more sides or less sides the colored ones they are closer to be regular hexagons with respect to the gray one and you see that if we start from a configuration like that and we make the configuration evolve under the gradient flow in some iteration we get an hexagonal pattern more or less apart from some one-dimensional errors these one-dimensional errors are due to the fact that first at a certain moment we stop doing iterations so it's not really asymptotical second from the boundary condition that I imposed on the programmer not exactly the perfect one and also it may have a deeper reason that comes from the fact that these problems have many local minima for example if a regular hexagonal pattern is a minimal energy state but if you rotate it it's still a minimal energy state so these are like clusters of minimal energy states and so we can have many exactly I put the point in an arbitrary way close to in some way close to an hexagonal pattern because I don't want to them randomly just in a small region I want to be spread in order to be close to an hexagonal pattern and then I see that I get a regular hexagonal pattern in some time so coming back to the math so this simulation gave up some op that this could work so now the strategy will be to look at configuration close to the minimal energy states so we will consider a triangular lattice calligraphical and if we take this triangular lattice we see that the Voronoi cells induced by the lattice are hexagonal and so as the beginning I told that I wanted to transform the discrete problem into a continuous one I needed to to reparametraize families of points with respect to a certain map X that is a map in a functional space like elsewhere, infinity but to do that I choose a grid, a reference grid so in this situation, so in 1D I chose i over n in this situation I choose the center of the hexagonal pattern so and in this context now we still would like to send n to infinity so we will do that by scaling the lattice of a factor epsilon positive epsilon 1 over n into the natural so we consider pi to be a fundamental domain, this is not that important the only really important thing is that we are in a domain that is compact and whose periodicity agrees with the periodicity of the hexagonal pattern we want this two periodicity to agree to be compatible and so the idea is that now X will be a deformorphism in R2, pi periodic close to the identity so and what we do is that here we have our scaled hexagonal pattern regular hexagonal pattern we use X as a deformation and we want to see who is the energy of this configuration and who is the energy of this configuration with respect to X so it's important that X a priori is close to one because we don't want to lose completely the structure we still want to have hexagons even if not regular but we don't want to go out from the perturbative regime because also the definition I mean when we will have to find the formulas for our functional these formulas are related to the fact that we are in a perturbative regime because we constructively find the formulas for F close to the hexagons so now the goal is to compute the energy of this continuous functional as epsilon goes to zero improve that under the gradient flow of calligraphic F the limit of the near hexagonal vornalization will convert to the regular tessellation so that not only the regular tessellation are optimal but they are optimally stable asymptotically stable so now let me explain a bit the formulas for the function now as you can see is not that nice and also I was kind of scary when I saw it because then doing gradient flow on this function is not easy but we can still do something on it so now let me just write here that capital F is the integrant of calligraphic F so this is not the function this is the integrant of the function so capital F is evaluated on matrices in R2 and we see that it is anisotropic functional that can be seen as a functional coming from nonlinear elasticity problems for anisotropic nonlinear elasticity problems but let's find another formula for that that maybe is better later so our functional will be calligraphic F equal to calligraphic F of X is equal to integral of capital X of grad X and now instead of having a parabolic PD we have a system of parabolic PDs so everything becomes more challenging because when I explained the one-dimensional problem I asked for some regularity assumption on row I asked for row to be C3alpha and this is because I wanted to use regularity for the parabolic equation that was governing the gradient flow to say that I could estimate well the higher terms of a Taylor expansion so I needed regularity theory for the parabolic equation but here we have a system so let's find a better formula for doing computation that is this one and here you can see that we have anisotropic degenerate functional because here we have the determinant of grad X that makes everything unknown convex so the next step is to see what happens close to the identity so here we have an expansion close to the identity and we see that the blue terms are nice, are fine because they are positive but we also have two terms that are bothering us in principle because we would like to say that at least the continuous functional is convex so at least in a region so luckily for us these two terms integrate by zero because determinant is in Lagrangian why periodic, axis periodic so these integrate by zero and so the standard I mean the clear idea is that now we want to convexify this problem using this fact so convexify the problem close to the identity where we have this expansion so now instead of considering calligraphic F of X to be integral of capital X grad X we will consider integral of F not of grad X where F not is uniformly convex close to the identity so at least for close to the identity we are fine because for this functional we can apply the standard gradient flow theory and so I mean we can say that if grad X is close to the identity for all time then X of t will convert to the identity exponentially fast in L2 using the classical gradient flow theory for convex function so now the main issue is proving one because this is still in the same spirit of the problem one dimension even if it's different but in the one dimensional problem we had theta in zero one and we wanted to say that the theta of X at time zero is positive then it remains positive for all time so the theta of X of t is positive for all time and so here we have a similar situation in which we want to prove that imposing a condition at time zero will tell us that we can so here is our main theorem still with Emanuele, Cagliotti and François Gauls so we prove that actually the hexagonal lattice is asymptotically optimal and dynamically stable it means that if we start from initial using initial the form of X initial that satisfies the condition that I told at the beginning so that it's pi periodic and it's zero mean but the pi periodicity and the closeness to the identity are essential so we want X initial to be close to the identity in some good stable space that will be enough for us to apply parabolic regularity theorem then we have that actually the L2 gradient flow for calligraphic F as a unique solution with initial data X initial and that it converts exponentially fast to the identity in L2 so we recover what was predicted by the simulations so just a few comments on the strategy of the proof so ok now we have calligraphic F and we know that calligraphic F is is uniformly convex in a neighbor of the identity but then we don't know what is doing calligraphic F so what we do this is calligraphic F what we do is that we built calligraphic G that coincide with calligraphic F in a neighbor of the identity and it is very nice everywhere else so we would like to do analysis on calligraphic G and then to say that actually the gradient flow induced by calligraphic G is the gradient flow induced by calligraphic F this is so now Y of t will be the gradient flow induced by by G then since calligraphic G is uniformly convex we can say that Y of t converts exponentially fast to the identity and now we want to prove that our condition so the gradient of Y remains close to the identity for all time for G so would imply that actually the two gradient flows are the same for uniqueness so how to do that we cannot use regularity for equation because here we have systems so we cannot say that immediately regularity propagates so we first use subval of regularity of the initial datum and so we have propagation of regularity to detach from zero for a small interval of time we can use propagation of regularity and then at this point we can combine the exponential convergenum of t to the identity so the fact that we have this convergence in L2 with some epsilon regularity theorem for parabolic system this is a reference to a paper by Duserem in journal 2005 to say that actually the gradient of Y remains close to the identity for all times so we can say that we have this hypothesis because in some sense we are always close to a graph we are always close to an affine function it's like for minimal surfaces the same idea of using epsilon regularity for that and this concludes the proof so in one dimension we have a quantitative result because we could actually perform the gradient flow in a general setting not in a perturbative setting while in dimension 2 we had to restrict ourselves to a perturbative setting to have some good properties because also to finding the definition of the functional I never wrote at the blackboard the discrete functional in 2 dimension close to exagonals is one page of indices so the better formula was given by the continuous one and so now what we would like to do is clearly to see if it is possible to go out from the perturbative regime and the hope is that we could find an Eulerian formulation of this problem as we did in the case of dimension 1 to have some more information but what I think is more doable in the immediate future is to try to prove this result for rho different from 1 so to make the same proof and see what change when rho is close to 1 but not exactly 1 and see if we can do the same things and what would be very interesting is to guess the optimal configurations in dimension higher than 2 because in dimension 2 we had the results by facial stop that could tell us that the exagonal patterns are the best but now we would like to say that we would like to find the optimal configuration for the 3D problem that is still very interesting for the application then I will not go on for dimension 4 or 5 but dimension 3 would be interesting and there is a guess about what can be a good configuration that is when you put the points on the vertices of a cubic lattice adding a point in the body center this is a guess but it is not proved so a lot of work to do still so thank you very much for your attention