Reduced Basis Methods for (some particular) HJB equations · [2]Yong, Zhou: Stochastic controls: Hamiltonian systems and HJB equations, 1999 page 6/27 RBM for HJB |RICAM 2016 Karsten

Karsten Urban

Ulm University (Germany)Institute for Numerical Mathematics

Reduced Basis Methodsfor (some particular)HJB equations

page 1/27 RBM for HJB | RICAM 2016 | Karsten Urban | Acknowledgements

Acknowledgements

I joint work withI Rudiger Kiesel (Duisburg-Essen)I Silke Glas, Sebastian Steck (Ulm)

I Funding:I Deutsche Forschungsgemeinschaft (DFG: GrK1100, Ur-63/9, SPP1324)I Federal Ministry of Economy (BMWT)

page 2/27 RBM for HJB | RICAM 2016 | Karsten Urban | Acknowledgements

Outline

1 “Particular” HJB: The EU-ETS

2 (A very short) Introduction to RBM

3 RBM for the EU-ETS-HJB

4 Conclusions and outlook

page 3/27 RBM for HJB | RICAM 2016 | Karsten Urban |“Particular” HJB: The EU-ETS






EU-ETS: European Union Emission Trading System

I anthropogenic global warming

I Kyoto protocol: limit of CO2-emissions

I according amount of emission permits are issued

I permits are traded at the exchange: EU-ETS

I penalty for emissions not covered by permits

I goal here: public control of EU-ETS: abate 5% of emissions

















page 5/27 RBM for HJB | RICAM 2016 | Karsten Urban | Model of the Emission Trading System

Modeling EU-ETS

I trading periods: [0,T ]

I equlibirium ≡ sum of costs of all market participants is minimal[1]

I state Yτ ∈ Rd , τ ∈ [0,T ]: amount of uncovered emissions (d : # companies)

I control πτ ∈ Rd : additional abatement compared to business as usual

I optimal abatement strategy π = (πτ )τ∈[0,T ]:should minimize the expected abatement costs (cost functional)

I f π: running abatement cost using strategy πI h: penalty at the end of the trading period

I stochastic model for Yτ :

dYτ = bπ(τ,Yτ )dτ + σπ(τ,Yτ )dWτ ,

I Wτ : a d-dimensional Wiener processI bπ, σπ: drift and volatility coefficients, bπ, σπ(σπ)T linear in π.

[1]Camora, Fehr, Hinz: Optimal Stochastic Control and Carbon Price Formation, 2009


Modeling EU-ETS












Modeling EU-ETS












Modeling EU-ETS






J(π) := E

[∫ T

0

f π (τ,Yτ ) dτ + h(YT )

]I f π: running abatement cost using strategy πI h: penalty at the end of the trading period






Modeling EU-ETS






J(π) := E

[∫ T

0

f π (τ,Yτ ) dτ + h(YT )



dYτ = bπ(τ,Yτ )dτ + σπ(τ,Yτ )dWτ , τ ∈ (0,T ],Y0 = y0




Modeling EU-ETS






J(t, x ; γ) := E

[∫ T

t

f γ (τ,Yτ ) dτ + h(YT )



dYτ = bπ(τ,Yτ )dτ + σπ(τ,Yτ )dWτ , τ ∈ (t,T ],Yt = x




(Parameterized) Hamilton Jacobi Bellman Equation 1/3

I value function (x ∈ Rd)

u(t, x) = infγ∈Γ

J(t, x ; γ) ∀t ∈ [0,T ), u(T , x) = h(x)

I Γ ⊂ L∞((0,T )× Rd ;Rd): set of admissible controls

I HJB[2]

∂tu(t, x) + supγ∈Γ

1

2tr(σγ(σγ)T ∇2u(t, x)) + bγ · ∇u(t, x)− f γ(t, x)

= 0

I parameters µ ∈ D ⊂ RP

e.g. regulatory constraints, market values, etc.f γ(µ), bγ(µ), σγ(µ), J(µ; t, x ; γ) u(µ) = u(µ; t, x)

I Goal: find “optimal” parameters (also in realtime)

[2]Yong, Zhou: Stochastic controls: Hamiltonian systems and HJB equations, 1999





J(t, x ; γ) ∀t ∈ [0,T ), u(T , x) = h(x)


I HJB[2]


1


= 0









J(t, x ; γ) ∀t ∈ [0,T ), u(T , x) = h(x)


I HJB[2]


1


= 0









J(t, x ; γ) ∀t ∈ [0,T ), u(T , x) = h(x)


I HJB[2]


1


= 0







I parameterized coefficients:

u 7→ Aγ(µ; u) := −aγ(µ) ∆u + bγ(µ) · ∇u + cγ(µ)u, µ ∈ D.

I parameterized Hamilton-type operator

H(µ; u) := supγ∈ΓAγ(µ; u)− f γ(µ)

I P-HJB

∂tu +H(µ; u) = 0, in ΩT , (1a)

∂

∂nu = ψ, on ∂ΩT = (0,T )× ∂Ω, (1b)

u(T ) = uT , on Ω, (1c)







I P-HJB

∂tu +H(µ; u) = 0, in ΩT , (1a)

∂

∂nu = ψ, on ∂ΩT = (0,T )× ∂Ω, (1b)

u(T ) = uT , on Ω, (1c)







I P-HJB

∂tu +H(µ; u) = 0, in ΩT , (1a)

∂

∂nu = ψ, on ∂ΩT = (0,T )× ∂Ω, (1b)

u(T ) = uT , on Ω, (1c)



I parameterized (linear) space-time differential operatorLγ(µ; u) := ∂tu + Aγ(µ; u)

I P-HJB revisitedsupγ∈ΓLγ(µ; u)− gγ(µ) = 0 on ΩT

u∗ = u∗(µ) ∈ U (solution space)

I optimal (parameter-dependent) control

Γ 3 γ∗(µ) = arg supγ∈ΓLγ(µ; u∗)− gγ(µ)

I couple x∗ = x∗(µ) = (γ∗(µ), u∗(µ)) ∈ Γ× U

I Goal: determine x∗(µ) for many values of µ in realtime




























page 9/27 RBM for HJB | RICAM 2016 | Karsten Urban | (A very short) Introduction to RBM






(A very short) Introduction to RBM

I 1st application: Thermal Fin(∼ 2000, Patera, Maday, et al.)

I variable length of ribs(∼ parameter)

I goal: optimizationI w.r.t. desired heat conductionI for fixed volumeI with size constraints

Picture courtesy of A.T. Patera, D.V. Rovas and L. Machiels, 2000

Mathematical Model: Parameterized PDE (PPDE) – linear

I µ ∈ D ⊂ RP : parameter; Ω ⊂ Rd : (spatial) domain for the pde

I X := X (Ω), Y := Y (Ω): function spaces (Sobolev space with BCs)

I A : X × Y ×D: parametric form (often bilinear in 1st and 2nd argument)

I seek u(µ) ∈ X : A(u(µ),w ;µ) = 0∀w ∈ Y (PPDE for many µ)

I Output: s(µ) := `(u(µ)), ` : X → R












I Output: s(µ) := `(u(µ)), ` : X → R












I Output: s(µ) := `(u(µ)), ` : X → R












I Output: s(µ) := `(u(µ)), ` : X → R












I Output: s(µ) := `(u(µ)), ` : X → R












I Output: s(µ) := `(u(µ)), ` : X → R



PPDEI u(µ) ∈ X : A(u(µ),w ;µ) = 0∀w ∈ Y (PPDE)

I Output: s(µ) := `(u(µ))

Common situation / observation / assumption

I PPDE must be solved for many parameters µ: many-query (like optimization)

I u(µ) (often) depends smoothly on µ

I detailed discretization XN , YN , dim(XN ) = dim(YN ) = N large;uN ≈ u indistinguishable (‘truth’)



PPDEI u(µ) ∈ X : A(u(µ),w ;µ) = 0∀w ∈ Y (PPDE)

I Output: s(µ) := `(u(µ))

Common situation / observation / assumption

I PPDE must be solved for many parameters µ: many-query (like optimization)

I u(µ) (often) depends smoothly on µ

I detailed discretization XN , YN , dim(XN ) = dim(YN ) = N large;uN ≈ u indistinguishable (‘truth’)


Introduction to RBM

‘Good’ RBM situation / Idea for RBM

I offline/online-decomposition possible:

I offline: pre-compute “snapshots” uN (µi ) = u(µi ), i = 1, . . . ,N for certain µi

(by e.g. FEM) – choice by error estimateI online: for new µ 6∈ µi : i = 1, . . . ,N compute (Petrov-)Galerkin projection

uN(µ) ≈ u(µ) onto XN := spanuN (µi ) : i = 1, . . . ,N (RB approx.)

I in some cases (or you have to assume/numerically check it):I ‖uN (µ)− uN(µ)‖ . e−αN (rapid decay) N N sufficesI online complexity independent of N (“online efficient”)

uN (µnew ) =?uN (µ)

uN (µ3)

uN (µ2)

uN (µ1)

I Form a reduced basis out of Nsolutions uN (µ1), . . . , uN (µN)offline


Introduction to RBM


I offline/online-decomposition possible:I offline: pre-compute “snapshots” uN (µi ) = u(µi ), i = 1, . . . ,N for certain µi

(by e.g. FEM) – choice by error estimate

I online: for new µ 6∈ µi : i = 1, . . . ,N compute (Petrov-)Galerkin projectionuN(µ) ≈ u(µ) onto XN := spanuN (µi ) : i = 1, . . . ,N (RB approx.)



uN (µ3)

uN (µ2)

uN (µ1)



Introduction to RBM







uN (µ3)

uN (µ2)

uN (µ1)



Introduction to RBM







uN (µ3)

uN (µ2)

uN (µ1)



When to use RBM ... and when not

Consequences

I RB-approximation can only be as good as truth approximation

I Benchmark: Kolmogorov N-width (1936) (for S ⊂ X )

dN(S) := infXN⊂XN ; dim(XN )=N

supuN∈S

supuN∈XN

‖uN − uN‖

Buffa, Maday, Patera,Haasdonk, Cohen, Dahmen, DeVore, ...

I complexity-offset by offline phase RBM (only) reasonable, ifI many evaluations required

(‘many-query context’)

I very fast or limited evaluationsrequired (‘realtime context’)



Consequences




supuN∈S

supuN∈XN

‖uN − uN‖


I complexity-offset by offline phase RBM (only) reasonable, ifI many evaluations required

(‘many-query context’)




Consequences




supuN∈S

supuN∈XN

‖uN − uN‖


I complexity-offset by offline phase RBM (only) reasonable, if

I many evaluations required(‘many-query context’)


total

complexity

offline

complexity

RBM

direct solution

for each µnew

# new parameters µnew



Consequences




supuN∈S

supuN∈XN

‖uN − uN‖


I complexity-offset by offline phase RBM (only) reasonable, if

I many evaluations required(‘many-query context’)


Use the right tool!


RBM – A posteriori error analysis / how to compute an RB

Well-posedness assumption for PPDE A(u(µ),w ;µ) = 0∀w ∈ Y

I βLB := infµ∈D

infv∈X

supw∈YA(v ,w ;µ) > 0

I γUB := supµ∈D

supv∈X

supw∈Y

A(v ,w ;µ)

‖v‖X ‖w‖Y<∞

I e.g.: A(w , v ;µ) = b(w , v ;µ)− 〈f (µ), v〉I Exact error: eN(µ) := u(µ)− uN(µ); residual: rN(v ;µ) := A(uN(µ), v ;µ)




I βLB := infµ∈D

infv∈X


I γUB := supµ∈D

supv∈X

supw∈Y

A(v ,w ;µ)

‖v‖X ‖w‖Y<∞

I e.g.: A(w , v ;µ) = b(w , v ;µ)− 〈f (µ), v〉

I Exact error: eN(µ) := u(µ)− uN(µ); residual: rN(v ;µ) := A(uN(µ), v ;µ)




I βLB := infµ∈D

infv∈X


I γUB := supµ∈D

supv∈X

supw∈Y

A(v ,w ;µ)

‖v‖X ‖w‖Y<∞

I e.g.: A(w , v ;µ) = b(w , v ;µ)− 〈f (µ), v〉I Exact error: eN(µ) := u(µ)− uN(µ); residual: rN(v ;µ) := A(uN(µ), v ;µ)


RBM – A posteriori error analysis

I e.g.: A(v ,w ;µ) = b(v ,w ;µ)− 〈f (µ), v〉;I Exact error: eN(µ) := u(µ)− uN(µ); residual: rN(w ;µ) := A(uN(µ), v ;µ)

Thus:

I inf-sup and definition of dual norm:

‖eN(µ)‖X ≤(βLB)−1

supw∈Y

A(eN(µ),w ;µ)

‖w‖Y

=(βLB)−1

supw∈Y

A(uN(µ),w ;µ)

‖w‖Y=

(βLB)−1 ‖rN(µ)‖Y ′

‖eN(µ)‖X ≤(βLB)−1 ‖rN(µ)‖Y ′ =: ∆N(µ)

⇒ ∆N(µ) should be efficiently computable and efficient!

Compute supw∈Y

rN(w ;µ)

‖w‖Y(wavelets; Ali, Steih, U., 2016) or use sup

wN∈YN

rN(wN ;µ)

‖wN ‖YN




Thus:



supw∈Y

A(eN(µ),w ;µ)

‖w‖Y

=(βLB)−1

supw∈Y

A(uN(µ),w ;µ)

‖w‖Y

=(βLB)−1 ‖rN(µ)‖Y ′

‖eN(µ)‖X ≤(βLB)−1 ‖rN(µ)‖Y ′ =: ∆N(µ)


Compute supw∈Y

rN(w ;µ)


wN∈YN

rN(wN ;µ)

‖wN ‖YN




Thus:



supw∈Y

A(eN(µ),w ;µ)

‖w‖Y

=(βLB)−1

supw∈Y

A(uN(µ),w ;µ)

‖w‖Y=

(βLB)−1 ‖rN(µ)‖Y ′

‖eN(µ)‖X ≤(βLB)−1 ‖rN(µ)‖Y ′ =: ∆N(µ)


Compute supw∈Y

rN(w ;µ)


wN∈YN

rN(wN ;µ)

‖wN ‖YN




Thus:



supw∈Y

A(eN(µ),w ;µ)

‖w‖Y

=(βLB)−1

supw∈Y

A(uN(µ),w ;µ)

‖w‖Y=

(βLB)−1 ‖rN(µ)‖Y ′

‖eN(µ)‖X ≤(βLB)−1 ‖rN(µ)‖Y ′ =: ∆N(µ)


Compute supw∈Y

rN(w ;µ)


wN∈YN

rN(wN ;µ)

‖wN ‖YN


RB determination (offline)

I use error estimator ∆N(µ)

I maximize ∆N(µ) over finite µ ∈ Dtest

(Greedy: Patera, Maday et. al.;Nonlinear optimization: Ghattas et. al., Volkwein, U., Zeeb)

parameters, snapshots, reduced spaces XN , YN

I precompute all µ-independent terms offline (EIM: Maday et. al.) N -independent online stage

I online certification via ∆N(µ)

⇒ error estimate ∆N(µ) is crucial:

I sharpness ∼ N

I if residual based, we need to compute the inf-sup-constant(e.g. SCM, Maday, Patera et. al.)

I instationary: POD-Greedy or space-time (Mayerhofer, Glas, U.)










I sharpness ∼ N












I sharpness ∼ N












I sharpness ∼ N












I sharpness ∼ N












I sharpness ∼ N



page 17/27 RBM for HJB | RICAM 2016 | Karsten Urban | RBM for the EU-ETS-HJB






RBM for HJB — detailed (“truth”) discretization 1/2

I Z := zi := (ti , xi ) ∈ ΩT = (0,T )× Ω : i = 1, . . . ,N, N 1:points in time-space

I u∗(µ) = (u∗i (µ))i=1,...,N ∈ RN ≈ u∗(µ) on Z optimization problems for each zi ∈ Z; result: γ∗i (µ) ∈ Rd

I let LγZ ∈ RN×N , gγ

Z ∈ RN : some discretization of Lγ , gγ on Z(e.g. collocation, nodal interpolation, finite differences, ...)

I determine u∗(µ) ∈ RN by solving

Find u(µ) ∈ RN : maxγi∈Rd

Lγi

Z (µ) u(µ)− gγi

Z (µ) = 0, ∀1 ≤ i ≤ N .

I find optimal control by:

Rd 3 γ∗i (µ) := arg maxγi∈Rd

Lγi

Z (µ)u(µ)− gγi

Z (µ), 1 ≤ i ≤ N .




I u∗(µ) = (u∗i (µ))i=1,...,N ∈ RN ≈ u∗(µ) on Z

optimization problems for each zi ∈ Z; result: γ∗i (µ) ∈ Rd





Lγi


Z (µ) = 0, ∀1 ≤ i ≤ N .



Lγi

Z (µ)u(µ)− gγi

Z (µ), 1 ≤ i ≤ N .









Lγi


Z (µ) = 0, ∀1 ≤ i ≤ N .



Lγi

Z (µ)u(µ)− gγi

Z (µ), 1 ≤ i ≤ N .









Lγi


Z (µ) = 0, ∀1 ≤ i ≤ N .



Lγi

Z (µ)u(µ)− gγi

Z (µ), 1 ≤ i ≤ N .









Lγi


Z (µ) = 0, ∀1 ≤ i ≤ N .



Lγi

Z (µ)u(µ)− gγi

Z (µ), 1 ≤ i ≤ N .









Lγi


Z (µ) = 0, ∀1 ≤ i ≤ N .



Lγi

Z (µ)u(µ)− gγi

Z (µ), 1 ≤ i ≤ N .


RB - detailed (“truth”) discretization 2/2

I write this as one large system:

Lγ∗(µ) u∗(µ)− gγ∗(µ) = 0 in RN

∂γ [Lγ∗(µ)u∗(µ)− gγ∗(µ)](δ) = 0, ∀δ ∈ RN×d

I or, as one system

0 = G(x;µ) :=

∂γ [Lγ(µ) u(µ)− gγ(µ)]

Lγ(µ)u(µ)− gγ(µ)

=:

G1(x;µ)

G2(x;µ)

I example: Howard’s algorithm

RN×d 3 γ(k+1)(µ) = arg maxγ∈RN×d

Lγ(µ) u(k)(µ)− gγ(µ)

find u(k+1)(µ) ∈ RN : Lγ(k+1)

(µ) u(k+1)(µ) = gγ(k+1)

(µ)

I (Quasi-)Newton: Hx(x;µ) := x− (DG(x;µ))−1G(x;µ)





∂γ [Lγ∗(µ)u∗(µ)− gγ∗(µ)](δ) = 0, ∀δ ∈ RN×d

I or, as one system

0 = G(x;µ) :=



=:

G1(x;µ)

G2(x;µ)





(µ) u(k+1)(µ) = gγ(k+1)

(µ)






∂γ [Lγ∗(µ)u∗(µ)− gγ∗(µ)](δ) = 0, ∀δ ∈ RN×d

I or, as one system

0 = G(x;µ) :=



=:

G1(x;µ)

G2(x;µ)





(µ) u(k+1)(µ) = gγ(k+1)

(µ)






∂γ [Lγ∗(µ)u∗(µ)− gγ∗(µ)](δ) = 0, ∀δ ∈ RN×d

I or, as one system

0 = G(x;µ) :=



=:

G1(x;µ)

G2(x;µ)





(µ) u(k+1)(µ) = gγ(k+1)

(µ)



Well-posedness / Error analysis

I Hx(x;µ) := x− (DG(x;µ))−1G(x;µ)

I Assume: ∃% > 0: ‖DG(x1;µ)− DG(x2;µ)‖L(X,Y) ≤ %‖x1 − x2‖X ∀µI βx(µ) := ‖(DG(x;µ))−1‖−1

L(Y,X) (inf-sup)

Lemma (Steck, U., 2015)

Suitable assumptions...Then, the mapping Hx(·;µ) : Bγ(µ)(x)→ Bγ(γ)(x) is a self-map

∀x ∈ X : τ(x;µ) :=2%

βx(µ)2‖G(x;µ)‖Y ≤ 1(indicator)

∀γ ∈ [γmin, γmax] :=βx(µ)

%

[1−

√1− τ(x;µ), 1 +

√1− τ(x;µ)

]Moreover,

‖x∗(µ)− x(µ)‖X ≤βx(µ)

%(1−

√1− τ(x;µ))



I Hx(x;µ) := x− (DG(x;µ))−1G(x;µ)

I Assume: ∃% > 0: ‖DG(x1;µ)− DG(x2;µ)‖L(X,Y) ≤ %‖x1 − x2‖X ∀µ

I βx(µ) := ‖(DG(x;µ))−1‖−1L(Y,X) (inf-sup)



∀x ∈ X : τ(x;µ) :=2%



%

[1−

√1− τ(x;µ), 1 +

√1− τ(x;µ)

]Moreover,

‖x∗(µ)− x(µ)‖X ≤βx(µ)

%(1−

√1− τ(x;µ))



I Hx(x;µ) := x− (DG(x;µ))−1G(x;µ)


L(Y,X) (inf-sup)



∀x ∈ X : τ(x;µ) :=2%



%

[1−

√1− τ(x;µ), 1 +

√1− τ(x;µ)

]Moreover,

‖x∗(µ)− x(µ)‖X ≤βx(µ)

%(1−

√1− τ(x;µ))



I Hx(x;µ) := x− (DG(x;µ))−1G(x;µ)


L(Y,X) (inf-sup)



∀x ∈ X : τ(x;µ) :=2%



%

[1−

√1− τ(x;µ), 1 +

√1− τ(x;µ)

]

Moreover,

‖x∗(µ)− x(µ)‖X ≤βx(µ)

%(1−

√1− τ(x;µ))



I Hx(x;µ) := x− (DG(x;µ))−1G(x;µ)


L(Y,X) (inf-sup)



∀x ∈ X : τ(x;µ) :=2%



%

[1−

√1− τ(x;µ), 1 +

√1− τ(x;µ)

]Moreover,

‖x∗(µ)− x(µ)‖X ≤βx(µ)

%(1−

√1− τ(x;µ))


Online computation of “inf-sup” Constant βN(µ)

I recall: we need (online efficient!)

βN(µ) := βx∗N

(µ)(µ) := ‖((DG(µ))(x∗N(µ)))−1‖−1L(Y,X) (2)

= infx∈X

‖(DG(µ)(x∗N(µ)))(x)‖Y‖x‖X

.

I lower bound: fix “anchor point” µ and prove

βN(µ) > βN(µ) · βµN(µ) =: βofflineLB (µ) · βonline

LB (µ),

Greedy selection of anchor points

1: choose µ1 ∈ D arbitrarily, N ← 1, SR := µ1, compute βofflineLB (µ1)

2: while minµ∈Ξanchor

train

βonlineLB (µ) 6 1

2 do

3: µN+1 ← arg minµ∈Ξanchor

train

βonlineLB (µ)

4: SR ← SR ∪ µN+1, compute βofflineLB (µN+1)

5: N ← N + 16: end while




βN(µ) := βx∗N

(µ)(µ) := ‖((DG(µ))(x∗N(µ)))−1‖−1L(Y,X) (2)

= infx∈X


.



LB (µ),




train

βonlineLB (µ) 6 1

2 do


train

βonlineLB (µ)






βN(µ) := βx∗N

(µ)(µ) := ‖((DG(µ))(x∗N(µ)))−1‖−1L(Y,X) (2)

= infx∈X


.



LB (µ),




train

βonlineLB (µ) 6 1

2 do


train

βonlineLB (µ)






βN(µ) := βx∗N

(µ)(µ) := ‖((DG(µ))(x∗N(µ)))−1‖−1L(Y,X) (2)

= infx∈X


.



LB (µ),




train

βonlineLB (µ) 6 1

2 do


train

βonlineLB (µ)






βN(µ) := βx∗N

(µ)(µ) := ‖((DG(µ))(x∗N(µ)))−1‖−1L(Y,X) (2)

= infx∈X


.



LB (µ),




train

βonlineLB (µ) 6 1

2 do


train

βonlineLB (µ)




Numerical Example 1/4: Inf-Sup Lower Bound

I diffusion of emissions σµ(t, x , α) := 1I drift of emissions bµ(t, x , α) := µ− αI abatement costs: f µ(t, x , α) := 0.5 · α2 · e0,05(t−T )

I penalty: hµ(x) := x+

0 20 40 60 80 100

−0.6

−0.4

−0.2

0

0.2

N = 10 µ

log

(βL

BN

(µ))





0 20 40 60 80 100

−0.6

−0.4

−0.2

0

0.2

N = 10 µ

log

(βL

BN

(µ))





0 20 40 60 80 100

−0.6

−0.4

−0.2

0

0.2

N = 10 µ

log

(βL

BN

(µ))





0 20 40 60 80 100

−0.6

−0.4

−0.2

0

0.2

N = 10 µ

log

(βL

BN

(µ))





0 20 40 60 80 100

−0.6

−0.4

−0.2

0

0.2

N = 10 µ

log

(βL

BN

(µ))


Numerical Example 2/4: Error vs. indicator/estimator (∼ N)

0 2 4 6 8 10 12 14

10−4

10−3

10−2

10−1

100

101

102

103

104

N

indicatorerror boundtrue errorresidual


Numerical Example 3/4: Error vs. indicator/estimator (∼ µ)

0 20 40 60 80 100

10−12

10−10

10−8

10−6

10−4

10−2

100

µ

indicatorerror boundtrue errorresidual


Numerical Example 4/4: Effectivity

0 20 40 60 80 100

2

3

4

5

6

7

8

µ

page 26/27 RBM for HJB | RICAM 2016 | Karsten Urban | Conclusions and outlook






Conclusions and outlook

Reduced Basis Methods for HJB ...

I ... are useful in parametric cases

I ... in multi-query and/or realtime contexts

I ... need a special error analysis(online efficiency)

I ... can be coupled with space-time methods(U., Patera; Glas, Mayerhofer, U.)

I ... can be coupled with adaptive schemes(Ali, Steih, U.)

I Extensions / ongoing work:

I Intraday power markets (Glas, Kiesel)

I use other error bounds (no inf-sup) (Hain, Radic)

I other (more challenging) versions of HJB

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I Extensions / ongoing work:I Intraday power markets (Glas, Kiesel)



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Documents

Reduced Basis Methods for (some particular) HJB equations · [2]Yong, Zhou: Stochastic controls: Hamiltonian systems and HJB equations, 1999 page 6/27 RBM for HJB |RICAM 2016 Karsten