Filtered schemes for Hamilton-Jacobi equations · Filtered schemes for Hamilton-Jacobi equations Tiago Salvador joint work with Adam Oberman Department of Mathematics and Statistics

Filtered schemes for Hamilton-Jacobi equations

Tiago Salvadorjoint work with Adam Oberman

Department of Mathematics and Statistics

Numerical methods for Hamilton-Jacobi equations

in optimal control and related fields

RICAM, Linz, November 22, 2016

Hamilton-Jacobi equations Filtered Schemes Numerical schemes Explicit formulas Results Conclusions

Outline

1 Hamilton-Jacobi equations

2 Filtered Schemes

3 Numerical schemesSchemes for the Eikonal equationSchemes for Hamilton-Jacobi equations

4 Explicit formulasEikonal equationHJ equations

5 Results

6 Conclusions

Tiago Salvador Filtered schemes for Hamilton-Jacobi equations RICAM, 21-25 November 2016


Contents


2 Filtered Schemes



5 Results

6 Conclusions



Hamilton-Jacobi equations

Consider the Hamilton-Jacobi (HJ) equationH(x ,∇u) = f (x), x ∈ Ω \ Γ,u(x) = g(x), x ∈ Γ ⊂ Ω,

where

∇u is the gradient of the function u,

Ω is a bounded open set,

Γ is a subset of Ω,

H is a nonlinear Lipschitz continuous function.



Eikonal equation

In particular, we will focus on the the Eikonal equation|∇u(x)| = f (x), for x outside Γ,

u(x) = g(x), for x on Γ,

where f > 0 and Γ is closed, bounded set.

Fact: Solutions are usually piecewise smooth and globally Lipschitz.



Related work

Semi-Lagrangian schemes (Falconi, Ferretti ’02 and Cristiani,Falconi ’07)

Central schemes (Lin, Tadmor ’00)

ENO and WENO (Osher, Shu ’91)

Compact upwind second order scheme (Benamou, Luo, Zhao ’03)

Abgrall blended scheme (Abgrall ’09)



Contents


2 Filtered Schemes



5 Results

6 Conclusions



Motivation

In the one-dimensional case we are interested in solving |ux | = f (x). Themonotone (upwind) scheme is given by

∣∣uhx ∣∣M = max

−u(x + h)− u(x)

h,u(x)− u(x − h)

h, 0

= max

−D+u(x),D−u(x), 0

This scheme is also consistent and stable and so, by the Barles andSouganidis theory, it converges to the unique viscosity solution.

Fact: Monotone schemes are provably convergent but only first orderaccurate.



We could consider the second order accurate centered scheme∣∣uhx ∣∣A =|u(x + h)− u(x − h)|

2h.

However this scheme is not monotone nor stable.

Fact: In general, higher order finite difference schemes for HJ equationsare neither monotone nor stable.

Up to a scaling in h, the difference of the schemes is the centered finitedifference approximation for |uxx |:∣∣∣∣∣uhx ∣∣A −

∣∣uhx ∣∣M ∣∣∣ = h

2

|u(x + h)− 2u(x) + u(x − h)|h2

=h

2uhxx(x).

We can then use this as a (local) criteria to decide whether or not to usethe accurate scheme.



We could consider the second order accurate centered scheme∣∣uhx ∣∣A =|u(x + h)− u(x − h)|

2h.

However this scheme is not monotone nor stable.

Fact: In general, higher order finite difference schemes for HJ equationsare neither monotone nor stable.

Up to a scaling in h, the difference of the schemes is the centered finitedifference approximation for |uxx |:∣∣∣∣∣uhx ∣∣A −

∣∣uhx ∣∣M ∣∣∣ = h

2

|u(x + h)− 2u(x) + u(x − h)|h2

=h

2uhxx(x).

We can then use this as a (local) criteria to decide whether or not to usethe accurate scheme.



Filtered Schemes

The filtered scheme, |∇uh|F , then uses the following simple formula:

∣∣uhx ∣∣F =

∣∣uhx ∣∣A , if∣∣∣∣∣uhx ∣∣A −

∣∣uhx ∣∣M ∣∣∣ ≤ √h∣∣uhx ∣∣M , otherwise.

The filtered scheme, which is consistent provided both underlyingschemes are consistent, is usually not monotone. However it is almostmonotone, since∣∣uhx ∣∣F =

∣∣uhx ∣∣M +O(√h).



Filtered Schemes

DefinitionS is a filter function if it is a bounded function that is equal to theidentity in a neighborhood of the origin and zero outside.

-2 -1 1 2

-1.0

-0.5

0.5

1.0

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

DefinitionThe filtered scheme is given by F h[u] = F h

M [u] + ϵ(h)S(

F hA [u]−FM [u]

ϵ(h)

).



Theorem (Froese, Oberman 2013)Let F h

M be a consistent, stable, monotone scheme. Assume S is acontinuous filter function and ϵ(h) → 0 as h → 0. For each h > 0, let uhbe a solution of F h[u] = 0, where the filtered scheme is given by

F h[u] = F hM [u] + ϵ(h)S

(F hA[u]− FM [u]

ϵ(h)

).

Then uh converges to the unique viscosity solution of the underlying PDE.

Why use filtered schemes?

Simplicity.

Provably convergent.

Achieve higher accuracy where the solution is smooth.

In practice, it avoids the use of a wide stencil in second orderequations (e.g. Monge-Ampere equation).






(F hA[u]− FM [u]

ϵ(h)

).



Simplicity.









(F hA[u]− FM [u]

ϵ(h)

).



Simplicity.









(F hA[u]− FM [u]

ϵ(h)

).



Simplicity.









(F hA[u]− FM [u]

ϵ(h)

).



Simplicity.









(F hA[u]− FM [u]

ϵ(h)

).



Simplicity.






Contents


2 Filtered Schemes



5 Results

6 Conclusions



Schemes for the Eikonal equation


Monotone scheme

|uhx |M = max−D+u(x),D−u(x), 0

Accurate scheme (centred scheme)

|uhx |C ,2 =|u(x + h)− u(x − h)|

2h

High-order upwind scheme

|uhx |U,n = max

− d

dxP+,n[u](x),

d

dxP−,n[u](x), 0

ENO schemes

|uhx |E ,n = max

− d

dxE n, 12 [u](x),

d

dxE n,− 1

2 [u](x), 0





Monotone scheme



|uhx |C ,2 =|u(x + h)− u(x − h)|

2h


|uhx |U,n = max

− d

dxP+,n[u](x),

d

dxP−,n[u](x), 0

ENO schemes

|uhx |E ,n = max

− d

dxE n, 12 [u](x),

d

dxE n,− 1

2 [u](x), 0





Monotone scheme



|uhx |C ,2 =|u(x + h)− u(x − h)|

2h


|uhx |U,n = max

− d

dxP+,n[u](x),

d

dxP−,n[u](x), 0

ENO schemes

|uhx |E ,n = max

− d

dxE n, 12 [u](x),

d

dxE n,− 1

2 [u](x), 0





Monotone scheme



|uhx |C ,2 =|u(x + h)− u(x − h)|

2h


|uhx |U,n = max

− d

dxP+,n[u](x),

d

dxP−,n[u](x), 0

ENO schemes

|uhx |E ,n = max

− d

dxE n, 12 [u](x),

d

dxE n,− 1

2 [u](x), 0



Schemes for Hamilton-Jacobi equations


Monotone scheme (Lax-Friedrichs)

HhLF [u](x) = Hh

LF (x ,D+u(x),D−u(x)) = H

(x ,

D+u(x) + D−u(x)

2

)−1

2σx (D

+u(x)−D−u(x))


HhC ,2[u](x) = H

(x ,

u(x + h)− u(x − h)

2h

)High-order upwind scheme

HhU,n[u](x) = Hh

LF

(x ,

d

dxP+,n[u](x),

d

dxP−,n[u](x)

)ENO schemes

HhE ,n[u](x) = Hh

LF

(x ,

d

dxEn, 1

2 [u](x),d

dxEn,− 1

2 [u](x)

)






HhLF [u](x) = Hh


(x ,

D+u(x) + D−u(x)

2

)−1

2σx (D

+u(x)−D−u(x))


HhC ,2[u](x) = H

(x ,

u(x + h)− u(x − h)

2h

)


HhU,n[u](x) = Hh

LF

(x ,

d

dxP+,n[u](x),

d

dxP−,n[u](x)

)ENO schemes

HhE ,n[u](x) = Hh

LF

(x ,

d

dxEn, 1

2 [u](x),d

dxEn,− 1

2 [u](x)

)






HhLF [u](x) = Hh


(x ,

D+u(x) + D−u(x)

2

)−1

2σx (D

+u(x)−D−u(x))


HhC ,2[u](x) = H

(x ,

u(x + h)− u(x − h)

2h


HhU,n[u](x) = Hh

LF

(x ,

d

dxP+,n[u](x),

d

dxP−,n[u](x)

)

ENO schemes

HhE ,n[u](x) = Hh

LF

(x ,

d

dxEn, 1

2 [u](x),d

dxEn,− 1

2 [u](x)

)






HhLF [u](x) = Hh


(x ,

D+u(x) + D−u(x)

2

)−1

2σx (D

+u(x)−D−u(x))


HhC ,2[u](x) = H

(x ,

u(x + h)− u(x − h)

2h


HhU,n[u](x) = Hh

LF

(x ,

d

dxP+,n[u](x),

d

dxP−,n[u](x)

)ENO schemes

HhE ,n[u](x) = Hh

LF

(x ,

d

dxEn, 1

2 [u](x),d

dxEn,− 1

2 [u](x)

)



Two-dimensional schemes


Eikonal equation∣∣∇uh∣∣ = N

(∣∣uhx ∣∣ , ∣∣uhy ∣∣) where N(x , y) =√x2 + y2


HhLF [u](x) = Hh

LF (x , p+, p−, q+, q−)

= H

(x ,

p+ + p−

2,q+ + q−

2

)− σx

p+ − p−

2− σy

q+ − q−

2

where p± = D±x u(x) and q± = D±

y u(x).





Eikonal equation∣∣∇uh∣∣ = N

(∣∣uhx ∣∣ , ∣∣uhy ∣∣) where N(x , y) =√x2 + y2


HhLF [u](x) = Hh

LF (x , p+, p−, q+, q−)

= H

(x ,

p+ + p−

2,q+ + q−

2

)− σx

p+ − p−

2− σy

q+ − q−

2

where p± = D±x u(x) and q± = D±

y u(x).



Contents


2 Filtered Schemes



5 Results

6 Conclusions



Eikonal equation

Explicit formulas for Eikonal equation

Solving∣∣uhx ∣∣M = f for the reference variable, u(x), leads to

∣∣uhx ∣∣M = f ⇐⇒ max

−u(x + h)− u(x)

h,u(x)− u(x − h)

h, 0

= f (x)

⇐⇒ max u(x)− u(x + h), u(x)− u(x − h), 0 = hf (x)

⇐⇒ u(x) = minu(x + h), u(x − h)+ hf (x)

Similarly, for the second order upwind scheme we have∣∣uhx ∣∣U,2= f ⇐⇒ max3u(x)− 4u(x ± h) + u(x ± 2h), 0 = 2hf (x)

⇐⇒ u(x) =1

3min4u(x ± h)− u(x ± 2h)+ 2

3hf (x)



Eikonal equation

Explicit formulas for Eikonal equation

Solving∣∣uhx ∣∣M = f for the reference variable, u(x), leads to

∣∣uhx ∣∣M = f ⇐⇒ max

−u(x + h)− u(x)

h,u(x)− u(x − h)

h, 0

= f (x)

⇐⇒ max u(x)− u(x + h), u(x)− u(x − h), 0 = hf (x)

⇐⇒ u(x) = minu(x + h), u(x − h)+ hf (x)

Similarly, for the second order upwind scheme we have∣∣uhx ∣∣U,2= f ⇐⇒ max3u(x)− 4u(x ± h) + u(x ± 2h), 0 = 2hf (x)

⇐⇒ u(x) =1

3min4u(x ± h)− u(x ± 2h)+ 2

3hf (x)



Eikonal equation

Two-dimensional case

Solving∣∣∇uh∣∣ = f ⇐⇒

∣∣uhx ∣∣2 + ∣∣uhy ∣∣2 = f (x , y)2

for the reference variable u(x , y) requires solving nonlinear equation

maxz − a, 02 +maxz − b, 02 = c2

for the unknown z where a, b and c > 0 are constants. The uniquesolution is given by

z =

mina, b+ c , |a− b| ≥ c ,a+b+

√2c2−(a−b)2

2 , |a− b| < c .



HJ equations

Explicit formulas for the HJ equations

Recall that

D+u(x) =u(x + h)− u(x)

h, D−u(x) =

u(x)− u(x − h)

h.

Solving HhLF [u] = f for the reference variable, u(x), leads to

H

(x ,

D+u(x) + D−u(x)

2

)−

1

2σx(D+u(x)− D−u(x)

)= f (x)

⇐⇒ H

(x ,

u(x + h)− u(x − h)

2h

)−

1

2σx

u(x + h)− 2u(x) + u(x − h)

h= f (x)

⇐⇒ u(x) =1

σx

[f (x)− H

(x ,

u(x + h)− u(x − h)

2h

)+ σx

u(x + h) + u(x − h)

2h

].



HJ equations

Explicit formulas for the HJ equations

Consider now the second order upwind scheme, HhU,2[u] = f , rewritten as

H

(x ,

ddxP+,2[u](x) + d

dxP−,2[u](x)

2

)−

1

2σx

(d

dxP+,2[u](x)−

d

dxP−,2[u](x)

)= f (x).

where

d

dxP+,2[u](x) =

−3u(x) + 4u(x + h)− u(x + 2h)

2h,

d

dxP−,2[u](x) =

3u(x)− 4u(x − h) + u(x − 2h)

2h.

Thus, solving for the reference variable, u(x), leads to

u(x) =2

3σx

[f (x)− H

(x ,

−u(x + 2h) + 4u(x + h)− 4u(x − h) + u(x − 2h)

4h

)+σx

−u(x + 2h) + 4u(x + h) + 4u(x − h)− u(x − 2h)

4h

].



Contents


2 Filtered Schemes



5 Results

6 Conclusions



One-dimensional examples

The 1st example is given by

|ux | = 1 + cos(x), u(x) = 3− |x + sin(x)|

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

Figure: Profile of the solution.



−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.42.5

2.55

2.6

2.65

2.7

2.75

2.8

2.85

2.9

2.95

3

Monotone

2ndupwind

Exact

Figure: Exact solution and solutions obtained with the monotone scheme andthe 2nd order upwind filtered scheme with 50 grid points.



−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure: Active scheme for the solutions of the 1st example: filtered scheme withcentred scheme (left) and filtered scheme with 2nd order upwind scheme (right).



101

102

103

104

10−12

10−10

10−8

10−6

10−4

10−2

100

MonotoneFiltered (2nd)Filtered (2ndUpwind)Filtered (3rdUpwind)Filtered (4thUpwind)Filtered (2ndENO)Filtered (3rdENO)Filtered (4thENO)

Figure: Loglog plot of the errors for the 1st Example.



The 2nd example is given by

|ux | = 1 + e|x|, u(x) = 10− |x | − e|x|

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

1

2

3

4

5

6

7

8

9




101

102

103

104

10−12

10−10

10−8

10−6

10−4

10−2

100


Figure: Loglog plot of the errors for the 2nd Example.



The 3rd example is given by

|ux | = 3x2 + a, u(x) =

x3 + ax , x ∈ [0, x0],

1 + a− ax − x3, x ∈ [x0, 1],

with a =1−2x3

0

2x0−1 , x0 =3√2+24 3√2

.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4




101

102

103

104

10−12

10−10

10−8

10−6

10−4

10−2

100


Figure: Loglog plot of the errors for the 3rd Example.



The 4th example is given by

|ux |2 = ex , u(x) =

2e

x2 + 16, x x ∈ [−2, 0],

−2ex2 + 20, x ∈ [0, 2].

−2 −1.5 −1 −0.5 0 0.5 1 1.5 214.5

15

15.5

16

16.5

17

17.5

18




101

102

103

10−3

10−2

10−1

100

MonotoneFiltered (2nd)Filtered (2ndUpwind)Filtered (2ndENO)

Figure: Loglog plot of the errors for the 4th Example.



For the last example we take

cos2(ux) + |ux | = cos2(e−|x|

)+ e−|x|, u(x) = e−|x|

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1




101

102

103

10−4

10−3

10−2

10−1

100

MonotoneFiltered (2nd)Filtered (2ndUpwind)Filtered (2ndENO)

Figure: Loglog plot of the errors for the 5th Example.



Two-dimensional examples

We consider three distinct examples all of them solution to the Eikonalequation

|∇u| = 1 for x outside Γ,

u = 0 for x on Γ

with the computational domain being [−2, 2]2 taking for Γ:

1 Γ =(x , y) ∈ R2 : x2 + y2 ≤ 1

,

2 Γ =(

12 ,

12

),(− 1

2 ,−12

),

3 Γ =(x , y) ∈ R2 : x2 + y2 ≤ 1, x ≥ 0

.



−2−1

01

2

−2

−1

0

1

20

0.5

1

1.5

2

−2−1

01

2 −2

0

20

0.5

1

1.5

2

2.5

3

−2

−1

0

1

2

−2

−1

0

1

20

0.5

1

1.5

2

2.5

Figure: Profile of the solutions of the three examples considered in 2D.



Distance to a circle

102

103

10−6

10−4

10−2

Monotone

Filtered (2nd)

Filtered (2ndUpwind)

Filtered (2ndENO)

Figure: Log-log plot of the errors for the first example in the L∞ norm.



Distance to two points

102

103

10−4

10−2

Monotone

Filtered (2nd)


Filtered (2ndENO)

102

103

10−4

10−2

100

Monotone

Filtered (2nd)


Filtered (2ndENO)

Figure: Log-log plot of the errors for the second example in the L∞ norm (left)and L1 norm (right).



Distance to a semi-circle

102

103

10−4

10−2

Monotone

Filtered (2nd)


Filtered (2ndENO)

102

103

10−6

10−4

10−2

Monotone

Filtered (2nd)


Filtered (2ndENO)

Figure: Log-log plot of the errors for the third example in the L∞ norm in[−2, 2] (left) and L∞ norm in

(x , y) ∈ R2 : x2 + y 2 ≥ 1, x > 0.1

(right).



Results summary

One-dimensional case

The order of convergence in the filtered upwind schemes is the orderof accuracy of the accurate scheme used (special result to theEikonal equation).We found an example where the error for the ENO was greater thanthe formal accuracy.


We obtain second order convergence for smooth solutions.In general, solutions are piecewise smooth and we only recoversecond order convergence in the smooth regions.



Results summary


The order of convergence in the filtered upwind schemes is the orderof accuracy of the accurate scheme used (special result to theEikonal equation).

We found an example where the error for the ENO was greater thanthe formal accuracy.





Results summary







Results summary







Results summary




We obtain second order convergence for smooth solutions.

In general, solutions are piecewise smooth and we only recoversecond order convergence in the smooth regions.



Results summary







Contents


2 Filtered Schemes



5 Results

6 Conclusions



Conclusions

Filtered schemes:

are provably convergent;

are simple and easy to implement;

allow for a wide choice of accurate discretizations;

achieve higher accuracy where the solution is smooth;

in practice, it avoids the use of a wide stencil in second orderequations (e.g. Monge-Ampere equation).



Conclusions

Filtered schemes:








Conclusions

Filtered schemes:








Conclusions

Filtered schemes:








Conclusions

Filtered schemes:








Conclusions

Filtered schemes:








References

Guy Barles and Panagiotis E. Souganidis, Convergence ofapproximation schemes for fully nonlinear second order equations,Asymptotic Anal. 4 (1991).

Brittany D. Froese and Adam M. Oberman, Convergent filteredschemes for the Monge-Ampere partial differential equation, SIAM J.Numer. Anal. 51 (2013).

Stanley Osher and Chi-Wang Shu, High-order essentiallynonoscillatory schemes for Hamilton-Jacobi equations, SIAM J.Numer. Anal. 28 (1991).

Hongkai Zhao, A fast sweeping method for eikonal equations, Math.Comp. 74 (2005).

Adam M. Oberman and Tiago Salvador, Filtered schemes forHamilton-Jacobi equations: a simple construction of convergentaccurate difference schemes, JCP 284 (2015).


Documents

Filtered schemes for Hamilton-Jacobi equations · Filtered schemes for Hamilton-Jacobi equations Tiago Salvador joint work with Adam Oberman Department of Mathematics and Statistics