Sum Of Dirac Masses Research Articles

A blob method for diffusion

As a counterpoint to classical stochastic particle methods for diffusion, we develop a deterministic particle method for linear and nonlinear diffusion. At first glance, deterministic particle methods are incompatible with diffusive partial differential equations since initial data given by sums of Dirac masses would be smoothed instantaneously: particles do not remain particles. Inspired by classical vortex blob methods, we introduce a nonlocal regularization of our velocity field that ensures particles do remain particles and apply this to develop a numerical blob method for a range of diffusive partial differential equations of Wasserstein gradient flow type, including the heat equation, the porous medium equation, the Fokker–Planck equation, and the Keller–Segel equation and its variants. Our choice of regularization is guided by the Wasserstein gradient flow structure, and the corresponding energy has a novel form, combining aspects of the well-known interaction and potential energies. In the presence of a confining drift or interaction potential, we prove that minimizers of the regularized energy exist and, as the regularization is removed, converge to the minimizers of the unregularized energy. We then restrict our attention to nonlinear diffusion of porous medium type with at least quadratic exponent. Under sufficient regularity assumptions, we prove that gradient flows of the regularized porous medium energies converge to solutions of the porous medium equation. As a corollary, we obtain convergence of our numerical blob method. We conclude by considering a range of numerical examples to demonstrate our method’s rate of convergence to exact solutions and to illustrate key qualitative properties preserved by the method, including asymptotic behavior of the Fokker–Planck equation and critical mass of the two-dimensional Keller–Segel equation.

Measure dynamics with Probability Vector Fields and sources

We introduce a new formulation for differential equation describing dynamics of measures on an Euclidean space, that we call Measure Differential Equations with sources. They mix two different phenomena: on one side, a transport-type term, in which a vector field is replaced by a Probability Vector Field, that is a probability distribution on the tangent bundle; on the other side, a source term. Such new formulation allows to write in a unified way both classical transport and diffusion with finite speed, together with creation of mass. The main result of this article shows that, by introducing a suitable Wasserstein-like functional, one can ensure existence of solutions to Measure Differential Equations with sources under Lipschitz conditions. We also prove a uniqueness result under the following additional hypothesis: the measure dynamics needs to be compatible with dynamics of measures that are sums of Dirac masses.

1-D cubic NLS with several Dirac masses as initial data and consequences

In this proceedings article we present a result on the 1-D cubic nonlinear Schrödinger equation with a sum of Dirac masses as initial data. We shall give a sketch of the proof. By using this result we show how to construct the evolution in time of a polygonal line through the binormal flow. This equation is a geometric flow for curves in ℝ 3 and it is used as a model for the evolution of a vortex filament in fluid mechanics. These results were obtained in collaboration with Luis Vega in [4].

A Hamilton–Jacobi approach to characterize the evolutionary equilibria in heterogeneous environments

In this work, we characterize the solution of a system of elliptic integro-differential equations describing a phenotypically structured population subject to mutation, selection and migration between two habitats. Assuming that the effects of the mutations are small but nonzero, we show that the population’s phenotypical distribution has at most two peaks and we give explicit conditions under which the population will be monomorphic (unimodal distribution) or dimorphic (bimodal distribution). More importantly, we provide a general method to determine the dominant terms of the population’s distribution in each case. Our work, which is based on Hamilton–Jacobi equations with constraint, goes further than previous works where such tools were used, for different problems from evolutionary biology, to identify the asymptotic solutions, while the mutations vanish, as a sum of Dirac masses. The main elements for the computation of the dominant terms of the population’s distribution are the convergence of the logarithmic transform of the solution to the unique solution of a Hamilton–Jacobi equation and the computation of the correctors. This method allows indeed to go further than the Gaussian approximation commonly used by biologists and makes a connection between the theories of adaptive dynamics and quantitative genetics. Our work being motivated by biological questions, the objective of this paper is to provide the mathematical details which are necessary for our biological results [S. Mirrahimi and S. Gandon, The equilibrium between selection, mutation and migration in spatially heterogeneous environments, in preparation].

A Hamilton–Jacobi method to describe the evolutionary equilibria in heterogeneous environments and with non-vanishing effects of mutations

In this note, we characterize the solution to a system of elliptic integro-differential equations describing a phenotypically structured population subject to mutation, selection, and migration. Generalizing an approach based on the Hamilton–Jacobi equations, we identify the dominant terms of the solution when the mutation term is small (but nonzero). This method was initially used, for different problems arisen from evolutionary biology, to identify the asymptotic solutions, while the mutations vanish, as a sum of Dirac masses. A key point is a uniqueness property related to the weak KAM theory. This method allows us to go further than the Gaussian approximation commonly used by biologists, and is an attempt to fill the gap between the theories of adaptive dynamics and quantitative genetics.

Inverse problem and concentration method of a continuous-in-time financial model

In a continuous-in-time model, there is an important financial quantity called Loan which cannot be determined directly in terms of algebraic spending but has a major impact on the financial strategy. In this paper, we use a mathematical framework to discuss an inverse problem of determining the implied Loan Measure from Algebraic Spending Measure when it is possible. In addition, we build a numerical method to concentrate a measure as a sum of Dirac masses.

A Numerical Algorithm forL2Semi-Discrete Optimal Transport in 3D

This paper introduces a numerical algorithm to compute the L2 optimal transport map between two measures μ and ν, where μ derives from a density ρ defined as a piecewise linear function (supported by a tetrahedral mesh), and where ν is a sum of Dirac masses. I first give an elementary presentation of some known results on optimal transport and then observe a relation with another problem (optimal sampling). This relation gives simple arguments to study the objective functions that characterize both problems. I then propose a practical algorithm to compute the optimal transport map between a piecewise linear density and a sum of Dirac masses in 3D. In this semi-discrete setting [Aurenhammer et al., Proc. of 8th Symposium on Computational Geometry (1992) 350–357] showed that the optimal transport map is determined by the weights of a power diagram. The optimal weights are computed by minimizing a convex objective function with a quasi-Newton method. To evaluate the value and gradient of this objective function, I propose an efficient and robust algorithm, that computes at each iteration the intersection between a power diagram and the tetrahedral mesh that defines the measure μ. The numerical algorithm is experimented and evaluated on several datasets, with up to hundred thousands tetrahedra and one million Dirac masses.

Sums of Dirac masses and conditioned ubiquity

Multifractal formalisms hold for certain classes of atomless measures μ obtained as limits of multiplicative processes. This naturally leads us to ask whether non trivial discontinuous measures obey such formalisms. This is the case for a new kind of measures, whose construction combines additive and multiplicative chaos. This class is defined by ν γ , σ = ∑ j ⩾ 1 b − j γ / j 2 ∑ k = 0 b j − 1 μ ( [ k b − j , ( k + 1 ) b − j ) ) σ δ k b − j ( supp ( μ ) = [ 0 , 1 ] , b integer ⩾ 2 , γ ⩾ 0 , σ ⩾ 1 ). Under suitable assumptions on the initial measure μ, ν γ , σ obeys some multifractal formalisms. Its Hausdorff multifractal spectrum h ↦ d ν γ , σ ( h ) is composed of a linear part for h smaller than a critical value h γ , σ , and then of a concave part when h ⩾ h γ , σ . The same properties hold for the Hausdorff spectrum of some function series f γ , σ constructed according to the same scheme as ν γ , σ . These phenomena are the consequences of new results relating ubiquitous systems to the distribution of the mass of μ. To cite this article: J. Barral, S. Seuret, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

Equidistribution of parabolic fixed points in the limit set of Kleinian groups

We show that the Patterson–Sullivan measure on the limit set of a geometrically finite Kleinian group with cusps can be recovered as a weak limit of sums of Dirac masses placed on an appropriate orbit of each parabolic fixed point. A corollary is a sharp asymptotic estimate for a natural counting function associated to a cuspidal subgroup. We also discuss the connection between the above counting and the Riemann hypothesis in some examples of arithmetical lattices.

Quasilinear problems with singularities

We solve here some quasilinear problems with a sum of Dirac masses at the right-hand side. For that purpose, we prove a regularity theorem for nonlinear systems of the Hodge-de Rham type, and we generalize de Giorgi's notion of perimeter to subsets of compacts manifolds.