Hamilton-Jacobi equation on networks: generalized Lax-Hopf formula

Hamilton-Jacobi equations on networks:
Generalized Lax-Hopf formula
Guillaume Costeseque
(PhD with supervisors R. Monneau & J-P. Lebacque)
Université Paris Est, Ecole des Ponts ParisTech & IFSTTAR
Présentation Séminaire “Réseaux” - Institut Henri Poincaré,
June 03, 2014
G. Costeseque (UPE - ENPC & IFSTTAR) HJ on networks IHP Paris, June 03, 2014 1 / 62

Flows on a network

Flows on a network
A network is like a (oriented) graph
made of edges and vertices
Examples: traﬃc ﬂow, gas pipelines,
blood vessels, shallow water, internet
communications...

Outline
1 Introduction
2 “First” attempt: Imbert-Monneau’s framework (2010-2014)
3 Lax-Hopf: classical approach
4 Introducing time and space into Hamiltonian
5 Discussions: back to traﬃc
6 Concluding remarks

Introduction
Motivation
Classical approaches:
Macroscopic modeling on (homogeneous) sections
Coupling conditions at (pointwise) junction
For instance, consider



ρt + (Q(ρ))x = 0, scalar conservation law,
ρ(., t = 0) = ρ0(.), initial conditions,
ψ(ρ(x = 0−, t), ρ(x = 0+, t)) = 0, coupling condition.
(1)
See Garavello, Piccoli [5], Lebacque, Khoshyaran [9] and Bressan et al. [2]

Introduction
Motivation
Our aim
Macroscopic modeling in a single framework
Implicit condition at junctions
Analytical solution & tractable numerical methods
Our tools: Hamilton-Jacobi equations (HJ)
ut + H(x, t, Du) = 0
Class of nonlinear ﬁrst-order Partial Diﬀerential Equations (PDEs) that
naturally arise from mechanics (see Arnold’s book)
Arnol’d, Vladimir Igorevich, Mathematical methods of classical mechanics,
Vol. 60. Springer, 1989.

“First” attempt: Imbert-Monneau’s framework (2010-2014)
Outline
1 Introduction

“First” attempt: Imbert-Monneau’s framework (2010-2014) Hamilton-Jacobi model
Motivation: the simple divergent road
x > 0
x > 0γl
γrx < 0
Il
Ir
γe
Ie



γe = 1,
0 ≤ γl , γr ≤ 1,
γl + γr = 1
LWR model [Lighthill, Whitham ’55; Richards ’56]:
ρt + (Q(ρ))x = 0

Qmax
ρcrit ρmax
Density ρ
Flow Q(ρ)
Q(ρ) = ρV (ρ) with V (ρ) = speed of the equilibrium ﬂow
DFs

Getting the Hamilton-Jacobi equation
LWR model on each branch (outside the junction point)
ρα
t + (Qα
(ρα
))x = 0 on branch α
Primitive:



Uα(x, t) = Uα(0, t) +
1
γα
x
0
ρα
(y, t)dy,
Uα(0, t) = g(t) = index of the single car at the junction point
x > 0
x > 09
11
8
10
12
6420 1 3 5
7
−1
x < 0
Uα
t +
1
γα
Qα
(γα
Uα
x ) = g′
(t) +
1
γα
Qα
(ρα
(0, t))
= 0 for a good choice of g

Settings
JN
J1
J2
branch Jα
x
x
0
x
x
New functions uα:
uα(x, t) = −Uα(x, t), x > 0, for outgoing roads
uα(x, t) = −Uα(−x, t), x > 0

Junction model
Proposition (Junction model)
That leads to the following junction model (see [8, 7])



uα
t + Hα(uα
x ) = 0, x > 0, α = 1, . . . , N
uα = uβ =: u, x = 0,
ut + H(u1
x , . . . , uN
x ) = 0, x = 0
(2)
with initial condition uα(0, x) = uα
0 (x) and
H(u1
x , . . . , uN
x ) = max L
Flux limiter
, max
α=1,...,N
H−
α (uα
x )
Minimum between
demand and supply
Imbert, C., and R. Monneau, Level-set convex Hamilton-Jacobi equations on
networks, preprint (2014).

Pros & Cons
Good mathematical properties (existence and uniqueness of the
viscosity solution)...
...under weak assumptions on the Hamiltonian (“bi-monotone”
instead of strictly convex)
Possibility to compute the solution thanks to convergent ﬁnite
diﬀerence scheme [4]

Pros & Cons
Good mathematical properties (existence and uniqueness of the
viscosity solution)...
...under weak assumptions on the Hamiltonian (“bi-monotone”
instead of strictly convex)
Possibility to compute the solution thanks to convergent finite
difference scheme [4]
No explicit / analytical solution (Lax-Hopf formula)
Not very satisfactory in traffic: (γα)α fixed!

“First” attempt: Imbert-Monneau’s framework (2010-2014) Numerical scheme
Presentation of the scheme
Proposition (Numerical Scheme)
Let us consider the discrete space and time derivatives:
pα,n
i :=
Uα,n
i+1 − Uα,n
i
∆x
and (DtU)α,n
i :=
Uα,n+1
i − Uα,n
i
∆t
Then we have the following numerical scheme:



(DtU)α,n
i + max{H+
α (pα,n
i−1), H−
α (pα,n
i )} = 0, i ≥ 1
Un
0 := Uα,n
0 , i = 0, α = 1, ..., N
(DtU)n
0 + max
α=1,...,N
H−
α (pα,n
0 ) = 0, i = 0
(3)
With the initial condition Uα,0
i := uα
0 (i∆x).
∆x and ∆t = space and time steps satisfying a CFL condition

“First” attempt: Imbert-Monneau’s framework (2010-2014) Numerical scheme
CFL condition
The natural CFL condition is given by:
∆x
∆t
≥ sup
α=1,...,N
i≥0, 0≤n≤nT
|H′
α(pα,n
i )| (4)

“First” attempt: Imbert-Monneau’s framework (2010-2014) Mathematical results
Basic assumptions
For all α = 1, . . . , N,
(A0) The initial condition uα
0 is Lipschitz continuous.
(A1) The Hamiltonians Hα are C1(R) and convex such that:
p
H−
α (p) H+
α (p)
pα
0

Gradient estimates
Theorem (Time and Space Gradient estimates)
Assume (A0)-(A1). If the CFL condition (4) is satisﬁed, then we have
that:
(i) Considering Mn = sup
α,i
(DtU)α,n
i and mn = inf
α,i
(DtU)α,n
i , we have the
following time derivative estimate:
m0
≤ mn
≤ mn+1
≤ Mn+1
≤ Mn
≤ M0
(ii) Considering pα
= (H−
α )−1(−m0) and pα = (H+
α )−1(−m0), we have
the following gradient estimate:
pα
≤ pα,n
i ≤ pα, for all i ≥ 0, n ≥ 0 and α = 1, ..., N
Proof

Stronger CFL condition
−m0
pα
p
Hα(p)
pα
As for any α = 1, . . . , N, we have that:
pα
≤ pα,n
i ≤ pα for all i, n ≥ 0
Then the CFL condition becomes:
∆x
∆t
≥ sup
α=1,...,N
pα∈[pα
,pα]
|H′
α(pα)| (5)

Existence and uniqueness
(A2) Technical assumption (Legendre-Fenchel transform)
Hα(p) = sup
q∈R
(pq − Lα(q)) with L′′
α ≥ δ > 0, for all index α

Existence and uniqueness
(A2) Technical assumption (Legendre-Fenchel transform)
Hα(p) = sup
q∈R
(pq − Lα(q)) with L′′
α ≥ δ > 0, for all index α
Theorem (Existence and uniqueness [IMZ, ’11])
Under (A0)-(A1)-(A2), there exists a unique viscosity solution u of (2) on
the junction, satisfying for some constant CT > 0
|u(t, y) − u0(y)| ≤ CT for all (t, y) ∈ JT .
Moreover the function u is Lipschitz continuous with respect to (t, y).

Convergence
Theorem (Convergence from discrete to continuous [CML, ’13])
Assume that (A0)-(A1)-(A2) and the CFL condition (5) are satisﬁed.
Then the numerical solution converges uniformly to u the unique viscosity
solution of (2) when ε → 0, locally uniformly on any compact set K:
lim sup
ε→0
sup
(n∆t,i∆x)∈K
|uα
(n∆t, i∆x) − Uα,n
i | = 0
Proof

“First” attempt: Imbert-Monneau’s framework (2010-2014) Traﬃc interpretation
Setting
J1
JNI
JNI +1
JNI +NO
x < 0 x = 0 x > 0
Jβ
γβ Jλ
γλ
NI incoming and NO outgoing roads

Car densities
The car density ρα solves the LWR equation on branch α:
ρα
t + (Qα
(ρα
))x = 0
By deﬁnition
ρα
= γα
∂x Uα
on branch α
And
uα(x, t) = −Uα(−x, t), x > 0, for incoming roads
uα(x, t) = −Uα(x, t), x > 0, for outgoing roads
where the car index uα solves the HJ equation on branch α:
uα
t + Hα
(uα
x ) = 0, for x > 0

Flow
Hα(p) :=



−
1
γα
Qα(γαp) for α = 1, ..., NI
−
1
γα
Qα(−γαp) for α = NI + 1, ..., NI + NO
Incoming roads Outgoing roads
ρcrit
γα
ρmax
γα
p
−
Qmax
γα
p
−
Qmax
γα
HαHα
H−
α H−
α H+
αH+
α
−
ρmax
γα
−
ρcrit
γα

Links with “classical” approach
Deﬁnition (Discrete car density)
The discrete car density ρα,n
i with n ≥ 0 and i ∈ Z is given by:
ρα,n
i :=



γαpα,n
|i|−1 for α = 1, ..., NI , i ≤ −1
−γαpα,n
i for α = NI + 1, ..., NI + NO, i ≥ 0
(6)
J1
JNI
JNI +1
JNI +NO
x < 0 x > 0
−2
−1
2
1
0
−2
−2
−1
−1
1
1
2
2
Jβ
Jλ
ρλ,n
1

Traﬃc interpretation
Proposition (Scheme for vehicles densities)
The scheme deduced from (3) for the discrete densities is given by:
∆x
∆t
{ρα,n+1
i − ρα,n
i } =



Fα(ρα,n
i−1, ρα,n
i ) − Fα(ρα,n
i , ρα,n
i+1) for i = 0, −1
Fα
0 (ρ·,n
0 ) − Fα(ρα,n
i , ρα,n
i+1) for i = 0
Fα(ρα,n
i−1, ρα,n
i ) − Fα
0 (ρ·,n
0 ) for i = −1
With



Fα(ρα,n
i−1, ρα,n
i ) := min Qα
D(ρα,n
i−1), Qα
S (ρα,n
i )
Fα
0 (ρ·,n
0 ) := γα min min
β≤NI
1
γβ
Qβ
D(ρβ,n
0 ), min
λ>NI
1
γλ
Qλ
S (ρλ,n
0 )
incoming outgoing
ρλ,n
0ρβ,n
−1ρβ,n
−2 ρλ,n
1
x
x = 0

Supply and demand functions
Remark
It recovers the seminal Godunov scheme with passing ﬂow = minimum
between upstream demand QD and downstream supply QS.
Density ρ
ρcrit ρmax
Supply QS
Qmax
Density ρ
ρcrit ρmax
Flow Q
Qmax
Density ρ
ρcrit
Demand QD
Qmax
From [Lebacque ’93, ’96]

Supply and demand VS Hamiltonian
H−
α (p) =



−
1
γα
Qα
D(γαp) for α = 1, ..., NI
−
1
γα
Qα
S (−γαp) for α = NI + 1, ..., NI + NO
And
H+
α (p) =



−
1
γα
Qα
S (γαp) for α = 1, ..., NI
−
1
γα
Qα
D(−γαp) for α = NI + 1, ..., NI + NO

Some references for conservation laws
ρt + (Q(x, ρ))x = 0 with Q(x, p) = 1{x<0}Qin
(p) + 1{x≥0}Qout
(p)
Uniqueness results only for restricted conﬁgurations:
See [Garavello, Natalini, Piccoli, Terracina ’07]
and [Andreianov, Karlsen, Risebro ’11]
Book of [Garavello, Piccoli ’06] for conservation laws on networks:
Construction of a solution using the “wave front tracking method”
No proof of the uniqueness of the solution on a general network

Numerics on networks
Godunov scheme mainly used for conservation laws:
[Bretti, Natalini, Piccoli ’06, ’07]: Godunov scheme compared to
kinetic schemes / fast algorithms
[Blandin, Bretti, Cutolo, Piccoli ’09]: Godunov scheme adapted for
Colombo model (only tested for 1 × 1 junctions)
For Hamilton-Jacobi equations on networks:
[G¨ottlich, Ziegler, Herty ’13]: Lax-Freidrichs scheme outside the
junction + coupling conditions (density) at the junction
[Han, Piccoli, Friesz, Yao ’12]: Lax-Hopf formula for HJ equation
coupled with a Riemann solver at junction
[Camilli, Festa, Schieborn ’13]: semi-Lagrangian scheme only
designed for Eikonal equations

Lax-Hopf: classical approach
Outline
1 Introduction

Basic idea
Consider the simplest ﬁrst order Hamilton-Jacobi equation
ut + H(Du) = 0 (7)
with u : Rn × (0, +∞) → R and H : Rn → R.

Basic idea
Family of simple linear functions
ˆu(x, t) = α.x − tH(α) + β, for any α ∈ Rn
, β ∈ R
which are solutions of (7).
Idea: build more general solutions,
taking an envelope of elementary solutions (E. Hopf 1965 [6])

Lax-Hopf formulæ
Consider the Cauchy problem
ut + H(Du) = 0, in Rn × (0, +∞),
u(., 0) = u0(.), on Rn.
(8)
Two cases according to the assumptions on the smoothness of
the Hamiltonian H and
the initial data u0

Lax-Hopf formulæ
Assumptions: case 1
(A1) H : Rn → R is convex
(A2) u0 : Rn → R is uniformly Lipschitz
Theorem (First Lax-Hopf formula)
If (A1)-(A2) hold true, then
u(x, t) := inf
z∈Rn
sup
y∈Rn
[u0(z) + y.(x − z) − tH(y)] (9)
is the unique uniformly continuous viscosity solution of (8).

Notice that analogue formula of (9) has been used by
Hopf in 1950 for a speciﬁc hyperbolic equation
ut + uux = µuxx
Lax in 1957 for hyperbolic systems
and Oleinik in 1963
From where the name “Lax-Hopf-Oleinik” formula

Lax-Hopf formulæ
(Continued)
Assumptions: case 2
(A3) H : Rn → R is continuous
(A4) u0 : Rn → R is uniformly Lipschitz and convex
Theorem (Second Lax-Hopf formula)
If (A3)-(A4) hold true, then
u(x, t) := sup
y∈Rn
inf
z∈Rn
[u0(z) + y.(x − z) − tH(y)] (10)
is the unique uniformly continuous viscosity solution of (8).

Proofs of Lax-Hopf formulæ are viscosity solutions
Classical results: existence and uniqueness of viscosity solution of (8)
Two ways of proving that (9) and (10) are viscosity solutions of (8)
PDE approach (sub- and supersolution inequalities)
Optimal control approach (dynamic programming equation)
Bardi, M., and L. C. Evans, On Hopf’s formulas for solutions of
Hamilton-Jacobi equations, Nonlinear Analysis: Theory, Methods &
Applications 8.11 (1984): 1373-1381.
Evans, L. C., Partial diﬀerential equations, Graduate studies in
mathematics, vol. 19, American Mathematics Society, 2009

Some naive remarks
First Lax-Hopf formula (9) can be recast as
u(x, t) := inf
z∈Rn
u0(z) − tH∗ x − z
t
thanks to Legendre-Fenchel transform
L(z) = H∗
(z) := sup
y∈Rn
(y.z − H(y)) .
If H is strictly convex, 1-coercive i.e. lim
|p|→∞
H(p)
|p|
= +∞,
then H∗ is also convex and
(H∗
)∗
= H.

Legendre-Fenchel transform
Illustration for the density-ﬂow Fundamental Diagram
F(p) = −H(−p), for any p ∈ R,
and
M(q) := sup
ρ
[F(ρ) − ρq] , for any q ∈ R
M(q)
u
w
Density ρ
q
q
Flow F
w u
q
Transform M
−wρmax

Interpretation of Lax-Hopf formula
Consider the cost function c that embeds
initial conditions (at time t = 0),
boundary conditions (at locations xmin, xmax),
internal conditions (along trajectories x∗(t) e.g.)
Theorem
If (A1), (A4) hold true
Then Lax-Hopf formulæ (9) and (10) are equivalent
and we have
u(xT , T) := inf
(x0,t0)∈Rn×(0,T)
c(x0, t0) − (T − t0)H∗ xT − x0
T − t0
. (11)

(Continued)
Lax-Hopf formula (11) reads as an optimal control problem
Time
Space
J
(T, xT )˙X(τ)
(t0, x0)

(Continued)
Time
Space
J
(T, xT )˙X(τ)
(t0, x0)
Lax-Hopf formula (11) reads as an optimal
control problem
u(xT , T) := inf
(x0,t0)∈Rn×(0,T)
inﬁmum over all
the feasible trajectories
starting from (x0,t0)
initial cost
c(x0, t0) −(T − t0)
actualized cost
H∗ xT − x0
T − t0
mean speed of
the trajectory

Introducing time and space into Hamiltonian
Outline
1 Introduction

Time-space dependent Hamiltonian
Simplicity: x ∈ R (i.e. n = 1)
but...
ut + H(x, t, ux ) = 0, on R × (0, +∞). (12)
Extension of the Lax-Hopf formula?
Classical formulæ (9) and (10) do not work
No simple linear solutions for (12)

Time-space dependent Hamiltonian
Simplicity: x ∈ R (i.e. n = 1)
but...
ut + H(x, t, ux ) = 0, on R × (0, +∞). (12)
Extension of the Lax-Hopf formula?
Classical formulæ (9) and (10) do not work
No simple linear solutions for (12)
Very recent generalization: Jean-Pierre Aubin & Luxi Chen [1, 3]
through viability tools

Survival kit in viability theory
Dynamical systems controlled by diﬀerential inclusions
Viability kernel Capture basin
Aubin, J.-P., and A. M. Bayen, and P. Saint-Pierre, Viability theory, new
directions, Springer, 2011.

Settings
Let (xT , T) the target point.
Denote Ω the aperture time between initial and ﬁnal points
Ω := T − t0 with t0 ∈ (0, T).

“Coarse” Lax-Hopf formula
We still can write
u(xT , T) := inf
Ω≥0
inf
{x(.)∈Al(xT ,T)}
c(x(T − Ω), T − Ω) +
T
T−Ω
l(τ, x(τ), x′
(τ))
which is an optimal control problem very hard to solve,
optimal trajectories (characteristics) are not straight lines

Settings
(Continued)
We set µ the average speed on the trajectory from initial to ﬁnal points
µ :=
1
Ω
T
T−Ω
x′
(τ)dτ,
and the average cost on the optimal trajectory x⋆
Γ(xT , T, Ω, µ) :=
1
Ω
T
T−Ω
l(τ, x⋆(τ), x′
⋆(τ))dτ.
(Recall that l = H∗)

Generalized Lax-Hopf formula
Theorem (Generalized Lax-Hopf formula)
If H : (x, t, p) → H(x, t, p) is convex and lower semi-continuous w.r.t. p,
Then
u(xT , T) := inf
Ω≥0, µ∈X
[c(x(T − Ω), T − Ω) + ΩΓ(xT , T, Ω, µ)]
is the (unique) lower semi-continuous solution of
ut + H(t, x, ux ) = 0, on R × (0, +∞),
u(t, x) ≤ c(t, x), for any (t, x)
(13)
(in Barron-Jensen / Frankowska sense).
Notice that we have deﬁned X the set of speeds such that l(µ) < ∞.

Drawback
Actually,
Γ(xT , T, Ω, µ) := inf
x(.)∈Al(xT ,T) T
T−Ω x′(τ)dτ=Ωµ
1
Ω
T
T−Ω
l(τ, x(τ), x′
(τ))dτ
is still a optimal control problem to solve...

Drawback
Actually,
Γ(xT , T, Ω, µ) := inf
x(.)∈Al(xT ,T) T
T−Ω x′(τ)dτ=Ωµ
1
Ω
T
T−Ω
l(τ, x(τ), x′
(τ))dτ
is still a optimal control problem to solve...
Alternatives?
Viability kernel algorithms (see Patrick Saint-Pierre works) more
useful for simple conditions c (e.g. only Cauchy problems)
Compute some characteristics equations (system of coupled ODEs)

Discussions: back to traﬃc
Outline
1 Introduction

Network Model
A simple case study: generalization of the three-detector problem
Shock information
u(t, x)
Kinematic information
Kinematic information = upstream demand advected by kinematic
waves moving forward
Shock information = downstream supply aﬀected by shock waves
moving backward

How to...
Shock information
u(t, x)
Kinematic information
Open question: How to compute the
Cumulative Flow defined as
u(t, x) := −
x
x0
ρ(t, y)dy
at point x, and time t > 0, knowing the
traffic states at different points of this
simple network?

Concluding remarks
Outline
1 Introduction

Concluding remarks
In brief
Heightened interest in traffic modeling for Lax-Hopf formula
Potentialities of the generalized Lax-Hopf formula
Deducing an implicit traffic flow model at junction is not trivial
A lot remains to do but the impact must be important...

Concluding remarks
The End
Thanks for your attention
guillaume.costeseque@cermics.enpc.fr
guillaume.costeseque@ifsttar.fr

Complements References
Some references I
J.-P. Aubin and L. Chen, Generalized lax-hopf formulas for cournot maps and
hamilton-jacobi-mckendrik equations, (2014).
A. Bressan, S. Canic, M. Garavello, M. Herty, and B. Piccoli, Flows
on networks: recent results and perspectives, EMS Surveys in Mathematical
Sciences, (2014).
L. Chen, Computation of the” enrichment” of a value function of an optimization
problem on cumulated transaction-costs through a generalized lax-hopf formula,
arXiv preprint arXiv:1401.1610, (2014).
G. Costeseque, J.-P. Lebacque, and R. Monneau, A convergent scheme for
hamilton-jacobi equations on a junction: application to traffic, arXiv preprint
arXiv:1306.0329, (2013).
M. Garavello and B. Piccoli, Traffic flow on networks, American institute of
mathematical sciences Springfield, MO, USA, 2006.

Some references II
E. Hopf, Generalized solutions of non-linear equations of first order, Journal of
Mathematics and Mechanics, 14 (1965), pp. 951–973.
C. Imbert and R. Monneau, Level-set convex hamilton-jacobi equations on
networks, (2014).
C. Imbert, R. Monneau, and H. Zidani, A hamilton-jacobi approach to
junction problems and application to traffic flows, ESAIM: Control, Optimisation
and Calculus of Variations, 19 (2013), pp. 129–166.
J.-P. Lebacque and M. M. Khoshyaran, First-order macroscopic traffic flow
models: Intersection modeling, network modeling, in Transportation and Traffic
Theory. Flow, Dynamics and Human Interaction. 16th International Symposium on
Transportation and Traffic Theory, 2005.

Fundamental diagram
Fundamental diagram: multi-valued in congested case
[S. Fan, M. Herty, B. Seibold, 2013], NGSIM dataset
Back

Complements Proofs of the main results
Sketch of the proof (gradient estimates):
Time derivative estimate:
1. Estimate on mα,n = inf
i
(DtU)α,n
i and partial result for mn = inf
α
mα,n
2. Similar estimate for Mn
3. Conclusion
Space derivative estimate:
1. New bounded Hamiltonian ˜Hα(p) for p ≤ pα
and p ≥ pα
2. Time derivative estimate from above
3. Lemma: if for any (i, n, α), (DtU)α,n
i ≥ m0 then
pα
≤ pα,n
i ≤ pα
4. Conclusion as ˜Hα = Hα on [pα
, pα]
Back

Convergence with uniqueness assumption
Sketch of the proof: (Comparison principle very helpful)
1. uα(t, x) := lim sup
ε
Uα,n
i is a subsolution of (2) (contradiction on
Deﬁnition inequality with a test function ϕ)
2. Similarly, uα is a supersolution of (2)
3. Conclusion: uα = uα viscosity solution of (2)
Back

Convergence without uniqueness assumption
Sketch of the proof: (No comparison principle)
1. Discrete Lipschitz bounds on uα
ε (n∆t, i∆x) := Uα,n
i
2. Extension by continuity of uα
ε
3. Ascoli theorem (convergent subsequence on every compact set)
4. The limit of one convergent subsequence (uα
ε )ε is super and
sub-solution of (2)

Hamilton-Jacobi equation on networks: generalized Lax-Hopf formula

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Similar to Hamilton-Jacobi equation on networks: generalized Lax-Hopf formula

Similar to Hamilton-Jacobi equation on networks: generalized Lax-Hopf formula (20)

More from Guillaume Costeseque

More from Guillaume Costeseque (20)

Recently uploaded

Recently uploaded (20)

Hamilton-Jacobi equation on networks: generalized Lax-Hopf formula