Numerical schemes for partial differential equations
related to optimal stopping time problems
Olivier Bokanowski
Laboratory Jacques Louis Lions
University Paris-Diderot (Paris 7)
and Ensta ParisTech
Joint work with
K. Debrabant (University of Southern Denmark)
O. Bokanowski (Ensta/LJLL)
Workshop AASS, IMPA, March 30, 2016
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Numerical schemes for optimal stoppting time PDEs
Olivier Bokanowski
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Outline
I. Diffusion equations with an obstacle term
II. Schemes ... and difficulties
III. A second order scheme
VI. Numerical examples
O. Bokanowski (Ensta/LJLL)
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Numerical schemes for optimal stoppting time PDEs
Olivier Bokanowski
/ 57
I. Motivation
Diffusion equation with an obstacle term:
(1)



 min ut −



σ(x)2
2 uxx
− b(x)ux + ru, u − ϕ(x)
= 0,
t > 0, x ∈ R
u(0, x) = ϕ(x),
x ∈R
Link with stochastic optimal stopping time problem:
−r τ
0,x
u(t, x) = sup E e ϕ(Xτ ) , t ≥ 0, x ∈ R
τ ∈T[0,t]
where
- T[0,t] is the set of "stopping times" with values in [0, t]
- Xθ = Xθ0,x satisfies the following SDE :
dXθ = b(Xθ )dθ + σ(Xθ )dWθ , θ ≥ 0
X0 = x.
O. Bokanowski (Ensta/LJLL)
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Numerical schemes for optimal stoppting time PDEs
Olivier Bokanowski
/ 57
I. Motivation
Diffusion equation with an obstacle term:
Au






}|
{
z
σ(x)2
min ut −
uxx − b(x)ux + ru, u − ϕ(x) = 0,
(1)
2



t > 0, x ∈ R


u(0, x) = ϕ(x), x ∈ R
Link with stochastic optimal stopping time problem:
−r τ
0,x
u(t, x) = sup E e ϕ(Xτ ) , t ≥ 0, x ∈ R
τ ∈T[0,t]
where
- T[0,t] is the set of "stopping times" with values in [0, t]
- Xθ = Xθ0,x satisfies the following SDE :
dXθ = b(Xθ )dθ + σ(Xθ )dWθ , θ ≥ 0
X0 = x.
O. Bokanowski (Ensta/LJLL)
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Numerical schemes for optimal stoppting time PDEs
Olivier Bokanowski
/ 57
I. Motivation
Diffusion equation with an obstacle term:
Au






}|
{
z
σ(x)2
min ut −
uxx − b(x)ux + ru, u − ϕ(x) = 0,
(1)
2



t > 0, x ∈ R


u(0, x) = ϕ(x), x ∈ R
Link with stochastic optimal stopping time problem:
−r τ
0,x
u(t, x) = sup E e ϕ(Xτ ) , t ≥ 0, x ∈ R
τ ∈T[0,t]
where
- T[0,t] is the set of "stopping times" with values in [0, t]
- Xθ = Xθ0,x satisfies the following SDE :
dXθ = b(Xθ )dθ + σ(Xθ )dWθ , θ ≥ 0
X0 = x.
O. Bokanowski (Ensta/LJLL)
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Numerical schemes for optimal stoppting time PDEs
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Financial problem: American put option
dXθ
Xθ
SDE X0 = x ≥ 0 and
= rdθ + σ(Xθ )dWθ , (θ ≥ 0)
COST ϕ(x) := max(K − x, 0).


x 2 σ 2 (x)
uxx − rxux + ru, u − ϕ(x)) = 0,
⇒
2
 u(0, x) = ϕ(x),
x ≥0
min(ut −
O. Bokanowski (Ensta/LJLL)
t ∈ [0, T ], x ≥ 0
Figure: american option
t=1
100
80
60
40
20
0
−20
0
20
40
60
80
100
120
140
160
180
200
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Numerical schemes for optimal stoppting time PDEs
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Dewynne, Howison, Rupf and Wilmott (1993): "Some mathematical
results in the pricing of American options";
• There exists a singular point xs (t) such that
x ≤ xs (t),
u(t, x) = ϕ(x),
ut (t, x) + Au(t, x) = 0,
(1)
x > xs (t),
(2)
u(t, x) = ϕ(t, x) and ux (t, x) = ϕx (t, x), for x = xs (t).
√
• xs (t) ' K − c0 log(t) t for small t.
• regularity : ut , uxx bounded, regular for x 6= xs (t)
O. Bokanowski (Ensta/LJLL)
(3)
t=1
30
Model 1
25
20
15
10
5
0
74
76
78
80
82
84
86
88
90
92
94
96
Figure: Zoom around the singular point xs
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Numerical schemes for optimal stoppting time PDEs
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Numerical difficulties
Non linear PDE
Singular: ut , uxx bounded but no more at x = xs (t) (for t > 0)
Singular initial data at t = 0 : max(K − x, 0)
√
Fast variation ( t) of the singularity xs (t) for t ' 0
A recent work of Reisinger and Forsyth shows that by using the
change of variable of the form v (t, x) = u(t 2 , x) allows to smoothen the
problem at t = 0 and partly remove the numerical problems coming
from the initial singularity.
O. Bokanowski (Ensta/LJLL)
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Numerical schemes for optimal stoppting time PDEs
Olivier Bokanowski
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Numerical difficulties
Non linear PDE
Singular: ut , uxx bounded but no more at x = xs (t) (for t > 0)
Singular initial data at t = 0 : max(K − x, 0)
√
Fast variation ( t) of the singularity xs (t) for t ' 0
A recent work of Reisinger and Forsyth shows that by using the
change of variable of the form v (t, x) = u(t 2 , x) allows to smoothen the
problem at t = 0 and partly remove the numerical problems coming
from the initial singularity.
O. Bokanowski (Ensta/LJLL)
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Numerical schemes for optimal stoppting time PDEs
Olivier Bokanowski
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An equivalent PDE (Thanks to R. Eymard)
Using the fact that ut ≥ 0, we can show that PDE (1) is equivalent to
an other PDE:
(2)


min ut −

u(0, x) = ϕ(x),
σ(x)2
2 uxx
− b(x)ux + ru, ut
= 0,
t > 0, x ∈ R
x ∈R
that is, the obstacle term is replaced by ut .
It is important that the coefficients r , b, σ does not depend of the time
for this equivalence to hold (in the financial context).
Equivalently,
(
2
ut + min(− σ(x)
2 uxx − b(x)ux + ru, 0) = 0,
(2) ⇔
u(0, x) = g(x), x ∈ R
O. Bokanowski (Ensta/LJLL)
t > 0, x ∈ R
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Numerical schemes for optimal stoppting time PDEs
Olivier Bokanowski
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An equivalent PDE (Thanks to R. Eymard)
Using the fact that ut ≥ 0, we can show that PDE (1) is equivalent to
an other PDE:
(2)


min ut −

u(0, x) = ϕ(x),
σ(x)2
2 uxx
− b(x)ux + ru, ut
= 0,
t > 0, x ∈ R
x ∈R
that is, the obstacle term is replaced by ut .
It is important that the coefficients r , b, σ does not depend of the time
for this equivalence to hold (in the financial context).
Equivalently,
(
2
ut + min(− σ(x)
2 uxx − b(x)ux + ru, 0) = 0,
(2) ⇔
u(0, x) = g(x), x ∈ R
O. Bokanowski (Ensta/LJLL)
t > 0, x ∈ R
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Numerical schemes for optimal stoppting time PDEs
Olivier Bokanowski
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O. Bokanowski (Ensta/LJLL)
All truth passes through three stages :
First, it is ridiculed
Second, it is violently opposed
Third, it is accepted as being self-evident
(Schopenhauer 1788-1860)
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Numerical schemes for optimal stoppting time PDEs
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Idea of the proof of (1) ⇔ (2)
1)
ut + Au ≥ 0.
Using the semi-martingale property (or semi-DPP) :
u(t + h, x) ≥ E(e−rh u(t, Xh0,x ))
2) Notice that ut ≥ 0. Indeed,
u(t + h, x) =
sup E[g(Xτ0,x )]
τ ∈T[0,t+h]
≥
sup E[g(Xτ0,x )] = u(t, x).
t∈T[0,t]
Hence
O. Bokanowski (Ensta/LJLL)
min(ut + Au, ut ) ≥ 0.
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Numerical schemes for optimal stoppting time PDEs
Olivier Bokanowski
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Idea of the proof of (1) ⇔ (2)
1)
ut + Au ≥ 0.
Using the semi-martingale property (or semi-DPP) :
u(t + h, x) ≥ E(e−rh u(t, Xh0,x ))
2) Notice that ut ≥ 0. Indeed,
u(t + h, x) =
sup E[g(Xτ0,x )]
τ ∈T[0,t+h]
≥
sup E[g(Xτ0,x )] = u(t, x).
t∈T[0,t]
Hence
O. Bokanowski (Ensta/LJLL)
min(ut + Au, ut ) ≥ 0.
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Numerical schemes for optimal stoppting time PDEs
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Idea of the proof of (1) ⇔ (2) (continued)
3) Suppose that ut (t, x) > 0 (with t > 0). Then for h > 0 small enough,
u(t, x) > u(t − h, x) ≥ ϕ(x).
Hence one can show that the optimal stopping time
∗ := max{0 ≤ θ ≤ t, u(θ, X 0,x ) = ϕ(X 0,x )} satisfies τ ∗ < t a.s.,
τt,x
t,x
θ
θ
from which we can conclude using a DPP and Ito’s formula that
ut (t, x) + (Au)(t, x) = 0.
4) Hence u (solution of (1)) satisfies also (2). By using a uniqueness
argument on the (viscosity) solutions of (2), solutions of both systems
coincide.
O. Bokanowski (Ensta/LJLL)
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Numerical schemes for optimal stoppting time PDEs
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Idea of the proof of (1) ⇔ (2) (continued)
3) Suppose that ut (t, x) > 0 (with t > 0). Then for h > 0 small enough,
u(t, x) > u(t − h, x) ≥ ϕ(x).
Hence one can show that the optimal stopping time
∗ := max{0 ≤ θ ≤ t, u(θ, X 0,x ) = ϕ(X 0,x )} satisfies τ ∗ < t a.s.,
τt,x
t,x
θ
θ
from which we can conclude using a DPP and Ito’s formula that
ut (t, x) + (Au)(t, x) = 0.
4) Hence u (solution of (1)) satisfies also (2). By using a uniqueness
argument on the (viscosity) solutions of (2), solutions of both systems
coincide.
O. Bokanowski (Ensta/LJLL)
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Numerical schemes for optimal stoppting time PDEs
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Application of PDE (2) :
ut + min(Au, 0) = 0, t > 0, x ∈ R
u(0, x) = ϕ(x), x ∈ R
• Related stochastic optimal control problem: u is also obtained as
R
t
u(t, x) = sup E e− 0 α(s)r ds ϕ(Xt0,x,α )
α(.)∈[0,1]
where Xθ = Xθ0,x,α is the solution of X0 = x and
dXθ = α(θ)(b(Xθ )dθ + σ(Xθ )dWθ ).
• In particular the following DPP holds:
R
h
u(tn+1 , x) = sup E e− 0 α(s)r ds u(tn , Xh0,x,α )
α(.)∈[0,1]
• Taking α ≡ 0, or α ≡ 1 leads to the approximation
0,x
−rh
u(tn+1 , x) ' max u(tn , x), E e u(tn , Xh ) .
O. Bokanowski (Ensta/LJLL)
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II. Schemes and difficulties
• finite difference schemes
• efficient tools for solving non-linear implicit schemes.
• the "1st order barrier"
O. Bokanowski (Ensta/LJLL)
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Numerical schemes for optimal stoppting time PDEs
Olivier Bokanowski
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Simplified obstacle problem: (σ 2 /2 = 1, b = 0, r = 0)
min(vt − vxx , v − ϕ(x)) = 0,
t ∈ (0, T ), x ∈ Ω = (Xmin , Xmax ),
v (0, x) = ϕ(x)
with dirichlet boundary conditions on ∂Ω.
Explicit finite difference scheme: Mesh (xi = ih), (tn = nτ ),
min
uin+1 − uin
−
τ
n − 2u n + u n ui−1
i
i−1
n+1
,
u
−
ϕ(x
)
= 0,
i
i
h2
1≤i ≤J
n+1
with u0n+1 = uJ+1
=0
... is really explicit
!:
⇒ uin+1 = max uin − τ
O. Bokanowski (Ensta/LJLL)
n − 2u n + u n ui−1
i
i−1
, ϕ(xi )
h2
Numerical schemes for optimal stoppting time PDEs
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Olivier Bokanowski
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Simplified obstacle problem: (σ 2 /2 = 1, b = 0, r = 0)
min(vt − vxx , v − ϕ(x)) = 0,
t ∈ (0, T ), x ∈ Ω = (Xmin , Xmax ),
v (0, x) = ϕ(x)
with dirichlet boundary conditions on ∂Ω.
Explicit finite difference scheme: Mesh (xi = ih), (tn = nτ ),
min
uin+1 − uin
−
τ
n − 2u n + u n ui−1
i
i−1
n+1
,
u
−
ϕ(x
)
= 0,
i
i
h2
1≤i ≤J
n+1
with u0n+1 = uJ+1
=0
... is really explicit
!:
⇒ uin+1 = max uin − τ
O. Bokanowski (Ensta/LJLL)
n − 2u n + u n ui−1
i
i−1
, ϕ(xi )
h2
Numerical schemes for optimal stoppting time PDEs
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Olivier Bokanowski
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• Linear case
vt − vxx = 0
• Explicit scheme:
uin+1 − uin
−
τ
hence with k :=
n + 2u n − u n
−ui−1
i
i−1
h2
=0
1≤i ≤J
τ
:
h2
n
n
uin+1 = kui−1
+ (1 − 2k )uin + kui+1
≡ (Su n )i
• CONSISTENCY:
v n+1 − Sv n
= (vt − vxx )(tn , xi ) + O(τ ) + O(h2 )
τ
• STABILITY : CFL condition
O. Bokanowski (Ensta/LJLL)
2τ
h2
≤ 1 ⇒ kU n+1 k∞ ≤ kU n k∞
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Olivier Bokanowski
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• Linear case
vt − vxx = 0
• Explicit scheme:
uin+1 − uin
−
τ
hence with k :=
n + 2u n − u n
−ui−1
i
i−1
h2
=0
1≤i ≤J
τ
:
h2
n
n
uin+1 = kui−1
+ (1 − 2k )uin + kui+1
≡ (Su n )i
• CONSISTENCY:
v n+1 − Sv n
= (vt − vxx )(tn , xi ) + O(τ ) + O(h2 )
τ
• STABILITY : CFL condition
O. Bokanowski (Ensta/LJLL)
2τ
h2
≤ 1 ⇒ kU n+1 k∞ ≤ kU n k∞
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• IMPLICIT scheme: (for linear case vt − vxx = 0)
uin+1 − uin
−
τ
n+1
n+1
−ui−1
+ 2uin+1 − ui−1
=0
h2
1≤i ≤J
⇒ AU n+1 = U n , with


u1n



U =  ... 
uJn
n


A=


1 + 2k
−k

−k
..
.
..
..
.
.
−k
−k
1 + 2k





and k :=
τ
≥ 0.
h2
• CONSISTENCY: idem, O(τ ) + O(h2 )
• STABILITY : NO CFL condition !
A "δ-diag. dominant" ⇒ kA−1 k∞ ≤
O. Bokanowski (Ensta/LJLL)
1
δ
≤ 1 ⇒ kU n+1 k∞ ≤ kU n k∞ ∀τ > 0
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• implicit for the nonlinear case: Can we do the same ?
Implicit finite difference scheme
min
uin+1 − uin
−
τ
n+1
n+1 −ui−1
+ 2uin+1 − ui−1
n+1
, ui
− ϕ(xi ) = 0,
h2
1≤i ≤J
After multiplication of the left part of the min by τ > 0, we get:
n+1
n+1
n+1
n+1
n
min (1 + 2k )ui
− ϕ(xi ) = 0
− kui−1 − kui+1 − ui , ui
| {z }
|
{z
} |{z}
=(Au n+1 )i
⇔
find x = U n+1 ,
min((Ax − b)i , xi − gi ) = 0,
• EXISTENCE, STABILITY :
- If Aii ≥ 0 & A: δ ≥ 1-diag. domi. ⇒ ∃!x
- Moreover kU n+1 k∞ ≤ max(kU n k∞ , kgk∞ )
O. Bokanowski (Ensta/LJLL)
≡gi
≡bi
1≤i ≤J
⇒ NO CFL condition.
Numerical schemes for optimal stoppting time PDEs
Lab
Olivier Bokanowski
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• implicit for the nonlinear case: Can we do the same ?
Implicit finite difference scheme
min
uin+1 − uin
−
τ
n+1
n+1 −ui−1
+ 2uin+1 − ui−1
n+1
, ui
− ϕ(xi ) = 0,
h2
1≤i ≤J
After multiplication of the left part of the min by τ > 0, we get:
n+1
n+1
n+1
n+1
n
min (1 + 2k )ui
− ϕ(xi ) = 0
− kui−1 − kui+1 − ui , ui
| {z }
|
{z
} |{z}
=(Au n+1 )i
⇔
find x = U n+1 ,
min((Ax − b)i , xi − gi ) = 0,
• EXISTENCE, STABILITY :
- If Aii ≥ 0 & A: δ ≥ 1-diag. domi. ⇒ ∃!x
- Moreover kU n+1 k∞ ≤ max(kU n k∞ , kgk∞ )
O. Bokanowski (Ensta/LJLL)
≡gi
≡bi
1≤i ≤J
⇒ NO CFL condition.
Numerical schemes for optimal stoppting time PDEs
Lab
Olivier Bokanowski
/ 57
• implicit for the nonlinear case: Can we do the same ?
Implicit finite difference scheme
min
uin+1 − uin
−
τ
n+1
n+1 −ui−1
+ 2uin+1 − ui−1
n+1
, ui
− ϕ(xi ) = 0,
h2
1≤i ≤J
After multiplication of the left part of the min by τ > 0, we get:
n+1
n+1
n+1
n+1
n
min (1 + 2k )ui
− ϕ(xi ) = 0
− kui−1 − kui+1 − ui , ui
| {z }
|
{z
} |{z}
=(Au n+1 )i
⇔
find x = U n+1 ,
min((Ax − b)i , xi − gi ) = 0,
• EXISTENCE, STABILITY :
- If Aii ≥ 0 & A: δ ≥ 1-diag. domi. ⇒ ∃!x
- Moreover kU n+1 k∞ ≤ max(kU n k∞ , kgk∞ )
O. Bokanowski (Ensta/LJLL)
≡gi
≡bi
1≤i ≤J
⇒ NO CFL condition.
Numerical schemes for optimal stoppting time PDEs
Lab
Olivier Bokanowski
/ 57
efficient tools for solving non-linear implicit schemes
PB: solve F (x) := min(Ax − b, x − g) = 0.
• Known also as the "linear complementary problem" (when g = 0).
• Exist. and Uniq. result based on
- positivity of all principal minors of A (A is a "P-matrix"1 )
- the eigenvalues of A (Iain Smears, Lecture Notes, ∼2013)
• From the numerical and financial math. point of view, the following
approaches have been considered (non exhaustive list)
(1) PSOR (A = L + U)
(2)
(3)
(4) Splitting (not solving F = 0 but rather x = max(A−1 b, g)).
1
K. Murty, 1972
O. Bokanowski (Ensta/LJLL)
Lab
Numerical schemes for optimal stoppting time PDEs
Olivier Bokanowski
/ 57
efficient tools for solving non-linear implicit schemes
PB: solve F (x) := min(Ax − b, x − g) = 0.
• Known also as the "linear complementary problem" (when g = 0).
• Exist. and Uniq. result based on
- positivity of all principal minors of A (A is a "P-matrix"1 )
- the eigenvalues of A (Iain Smears, Lecture Notes, ∼2013)
• From the numerical and financial math. point of view, the following
approaches have been considered (non exhaustive list)
(1) PSOR (A = L + U)
(2)
(3)
(4) Splitting (not solving F = 0 but rather x = max(A−1 b, g)).
1
K. Murty, 1972
O. Bokanowski (Ensta/LJLL)
Lab
Numerical schemes for optimal stoppting time PDEs
Olivier Bokanowski
/ 57
efficient tools for solving non-linear implicit schemes
PB: solve F (x) := min(Ax − b, x − g) = 0.
• Known also as the "linear complementary problem" (when g = 0).
• Exist. and Uniq. result based on
- positivity of all principal minors of A (A is a "P-matrix"1 )
- the eigenvalues of A (Iain Smears, Lecture Notes, ∼2013)
• From the numerical and financial math. point of view, the following
approaches have been considered (non exhaustive list)
(1) PSOR (A = L + U)
(2) Brennan Schwartz (A = UL)
(3)
(4) Splitting (not solving F = 0 but rather x = max(A−1 b, g)).
1
K. Murty, 1972
O. Bokanowski (Ensta/LJLL)
Lab
Numerical schemes for optimal stoppting time PDEs
Olivier Bokanowski
/ 57
efficient tools for solving non-linear implicit schemes
PB: solve F (x) := min(Ax − b, x − g) = 0.
• Known also as the "linear complementary problem" (when g = 0).
• Exist. and Uniq. result based on
- positivity of all principal minors of A (A is a "P-matrix"1 )
- the eigenvalues of A (Iain Smears, Lecture Notes, ∼2013)
• From the numerical and financial math. point of view, the following
approaches have been considered (non exhaustive list)
(1) PSOR (A = L + U)
(2) Brennan Schwartz (A = UL)
(3) Newton / Howard
(4) Splitting (not solving F = 0 but rather x = max(A−1 b, g)).
1
K. Murty, 1972
O. Bokanowski (Ensta/LJLL)
Lab
Numerical schemes for optimal stoppting time PDEs
Olivier Bokanowski
/ 57
efficient tools for solving non-linear implicit schemes
PB: solve F (x) := min(Ax − b, x − g) = 0.
• Known also as the "linear complementary problem" (when g = 0).
• Exist. and Uniq. result based on
- positivity of all principal minors of A (A is a "P-matrix"1 )
- the eigenvalues of A (Iain Smears, Lecture Notes, ∼2013)
• From the numerical and financial math. point of view, the following
approaches have been considered (non exhaustive list)
(1) PSOR (A = L + U)
(2) Brennan Schwartz (A = UL)
(3) Newton / Howard
(4) Splitting (not solving F = 0 but rather x = max(A−1 b, g)).
1
K. Murty, 1972
O. Bokanowski (Ensta/LJLL)
Lab
Numerical schemes for optimal stoppting time PDEs
Olivier Bokanowski
/ 57
efficient tools for solving non-linear implicit schemes
PB: solve F (x) := min(Ax − b, x − g) = 0.
• Known also as the "linear complementary problem" (when g = 0).
• Exist. and Uniq. result based on
- positivity of all principal minors of A (A is a "P-matrix"1 )
- the eigenvalues of A (Iain Smears, Lecture Notes, ∼2013)
• From the numerical and financial math. point of view, the following
approaches have been considered (non exhaustive list)
(1) PSOR (A = L + U)
(2) Brennan Schwartz (A = UL) ⇐
(3) Newton / Howard ⇐
(4) Splitting (not solving F = 0 but rather x = max(A−1 b, g)).
1
K. Murty, 1972
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PB: solve F (x) := min(Ax − b, x − g) = 0.
Approach (2): Brennan and Schwartz : based on A = UL
decomposition with
U = tridiag(0, 1, ui ),
and L = tridiag(li , di , 0), (di > 0)
Formally assumes that
min(ULx − b, x − g) = 0
(∗)
⇐⇒
min(Lx − U −1 b, x − g) = 0.
Then uses the descent algorithm (cost O(J))
Proposition
(i) The equivalence (∗) is true for "profiled" solutions x ∈ RJ such that
∃i0 , xi = gi for i ≤ i0 , and xi > gi otherwise
(ii) Otherwise false (or should be adapted...)
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PB: solve F (x) := min(Ax − b, x − g) = 0.
Approach (3): semi-smooth Newton’s method
• Definition. For F (x)i = min((Ax − b)i , xi − gi ), let
Aij if (Ax − b)i ≤ (x − g)i
0
F (x)ij =
Iij = δij otherwise.
• Newton’s algorithm:
O. Bokanowski (Ensta/LJLL)
x k +1 = x k − F (x k )−1 F (x k ).
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• More generally, consider for instance Merton’s portfolio problem:
v (T − t, x) := ess sup E[ϕ(XTt,x,α )|Ft ],
K compact
α:(t,T )→K
with dXθ /Xθ = b(Xθ , αθ )dθ + σ(Xθ , αθ )dWθ , leads to
1 2
2
min vt − σ (x, a)x vxx − b(x, a)x vx = 0.
2
a∈K
Implicit finite difference scheme : leads to a matrix Aa and vector ba
depending of the parameter a, and the implicit scheme to solve:
O. Bokanowski (Ensta/LJLL)
mina∈K (Aa x − ba ) = 0,
x ∈ RJ
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• More generally, consider for instance Merton’s portfolio problem:
v (T − t, x) := ess sup E[ϕ(XTt,x,α )|Ft ],
K compact
α:(t,T )→K
with dXθ /Xθ = b(Xθ , αθ )dθ + σ(Xθ , αθ )dWθ , leads to
1 2
2
min vt − σ (x, a)x vxx − b(x, a)x vx = 0.
2
a∈K
Implicit finite difference scheme : leads to a matrix Aa and vector ba
depending of the parameter a, and the implicit scheme to solve:
O. Bokanowski (Ensta/LJLL)
mina∈K (Aa x − ba ) = 0,
x ∈ RJ
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• Definition 1. B is a monotone matrix if, equivalently
(i) B −1 ≥ 0 componentwise
(ii) BX ≥ 0 ⇒ X ≥ 0 for all X .
• Definition 2. B is an M matrix if
B is diagonal dominant,
Bii ≥ 0
Bij ≤ 0.
• Definition 3. (Mixed matrices) For a = (a1 , . . . , aJ ) ∈ {0, 1}J , let
(
Bija := Aij , bia := bi if ai = 0
Bija := Iij , bia := gi if ai = 1
Then F (x) = min(Ax − b, x − g) = mina∈{0,1}J (B a x − ba )
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• Monotony assumption (Mono):
∀a ∈ {0, 1}J , B a is a monotone matrix
• Example: A is an M-matrix ⇒ (Mono)
Theorem ( (Rust & Santos 04’) (O.B., Maroso, Zidani 09’))
Assume (Mono),
(i) there exists a unique x ∈ RJ s.t. F (x) = 0;
(ii) ∀x 0 , lim x k = x. (Furthermore x k ≤ x k +1 )
k →∞
(iii) The convergence is in at most 2J iterations.
Proof: equivalence of Howard’s algorithm (1958’) and Newton’s
method.
Theorem ((Ito,Hintermuller,Kunisch) (O.B., Maroso ,Zidani))
Obstacle pb: the convergence is in at most J iterations !
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• Monotony assumption (Mono):
∀a ∈ {0, 1}J , B a is a monotone matrix
• Example: A is an M-matrix ⇒ (Mono)
Theorem ( (Rust & Santos 04’) (O.B., Maroso, Zidani 09’))
Assume (Mono),
(i) there exists a unique x ∈ RJ s.t. F (x) = 0;
(ii) ∀x 0 , lim x k = x. (Furthermore x k ≤ x k +1 )
k →∞
(iii) The convergence is in at most 2J iterations.
Proof: equivalence of Howard’s algorithm (1958’) and Newton’s
method.
Theorem ((Ito,Hintermuller,Kunisch) (O.B., Maroso ,Zidani))
Obstacle pb: the convergence is in at most J iterations !
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• Monotony assumption (Mono):
∀a ∈ {0, 1}J , B a is a monotone matrix
• Example: A is an M-matrix ⇒ (Mono)
Theorem ( (Rust & Santos 04’) (O.B., Maroso, Zidani 09’))
Assume (Mono),
(i) there exists a unique x ∈ RJ s.t. F (x) = 0;
(ii) ∀x 0 , lim x k = x. (Furthermore x k ≤ x k +1 )
k →∞
(iii) The convergence is in at most 2J iterations.
Proof: equivalence of Howard’s algorithm (1958’) and Newton’s
method.
Theorem ((Ito,Hintermuller,Kunisch) (O.B., Maroso ,Zidani))
Obstacle pb: the convergence is in at most J iterations !
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• Monotony assumption (Mono):
∀a ∈ {0, 1}J , B a is a monotone matrix
• Example: A is an M-matrix ⇒ (Mono)
Theorem ( (Rust & Santos 04’) (O.B., Maroso, Zidani 09’))
Assume (Mono),
(i) there exists a unique x ∈ RJ s.t. F (x) = 0;
(ii) ∀x 0 , lim x k = x. (Furthermore x k ≤ x k +1 )
k →∞
(iii) The convergence is in at most 2J iterations.
Proof: equivalence of Howard’s algorithm (1958’) and Newton’s
method.
Theorem ((Ito,Hintermuller,Kunisch) (O.B., Maroso ,Zidani))
Obstacle pb: the convergence is in at most J iterations !
O. Bokanowski (Ensta/LJLL)
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Application to american options: (OB, Maroso, Zidani 2009)
Limitation of the total number’s of Newton’s iteration := bounded by the
number of mesh points where the value "takes off" the payoff function.
Figure: Two successive iteration for the american option
O. Bokanowski (Ensta/LJLL)
100
Payoff
IE scheme: U^n
90
80
IE scheme: U^{n+1}
70
60
50
40
30
20
10
0
0
20
40
60
80
100
120
140
160
180
200
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The first order "barrier"
• Can we improve the error in time from O(τ ) to O(τ 2 ) or better ?
• (Godunov’s
Th.) For ut + bux = 0, monotone linear schemes
P
n+1
n
ui
= aj ui+j (aj ≥ 0) are limited to at most first order.
• For other reasons (see later on..) for non linear diffusion problems
some limitations hold, and we are led to try implicit schemes.
• Let us try a Crank - Nicolson scheme (RK2):
Sj1,n (u)
:≡ min
uin+1 − uin 1
+ (Au n + Au n+1 )i , uin+1 − gi
τ
2
1≤i ≤J
= 0,
For A: M-matrix, we can solve it by a Newton’s Algorithm.
• Numerical experiments show
- using τ = h, stable, second order (ObservationA)
- using τ = 10h, stable, but only first order (ObservationB)
What is wrong ??
O. Bokanowski (Ensta/LJLL)
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The first order "barrier"
• Can we improve the error in time from O(τ ) to O(τ 2 ) or better ?
• (Godunov’s
Th.) For ut + bux = 0, monotone linear schemes
P
n+1
n
ui
= aj ui+j (aj ≥ 0) are limited to at most first order.
• For other reasons (see later on..) for non linear diffusion problems
some limitations hold, and we are led to try implicit schemes.
• Let us try a Crank - Nicolson scheme (RK2):
Sj1,n (u)
:≡ min
uin+1 − uin 1
+ (Au n + Au n+1 )i , uin+1 − gi
τ
2
1≤i ≤J
= 0,
For A: M-matrix, we can solve it by a Newton’s Algorithm.
• Numerical experiments show
- using τ = h, stable, second order (ObservationA)
- using τ = 10h, stable, but only first order (ObservationB)
What is wrong ??
O. Bokanowski (Ensta/LJLL)
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The first order "barrier"
• Can we improve the error in time from O(τ ) to O(τ 2 ) or better ?
• (Godunov’s
Th.) For ut + bux = 0, monotone linear schemes
P
n+1
n
ui
= aj ui+j (aj ≥ 0) are limited to at most first order.
• For other reasons (see later on..) for non linear diffusion problems
some limitations hold, and we are led to try implicit schemes.
• Let us try a Crank - Nicolson scheme (RK2):
Sj1,n (u)
:≡ min
uin+1 − uin 1
+ (Au n + Au n+1 )i , uin+1 − gi
τ
2
1≤i ≤J
= 0,
For A: M-matrix, we can solve it by a Newton’s Algorithm.
• Numerical experiments show
- using τ = h, stable, second order (ObservationA)
- using τ = 10h, stable, but only first order (ObservationB)
What is wrong ??
O. Bokanowski (Ensta/LJLL)
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The first order "barrier"
• Can we improve the error in time from O(τ ) to O(τ 2 ) or better ?
• (Godunov’s
Th.) For ut + bux = 0, monotone linear schemes
P
n+1
n
ui
= aj ui+j (aj ≥ 0) are limited to at most first order.
• For other reasons (see later on..) for non linear diffusion problems
some limitations hold, and we are led to try implicit schemes.
• Let us try a Crank - Nicolson scheme (RK2):
Sj1,n (u)
:≡ min
uin+1 − uin 1
+ (Au n + Au n+1 )i , uin+1 − gi
τ
2
1≤i ≤J
= 0,
For A: M-matrix, we can solve it by a Newton’s Algorithm.
• Numerical experiments show
- using τ = h, stable, second order (ObservationA)
- using τ = 10h, stable, but only first order (ObservationB)
What is wrong ??
O. Bokanowski (Ensta/LJLL)
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The first order "barrier"
• Can we improve the error in time from O(τ ) to O(τ 2 ) or better ?
• (Godunov’s
Th.) For ut + bux = 0, monotone linear schemes
P
n+1
n
ui
= aj ui+j (aj ≥ 0) are limited to at most first order.
• For other reasons (see later on..) for non linear diffusion problems
some limitations hold, and we are led to try implicit schemes.
• Let us try a Crank - Nicolson scheme (RK2):
Sj1,n (u)
:≡ min
uin+1 − uin 1
+ (Au n + Au n+1 )i , uin+1 − gi
τ
2
1≤i ≤J
= 0,
For A: M-matrix, we can solve it by a Newton’s Algorithm.
• Numerical experiments show
- using τ = h, stable, second order (ObservationA)
- using τ = 10h, stable, but only first order (ObservationB)
What is wrong ??
O. Bokanowski (Ensta/LJLL)
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Olivier Bokanowski
/ 57
The first order "barrier"
• Can we improve the error in time from O(τ ) to O(τ 2 ) or better ?
• (Godunov’s
Th.) For ut + bux = 0, monotone linear schemes
P
n+1
n
ui
= aj ui+j (aj ≥ 0) are limited to at most first order.
• For other reasons (see later on..) for non linear diffusion problems
some limitations hold, and we are led to try implicit schemes.
• Let us try a Crank - Nicolson scheme (RK2):
Sj1,n (u)
:≡ min
uin+1 − uin 1
+ (Au n + Au n+1 )i , uin+1 − gi
τ
2
1≤i ≤J
= 0,
For A: M-matrix, we can solve it by a Newton’s Algorithm.
• Numerical experiments show
- using τ = h, stable, second order (ObservationA)
- using τ = 10h, stable, but only first order (ObservationB)
What is wrong ??
O. Bokanowski (Ensta/LJLL)
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Consistency analysis:
Notice that for a regular v and vin = v (tn , xi ):
vin+1 − vin 1
+ (Av n + Av n+1 )i
τ
2
= vt (tn+1/2 , xi ) + Av (tn+1/2 , xi ) + O(τ 2 )
= (vt − vxx )(tn+1/2 , xi ) + O(τ 2 + h2 )
Hence
n+1/2
+ O(τ 2 + h2 ), vin+1 − gi )
n+1/2
, vin+1 − gi ) + O(τ 2 + h2 )
Sj1,n (v ) = min((vt − vxx )i
= min((vt − vxx )i
But we cannot assume that vt − vxx ≡ 0 !
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In general
n+1/2
Sj1,n (v ) = min((vt − vxx )i
, vin+1 − gi ) + O(τ 2 + h2 )
τ
= min((vt − vxx )n+1
− ∂t (vt − vxx ), , vin+1 − gi ) + O(τ 2 + h2 )
i
2
so we have in general only first order consistency, in the sense
Sj1,n (v ) = min(vt − vxx , v − ϕ(xj ))(tn+1 ,xi ) + O(τ + h2 )
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More generally, consider a high order scheme for the diffusion part, i.e.
such that, for instance,
v n+1 − S(u n )
= (vt − vxx )(tn , xi ) + O(τ q ) + O(h2 )
τ
for some q ≥ 2. (Ex: Weak Taylor schemes, Platen’s scheme, Kloeden
and Platen 1995,...)
The corresponding scheme for the obstacle equation,
uin+1 = max(S(u n )i , gi )
and equivalent to
min
v n+1 − S(u n ) n+1
, ui
− gi
τ
=0
will not be high order in time for the obstacle PDE.
Then how the CN scheme could be of second order ?!!
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Looking at PDE (2), an other scheme can be (Scheme (2) for PDE (2))
Sj2,n := min
ujn+1 − ujn
τ
n+1
n
uj
− uj
1
+ (Au n+1 + Au n )j ,
2
τ
1≤j ≤J
=0
(4)
This is second order in time consistent, at time t = tn+1/2 !
Numerical experiments show:
- using τ = h, stable, second order
- using τ = 10h, switch to first order (ObservationC)
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Some explanations...
Related to observations (A) and (B):
Lemma
(i) If u n+1 ≥ u n for the scheme (1) (CN-obstacle scheme for PDE (1),
with the constraint ujn ≥ gj ), then u n+1 is also solution of the scheme
(2) : hence both schemes give identical values.
(ii) This is the case in particular when
- the matrix I − τ2 A is positive componentwise (which is the case
under an appropriate CFL condition of the form hτ2 ≤ c0 )
- the matrix I + τ2 A is an M-matrix (always the case here).
Related to observation (C): the lost of the second order behavior can
be explained by the fact that the CN scheme does not have very nice
stability behavior for bad CFL numbers.
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Some explanations...
Related to observations (A) and (B):
Lemma
(i) If u n+1 ≥ u n for the scheme (1) (CN-obstacle scheme for PDE (1),
with the constraint ujn ≥ gj ), then u n+1 is also solution of the scheme
(2) : hence both schemes give identical values.
(ii) This is the case in particular when
- the matrix I − τ2 A is positive componentwise (which is the case
under an appropriate CFL condition of the form hτ2 ≤ c0 )
- the matrix I + τ2 A is an M-matrix (always the case here).
Related to observation (C): the lost of the second order behavior can
be explained by the fact that the CN scheme does not have very nice
stability behavior for bad CFL numbers.
O. Bokanowski (Ensta/LJLL)
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III. A BDF Scheme
In order to get the right consistency error, we use the following 3-point
formula:
3vin+1 − 4vin + vin−1
' vt (tn+1 , xi ) + O(τ 2 )
2τ
Known also as a "Backward Difference Formula" of second order
Hence the corresponding implicit BDF2 scheme, for n ≥ 1:
Hjn+1 (u)
:≡ min
3ujn+1 − 4ujn + ujn−1
2τ
+ (Au
n+1
)j ,
ujn+1
− gj
=0
1≤j ≤J
• Multi Step scheme, needs u 0 and u 1 estimate to start
• For the linear part, known also as the "Gear" Scheme
• Scheme already proposed by Oosterlee (2003) together with
multigrid idea for 2d problems.
O. Bokanowski (Ensta/LJLL)
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III. A BDF Scheme
In order to get the right consistency error, we use the following 3-point
formula:
3vin+1 − 4vin + vin−1
' vt (tn+1 , xi ) + O(τ 2 )
2τ
Known also as a "Backward Difference Formula" of second order
Hence the corresponding implicit BDF2 scheme, for n ≥ 1:
Hjn+1 (u)
:≡ min
3ujn+1 − 4ujn + ujn−1
2τ
+ (Au
n+1
)j ,
ujn+1
− gj
=0
1≤j ≤J
• Multi Step scheme, needs u 0 and u 1 estimate to start
• For the linear part, known also as the "Gear" Scheme
• Scheme already proposed by Oosterlee (2003) together with
multigrid idea for 2d problems.
O. Bokanowski (Ensta/LJLL)
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Proposition
(i) Scheme is second-order consistent in time and space.
(ii) Scheme is fastly implementable for B = A + 2τ
3 : M-matrix.
Proof of (ii): Equivalent scheme:
2τ
4 n 1 n−1 n+1
n+1
min (I +
A)u
− u + u
, u
−g =0
3
3
3
1≤j ≤J
• Other temptative consistent schemes can be proposed, but the
miracle here is that BDF2 for the obstacle problem can be shown to be
stable - and convergent ! (although non monotone)
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A-stability : for the ODE
ẏ = λy
we look for which λ ∈ C the scheme is stable. If it is stable for all
Re(λ) ≤ 0 then we say the scheme is A-stable.
BDF2 is A-stable
BDFk , k ≥ 3 are not A-stable.
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Wikipedia - Stability region for BDF2:
O. Bokanowski (Ensta/LJLL)
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Ernst Hairer and Gerhard Wanner (2010), Scholarpedia, 5(4):4591.
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Numerical experiments show:
using τ = h: stable, second order
using τ = 10h: (incondtionnally) stable, second order
⇒ very efficient !
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STABILITY AND CONVERGENCE RESULT
For a given vector x = (xj )1≤j≤J , let
J
X
kxk2 = (
|xj |2 )1/2
(5)
j=1
1/2
J+1
X
xj − xj−1 2
| 
Nh (x) := 
|
h

(6)
j=1
(with the convention x0 := 0 and xJ+1 := 0).
• Assume the following coercitivity on A: there exists constants η > 0,
γ ≥ 0 such that:
he, Aei ≥ ηNh (e)2 − γkek22 .
(7)
• Coercivity holds for the matrix A coming from the F.D. approximation
of Av = −a(x)vxx + b(x)vx + c(x)v with a(x) ≥ η > 0 and a ∈ Lip,
b, c ∈ L∞ .
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Scheme
Hjn+1 (u) :≡ min
3ujn+1 − 4ujn + ujn−1
2τ
+ (Au n+1 )j , ujn+1 − gj
=0
Perturbed scheme for the exact solution vjn = v (tn , xj ):
n+1
3vj
− 4vjn + vjn−1
n+1
n+1
n+1
n
Hj (u) :≡ min
+ (Av
)j −j , vj
− gj = 0
2τ
Error: en := v n − u n
Proposition (Stability estimate)
Let τ > 0 be sufficiently small. There exists C1 independent of n, for all
tn ≤ T
ken k22
O. Bokanowski (Ensta/LJLL)
+τ
n
X
Nh (ek )2
k =1
≤ C1 ke0 k22 + ke1 k22 + τ
X
kn k22 .
k =1,...,n
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For solution v with bounded vxx and vt (typical regularity for t > 0),
with an "isolated" singularity x = xs (t) we can show that nj = O(1) in
worst case (otherwise high-order), therefore kn k22 = O(1) and in the
end, ken k22 = O(1).
As a consequence:
Theorem
For the BDF2 implicit obstacle scheme, it holds
O. Bokanowski (Ensta/LJLL)
ken kL2 ' (hken k22 )1/2 = O(h1/2 )
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STABILITY : ELEMENTS OF THE PROOF
Let hx, y i denote the usual scalar product on RJ .
Lemma
For any matrix B, the following equivalence holds:
min(Bx − b, x − g) = 0 ⇔ x ≥ g and hBx − b, v − xi ≥ 0, ∀v ≥ g
Proof in the case B is a positive definite symmetric matrix:
1
min(Bx − b, x − g) = 0 ⇔ x solves min hx, Bxi − hb, xi
x≥g 2
⇔ x ≥ g and hBx − b, v − xi ≥ 0, ∀v ≥ g
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Let
B := I +
2τ
A,
3
and vectors
4 n
u −
3
4
bv := v n −
3
bu :=
1 n−1
u
3
1 n−1
v
3
Equation in u n+1 is equivalent to
min(Bu n+1 − bu , u n+1 − g) = 0
By the Lemma, this is equivalent to u n+1 ≥ g and
n+1
Bu
− bu , w − u n+1 ≥ 0, ∀w ≥ g.
(8)
Similarly, equation for v n+1 is equivalent to v n+1 ≥ g and
n+1
Bv
− (bv + τ n ), w − v n+1 ≥ 0, ∀w ≥ g.
(9)
O. Bokanowski (Ensta/LJLL)
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Plugging w = v n+1 in (8) and w = u n+1 into (9) gives
n+1 4 n 1 n−1 2τ n n+1 Be
− e + e
−
, e
≤0
3
3
3
and therefore
n+1
3e
− 4en + en−1 , en+1 + 2τ hen+1 , Aen+1 i ≤ 2τ hn , en+1 i.
Using the coercivity of A, the argument is then technical but "classical"
as for the analysis of BDF2 scheme for a parabolic linear problem.
DONE !
O. Bokanowski (Ensta/LJLL)
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Olivier Bokanowski
/ 57
Plugging w = v n+1 in (8) and w = u n+1 into (9) gives
n+1 4 n 1 n−1 2τ n n+1 Be
− e + e
−
, e
≤0
3
3
3
and therefore
n+1
3e
− 4en + en−1 , en+1 + 2τ hen+1 , Aen+1 i ≤ 2τ hn , en+1 i.
Using the coercivity of A, the argument is then technical but "classical"
as for the analysis of BDF2 scheme for a parabolic linear problem.
DONE !
O. Bokanowski (Ensta/LJLL)
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Olivier Bokanowski
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IV. Numerical examples
MODEL TEST PROBLEMS:
λ2 2
min vt − x vxx − rxvx + rv , v − ϕ(x) = f (t, x)
2
v (0, x) = ϕ(x).
√
• Set xs (t) := K − c0 t
• We construct explicit v (t, x) on x ∈ [0, Xmax ] s.t.
(i) v (t, x) = ϕ(x) = K − x for x ≤ xs (t),
(ii) v (t, x) > ϕ(x) = max(K − x, 0) for x ∈ ]xs (t), Xmax ],
(iii) v is at least C 1 on [0, Xmax ],
(iv ) v (t, Xmax ) = 0.
• MODEL 1: s.t. uxx bounded, regular away from x = xs (t).
• MODEL 2: s.t. u3x bounded, regular away from x = xs (t).
O. Bokanowski (Ensta/LJLL)
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Olivier Bokanowski
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IV. Numerical examples
MODEL TEST PROBLEMS:
λ2 2
min vt − x vxx − rxvx + rv , v − ϕ(x) = f (t, x)
2
v (0, x) = ϕ(x).
√
• Set xs (t) := K − c0 t
• We construct explicit v (t, x) on x ∈ [0, Xmax ] s.t.
(i) v (t, x) = ϕ(x) = K − x for x ≤ xs (t),
(ii) v (t, x) > ϕ(x) = max(K − x, 0) for x ∈ ]xs (t), Xmax ],
(iii) v is at least C 1 on [0, Xmax ],
(iv ) v (t, Xmax ) = 0.
• MODEL 1: s.t. uxx bounded, regular away from x = xs (t).
• MODEL 2: s.t. u3x bounded, regular away from x = xs (t).
O. Bokanowski (Ensta/LJLL)
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Olivier Bokanowski
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IV. Numerical examples
MODEL TEST PROBLEMS:
λ2 2
min vt − x vxx − rxvx + rv , v − ϕ(x) = f (t, x)
2
v (0, x) = ϕ(x).
√
• Set xs (t) := K − c0 t
• We construct explicit v (t, x) on x ∈ [0, Xmax ] s.t.
(i) v (t, x) = ϕ(x) = K − x for x ≤ xs (t),
(ii) v (t, x) > ϕ(x) = max(K − x, 0) for x ∈ ]xs (t), Xmax ],
(iii) v is at least C 1 on [0, Xmax ],
(iv ) v (t, Xmax ) = 0.
• MODEL 1: s.t. uxx bounded, regular away from x = xs (t).
• MODEL 2: s.t. u3x bounded, regular away from x = xs (t).
O. Bokanowski (Ensta/LJLL)
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Olivier Bokanowski
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IV. Numerical examples
MODEL TEST PROBLEMS:
λ2 2
min vt − x vxx − rxvx + rv , v − ϕ(x) = f (t, x)
2
v (0, x) = ϕ(x).
√
• Set xs (t) := K − c0 t
• We construct explicit v (t, x) on x ∈ [0, Xmax ] s.t.
(i) v (t, x) = ϕ(x) = K − x for x ≤ xs (t),
(ii) v (t, x) > ϕ(x) = max(K − x, 0) for x ∈ ]xs (t), Xmax ],
(iii) v is at least C 1 on [0, Xmax ],
(iv ) v (t, Xmax ) = 0.
• MODEL 1: s.t. uxx bounded, regular away from x = xs (t).
• MODEL 2: s.t. u3x bounded, regular away from x = xs (t).
O. Bokanowski (Ensta/LJLL)
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Olivier Bokanowski
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IV. Numerical examples
MODEL TEST PROBLEMS:
λ2 2
min vt − x vxx − rxvx + rv , v − ϕ(x) = f (t, x)
2
v (0, x) = ϕ(x).
√
• Set xs (t) := K − c0 t
• We construct explicit v (t, x) on x ∈ [0, Xmax ] s.t.
(i) v (t, x) = ϕ(x) = K − x for x ≤ xs (t),
(ii) v (t, x) > ϕ(x) = max(K − x, 0) for x ∈ ]xs (t), Xmax ],
(iii) v is at least C 1 on [0, Xmax ],
(iv ) v (t, Xmax ) = 0.
• MODEL 1: s.t. uxx bounded, regular away from x = xs (t).
• MODEL 2: s.t. u3x bounded, regular away from x = xs (t).
O. Bokanowski (Ensta/LJLL)
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Olivier Bokanowski
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t=1
t=1
30
30
Model 1
Model 2
25
25
20
20
15
15
10
10
5
5
0
0
74
76
78
80
82
84
86
88
90
92
94
96
74
76
78
80
82
84
86
88
90
92
94
96
Figure: Zooming around the singular point (xs , g(xs )) for model 1 (left) and 2
(right).
O. Bokanowski (Ensta/LJLL)
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Olivier Bokanowski
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t=1
100
80
60
40
20
0
−20
0
O. Bokanowski (Ensta/LJLL)
20
40
60
80
100
120
140
160
180
200
Figure: Model 2.
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Olivier Bokanowski
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Numerical examples
Crank Nicolson schemes
Mesh
J
N
80
80
160
160
320
320
640
640
1280
1280
2560
2560
Error L1
error
order
1.74E-02
3.15E-03
2.47
8.42E-04
1.91
2.23E-04
1.92
6.06E-05
1.88
1.66E-05
1.86
Error L2
error
order
2.16E-02
3.73E-03
2.54
9.87E-04
1.92
2.58E-04
1.93
6.96E-05
1.89
1.91E-05
1.86
Error L∞
error
order
4.49E-02
5.25E-03
3.10
1.39E-03
1.92
3.56E-04
1.97
9.15E-05
1.96
2.69E-05
1.77
CPU time
0.14
0.25
0.52
1.54
6.65
37.66
Table: EDP (1), Crank-Nicolson scheme with N = J (τ ≡ h)
O. Bokanowski (Ensta/LJLL)
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Mesh
J
N
80
160
320
640
1280
2560
8
16
32
64
128
256
Error L1
error
order
1.83E-02
4.67E-03
1.68E-03
6.89E-04
2.99E-04
1.37E-04
1.97
1.48
1.28
1.20
1.13
Error L2
error
order
2.47E-02
7.23E-03
2.97E-03
1.38E-03
6.59E-04
3.22E-04
1.77
1.28
1.11
1.06
1.03
Error L∞
error
order
4.90E-02
2.18E-02
9.90E-03
4.75E-03
2.31E-03
1.14E-03
1.17
1.14
1.06
1.04
1.02
time(s)
0.05
0.07
0.13
0.38
1.99
16.04
Table: EDP (1), Crank-Nicolson scheme with N = J/10 (τ >> h)
O. Bokanowski (Ensta/LJLL)
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Mesh
J
N
80
160
320
640
1280
2560
8
16
32
64
128
256
Error L1
error
order
2.68E-02
6.91E-03
2.53E-03
9.35E-04
3.74E-04
1.56E-04
1.96
1.45
1.43
1.32
1.26
Error L2
error
order
3.11E-02
8.28E-03
2.97E-03
1.11E-03
4.44E-04
1.86E-04
1.91
1.48
1.43
1.32
1.26
Error L∞
error
order
4.90E-02
1.14E-02
3.91E-03
1.50E-03
6.07E-04
2.56E-04
2.11
1.54
1.38
1.31
1.25
time(s)
0.25
0.13
0.18
0.41
1.66
11.36
Table: EDP (2) - Crank-Nicolson scheme, N = J/10
O. Bokanowski (Ensta/LJLL)
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BDF schemes (American option problem)
BDF2 implicit obstacle scheme:
min
3ujn+1 − 4ujn + ujn−1
O. Bokanowski (Ensta/LJLL)
2τ
+ (Au
n+1
+ q(tn+1 ))j ,
ujn+1
− gj
=0
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Olivier Bokanowski
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Mesh
J
N
80
160
320
640
1280
2560
80
160
320
640
1280
2560
O. Bokanowski (Ensta/LJLL)
Error L1
error
order
1.72E-02
3.02E-03
7.93E-04
2.06E-04
5.46E-05
1.44E-05
2.51
1.93
1.94
1.92
1.93
Error L2
error
order
2.14E-02
3.57E-03
9.30E-04
2.40E-04
6.30E-05
1.67E-05
2.58
1.94
1.96
1.93
1.92
Error L∞
error
order
4.48E-02
5.06E-03
1.32E-03
3.39E-04
8.75E-05
2.64E-05
3.15
1.93
1.97
1.95
1.73
time(s)
0.13
0.26
0.49
1.54
6.43
36.70
Table: BDF2-implicit scheme (N = J)
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Olivier Bokanowski
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Mesh
J
N
80
160
320
640
1280
2560
8
16
32
64
128
256
Error L1
error
order
8.23E-03
9.61E-04
4.18E-04
1.54E-04
4.77E-05
1.14E-05
3.10
1.20
1.44
1.69
2.07
Error L2
error
order
1.25E-02
1.28E-03
5.41E-04
1.92E-04
5.82E-05
1.39E-05
3.29
1.24
1.49
1.73
2.06
Error L∞
error
order
3.59E-02
2.20E-03
8.83E-04
2.99E-04
8.71E-05
2.01E-05
4.02
1.32
1.56
1.78
2.12
time(s)
0.06
0.09
0.13
0.32
1.14
6.21
Table: BDF2-implicit scheme, with high CFL numbers.
O. Bokanowski (Ensta/LJLL)
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BDF3 implicit obstacle scheme:
11
min
6
ujn+1 − 3ujn + 23 ujn−1 − 31 ujn−2
τ
+ (Au n+1 ))j ,
ujn+1
− gj
= 0,
for n ≥ 2
• No stability estimate !
• test on a more regular data (MODEL 2) to see the order
• changing A matrix to get 4-th order approximation in space
• Use smoother initial data ϕ(x) = v (t0 , x), t0 > 0 to avoid initial singularity
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Olivier Bokanowski
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Mesh
J
N
80
160
320
640
1280
2560
5120
8
16
32
64
128
256
512
Error L1
error
order
2.56E+00
6.48E-01
1.77E-01
4.72E-02
1.25E-02
3.24E-03
8.06E-04
1.98
1.87
1.90
1.92
1.95
2.01
Error L2
error
order
5.93E-01
1.48E-01
4.00E-02
1.06E-02
2.77E-03
7.14E-04
1.78E-04
2.00
1.89
1.92
1.94
1.96
2.00
Error L∞
error
order
1.94E-01
5.11E-02
1.33E-02
3.41E-03
8.60E-04
2.16E-04
5.39E-05
1.93
1.95
1.96
1.99
2.00
2.00
time(s)
0.10
0.11
0.24
0.48
1.28
4.64
19.30
Table: (Model 1) BDF2 implicit scheme with 4th order spatial approximation,
using high CFL numbers.
O. Bokanowski (Ensta/LJLL)
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Olivier Bokanowski
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Mesh
J
N
80
160
320
640
1280
2560
5120
8
16
32
64
128
256
512
Error L1
error
order
2.50E+00
6.42E-01
1.69E-01
4.41E-02
1.15E-02
2.96E-03
7.33E-04
1.96
1.93
1.94
1.94
1.96
2.01
Error L2
error
order
5.99E-01
1.48E-01
3.85E-02
9.96E-03
2.57E-03
6.55E-04
1.63E-04
2.01
1.95
1.95
1.96
1.97
2.01
Error L∞
error
order
2.09E-01
5.34E-02
1.35E-02
3.36E-03
8.43E-04
2.11E-04
5.28E-05
1.96
1.99
2.00
2.00
2.00
2.00
time(s)
0.09
0.13
0.24
0.53
1.66
7.27
39.23
Table: (Model 1) BDF3 - third order - implicit scheme with 4th order spatial
approximation, using high CFL numbers.
O. Bokanowski (Ensta/LJLL)
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Mesh
J
N
80
160
320
640
1280
2560
5120
8
16
32
64
128
256
512
Error L1
error
order
2.53E-01
5.02E-02
1.09E-02
2.58E-03
6.24E-04
1.54E-04
3.81E-05
2.34
2.20
2.08
2.05
2.02
2.01
Error L2
error
order
4.23E-02
8.54E-03
1.86E-03
4.39E-04
1.06E-04
2.62E-05
6.48E-06
2.31
2.20
2.08
2.05
2.02
2.01
Error L∞
error
order
9.90E-03
2.02E-03
4.37E-04
1.04E-04
2.50E-05
6.16E-06
1.53E-06
2.29
2.21
2.07
2.05
2.02
2.01
time(s)
0.10
0.15
0.26
0.60
1.60
5.65
24.37
Table: (Model 2) BDF2 implicit scheme with 4th order spatial approximation,
using high CFL numbers, and t0 = 0.3. (SMOOTHER INITIAL DATA)
O. Bokanowski (Ensta/LJLL)
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Olivier Bokanowski
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Mesh
J
N
80
160
320
640
1280
2560
5120
8
16
32
64
128
256
512
Error L1
error
order
2.51E-02
3.65E-03
6.00E-04
8.56E-05
1.09E-05
1.42E-06
1.92E-07
2.78
2.61
2.81
2.97
2.94
2.89
Error L2
error
order
4.42E-03
6.48E-04
1.04E-04
1.46E-05
1.83E-06
2.32E-07
3.02E-08
2.77
2.64
2.83
3.00
2.98
2.94
Error L∞
error
order
1.21E-03
1.76E-04
2.62E-05
3.55E-06
4.40E-07
5.52E-08
7.00E-09
2.78
2.75
2.89
3.01
2.99
2.98
time(s)
0.11
0.14
0.25
0.65
2.11
8.96
47.05
Table: (Model 2) BDF3 implicit scheme with 4th order spatial approximation,
using high CFL numbers, and t0 = 0.3. (SMOOTHER INITIAL DATA)
O. Bokanowski (Ensta/LJLL)
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CONCLUSION
Errors in terms of number of operations Nopt :
BDF2 using N ≡ J:
Nopt ≡ N · J ≡ N 2 ,
observed error : err = O(τ 2 ) = O( N12 ) = O( N1opt ).
theoretical bound : errL2 = O(h1/2 ) = O(
1
1/4 ).
Nopt
Tree method with N levels (Lamberton 2015), constant coefficient
case:
N levels ⇒ Nopt ≡ N 2 /2
observed error : err = O( N1 ) = O(
1
1/2 ).
Nopt
α
theoretical bound : errL∞ = O( log(N)
) ≡ O(
N
O. Bokanowski (Ensta/LJLL)
log(Nopt )α
1/2
Nopt
)
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Olivier Bokanowski
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CONCLUSION
Errors in terms of number of operations Nopt :
BDF2 using N ≡ J:
Nopt ≡ N · J ≡ N 2 ,
observed error : err = O(τ 2 ) = O( N12 ) = O( N1opt ).
theoretical bound : errL2 = O(h1/2 ) = O(
1
1/4 ).
Nopt
Tree method with N levels (Lamberton 2015), constant coefficient
case:
N levels ⇒ Nopt ≡ N 2 /2
observed error : err = O( N1 ) = O(
1
1/2 ).
Nopt
α
theoretical bound : errL∞ = O( log(N)
) ≡ O(
N
O. Bokanowski (Ensta/LJLL)
log(Nopt )α
1/2
Nopt
)
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Olivier Bokanowski
/ 57
CONCLUSION
Errors in terms of number of operations Nopt :
BDF2 using N ≡ J:
Nopt ≡ N · J ≡ N 2 ,
observed error : err = O(τ 2 ) = O( N12 ) = O( N1opt ).
theoretical bound : errL2 = O(h1/2 ) = O(
1
1/4 ).
Nopt
Tree method with N levels (Lamberton 2015), constant coefficient
case:
N levels ⇒ Nopt ≡ N 2 /2
observed error : err = O( N1 ) = O(
1
1/2 ).
Nopt
α
theoretical bound : errL∞ = O( log(N)
) ≡ O(
N
O. Bokanowski (Ensta/LJLL)
log(Nopt )α
1/2
Nopt
)
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CONCLUSION 2
Related ongoing works:
- Discontinuous Galerkin approaches for obstacle problems (with
C.-W Shu, Y. Cheng)
- Implicit schemes for nonlinear diffusion problems (with A. Picarelli,
C. Reisinger)
O. Bokanowski (Ensta/LJLL)
Thanks !
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Olivier Bokanowski
/ 57
CONCLUSION 2
Related ongoing works:
- Discontinuous Galerkin approaches for obstacle problems (with
C.-W Shu, Y. Cheng)
- Implicit schemes for nonlinear diffusion problems (with A. Picarelli,
C. Reisinger)
O. Bokanowski (Ensta/LJLL)
Thanks !
Lab
Numerical schemes for optimal stoppting time PDEs
Olivier Bokanowski
/ 57