The Kuratowski Ryll-Nardzewski Theorem and semismooth ... · The Kuratowski{Ryll-Nardzewski Theorem and semismooth Newton methods for Hamilton{Jacobi{Bellman equations Iain Smears

The Kuratowski–Ryll-Nardzewski Theorem

and semismooth Newton methods for

Hamilton–Jacobi–Bellman equations

Iain Smears

INRIA Paris

Linz, November 2016

joint work with

Endre Suli, University of Oxford

Overview

Talk outline

1. Introduction: Howard’s algorithm / policy iteration for

Hamilton–Jacobi–Bellman equations.

2. Semismoothness of HJB operators in function spaces.

3. Applications to discontinuous Galerkin FEM approximations of HJB equations

with Cordes coefficients.

Overview

Talk outline






1. Hamilton–Jacobi–Bellman Equation

F [u] := supα∈Λ

[Lαu − f α] = 0 in Ω,

u = 0 on ∂Ω,(HJB)

where Lαu := aα(x) : D2u + bα(x) · ∇u − cα(x) u.

Notation: aα(x) : D2u =d∑

i,j=1

aαij (x)uxi xj , bα(x) · ∇u =d∑

i=1

bαi (x)uxi .

Assumptions:

• bounded domain Ω,

• control set Λ is a compact metric space,

• continuous functions a, b, c and f in x ∈ Ω and α ∈ Λ.

Remark: Further assumptions are required for well-posedness of the problem, but

not for the semismoothness discussed here.

1/28

1. Motivation

Howard’s algorithm / policy iteration

Formal structure

1. Choose an initial guess u0.

2. For each k ≥ 0, choose αk : Ω→ Λ such that

αk(x) ∈ argmaxα∈Λ (Lαuk − f α)(x), ∀ x ∈ Ω.

3. Then, find uk+1 as a solution of the PDE

Lαkuk+1 = f αk in Ω, with uk+1 = 0 on ∂Ω,

where Lαk v := aαk (x)(x) : D2v + bαk (x)(x) · ∇v − cαk (x)v

In practice: used in a discrete context after discretization by a numerical method.

2/28

1. Motivation


Formal structure








2/28

1. Background

Classical works:

[Bellman,Dynamic Programming, 1957]

[Howard, Dynamic Programming and Markov Processes, 1960].

Historical summary from [Puterman & Brumelle, 1979]:

Policy iteration is usually attributed to Bellman [...] and Howard [...] Bellman

developed the technique, which he called iteration in policy space, to solve several

dynamic programming problems.

Howard [16] later developed a version of this procedure for Markovian decision

problems which he called the policy-iteration method.

[Puterman & Brumelle, 1979]: interpretation as Newton–Kantorovich method &

convergence rates assuming: there is δ ∈ (0, 1] such that, for all functions u and v ,

‖Lαv − Lαu‖L(X ,Y ) . ‖v − u‖δX

where αv and αu are arg-maximisers for v and u.

NB: this cannot hold when arg-max operation is non-unique or not continuous.

3/28

1. Background

On solver algorithms for HJB: [Santos & Rust, 2004] Analysis of policy iteration

for finite dimensional MDP problems.

[Bokanowski, Maroso, Zidani, 2009]: Superlinear convergence and

semismoothness of finite dimensional HJB operators of form

minα∈AN

[Bαx− cα] = 0

with matrices Bα ∈ RN×N and vectors x, c ∈ RN , (see also discussion of

Bellman–Isaacs).

Variant algorithms and applications: penalty methods [Reisinger & Witte, 2011,

2012], coupled value-policy iteration [Alla, Falcone, Kalise, 2015]

Semismooth Newton methods [Ulbrich, 2002], [Hintermuller, Ito, Kunisch, 2002]

(primal-dual active set method as as semismooth Newton method)

4/28

1. Semismooth Newton methods

Notation: Let X and Y be sets. We write G : X ⇒ Y if G is a set-valued map

that maps X into the subsets of Y .

Definition of semismoothness [Ulbrich, 2002]

Let X and Y be Banach spaces.

Let F : X → Y .

Let DF : X ⇒ L(X ,Y ) with non-empty images.

We say that F is DF -semismooth on U if, for all x ∈ U,

lim‖e‖X→0

1

‖e‖Xsup L∈DF [u+e]‖F [u + e]− F [u]− L e‖Y = 0.

Then DF is then called a generalised differential of F on U.

Semismoothness + uniform stability of linearizations:

supL∈DF [v ],v∈X

‖L−1‖L(Y ,X ) <∞

=⇒ local superlinear convergence of semismooth Newton method.5/28

1. Semismoothness of max(v , 0) and norm-gap

Important example from [Ulbrich, 2002], [Hintermuller, Ito, Kunisch, 2002]

Let 1 ≤ q < r ≤ ∞.

Let G : Lr (Ω)→ Lq(Ω) be defined by G : u 7→ max(u, 0).

Then G is semismooth from Lr (Ω) to Lq(Ω) with differential DF [v ] the set

of all L ∈ L∞(Ω) of the form:

L(x) =

1 if v(x) > 0

0 if v(x) < 0

an arbitrary fixed value if v(x) = 0

Norm gap: the restriction q < r cannot be removed (counter-examples).

How to generalise this to HJB operators?

6/28

Overview

Talk outline






7/28

1. Motivation


Formal structure








8/28

2. Semismoothness of HJB operators

For FEM applications: let Th be a mesh on Ω.

Space X = W 2,r (Ω, Th), 1 ≤ r ≤ ∞, with norm:

‖u‖W 2,r (Ω;Th) =

∑K∈Th

‖u‖rW 2,r (K)

1r

.

Function u ∈W 2,r (Ω, Th) have element-wise gradient ∇hu and Hessian D2hu.

For Λ compact and continuous coefficients, F : W 2,r (Ω, Th)→ Lr (Ω) is well

defined and Lipschitz continuous

F [u] := supα∈Λ

[Lαu − f α].

9/28

2. Semismoothness: argmax set-valued map

For each u ∈ X , we define u =(u,∇hu,D

2hu)∈ Lr (Ω;Rm) for suitable m.

We then view the differential operator F [u] as a composition of x 7→ u(x) with the

scalar function F : Ω× Rm → R defined by

F (x , v) = supα∈Λ

[aα(x) : M + bα(x) · p − cα(x)z − f α(x)], v = (z , p,M)

Define the set-valued map Ω× Rm 3 (x , v) 7→ Λ(x , v) ⊂ Λ by

Λ(x , v) := argmaxα∈Λ[aα(x) : M + bα(x) · p − cα(x)z − f α(x)]

Straightforward: Λ(x , v) is non-empty and closed in Λ.

10/28

2. Semismoothness: argmax set-valued map

Important lemma:

The mapping Λ(·, ·) : Ω× Rm ⇒ Λ is upper semicontinuous:

For every (x , v) ∈ Ω×Rm, and any open neighbourhood U of Λ(x , v),

there is an open neighbourhood V of (x , v) such that Λ(y ,w) ⊂ U

for all (y ,w) ∈ V .

(x,v)

(xn,vn)

(xn,vn) ! (x,v)

10/28

2. Kuratowski–Ryll-Nardzewski Theorem

Kuratowski–Ryll-Nardzewski

Let Ω ⊂ Rd be a bounded open set,

let Λ be a compact metric space,

let Λ(·, ·) : Ω × Rm ⇒ Λ be an upper semicontinuous set-valued function,

such that Λ(x , v) is non-empty and closed for every (x , v) ∈ Ω× Rm.

Then, for any Lebesgue measurable function u : Ω → Rm, there exists a

Lebesgue measurable selection α : Ω→ Λ such that

α(x) ∈ Λ(x , u(x)

)for a.e. x ∈ Ω.

(Presented here in the form needed for our purposes - original result is rather more general)

Kuratowski & Ryll-Nardzewski, Bull. Acad. Polon. Sci., 1965:

A general theorem on selectors.

A (specialised) proof in Aubin & Cellina, Differential Inclusions, 1984.

11/28

2. The generalized differential of HJB operators

Recall u(x) = (u(x),∇hu(x),D2hu(x)) for u ∈W 2,r (Ω; Th).

Define the set of measurable selections Λ[u]:

Λ[u] = α : Ω→ Λ; α Lebesgue measurable, α(x) ∈ Λ(x , u(x)) a.e. in Ω .

Kuratowski–Ryll-Nardzewski Thm =⇒ Λ[u] is non-empty for all u ∈W 2,r (Ω; Th).

Define the differential

DF [u] := Lα = aα : D2h + bα · ∇h − cα, α ∈ Λ[u]

The measurability of α ∈ Λ[u] implies that Lα is well defined in

L(W 2,r (Ω; Th), Lr (Ω)).

12/28

2. The generalized differential of HJB operators

Recall u(x) = (u(x),∇hu(x),D2hu(x)) for u ∈W 2,r (Ω; Th).

Define the set of measurable selections Λ[u]:

Λ[u] = α : Ω→ Λ; α Lebesgue measurable, α(x) ∈ Λ(x , u(x)) a.e. in Ω .

Kuratowski–Ryll-Nardzewski Thm =⇒ Λ[u] is non-empty for all u ∈W 2,r (Ω; Th).

Define the differential


The measurability of α ∈ Λ[u] implies that Lα is well defined in

L(W 2,r (Ω; Th), Lr (Ω)).

12/28



Theorem [S. & Suli, SINUM, 2014]

Let 1 ≤ q < r ≤ ∞.

The operator F : W 2,r (Ω; Th)→ Lq(Ω) is DF -semismooth on W 2,r (Ω; Th):

lim‖e‖

W 2,r (Ω;Th)→0

1

‖e‖W 2,r (Ω;Th)

supLα∈DF [u+e]

‖F [u + e]− F [u]− Lαe‖Lq(Ω) = 0.

Remark: The sup implies any choice of measurable selection is sufficient.

I. S. & E. Suli, SIAM J. Numer. Anal. 2014:

Discontinuous Galerkin finite element approximation of Hamilton–Jacobi–Bellman

equations with Cordes coefficients.

13/28

2. Semismoothness of HJB operators: Proof

Suppose the claim is false: there exists ej∞j=0, ‖ej‖W 2,r (Ω;Th) → 0, αj ∈ Λ[u + ej ],

and ρ > 0 such that, ∀j ≥ 0:

1

‖ej‖W 2,r (Ω;Th)

‖F [u + ej ]− F [u]− Lαj ej‖Lq(Ω) > ρ (?)

We will find a subsequence of ej∞j=0 such that (?) is contradicted.

14/28


1. Passing to a subsequence, we have (ej ,∇hej ,D2hej)→ 0 pointwise a.e. in Ω.

2. Key inequality: using the definition of F , we can show (pointwise):

|F [u+ej ](x)−F [u](x)−Lαj ej(x)| ≤ |Lαej(x)−Lαj ej(x)| ∀α ∈ Λ(x , u(x)), a.e. x .

This implies

|F [u + ej ]− F [u]− Lαj ej | . Gj (|ej |+ |∇ej |+ |D2hej |), (??)

where the function Gj is defined by:

Gj := infα∈Λ(·,u(·))

|aα − aαj |+ |bα − bαj |+ |cα − cαj | .

Remark: Gj is measurable (can be written as a composition of a lower

semicontinous function with measurable functions).

15/28


Recall Gj := infα∈Λ(·,u(·)) |aα − aαj |+ |bα − bαj |+ |cα − cαj |.Recall also αj(x) ∈ Λ(x , u(x) + ej(x)) for a.e. x ∈ Ω.

(x,u)

(x,u(x) + ej(x))

(x,u(x) + ej(x)) ! (x,u(x))

Upper semi-continuity of Λ(·, ·) leads to

Gj → 0 pointwise a.e. in Ω.

16/28


Also, we have a uniform bound supj≥0‖Gj‖L∞(Ω) ≤ C because all coefficients are

uniformly bounded on Ω× Λ.

Lebesgue’s Dominated Convergence Theorem:

limj→∞‖Gj‖Ls (Ω) = 0 for any 1 ≤ s <∞.

Recall (??):


If q < r , Holder’s inequality implies that we can find s ∈ [1,∞) s.t.

0 < ρ ≤ 1


‖F [u + ej ]− F [u]− Lαj ej‖Lq(Ω) ≤ ‖Gj‖Ls (Ω) → 0.

Remarks: the norm gap appears because there are cases where ‖Gj‖L∞(Ω) 6→ 0.

17/28


Also, we have a uniform bound supj≥0‖Gj‖L∞(Ω) ≤ C because all coefficients are

uniformly bounded on Ω× Λ.

Lebesgue’s Dominated Convergence Theorem:

limj→∞‖Gj‖Ls (Ω) = 0 for any 1 ≤ s <∞.

Recall (??):


If q < r , Holder’s inequality implies that we can find s ∈ [1,∞) s.t.

0 < ρ ≤ 1


‖F [u + ej ]− F [u]− Lαj ej‖Lq(Ω) ≤ ‖Gj‖Ls (Ω) → 0.

Remarks: the norm gap appears because there are cases where ‖Gj‖L∞(Ω) 6→ 0.

17/28



Theorem [S. & Suli, SINUM, 2014]

Let 1 ≤ q < r ≤ ∞.

The operator F : W 2,r (Ω; Th)→ Lq(Ω) is DF -semismooth on W 2,r (Ω; Th):

lim‖e‖

W 2,r (Ω;Th)→0

1

‖e‖W 2,r (Ω;Th)

supLα∈DF [u+e]

‖F [u + e]− F [u]− Lαe‖Lq(Ω) = 0.

Remark: The sup implies any choice of measurable selection is sufficient.

I. S. & E. Suli, SIAM J. Numer. Anal. 2014:

Discontinuous Galerkin finite element approximation of Hamilton–Jacobi–Bellman

equations with Cordes coefficients.

18/28

Overview

Talk outline






19/28

3. Cordes condition

From now on, we suppose

• Ω is bounded Lipschitz domain

• aα uniformly elliptic, cα ≥ 0, uniformly over Ω×.

Policy iteration:

Lαkuk+1 = f αk , αk ∈ Λ[uk ]

Is this linear PDE well-posed for general uniformly elliptic coefficients aα?

In general: no! Due to discontinuities in aα:

• Non-uniqueness of strong solutions [Gilbarg & Trudinger, 2001]

• Non-uniquess of viscosity solutions [Nadirashvili, 1997], [Safonov, 1999].

However: yes under a further assumption:

• Cordes condition (next slide) : well-posedness in H2(Ω) ∩ H10 (Ω).

20/28

3. Cordes condition




Policy iteration:








20/28

3. Cordes condition




Policy iteration:








20/28

3. Cordes condition

From now on, we assume that Ω is convex:

Cordes condition: Case 1: without advection and reaction

Assume that there exists ε ∈ (0, 1] s. t.

|aα(x)|2(Tr aα(x))2 ≤

1

d − 1 + ε∀ x ∈ Ω, α ∈ Λ. (Cordes0)

Cordes condition: Case 2: extension to bα 6= 0 and cα 6= 0

Assume that there exist λ > 0 and ε ∈ (0, 1] s. t.

|aα|2 + |bα|2/2λ+ (cα/λ)2

(Tr aα + cα/λ)2≤ 1

d + εin Ω, ∀α ∈ Λ. (Cordes1)

Thm: [Cordes 1956], [Maugeri, Palagachev, Softova 2000]

There is C = C(ε) s.t. for any measurable α→ Λ:

‖(Lα)−1‖L2→H2∩H10≤ C .

21/28

3. Cordes condition

Well-posedness theorem [S. & Suli, SINUM, 2014]

Under these assumptions, there exists a unique u ∈ H2(Ω) ∩ H10 (Ω) that

solves (HJB) pointwise a.e. in Ω.

Remarks

• If dimension d = 2, (Cordes0) ⇐⇒ uniform ellipticity.

• Proof is based on Cordes condition and Miranda–Talenti inequality for convex

domains:

|u|H2(Ω) := ‖D2u‖L2(Ω) ≤ ‖∆u‖L2(Ω) ∀ u ∈ H2(Ω) ∩ H10 (Ω).

22/28

3. Applications to numerical scheme

Construction an hp-version discontinuous Galerkin finite element scheme in [S. &

Suli, SINUM 2013 + SINUM 2014 + Num. Math. 2016]:

• Discrete Stability in H2: uh is uniquely defined, and stable in discrete

norm ‖·‖H2(Ω;Th) + |·|Jh .

• Consistency for true solution : if u ∈ Hs(Ω, Th) with s > 5/2.

• Near-best approximation w.r.t H2-conforming subspaces.

• Convergence rates: if u ∈ Hs(Ω, Th) with s > 5/2:

‖u − uh‖H2(Ω;Th) .hmin(p+1,s)−2

ps−5/2‖u‖Hs (Ω).

Main idea: based on approximating a strongly monotone operator formulation of

the PDE:

A(u; v) =

∫Ω

supα∈Λ

[γα(Lαu − f α)](∆v − λv)dx = 0 ∀v ∈ H2(Ω) ∩ H10 (Ω);

23/28

3. Applications to numerical scheme

Discrete linearised problems:

Akh(uk+1

h , vh) =∑K∈Th

〈γαk f αk , Lλvh〉K ∀ vh ∈ Vh,p,

where the bilinear form Akh : Vh,p × Vh,p → R is defined by

Akh(wh, vh) :=

∑K∈Th

〈(γαkLαkwh, Lλvh〉K + linear stabilization terms from Ah(·; ·).

Superlinear convergence [S. & Suli, SINUM, 2014]

There exists R > 0, possibly depending on h and p, such that

if ‖uh − u0h‖H2(Ω;Th) < R, then superlinear convergence:

limk→∞

‖uh − uk+1h ‖H2(Ω;Th)

‖uh − ukh‖H2(Ω;Th)

= 0

24/28

3. Numerics: experiment 1: h-refinement

Experiment 1 : Test of high order convergence rates under h-refinement, fixed p.

aα =1

2R>(

1 + sin2 θ sin θ cos θ

sin θ cos θ cos2 θ

)R

with α = (θ,R) ∈ Λ = [0, π3

]× SO(2).

Remark: aα becomes increasingly anisotropic as θ → π/3; rotation matrices

R ∈ SO(2) prevent monotone schemes from aligning the grid with the anisotropy.

1/21/41/81/161/321/6410−7

10−5

10−3

10−1

101

Mesh size

‖u−

uh‖ H

2(Ω

;Th

)

p = 2p = 3p = 4p = 5

1 2 3 4 5 6 7Converged

10−12

10−8

10−4

1

Iteration number k

‖uh−uk h‖ H

2(Ω

;Th

)

h = 1/4

h = 1/8

h = 1/16

h = 1/32

h = 1/64

25/28

3. Numerics: experiment 2: hp-refinement

Experiment 2: test of exponential convergence rates under hp-refinement

Let Ω = (0, 1)2, bα := (0, 1), cα := 10 and define

aα := α>(

20 1

1 0.1

)α, α ∈ Λ := SO(2), λ =

1

2.

(Cordes1) holds with ε ≈ 0.0024 (nearly degenerate, strongly anisotropic).

Solution:

u(x , y) = (2x − 1)(e1−|2x−1| − 1

)(y +

1− ey/δ

e1/δ − 1

), δ := 0.005 = O(ε)

• Near-degenerate and anisotropic diffusion.

• Sharp boundary layer.

• Non-smooth solution.

26/28


Experiment 2: test of exponential convergence rates under hp-refinement

Let Ω = (0, 1)2, bα := (0, 1), cα := 10 and define

aα := α>(

20 1

1 0.1

)α, α ∈ Λ := SO(2), λ =

1

2.

(Cordes1) holds with ε ≈ 0.0024 (nearly degenerate, strongly anisotropic).

Solution:

u(x , y) = (2x − 1)(e1−|2x−1| − 1

)(y +

1− ey/δ

e1/δ − 1

), δ := 0.005 = O(ε)

Boundary layer adapted meshes with p-refinement: 2 ≤ pK ≤ 10, from 100 to

1320 DoFs.

26/28


5 6 7 8 9 10 11

10−6

10−5

10−4

10−3

10−2

10−1

100

3√DoF

RelativeError

Broken H2 norm

Broken H1 norm

27/28


1 3 5 7 9 11 13

10−14

10−10

10−6

10−2

102

Iteration number k

‖uh−uk h‖ H

2(Ω

;Th

)

‖uh‖ H

2(Ω

;Th

)

p = 3p = 4p = 5p = 6p = 7

27/28

Conclusions

Conclusions:

• General semismoothness result for HJB operators with Λ a general compact

metric space and continuous coefficients.

• Usage of the measurable selection theorem of Kuratowski–Ryll-Nardzewski.

• Application to DGFEM for HJB equations with Cordes coefficients: superlinear

convergence of the semismooth Newton method

• Numerical experiments showing fast convergence and weak dependence of the

iteration counts on h and p.

Linear nondivergence form PDE: Discontinuous Galerkin finite element

approximation of nondivergence form elliptic equations with Cordes coefficients, I. S.

& E. Suli, SIAM J. Numer. Anal. 2013.

Elliptic HJB: Discontinuous Galerkin finite element approximation of

Hamilton–Jacobi–Bellman equations with Cordes coefficients, I. S. & E. Suli, SIAM

J. Numer. Anal. 2014.

Parabolic HJB: Discontinuous Galerkin finite element methods for time-dependent

Hamilton–Jacobi–Bellman equations with Cordes coefficients, I. S. & E. Suli,

Numerische Mathematik 2016.

Thank you!28/28

3. Numerical scheme

Numerical scheme: solve Ah(uh; vh) = 0 for all vh ∈ Vh,p

Ah(uh; vh) :=∑K∈Th

〈Fγ [uh], Lλvh〉K + Jh(uh, vh)

+1

2

Bh(uh, vh)−∑K∈Th

〈Lλuh, Lλvh〉K

.

28/28

3. Numerical scheme




+1

2


〈Lλuh, Lλvh〉K

.

〈Fγ [uh], Lλvh〉K :=

∫K

supα∈Λ

[γα(Lαuh − f α)] (∆vh − λvh) dx .

Remark: the term γα rescales the operators Lα without changing the true solution.

28/28

3. Numerical scheme




+1

2


〈Lλuh, Lλvh〉K

.

Jump penalisation with µF ' p2K/hK and ηF ' p4

K/h3K for F ⊂ ∂K :

Jh(uh, vh) :=∑

F∈F i,bh

[µF 〈J∇T uhK, J∇T vhK〉F + ηF 〈JuhK, JvhK〉F

]+∑F∈F i

h

µF 〈J∇uh · nF K, J∇vh · nF K〉F .

28/28

3. Numerical scheme




+1

2


〈Lλuh, Lλvh〉K

.

〈Lλuh, Lλvh〉K :=

∫K

(∆uh − λuh) (∆vh − λvh) dx .

28/28

3. Numerical scheme




+1

2


〈Lλuh, Lλvh〉K

.

Weak enforcement of Miranda–Talenti inequality:

Bh(uh, vh) :=∑K∈Th

[〈D2uh,D

2vh〉K + 2λ〈∇uh,∇vh〉K + λ2〈uh, vh〉K]

+∑F∈F i

h

[〈divT∇Tuh, J∇vh · nF K〉F + 〈divT∇Tvh, J∇uh · nF K〉F

]−

∑F∈F i,b

h

[〈∇T∇uh · nF , J∇T vhK〉F + 〈∇T∇vh · nF , J∇T uhK〉F

]− λ

∑F∈F i,b

h

[〈∇uh · nF , JvhK〉F + 〈∇vh · nF , JuhK〉F ]

− λ∑F∈F i

h

[〈uh, J∇vh · nF K〉F + 〈vh, J∇uh · nF K〉F ]

28/28

3. Numerical scheme




+1

2


〈Lλuh, Lλvh〉K

.

Consistency, Stability, Convergence rates [S. & Suli, SINUM, 2014]

• Discrete Stability: uh is uniquely defined, and stable in discrete norm

‖·‖H2(Ω;Th) + |·|Jh .

• Consistency: Ah(u; vh) = 0 for the solution u, if u ∈ Hs(Ω, Th) with

s > 5/2.

• Near-best approximation w.r.t H2-conforming subspaces.

• Convergence rates: ‖u − uh‖H2(Ω;Th) .hs

ps−1/2 ‖u‖Hs (Ω) if

u ∈ Hs(Ω, Th) with s > 5/2.

28/28

3. Discrete Semismooth Newton method

Algorithm

1. Choose an initial guess u0h ∈ Vh,p.

2. Given ukh ∈ Vh,p, k ∈ N, choose αk ∈ Λ[uk

h ].

3. Solve

Akh(uk+1

h , vh) =∑K∈Th



Akh(wh, vh) :=

∑K∈Th


Well-posedness : the bilinear forms Akh(·, ·) have a uniform coercivity constant.



if ‖uh − u0h‖H2(Ω;Th) < R, then uk

h → uh superlinearly.

28/28

3. Discrete Semismooth Newton method

Algorithm

1. Choose an initial guess u0h ∈ Vh,p.

2. Given ukh ∈ Vh,p, k ∈ N, choose αk ∈ Λ[uk

h ].

3. Solve

Akh(uk+1

h , vh) =∑K∈Th



Akh(wh, vh) :=

∑K∈Th


Well-posedness : the bilinear forms Akh(·, ·) have a uniform coercivity constant.



if ‖uh − u0h‖H2(Ω;Th) < R, then uk

h → uh superlinearly.

28/28

Documents

The Kuratowski Ryll-Nardzewski Theorem and semismooth ... · The Kuratowski{Ryll-Nardzewski Theorem and semismooth Newton methods for Hamilton{Jacobi{Bellman equations Iain Smears