Deterministic particle approximation for scalar - PowerPoint PPT Presentation

Deterministic particle approximation for scalar aggregation-diffusion equations with nonlinear mobility Emanuela Radici joint work with M. Di Francesco & S. Fagioli Universit` a degli Studi dell’Aquila Crowds: models and control 4 th June 2019 , CIRM Marseille E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 1 / 29

Aggregation case Introduction Introduction N particles located at energetical setting: positions x 1 ( t ) , . . . , x N ( t ) nonlocal interaction potential W depending on the relative distance of the particles no inertia (negligible in many socio-biological aggregation phenomena) x i ( t ) E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 2 / 29

Aggregation case Introduction Introduction N particles located at energetical setting: positions x 1 ( t ) , . . . , x N ( t ) nonlocal interaction potential W depending on the relative distance of the particles no inertia (negligible in many socio-biological aggregation phenomena) x i ( t ) x i ( t ) = − 1 � j � = i ∇ W ( x i ( t ) − x j ( t )) ˙ N E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 2 / 29

Aggregation case Introduction Bertozzi , Carrillo , Laurent , Rosado and Brandman : L p theory x i = − 1 ˙ � j � = i ∇ W ( x i − x j ) N Ambrosio , Gigli and Savar´ e : optimal transport with smooth potentials � Carrillo , Choi , Di Francesco , Figalli , ∂ t ρ = ∇ · ( ρ ∇ W ∗ ρ ) Hauray , Laurent and Slepˇ cev : optimal transport with midly-singular potentials E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 3 / 29

Aggregation case Introduction If the potential W is attractive then the particles tend to concentrate x i = − 1 ˙ � j � = i ∇ W ( x i − x j ) N � ∂ t ρ = ∇ · ( ρ ∇ W ∗ ρ ) E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 4 / 29

Aggregation case Introduction If the potential W is attractive then the particles tend to concentrate x i = − 1 ˙ � j � = i ∇ W ( x i − x j ) (the density ρ blows up) N � ∂ t ρ = ∇ · ( ρ ∇ W ∗ ρ ) E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 4 / 29

Aggregation case Introduction Non linear mobility one way to prevent the overcrowing effect is to let the mobility depend also on a velocity term that decreases where the concentration is higher E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 5 / 29

Aggregation case Introduction Non linear mobility one way to prevent the overcrowing effect is to let the mobility depend also on a velocity term that decreases where the concentration is higher not a new idea (see Hillen , Painter , Dolak , Schmeiser , Burger , Di Francesco . . . ) E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 5 / 29

Aggregation case Introduction Non linear mobility one way to prevent the overcrowing effect is to let the mobility depend also on a velocity term that decreases where the concentration is higher not a new idea (see Hillen , Painter , Dolak , Schmeiser , Burger , Di Francesco . . . ) the continuum model becomes ∂ t ρ = ∇ · ( ρ v ( ρ ) ∇ W ∗ ρ ) E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 5 / 29

Aggregation case Introduction Non linear mobility one way to prevent the overcrowing effect is to let the mobility depend also on a velocity term that decreases where the concentration is higher not a new idea (see Hillen , Painter , Dolak , Schmeiser , Burger , Di Francesco . . . ) the continuum model becomes ∂ t ρ = ∇ · ( ρ v ( ρ ) ∇ W ∗ ρ ) Is there a microscopic counterpart of this PDE? E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 5 / 29

Aggregation case Introduction Non linear mobility one way to prevent the overcrowing effect is to let the mobility depend also on a velocity term that decreases where the concentration is higher not a new idea (see Hillen , Painter , Dolak , Schmeiser , Burger , Di Francesco . . . ) the continuum model becomes ∂ t ρ = ∂ x ( ρ v ( ρ ) W ′ ∗ ρ ) Is there a microscopic counterpart of this PDE? E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 5 / 29

Aggregation case Introduction Setting initial density ρ ∈ BV ( R , [0 , 1]) with compact ¯ support and � ¯ ρ � L 1 = 1 E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 6 / 29

Aggregation case Introduction Setting initial density ρ ∈ BV ( R , [0 , 1]) with compact ¯ support and � ¯ ρ � L 1 = 1 velocity v ∈ C 1 ([0 , ∞ )) monotone decreasing and such that it takes the value 0, assume v ( ρ ) = (1 − ρ ) + E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 6 / 29

Aggregation case Introduction Setting initial density ρ ∈ BV ( R , [0 , 1]) with compact ¯ support and � ¯ ρ � L 1 = 1 velocity v ∈ C 1 ([0 , ∞ )) monotone decreasing and such that it takes the value 0, assume v ( ρ ) = (1 − ρ ) + potential W ∈ C 2 ( R ) attractive, radially symmetric with W ′′ locally Lipschitz E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 6 / 29

Aggregation case Introduction Setting initial density ρ ∈ BV ( R , [0 , 1]) with compact ¯ support and � ¯ ρ � L 1 = 1 velocity v ∈ C 1 ([0 , ∞ )) monotone Continuum Problem decreasing and such that it � ∂ t ρ = ∂ x ( ρ v ( ρ ) W ′ ∗ ρ ) takes the value 0, assume ρ (0 , · ) = ¯ ρ ( · ) v ( ρ ) = (1 − ρ ) + potential W ∈ C 2 ( R ) attractive, radially symmetric with W ′′ locally Lipschitz E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 6 / 29

Aggregation case Introduction Plan of the talk 1 Find a candidate discrete model E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 7 / 29

Aggregation case Introduction Plan of the talk 1 Find a candidate discrete model 2 Study the deterministic many particle limit E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 7 / 29

Aggregation case Introduction Plan of the talk 1 Find a candidate discrete model 2 Study the deterministic many particle limit 3 Aggregation-Diffusion case E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 7 / 29

Aggregation case Introduction Plan of the talk 1 Find a candidate discrete model 2 Study the deterministic many particle limit 3 Aggregation-Diffusion case 4 Numerical simulations E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 7 / 29

Aggregation case Discrete model Lagrangian description via pseudo inverse function � x distribution function R ( t , x ) = −∞ ρ ( t , y ) dy X ( t , z ) = inf { x ∈ R : R ( t , x ) > z } pseudo inverse function E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 8 / 29

Aggregation case Discrete model Lagrangian description via pseudo inverse function � x distribution function R ( t , x ) = −∞ ρ ( t , y ) dy X ( t , z ) = inf { x ∈ R : R ( t , x ) > z } pseudo inverse function (see Gosse , Toscani , Russo , Matthes , Osberger , Di Francesco , Rosini , Fagioli . . . ) E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 8 / 29

Aggregation case Discrete model Lagrangian description via pseudo inverse function � x distribution function R ( t , x ) = −∞ ρ ( t , y ) dy X ( t , z ) = inf { x ∈ R : R ( t , x ) > z } pseudo inverse function (see Gosse , Toscani , Russo , Matthes , Osberger , Di Francesco , Rosini , Fagioli . . . ) formal change of variables in the PDE: ∂ t ρ = ∂ x ( ρ v ( ρ ) W ′ ∗ ρ ) density � 1 1 pseudo inverse ∂ t X = − v ( ∂ z X ) 0 W ′ ( X ( z ) − X ( ξ )) d ξ E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 8 / 29

Aggregation case Discrete model Lagrangian description via pseudo inverse function � 1 � 1 � � z � � 1 W ′ ( X ( z ) − X ( ξ )) d ξ − v W ′ ( X ( z ) − X ( ξ )) d ξ ∂ t X = − v ∂ z X ∂ z X 0 z discretization of ∂ z X ( t , z ) via forward and backward finite differences ∂ z X ( t , z ) ≈ ± X ( t , z ± 1 N ) − X ( t , z ) 1 N take X ( t , z ) piecewise constant N − 1 � X ( t , z ) = x i ( t ) χ [ i N ) ( z ) N , i +1 i =0 then, for every i = 0 , . . . , N , it is immediate to obtain � � � W ′ ( x i − x j ) � � � W ′ ( x i − x j ) 1 / N 1 / N x i ( t ) = − v ˙ − v x i +1 − x i x i − x i − 1 N N j > i j < i E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 9 / 29

Aggregation case Discrete model Discretization of the initial condition Let us call [¯ x min , ¯ x max ] the minimal interval containing supp [¯ ρ ], then we discretize the initial condition in N intervals of measure 1 / N ρ ¯ x 0 ¯ x 1 ¯ x N − 1 ¯ x N ¯ R E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 10 / 29

Aggregation case Discrete model Discretization of the initial condition Let us call [¯ x min , ¯ x max ] the minimal interval containing supp [¯ ρ ], then we discretize the initial condition in N intervals of measure 1 / N ρ ¯ x 0 ¯ x 1 ¯ x N − 1 ¯ x N ¯ R x 0 = ¯ ¯ x min , ¯ x N = ¯ x max E.Radici (UNIVAQ) Deterministic Particle... CIRM, 04/06/2019 10 / 29