The KuratowskiRyll-Nardzewski Theorem and semismooth Newton methods - - PowerPoint PPT Presentation

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 S¨ uli, 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. 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.

• 1. Hamilton–Jacobi–Bellman Equation

F[u] := sup

α∈Λ

[Lαu − f α] = 0 in Ω, u = 0

• n ∂Ω,

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

d

• i,j=1

aα

ij (x)uxi xj ,

bα(x) · ∇u =

d

• i=1

bα

i (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

Lαk uk+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

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

Lαk uk+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. 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 − LαuL(X,Y ) v − uδ

X

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 min

α∈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], [Hinterm¨ uller, 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

eX →0

1 eX sup L∈DF[u+e]F[u + e] − F[u] − L eY = 0. Then DF is then called a generalised differential of F on U. Semismoothness + uniform stability of linearizations: sup

L∈DF[v],v∈X

L−1L(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], [Hinterm¨ uller, Ito, Kunisch, 2002] Let 1 ≤ q < r ≤ ∞. Let G : Lr(Ω) → Lq(Ω) be defined by G : u → max(u, 0). Then G is semismooth from Lr(Ω) to Lq(Ω) with differential DF[v] the set

• f all L ∈ L∞(Ω) of the form:

L(x) =        1 if v(x) > 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

• 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.

7/28

• 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

Lαk uk+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.

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: uW 2,r (Ω;Th) =   

• K∈Th

ur

W 2,r (K)

  

1 r

. Function u ∈ W 2,r(Ω, Th) have element-wise gradient ∇hu and Hessian D2

hu.

For Λ compact and continuous coefficients, F : W 2,r(Ω, Th) → Lr(Ω) is well defined and Lipschitz continuous F[u] := sup

α∈Λ

[Lαu − f α].

9/28

• 2. Semismoothness: argmax set-valued map

For each u ∈ X, we define u =

• u, ∇hu, D2

hu

• ∈ Lr(Ω; Rm) for suitable m.

We then view the differential operator F[u] as a composition of x → u(x) with the scalar function F : Ω × Rm → R defined by F(x, v) = sup

α∈Λ

[aα(x) : M + bα(x) · p − cα(x)z − f α(x)], v = (z, p, M) Define the set-valued map Ω × Rm ∋ (x, v) → 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 . Λ Λ(x, v) Λ(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), D2

hu(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 : D2

h + 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), D2

hu(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 : D2

h + 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. Semismoothness of HJB operators

DF[u] := {Lα = aα : D2

h + bα · ∇h − cα,

α ∈ Λ[u]} Theorem [S. & S¨ uli, 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

eW 2,r (Ω;Th)→0

1 eW 2,r (Ω;Th) sup

Lα∈DF[u+e]

F[u + e] − F[u] − LαeLq(Ω) = 0. Remark: The sup implies any choice of measurable selection is sufficient.

• I. S. & E. S¨

uli, 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, ejW 2,r (Ω;Th) → 0, αj ∈ Λ[u + ej],

and ρ > 0 such that, ∀j ≥ 0: 1 ejW 2,r (Ω;Th) F[u + ej] − F[u] − Lαj ejLq(Ω) > ρ (⋆) We will find a subsequence of {ej}∞

j=0 such that (⋆) is contradicted.

14/28

• 2. Semismoothness of HJB operators: Proof
• 1. Passing to a subsequence, we have (ej, ∇hej, D2

hej) → 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| + |D2

hej|),

(⋆⋆) where the function Gj is defined by: Gj := inf

α∈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

• 2. Semismoothness of HJB operators: Proof

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. Λ Λ(x, u) Λ(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

• 2. Semismoothness of HJB operators: Proof

Also, we have a uniform bound supj≥0GjL∞(Ω) ≤ C because all coefficients are uniformly bounded on Ω × Λ. Lebesgue’s Dominated Convergence Theorem: lim

j→∞GjLs(Ω) = 0

for any 1 ≤ s < ∞. Recall (⋆⋆): |F[u + ej] − F[u] − Lαj ej| Gj (|ej| + |∇ej| + |D2

hej|),

(⋆⋆) If q < r, H¨

• lder’s inequality implies that we can find s ∈ [1, ∞) s.t.

0 < ρ ≤ 1 ejW 2,r (Ω;Th) F[u + ej] − F[u] − Lαj ejLq(Ω) ≤ GjLs(Ω) → 0.

Remarks: the norm gap appears because there are cases where GjL∞(Ω) → 0. 17/28

• 2. Semismoothness of HJB operators: Proof

Also, we have a uniform bound supj≥0GjL∞(Ω) ≤ C because all coefficients are uniformly bounded on Ω × Λ. Lebesgue’s Dominated Convergence Theorem: lim

j→∞GjLs(Ω) = 0

for any 1 ≤ s < ∞. Recall (⋆⋆): |F[u + ej] − F[u] − Lαj ej| Gj (|ej| + |∇ej| + |D2

hej|),

(⋆⋆) If q < r, H¨

• lder’s inequality implies that we can find s ∈ [1, ∞) s.t.

0 < ρ ≤ 1 ejW 2,r (Ω;Th) F[u + ej] − F[u] − Lαj ejLq(Ω) ≤ GjLs(Ω) → 0.

Remarks: the norm gap appears because there are cases where GjL∞(Ω) → 0. 17/28

• 2. Semismoothness of HJB operators

DF[u] := {Lα = aα : D2

h + bα · ∇h − cα,

α ∈ Λ[u]} Theorem [S. & S¨ uli, 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

eW 2,r (Ω;Th)→0

1 eW 2,r (Ω;Th) sup

Lα∈DF[u+e]

F[u + e] − F[u] − LαeLq(Ω) = 0. Remark: The sup implies any choice of measurable selection is sufficient.

• I. S. & E. S¨

uli, SIAM J. Numer. Anal. 2014: Discontinuous Galerkin finite element approximation of Hamilton–Jacobi–Bellman equations with Cordes coefficients. 18/28

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.

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: Lαk uk+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) ∩ H1

0(Ω).

20/28

• 3. Cordes condition

From now on, we suppose

• Ω is bounded Lipschitz domain
• aα uniformly elliptic, cα ≥ 0, uniformly over Ω×.

Policy iteration: Lαk uk+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) ∩ H1

0(Ω).

20/28

• 3. Cordes condition

From now on, we suppose

• Ω is bounded Lipschitz domain
• aα uniformly elliptic, cα ≥ 0, uniformly over Ω×.

Policy iteration: Lαk uk+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) ∩ H1

0(Ω).

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 = 0 and cα = 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 + ε in Ω, ∀ α ∈ Λ. (Cordes1) Thm: [Cordes 1956], [Maugeri, Palagachev, Softova 2000] There is C = C(ε) s.t. for any measurable α → Λ: (Lα)−1L2→H2∩H1

0 ≤ C.

21/28

• 3. Cordes condition

Well-posedness theorem [S. & S¨ uli, SINUM, 2014] Under these assumptions, there exists a unique u ∈ H2(Ω) ∩ H1

0(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) := D2uL2(Ω) ≤ ∆uL2(Ω) ∀ u ∈ H2(Ω) ∩ H1

0(Ω).

22/28

• 3. Applications to numerical scheme

Construction an hp-version discontinuous Galerkin finite element scheme in [S. & S¨ uli, 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 − uhH2(Ω;Th) hmin(p+1,s)−2 ps−5/2 uHs(Ω). Main idea: based on approximating a strongly monotone operator formulation of the PDE: A(u; v) =

• Ω

sup

α∈Λ

[γα(Lαu − f α)](∆v − λv) dx = 0 ∀v ∈ H2(Ω) ∩ H1

0(Ω);

23/28

• 3. Applications to numerical scheme

Discrete linearised problems: Ak

h(uk+1 h

, vh) =

• K∈Th

γαk f αk , LλvhK ∀ vh ∈ Vh,p, where the bilinear form Ak

h : Vh,p × Vh,p → R is defined by

Ak

h(wh, vh) :=

• K∈Th

(γαk Lαk wh, LλvhK + linear stabilization terms from Ah(·; ·). Superlinear convergence [S. & S¨ uli, SINUM, 2014] There exists R > 0, possibly depending on h and p, such that if uh − u0

hH2(Ω;Th) < R, then superlinear convergence:

lim

k→∞

uh − uk+1

h

H2(Ω;Th) uh − uk

hH2(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, π

3 ] × 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.

1/2 1/4 1/8 1/16 1/32 1/64 10−7 10−5 10−3 10−1 101 Mesh size u − uhH2(Ω;Th) p = 2 p = 3 p = 4 p = 5 1 2 3 4 5 6 7 Converged 10−12 10−8 10−4 1 Iteration number k uh − uk

hH2(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

• 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) Boundary layer adapted meshes with p-refinement: 2 ≤ pK ≤ 10, from 100 to 1320 DoFs.

26/28

• 3. Numerics: experiment 2: hp-refinement

5 6 7 8 9 10 11 10−6 10−5 10−4 10−3 10−2 10−1 100

3

√ DoF Relative Error Broken H2 norm Broken H1 norm

27/28

• 3. Numerics: experiment 2: hp-refinement

1 3 5 7 9 11 13 10−14 10−10 10−6 10−2 102 Iteration number k

uh−uk

hH2(Ω;Th)

uhH2(Ω;Th)

p = 3 p = 4 p = 5 p = 6 p = 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. S¨ uli, SIAM J. Numer. Anal. 2013. Elliptic HJB: Discontinuous Galerkin finite element approximation of Hamilton–Jacobi–Bellman equations with Cordes coefficients, I. S. & E. S¨ uli, 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. S¨ uli, 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], LλvhK + Jh(uh, vh) + 1 2  Bh(uh, vh) −

• K∈Th

Lλuh, LλvhK   .

28/28

• 3. Numerical scheme

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

• K∈Th

Fγ[uh], LλvhK + Jh(uh, vh) + 1 2  Bh(uh, vh) −

• K∈Th

Lλuh, LλvhK   .

Fγ[uh], LλvhK :=

• K

sup

α∈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

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

• K∈Th

Fγ[uh], LλvhK + Jh(uh, vh) + 1 2  Bh(uh, vh) −

• K∈Th

Lλuh, LλvhK   .

Jump penalisation with µF ≃ p2

K /hK and ηF ≃ p4 K /h3 K for F ⊂ ∂K:

Jh(uh, vh) :=

• F∈Fi,b

h

• µF ∇T uh, ∇T vhF + ηF uh, vhF
• +
• F∈Fi

h

µF ∇uh · nF , ∇vh · nF F . 28/28

• 3. Numerical scheme

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

• K∈Th

Fγ[uh], LλvhK + Jh(uh, vh) + 1 2  Bh(uh, vh) −

• K∈Th

Lλuh, LλvhK   .

Lλuh, LλvhK :=

• K

(∆uh − λuh) (∆vh − λvh) dx. 28/28

• 3. Numerical scheme

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

• K∈Th

Fγ[uh], LλvhK + Jh(uh, vh) + 1 2  Bh(uh, vh) −

• K∈Th

Lλuh, LλvhK   .

Weak enforcement of Miranda–Talenti inequality: Bh(uh, vh) :=

• K∈Th
• D2uh, D2vhK + 2λ∇uh, ∇vhK + λ2uh, vhK
• +
• F∈Fi

h

• divT ∇T{uh}, ∇vh · nF F + divT ∇T{vh}, ∇uh · nF F
• −
• F∈Fi,b

h

• ∇T{∇uh · nF }, ∇T vhF + ∇T{∇vh · nF }, ∇T uhF
• − λ
• F∈Fi,b

h

[{∇uh · nF }, vhF + {∇vh · nF }, uhF ] − λ

• F∈Fi

h

[{uh}, ∇vh · nF F + {vh}, ∇uh · nF F ] 28/28

• 3. Numerical scheme

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

• K∈Th

Fγ[uh], LλvhK + Jh(uh, vh) + 1 2  Bh(uh, vh) −

• K∈Th

Lλuh, LλvhK   . Consistency, Stability, Convergence rates [S. & S¨ uli, 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 − uhH2(Ω;Th)

hs ps−1/2 uHs(Ω) if

u ∈ Hs(Ω, Th) with s > 5/2.

28/28

• 3. Discrete Semismooth Newton method

Algorithm

• 1. Choose an initial guess u0

h ∈ Vh,p.

• 2. Given uk

h ∈ Vh,p, k ∈ N, choose αk ∈ Λ[uk h].

• 3. Solve

Ak

h(uk+1 h

, vh) =

• K∈Th

γαk f αk , LλvhK ∀ vh ∈ Vh,p, where the bilinear form Ak

h : Vh,p × Vh,p → R is defined by

Ak

h(wh, vh) :=

• K∈Th

(γαk Lαk wh, LλvhK + linear stabilization terms from Ah(·; ·). Well-posedness : the bilinear forms Ak

h(·, ·) have a uniform coercivity constant.

Superlinear convergence [S. & S¨ uli, SINUM, 2014] There exists R > 0, possibly depending on h and p, such that if uh − u0

hH2(Ω;Th) < R, then uk h → uh superlinearly.

28/28

• 3. Discrete Semismooth Newton method

Algorithm

• 1. Choose an initial guess u0

h ∈ Vh,p.

• 2. Given uk

h ∈ Vh,p, k ∈ N, choose αk ∈ Λ[uk h].

• 3. Solve

Ak

h(uk+1 h

, vh) =

• K∈Th

γαk f αk , LλvhK ∀ vh ∈ Vh,p, where the bilinear form Ak

h : Vh,p × Vh,p → R is defined by

Ak

h(wh, vh) :=

• K∈Th

(γαk Lαk wh, LλvhK + linear stabilization terms from Ah(·; ·). Well-posedness : the bilinear forms Ak

h(·, ·) have a uniform coercivity constant.

Superlinear convergence [S. & S¨ uli, SINUM, 2014] There exists R > 0, possibly depending on h and p, such that if uh − u0

hH2(Ω;Th) < R, then uk h → uh superlinearly.

28/28