A Fully-Discrete Scheme for the Value Function of Pursuit ... · Pursuit-Evasion games Convergence of the fully-discrete scheme with SC vk h!vh vh!v ([Bardi, Koike, Soravia, 2000])

A Fully-Discrete Scheme for the ValueFunction of Pursuit-Evasion Games with

State Constraints

E. Cristiani, M. Falcone

Dip. di Matematica

Universita ”La Sapienza” - Roma

28 novembre 2006

A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 1/28

Outline

Pursuit-Evasion games

Convergence of the fully-discrete scheme with SCvk

h → vh

vh → v ([Bardi, Koike, Soravia, 2000])

Capturability in Tag-Chase game if VP = VE

Some hints for the implementationInterpolation in high dimensionReducing the size of the problem

Numerical tests


Outline


Convergence of the fully-discrete scheme with SC

vkh → vh




Numerical tests


Outline



h → vh




Numerical tests


Outline



h → vh




Numerical tests


Outline



h → vh




Numerical tests


Outline



h → vh



Some hints for the implementation

Interpolation in high dimensionReducing the size of the problem

Numerical tests


Outline



h → vh



Some hints for the implementationInterpolation in high dimension

Reducing the size of the problem

Numerical tests


Outline



h → vh




Numerical tests


Outline



h → vh




Numerical tests



f(x, a, b) = f(xP , xE, a, b) =

(

fP (xP , a)

fE(xE, b)

)

, fP , fE ∈ Rn

y = (yP , yE) , x = (xP , xE)

v +minb∈B−∇xE

v · fE(xE, b)+

maxa∈A

−∇xPv · fP (xP , a) − 1 = 0 x ∈ Ω\T


Time-discrete scheme for P-E gameswith State Constraints

Ω = Ω1 × Ω2 , yP ∈ Ω1 , yE ∈ Ω2

Ah(x) :=

a ∈ A : xP + hfP (xP , a) ∈ Ω1

, x ∈ Ω

Bh(x) :=

b ∈ B : xE + hfE(xE, b) ∈ Ω2

, x ∈ Ω.

vh(x) = maxb∈Bh(x)

mina∈Ah(x)

βvh(x+ hf(x, a, b))+ 1− β x ∈ Ω\T

vh(x) = 0 x ∈ T

β = e−h.A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 4/28

Fully-discrete scheme

We build a regular triangulation of Ω denoting by X the setof its nodes xi, i = 1, . . . , N and by S the set of simplicesSj, j = 1, . . . , L. V (S) will denote the set of the vertices ofa simplex S and the space discretization step will bedenoted by k where k := maxjdiam(Sj).

vkh(xi) = max

b∈Bh(xi)min

a∈Ah(xi)

βvkh(xi + hf(xi, a, b))

+ 1− β xi ∈ (X\T )

vkh(xi) = 0 xi ∈ T ∩X

vkh(x) =

∑

j λj(x)vkh(xj) , 0 ≤ λj(x) ≤ 1 ,

∑

j λj(x) = 1 x ∈ Ω


Reachable set

R0 := T

Rn :=

x ∈ Ω\n−1⋃

j=0

Rj : for all bx ∈ Bh(x) there exists

ax(bx) ∈ Ah(x) such that x+hf(x, ax(bx), bx) ∈ Rn−1

, n ≥ 1.


Main theorem

Theorem Let Ω an open bounded set. Let f be continuousand Lipschitz continuous w.r. to x. Assume Ω = ∪∞j=0Rj.Moreover assume that minx,a,b |f(x, a, b| ≥ f0 > 0 and0 < k ≤ f0h. Then, for n ≥ 1

a) vh(x) ≤ vh(y) , for any x ∈n⋃

j=0

Rj , for any y ∈ Ω\n⋃

j=0

Rj;

b) vh(x) = 1− e−nh , for any x ∈ Rn;

c) vkh(x) = 1− e−nh +O(k)

n∑

j=0

e−jh for any x ∈ Rn;

d) There exists a constant C > 0 such that|vh(x)− vk

h(x)| ≤Ck

1−e−h , for any x ∈ Rn.


Convergence for P-E games

In order to obtain the convergence of vkh to the value

function v we can couple our result with that in [Bardi,Koike, Soravia, 2000]. In addition to our and theirhypotheses, we have to assume that

fP (xP , A(xP )) and fE(xE, B(xE))

are convex sets.


Convergence for differential games

We can generalize our convergence result to any kind ofdifferential games. To do this, we have to modify thedefinition of admissible sets properly.


Tag-Chase game

We consider two boys P and E which run one after theother in the same 2-dimensional domain, so that the gameis set in Ω = Ω

2

1 ⊂ R4 where Ω1 is an open bounded set of

R2. We denote by (xP , xE) the coordinates of Ω where

xP , xE ∈ Ω1. P and E can run in every direction withvelocity VP and VE respectively.

xP = VP a a ∈ B2(0, 1)

xE = VE b b ∈ B2(0, 1)

The case VP > VE was completely studied by Alziary deRoquefort.


Capturability in Tag-Chase game

PropositionLet the target be

T = (xP , xE) ∈ R2n : d(xP , xE) ≤ ε , ε ≥ 0.

and Ω1 an open bounded set. Then,

If VP > VE then the capture timetc = T (xP , xE) = − ln(1− v(xP , xE)) is finite andbounded by

tc ≤|xP − xE|

VP − VE

.

If VP = VE, ε 6= 0 and Ω1 is convex then the capturetime tc is finite.


Interpolation in high dimension

x

x

x

P

P

PP

PP

P

P

P

Px

P

1

23

3

3

1

1

2

34

11

2

22

38

24

37

12

23

P 33

P36

P35


Error estimate

Theorem Let Qn := [a1, b1]× . . .× [an, bn] ⊂ Rn and

x = (x1, . . . , xn).Assume f ∈ C2(Qn;R) and let q(x), x ∈ Qn be theapproximate value of f(x) obtained by the n-dimensionallinear interpolation described above.Then, the error E(x) := f(x)− q(x) is bounded by

|E(x)| ≤n∑

i=1

∆2i

8Mi , for all x ∈ Qn

where Mi = maxx∈Qn

|∂2f(x)

∂x2

i

| and ∆i = bi − ai.


Reducing the size of the problem

As already noted by Pesch and Alziary de Roquefort, dueto the state constraints it seems not possible to usereduced coordinates or a similar approach. In fact, usingreduced coordinates we loose every information about thereal positions of the two players, so that we can not detectwhen they touch the boundary of the domain (and thenchange the dynamics accordingly).

Nevertheless, we can make use of the symmetries of theproblem, if any.


Example in 1D

Unidimensional Tag-Chase game. VP = 2, VE = 1.

P PE E

xP

x E

−1.5 −1 −0.5 0 0.5 1 1.5 2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Target


Example in 2D

P P

P

E

E

E

P

E


Test 1, VP > VE

ε = 10−3, VP = 2, VE = 1, n = 50, nc = 48+ 1. Convergencewas reached in 85 iterations. The CPU time (IBM - 8 procs)was 17h 36m 16s, the wallclock time was 2h 47m 37s.

−2−1

01

2

−2

−1

0

1

20

0.5

1

1.5

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Value function T (0, 0, xE, yE).


Test 1, VP > VE

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2


Test 2, VP > VE

architecture wallclock time speed-up efficiencyIBM serial 26m 47s - -

IBM 2 procs 14m 19s 1.87 0.93IBM 4 procs 8m 09s 3.29 0.82IBM 8 procs 4m 09s 6.45 0.81

PC dual core, ser 1h 08m 44s - -PC dual core, par 34m 51s 1.97 0.99

speed-up :=Tser

Tpar

efficiency :=speed-up

np

.


Test 3, VP > VE

In this test the domain has a square hole in the center. Theside of the square is 1.06. ε = 10−4, VP = 2, VE = 1,n = 50, nc = 48 + 1. Convergence: 109 iterations. CPUtime: 1d 00h 34m 18s, wallclock time: 3h 54m 30s.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2

−1

0

1

20

0.5

1

1.5

2

2.5

3

Value function T (−1.5,−1.5, xE, yE).


Test 3, VP > VE

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2


Test 4, VP > VE

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2


Test 5, VP = VE

v is discontinuous on ∂T . No convergence results butv < 1 by our proposition.ε = 10−3, VP = 1, VE = 1, n = 50, nc = 36. Convergencewas reached in 66 iterations.

−2 −1 0 1 2

−2−10120

0.5

1

1.5

2

2.5

3

3.5

Value function T (0, 0, xE, yE).


Test 5, VP = VE

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2


Test 6, VP = VE

The domain has a circular hole in the center. Non-convexdomain, then no guarantee capture occurs. v is equal to 1in a large part of the domain. Strange behavior of someoptimal trajectories.ε = 10−4, VP = 1, VE = 1, n = 50, nc = 48 + 1.Convergence: 94 iterations. CPU time: 1d 12h 05m 22s.

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2


Test 7, VP < VE

v is discontinuous on ∂T . No guarantee capture occurs. v

is equal to 1 in a large part of the domain. ε = 10−3, VP = 1,VE = 1.25, n = 50, nc = 48 + 1. Convergence: 53 iterations.CPU time: 12h 43m 02s, wallclock time: 2h 18h 06s.

−2−1

01

2 −2−1

01

20

2

4

6

8

10

12

14

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Value function T (−1,−1, xE, yE).


Test 7, VP < VE

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2


Test 8, VP < VE

ε = 10−4, VP = 1, VE = 1.5, n = 50, nc = 36. Convergence:65 iterations. CPU time: 15h 48m 46s, wallclock time: 2h30m 19s.

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2


Documents

A Fully-Discrete Scheme for the Value Function of Pursuit ... · Pursuit-Evasion games Convergence of the fully-discrete scheme with SC vk h!vh vh!v ([Bardi, Koike, Soravia, 2000])