Numerical solution of an Eikonal equation on a Graph

Transcription

1 Numerical solution of an Eikonal equation on a Graph Adriano Festa Sapienza" Univ. of Roma Math. Dep. oint work with Fabio Camilli ( Sapienza University, Rome) London. September 2011 SADCO Workshop

2 Outline 1 Motivation 2 Model 3 Viscosity solutions on a graph 4 Numerical approximation 5 Tests

3 Walking on a graph Problem Optimization problems on a graph (shortest path, cheapest path etc. etc.) Some classical approches Combinatory (Dikstra algorithm) Differential (Eikonal equation with constraints)

4 Aim Introduce a concept of viscosity solution on an undirected graph. This solution should preserve the main features of viscosity theory: uniqueness, existence, and stability: sufficiently weak to yield existence, while sufficiently selective to ensure uniqueness and stability with respect to uniform convergence. Difficulties How to modelize the differential structure of a function on the graph (which is not a regular manifold). Which condition impose at the vertices (transition condition).

5 Aim Introduce a concept of viscosity solution on an undirected graph. This solution should preserve the main features of viscosity theory: uniqueness, existence, and stability: sufficiently weak to yield existence, while sufficiently selective to ensure uniqueness and stability with respect to uniform convergence. Difficulties How to modelize the differential structure of a function on the graph (which is not a regular manifold). Which condition impose at the vertices (transition condition).

6 The undirected graph A graph Γ is couple (V, E) where V := {v i, i I} is a finite collection of pairwise different points in R N ; E := {e : J} is a finite collection of differentiable curves in R N given by e := π ((0, l )) with π : [0, l ] R N J. Furthermore i) π (0), π (l ) V for all J, ii) (ē V ) = 2 for all J, iii) ē ē k V, and #(ē ē k ) 1 for all, k J, k. iv) For all v, w V there is a path with end points v and w (i.e. a sequence of edges {e } N =1 such that #(ē ē +1 ) = 1 and v ē 1, w ē N ) (the graph is connected).

7 The undirected graph A graph Γ is couple (V, E) where V := {v i, i I} is a finite collection of pairwise different points in R N ; E := {e : J} is a finite collection of differentiable curves in R N given by e := π ((0, l )) with π : [0, l ] R N J. Furthermore i) π (0), π (l ) V for all J, ii) (ē V ) = 2 for all J, iii) ē ē k V, and #(ē ē k ) 1 for all, k J, k. iv) For all v, w V there is a path with end points v and w (i.e. a sequence of edges {e } N =1 such that #(ē ē +1 ) = 1 and v ē 1, w ē N ) (the graph is connected).

8 Some definitions Inc i := { J : v i e } is the set of arcs incident the vertex v i. The parametrization of the arcs e induces an orientation on the edges, expressed by the signed incidence matrix A = {a i } i, J 1 if v i ē and π (0) = v i, a i := 1 if v i ē and π (l ) = v i, (1) 0 otherwise. Given a nonempty set I B I, we define Γ := {v i, i I B } to be the set of boundaries vertices, while for I T := I \ I B is the set of transition vertices.

9 Some definitions Inc i := { J : v i e } is the set of arcs incident the vertex v i. The parametrization of the arcs e induces an orientation on the edges, expressed by the signed incidence matrix A = {a i } i, J 1 if v i ē and π (0) = v i, a i := 1 if v i ē and π (l ) = v i, (1) 0 otherwise. Given a nonempty set I B I, we define Γ := {v i, i I B } to be the set of boundaries vertices, while for I T := I \ I B is the set of transition vertices.

10 Continuity Given u : Γ R, u the restriction of u to ē, i.e. u := u π : [0, l ] R. u is continuous in Γ if u C([0, l ] for any J and u (π 1 (v i )) = u k (π 1 k (v i)) for any i I,, k Inc i. Differentiation We define differentiation along an edge e by u(x) := u (π 1 (x)) = x u (π 1 (x)), for all x e, and at a vertex v i by u(v i ) := u (π 1 (v i )) = x u (π 1 (v i )) for Inc i.

11 Continuity Given u : Γ R, u the restriction of u to ē, i.e. u := u π : [0, l ] R. u is continuous in Γ if u C([0, l ] for any J and u (π 1 (v i )) = u k (π 1 k (v i)) for any i I,, k Inc i. Differentiation We define differentiation along an edge e by u(x) := u (π 1 (x)) = x u (π 1 (x)), for all x e, and at a vertex v i by u(v i ) := u (π 1 (v i )) = x u (π 1 (v i )) for Inc i.

12 The eikonal equation We consider the eikonal equation { u = f (x) x Γ u(x) = g(x) x Γ (2) where f C 0 ( Γ), moreover we assume that f (x) η > 0 x Γ (3)

13 Test Functions Definition φ is differentiable at x e, if φ := φ π : [0, l ] R is differentiable at t = π 1 (x). Let x = v i, i I T,, k Inc i, k. φ is differentiable at x if a i φ (π 1 where (a i ) is the incidence matrix. (x)) a ik k φ k (π 1 (x)) = 0, (4) k Remark Condition (4) demands that the derivatives in the direction of the incident edges and k at the vertex v i coincide, taking into account the orientation of the edges.

14 A function u is called a viscosity subsolution of (2) in Γ If x e, J, and for any φ C(Γ) which is differentiable at x and for which u φ attains a local maximum at x, we have φ(x) f (x) := φ (π 1 (x)) f (π 1 (x)) 0. (5) If x = v i, i I T, inc i, for any φ which is differentiable at x and for which u φ attains a local maximum at x φ(x) f (x) 0. (6) A function u is called a viscosity supersolution of (2) in Γ If x e, J, and for any φ C(Γ) which is differentiable at x and for which u φ attains a local minimum at x φ(x) f (x) 0. (7) If x = v i, i I T, inc i, there exists k Inc i, k such that for any φ C(Γ) which is differentiable at x and for which u φ attains a local minimum at x φ(x) f (x) 0. (8)

15 A function u is called a viscosity subsolution of (2) in Γ If x e, J, and for any φ C(Γ) which is differentiable at x and for which u φ attains a local maximum at x, we have φ(x) f (x) := φ (π 1 (x)) f (π 1 (x)) 0. (5) If x = v i, i I T, inc i, for any φ which is differentiable at x and for which u φ attains a local maximum at x φ(x) f (x) 0. (6) A function u is called a viscosity supersolution of (2) in Γ If x e, J, and for any φ C(Γ) which is differentiable at x and for which u φ attains a local minimum at x φ(x) f (x) 0. (7) If x = v i, i I T, inc i, there exists k Inc i, k such that for any φ C(Γ) which is differentiable at x and for which u φ attains a local minimum at x φ(x) f (x) 0. (8)

16 The distance function { t } S(x, y) = inf f (γ(s))ds : t > 0 γ Bx,y t, x, y Γ (9) 0 where Bx,y t is the set of paths γ : [0, t] Γ connecting x to y and piecewise differentiable. Moreover we have γ(s) = d m γ)(s) = 1 (10) ds (π 1

17 Existence and uniqueness Theorem Let g : Γ R be a continuous function satisfying g(x) g(y) S(y, x) x, y Γ = I B then u(x) := min{g(y) + S(y, x) : y Γ} is the unique viscosity solution of equation (2)

18 Discretization in time For h > 0, we define A semi-lagrangian approximation scheme M u h (x) = inf{ hf (γm) q h m + g(y) : γ h Bx,y, h y Γ} m=0 x Γ where: An admissible traectory γ h = {γm} h M m=0 Γ is a finite number of points γm h = π m (t m ) Γ such that for any m = 0,..., M, the arc γ h mγ h m+1 ē m for some m J and q m := t m+1 t m h 1 B h x,y is the set of all such paths with γ h 0 = x, γh M = y.

19 Discretization in time For h > 0, we define A semi-lagrangian approximation scheme M u h (x) = inf{ hf (γm) q h m + g(y) : γ h Bx,y, h y Γ} m=0 x Γ where: An admissible traectory γ h = {γm} h M m=0 Γ is a finite number of points γm h = π m (t m ) Γ such that for any m = 0,..., M, the arc γ h mγ h m+1 ē m for some m J and q m := t m+1 t m h 1 B h x,y is the set of all such paths with γ h 0 = x, γh M = y.

20 Set x hq := π (t hq). u h is the unique Lipschitz-continuous solution of u h (x) = S(h, x, u h ) (11) where the scheme S : R + Γ B(Γ) R is defined by S(h, x, φ) := inf q [ 1,1]:xhq ē {φ(x hq ) + hf (x) q } ] inf k Inci [inf q [ 1,1]:xhq ē k {φ(x hq ) + hf (x) q } g(x) if x = π (t) e if x = v i I T if x I B

21 Discretization in space For J, given x > 0 consider a partition P = {t 0 = 0 <... < t m <... < t M = l } of [0, l ] such that max 1,...,M (t m t m 1 ) x. Set x m = π (t m) and consider W x = {w C(ē : w(x) is constant in (x m 1, x m), m = 1,..., M } Every element w W x can be expressed as w(x) = M m=1 β m(x)w (x m), x e for β (x) = β (π 1 (x)) and β tent functions for the partition P J.

22 Set x,m hq = π (tm hq) and x = max J x. We consider the approximation scheme where the scheme S is given by U = S( x, h, U) (12) S m(k, h, W ) := inf q [ 1,1]:x m ē {I [W ](xm(q)) + hf (xm) q } if xm e ] inf k Inci [inf q [ 1,1]:xhq ē k {I [W ](xm(q)) + hf (xm) q } if xm = v i I T g(v i ) if xm = v i I B for x m(q) = π (t m hq) and I(W )(x) the linear interpolation of W in x.

23 Convergence Theorem For any x > 0 with x h/2, there exists a unique solution U R M of (12). Moreover, if x = o(h) for h 0, then U converges to the unique solution u of (2) uniformly on Γ.

24 Tests Computational difficulties In whitch order to compute edges. How to initialize approximated solution. Our choices Fixed order with a re-initialization procedeure at the end of every step. An initial guess U 0 greater than the correct solution.

25 Test 1: structure of the graph u(x) = 0 f (x) 1 x Γ x Γ

26 x = h Ord(L ) 2 Ord(L 2 )

27 Test 2: structure of the graph u(x) = 0 f (x) 1 x Γ x Γ

28 x = h Ord(L ) 2 Ord(L 2 )

29 Test 3: structure of the graph u(x) = 0 x Γ f (x) = 1 x 1 x = (x 1, x 2 ) Γ

30

31 A kid s dream..

32 Thank you for your attention.