Aachen Institute for Advanced Study in Computational Engineering Science
Preprint: AICES-2011/09-1
06/September/2011
Towards an Adaptive One Shot Method for Optimal
Control Problems
L. Kaland, J. C. De Los Reyes and N. R. Gauger
Financial support from the Deutsche Forschungsgemeinschaft (German Research Association) through
grant GSC 111 is gratefully acknowledged.
©L. Kaland, J. C. De Los Reyes and N. R. Gauger 2011. All rights reserved
List of AICES technical reports: http://www.aices.rwth-aachen.de/preprints
Towards an Adaptive One Shot Method for
Optimal Control Problems
Lena Kaland
∗
Juan Carlos De Los Reyes
Nicolas R. Gauger
†
‡
Abstract
We consider two model problems for the One Shot approach in Hilbert
spaces, the stationary solid fuel ignition model and the stationary viscous
Burgers equation. For these two problems we present a convergence analysis for the One Shot method, where we evaluate one step in the state,
adjoint and design equation simultaneously. We experience mesh independence in the numerics, which motivates an adaptive One Shot approach.
1
Introduction
We study optimal control problems of the following type
min f (y, u)
s.t. e(y, u) = 0,
(P)
where e(y, u) displays a partial differential equation with a state vector y ∈ Y
and the control u ∈ U . Y and U are Hilbert spaces and f : Y × U → is the
objective function. These kind of problems are studied in many textbooks (see
e.g. [14] ) and can be solved numerically in numerous ways. One essential difference in treating the optimization problem is to first discretize the problem and
then use a finite dimensional optimization method, or to apply an optimization
method in a Hilbert space setting and finally discretize. For both approaches
the optimality system plays an important role. By the assumption that the discretized state and adjoint equation can be solved iteratively, Ta’asan suggested
in a multigrid framework for elliptic PDEs in [13] to solve the corresponding
optimality system simultaneously, i.e. one step in the state equation, one step
in the adjoint equation and finally one control update. This so-called One Shot
Method was employed in many applications, especially aerodynamics, see e.g.
[11] and [12] and references therein. In [5] and [6] a convergence proof for the
finite dimensional setting and a suitable preconditioner for the control update
is given.
To further improve the One Shot approach the question of adaptivity and therefore the analysis of the method in a Hilbert space setting arises. We present
a convergence result in Hilbert spaces following the ideas of the proof in [5].
∗ RWTH
Aachen, [email protected]
Quito, [email protected]
‡ RWTH Aachen, [email protected]
† EPN
1
In this context we study two distributed optimal control problems. On the one
hand we consider a tracking type functional and the following partial differential
equation
∆y + δ ey + u = 0
y=0
in Ω,
on Γ.
This equation serves for instance as a model for combustion and is often called
the steady state solid fuel ignition model. Additionally, due to the strong nonlinearity, it presents a good example for an optimal control problem with nonlinear
partial differential equations. The main difficulty is to handle this nonlinearity.
In fact just by changing the sign and working with −ey instead, one could easily
apply the standard theory for semilinear elliptic equations ([14]).
As a second example and simple model for convection-diffusion problems we
consider the stationary viscous Burgers equation given by
−νy !! + yy ! = u
y(0) = 0, y(1) = 0
in (0, 1),
in one space dimension. The instationary analogue was introduced by Burgers
in [1]. The objective function is again chosen to be a tracking type functional.
In two space dimensions the Burgers equation is given by
−ν∆y + (y · ∇)y = u,
y|Γ = 0.
The paper is structured as follows. In section 2 we introduce the setting of both
model problems and answer the question about the existence of an optimal
solution. In section 3 we state the general form of the One Shot Method. We
continue in section 4 with the convergence analysis for the two model problems.
Here we adopt the approach in [5] and show that the iterates of the One Shot
method serve as a descent direction for a doubly augmented Lagrange function.
Finally in section 5 we show numerical results and observe mesh independence.
We extend the One Shot method algorithmically by an additional adaptive step.
According to an error indicator the mesh is refined or coarsened and the solution
is interpolated to the new mesh.
Notation. Throughout this paper the norm & · & without any indices denotes
the operator norm. Additionally we will not differ between an operator and
its Riesz representative. g ∗ ∈ L(Y ∗ , X ∗ ) denotes the dual of some operator
g ∈ L(X, Y ) with given Banach spaces X and Y . If not stated otherwise C will
be a generic constant.
2
2.1
Problem statement
The solid fuel ignition model
We study (P) with the tracking type cost functional
!
!
µ
1
|y − zd |2 dx +
|u|2 dx
f (y, u) :=
2 Ω
2 Ω
2
(1)
and the partial differential equation
∆y + δ ey + u = 0
y=0
in Ω,
on Γ.
(2)
Here Ω is an open, bounded domain with Lipschitz boundary Γ := ∂Ω, y ∈ Y =
H01 = H01 (Ω), u ∈ U = L2 = L2 (Ω). Moreover, we assume that zd ∈ L2 (Ω) and
δ, µ ∈ + . Finally the optimal control problem reads
min f (y, u)
s.t. (y, u) satisfies (2).
(P1)
The existence and uniqueness of the uncontrolled problem of (2) is a complicated
task and depends on the domain Ω and the positive constant δ. More precisely
the problem may admit multiple solutions, exactly one solution or no solution
at all. For some values of δ, for which the uncontrolled case did not have a
solution, the controlled equation (2) may admit control-state solution pairs. We
can assume the existence for adequate Ω and δ (see [2]), but because of the
nonlinearity we need to state an appropriate concept of a solution.
Definition 1. For u ∈ L2 (Ω) a function y ∈ H01 (Ω) is called a solution of (2)
if ey ∈ L1 (Ω) and it satisfies (2) in the distributional sense.
The optimal control of the solid fuel ignition model was studied by several authors. To handle the exponential term in [8] the term ∇y is included in the cost
functional. This technique is also applied in [2], here the authors alternatively
include the exponential term. The optimal control of the instationary solid fuel
ignition model is handled e.g. in [9], [10] or [8].
In this paper we bound the nonlinearity explicitely to get existence of an solution
for problem (P1).
Theorem 1. If
ey(x) ≤ M
holds for a.e. x ∈ Ω, y(x) ∈
(3)
∗
∗
, then (P) admits a solution (y , u ) ∈
H01
× L2 .
Proof. There exists (y, u) ∈ H01 × L2 satisfying (2), i.e. the feasible set is
nonempty. Since the cost functional is bounded from below, there exists a
minimizing sequence (yn , un ) ∈ H01 × L2 with
lim f (yn , un ) = inf f (y, u).
n→∞
u
Since f (y, u) → ∞ for &u&L2 → ∞, {un } is bounded and also relatively weakly
sequentially compact because L2 is reflexive. Therefore there exists a subsequence that converges weakly to a limit u∗ in L2 . Now we need the convergence
of a subsequence of {yn } in H01 . From (3) we get that {eyn } is bounded in
L2 and a subsequence converges weakly to an element w∗ . In the following we
denote the subsequence of {(yn , un , eyn )} with the same symbol. Since (yn , un )
is a solution to (2) for every n, yn solves the linear boundary value problem
−∆yn = Rn
in Ω
yn = 0
on Γ
with Rn = eyn + un . {Rn } converges weakly to w∗ + u∗ in L2 . Additionally
Rn )→ yn is a linear and continuous mapping from L2 to H01 and therefore
3
weakly sequentially continuous. That is why we obtain the weak convergence of
{yn } to y ∗ in H01 . At all we obtain yn $ y ∗ in H01 , un $ u∗ in L2 and eyn $ w∗
in L2 .
We still need to show that (y ∗ , u∗ ) is a solution of (2). H01 is compactly embedded in L2 and it follows yn → y ∗ in L2 . Using the assumption (3), the
nonlinearity is Lipschitz for y(x) ∈ (−∞, ln M ], i.e.
∗
&eyn − ey &L2 ≤ M &yn − y ∗ &L2
∗
and therefore eyn → ey in L2 . For every v ∈ H01 the expression
(yn , v)H01 − δ(eyn , v)L2 = (un , v)L2
converges to
∗
(y ∗ , v)H01 − δ(ey , v)L2 = (u∗ , v)L2 .
Because f is weakly lower semicontinuous it follows that (y ∗ , u∗ ) is optimal.
We can reformulate the state equation (2) as a fixed point equation
y = G(y, u) := (−∆)−1 (δ ey + u)
(4)
and assume because of (3) a contraction factor of ρ, i.e.
&Gy (y, u)& ≤ ρ < 1.
(5)
We need this strong assumption for the subsequent analysis and are going to
verify it in section 5.
Formally we can derive the optimality system for problem (P1) as
∆y + δ ey + u = 0,
∆p + δ pey + y − zd = 0,
y|Γ = 0
p|Γ = 0
(6a)
(6b)
µu + p = 0
a.e.in Ω
(6c)
Note that due to (5) there exists a solution of the linearized fixed point equation
x = Gy (y, u)x.
Consequently there exists an unique adjoint p that solves (6b).
2.2
The stationary viscous Burgers equation
As a second model problem we consider again the tracking type cost functional
(1) with the stationary Burgers equation. In the two-dimensional case it is given
by
−ν∆y + (y · ∇)y = u
y=0
in Ω,
on Γ.
(7)
In one space dimension it reduces to:
−νy !! + yy ! = u
in Ω,
y=0
on Γ.
4
(8)
Here ν > 0 denotes the viscosity parameter and u ∈ L2 (Ω). Originally Burgers
introduced the equation
yt + yy ! = νy !!
(9)
in [1] as the simplest model for fluid flow and ever since it was studied analytically and numerically (see e.g. [7]). The optimal control problem reads
min f (y, u)
s.t. (y, u) satisfies (7) resp. (8).
(P2)
In [15] the one-dimensional stationary and instationary case was considered in
the context of optimal control. Therefore we refer to [15] for the aspects of existence and uniqueness of the Burgers equation and the existence of an optimal
solution of (P2). Additionally in [4] the optimal control of (8) with pointwise
control constraints was studied.
In the following we set without loss of generality the viscosity parameter ν = 1.
Formally we find an adjoint p ∈ H01 (Ω) to the optimal solution (y, u) ∈ H01 (Ω) ×
L2 (Ω), so that the following Lemma holds.
Lemma 1. The optimality system for (P2) is given in the two dimensional case
by
−∆y + (y · ∇)y − u = 0,
T
−∆p − (y · ∇)p − div(y)p + (∇y) p − y + zd = 0,
µu + p = 0
y|Γ = 0
(10a)
p|Γ = 0
(10b)
a.e. in Ω.
(10c)
Proof. See Remark 4.
In one dimension this reduces to the system:
−y !! + yy ! − u = 0,
!!
!
−p − yp − y + zd = 0,
µu + p = 0
y|Γ = 0
p|Γ = 0
a.e. in Ω.
In the subsequent analysis we deal with fixed point operators. To obtain a
numerical stable operator we choose
G(y, u) := (−∆ + y · ∇)−1 (u).
(12)
Several times we will need the derivatives of G(y, u). We obtain the derivative
with respect to u directly by
Gu (y, u)h := (−∆ + y · ∇)−1 (h).
(13)
For the derivative with respect to y we need the following lemma
Lemma 2. Let f be the inverse operator f : x )→ x−1 . Then the derivative of
f at x0 in direction h is given by
−1
f ! (x0 )h = x−1
0 (−h)x0 .
5
Proof. By definition and with a Neumann series we obtain
f (x0 + th) − f (x0 ) = (x0 + th)−1 − x−1
0
−1
= [(I + thx−1
− x−1
0 )x0 ]
0
∞
"
−1
n
= x−1
(−thx−1
0
0 ) − x0
=
n=0
−1
−tx0 hx−1
0
+ O(t2 )
Deviding by t and t → 0 finishes the proof.
For the more general form f (g(y)) we obtain
fy (g(y))w = f ! (g(y))[g ! (y)w]
It follows with g(y) = −∆ + y · ∇ and Lemma 2 that
Gy (y, u)w = (−∆ + y · ∇)−1 (−(w · ∇)(−∆ + y · ∇)−1 (u))
= (−∆ + y · ∇)−1 (−(w · ∇)G(y, u)).
(14)
Here again we need the general assumption
&Gy (y, u)& ≤ ρ < 1
(15)
for the fixed point equation to converge. Notice that in a stationary point (14)
yields
Gy (y, u)w = (−∆ + y · ∇)−1 (−(w · ∇)y).
3
General setting of the One Shot Method
To explain the idea of the One Shot Method we consider the general optimization
problem
min f (y, u) s.t. y = G(y, u),
(16)
with f (y, u) : Y × U → , G : Y × U → Y and Y and U appropriate Hilbert
spaces. For the optimality conditions let us define the Lagrange function
L(y, p, u) := f (y, u)− < ϕ, y − G(y, u) >Y ∗ ,Y .
(17)
Here ϕ is the problem specific adjoint operator as explained in the following.
For a given partial differential equation e(y, u) it is possible to reformulate it in
terms of a fixed point operator G(y, u)
e(y, u) = F (y, u)[y − G(y, u)]
with F (y, u) : Y → Y ∗ and provided that F (y, u)−1 exists . Now
< p, e(y, u) >Y,Y ∗ =< F (y, u)∗ p, y − G(y, u) >Y ∗ ,Y .
Finally we define ϕ := F (y, u)∗ p. Therefore the Lagrangian (17) is equivalent
to the standard Lagrangian.
As a first example consider the solid fuel ignition model with
e(y, u) = −∆y − δ ey − u = −∆(y − (−∆)−1 (δ ey + u))
6
With G(y, u) as defined in (4) it follows F (y, u) = F = −∆ and ϕ := −∆p. For
the Burgers equation with
e(y, u) = −∆y + (y · ∇)y − u = (−∆ + y · ∇)(y − (−∆ + y · ∇)−1 (u))
and (12) it follows F (y, u) = F (y) = (−∆+y·∇) and ϕ := (−∆−y·∇−div(y))p.
Because the operator F (y, u) depends in our examples at most on y we will
write F (y) in the following. With the help of the Lagrangian (17) we obtain the
optimality conditions by differentiation. For the state equation if follows
Lp (y, p, u)v = − < F (y)∗ v, y − G(y, u) >Y ∗ ,Y = − < v, F (y)[y − G(y, u)] >Y ∗ ,Y .
For the adjoint equation we need to take the dependencies on the state into
account. Therefore
Ly (y, p, u)w = fy (y, u)w− < Fy (y)∗ w p, y − G(y, u) >Y ∗ ,Y
− < F (y)∗ p, w − Gy (y, u)w >Y ∗ ,Y
= fy (y, u)w− < p, Fy (y)w(y − G(y, u)) >Y,Y ∗
+ < F (y)∗ p, Gy (y, u)w >Y ∗ ,Y − < F (y)∗ p, w >Y ∗ ,Y
If we now define the operator Ny (y, p, u) by
< F (y)∗ Ny (y, p, u), w >Y ∗ ,Y :=< fy (y, u), w >Y ∗ ,Y
(18)
− < p, Fy (y)w(y − G(y, u)) >Y,Y ∗
+ < F (y)∗ p, Gy (y, u)w >Y ∗ ,Y
it follows that
< Ly (y, p, u), w >Y ∗ ,Y =< F (y)∗ Ny (y, p, u), w >Y ∗ ,Y − < F (y)∗ p, w >Y ∗ ,Y .
Finally the KKT system for problem (16) reads
F (y)[y − G(y, u)] = 0 in Y ∗ ,
(19a)
∗
(19b)
∗
F (y) [Ny (y, p, u) − p] = 0 in Y ,
Lu (y, p, u) = 0 in U.
(19c)
Ny is the iteration operator for the adjoint problem. We use this notation to
be consistent with [5], where it was introduced as the derivative of the so-called
shifted Lagrangian. We note that we define it here directly in a linearized
manner.
A standard optimization technique would solve (19a) for some initial u = u0 to
get y, calculate p with these values using (19b) and finally update u using some
optimization strategy involving (19c). Following the approach in [5] we instead
introduce the iterative One Shot method
yk+1 = G(yk , uk )
pk+1 = Ny (yk , uk , pk )
(20a)
(20b)
uk+1 = uk − Bk−1 Lu (yk , uk , pk ),
(20c)
with some preconditioning operator Bk , that needs to be determined. Notice
that (20a), (20b) and (20c) operate fully independently in each iteration step.
7
Remark 1. If the contraction factor of the primal iteration is given by
&Gy (y, u)& ≤ ρ < 1 it is obvious by (18) that the contraction factor of the
dual iteration is the same in the case if Fy (y) = 0, for example for the solid fuel
ignition model.
Now we are able to formulate the following algorithm:
Algorithm 1 One Shot Method
1:
2:
3:
4:
5:
6:
7:
Choose u0 , y0 , p0 , tol > 0
for k = 0, 1, 2, . . . do
if res < tol then
STOP
end if
calculate yk+1 , pk+1 and uk+1 using (20a), (20b) and (20c)
end for
4
Convergence Analysis
For the convergence analysis of the One Shot Method we can exploit the special
structure of the two model problems. Especially for the solid fuel ignition model
the operator F (y) has a simple structure, in detail F (y) = −∆. Here the dual
product reduces to the H01 scalar product. For the Burgers equation F (y) also
depends on y and we cannot do this simplification.
4.1
The solid fuel ignition model
For the subsequent analysis we consider problem (P1) again. Because G(y, u) =
(−∆)−1 (δ ey + u), we can define the Lagrangian for p ∈ H01 (Ω) and ϕ := −∆p
according to (17) as
L(y, p, u) := f (y, u) − (p, y − G(y, u))H01 .
(21)
For this problem we can define the shifted Lagrangian directly by
N (y, p, u) := f (y, u) + (p, G(y, u))H01 .
Note that
p = Ny (y, p, u)
is the adjoint iteration.
Remark 2. Notice that since &Gy (y, u)& ≤ ρ < 1 holds, the adjoint equation
has the same contraction factor ρ, see Remark 1. In fact, from
Ny (y, p, u) = fy (y, u) + Gy (y, u)∗ p
we get
Nyp (y, p, u) = Gy (y, u)∗
and therefore &Nyp (y, p, u)& = &Gy (y, u)∗ & = &Gy (y, u)& ≤ ρ < 1.
8
According to (20) the One Shot iteration can be written as
yk+1 = G(yk , uk ) = (−∆)−1 (δ eyk + uk )
−1
yk
pk+1 = Ny (yk , pk , uk ) = (−∆) (δ pk e + yk − zd )
1
1
uk+1 = uk − Nu (yk , pk , uk ) = uk − (µuk + pk ).
γ
γ
(22a)
(22b)
(22c)
We get the iterative method (22) directly from the optimality system (6) by
introducing the preconditioning constant γ > 0. How this constant can be set,
will be determined in the subsequent analysis.
For the convergence analysis we introduce the augmented Lagrangian
α
β
&G(y, u) − y&2H 1 + &Ny (y, p, u) − p&2H 1
0
0
2
2
1
µ
= &y − zd &2L2 + &u&2L2 − (p, y − G(y, u))H01
2
2
β
α
+ &G(y, u) − y&2H 1 + &Ny (y, p, u) − p&2H 1 .
0
0
2
2
La (y, p, u) := L(y, p, u) +
(23)
The idea is to show that the augmented Lagrangian is an exact penalty function.
That means that every local minimizer of problem (16) is also a local minimizer
of the augmented Lagrangian. We want to find conditions on α and β, so that
this is fulfilled.
Lemma 3. If there exist constants α > 0 and β > 0 such that the following
condition is fulfilled
αβ(1 − ρ)2 > 1 + β&Nyy (y, p, u)&
(24)
then a point is a stationary point of La as defined in (23) if and only if it is a
root of s as defined in (25).
Proof. We start by deriving the gradient of La . For v, w ∈ H01 and h ∈ L2 we
get
Lay (y, p, u)w = (y − zd , w)L2 − (p, w)H01 + (p, (−∆)−1 (ey w))H01
+ α((−∆)−1 (ey + u) − y, (−∆)−1 (ey w) − w)H01
+ β((−∆)−1 (pey + y − zd ) − p, (−∆)−1 (pey w + w))H01
= (Ny (y, p, u) − p, w)H01 + α(G(y, u) − y, (Gy (y, u) − I)w)H01
+ β(Ny (y, p, u) − p, Nyy (y, p, u)w)H01
Lap (y, p, u)v = ((−∆)−1 (ey + u) − y, v)H01
+ β((−∆)−1 (pey + y − zd ) − p, (−∆)−1 (vey ) − v)H01
= (G(y, u) − y, v)H01 + β(Ny (y, p, u) − p, (Nyp (y, p, u) − I)v)H01
Lau (y, p, u)h = µ(u, h)L2 + (p, h)L2 + α((−∆)−1 (ey + u) − y, (−∆)−1 h)H01
= Nu (y, p, u)h + α(G(y, u) − y, Gu (y, u)h)H01 .
9
Since we aim to find a descent direction later we write this in the form
−Lay (y, p, u)w = −(s2 , w)H01 + α(s1 , (I − Gy (y, u))w)H01 − β(s2 , Nyy (y, p, u)w)H01
−Lap (y, p, u)v = −(s1 , v)H01 + β(s2 , (I − Nyp (y, p, u))v)H01
−Lau (y, p, u)h = γ(s3 , h)L2 − α(s1 , Gu (y, u)h)H01
with

  
G(y, u) − y
s1
s = s2  = Ny (y, p, u) − p .
− γ1 Nu (y, p, u)
s3
(25)
If s = 0, it follows immediately that −∇La = 0. For the other direction we
interpret −∇La = 0 as the weak formulation of a system of partial differential
equations
∆s2 − αs1 ey − α∆s1 − βs2 pey − βs2 = 0
∆s1 − β∆s2 − βs2 ey = 0
γs3 − αs1 = 0.
If the related bilinear form is coercive for all (w, v, h) ∈ H01 × H01 × L2 , then
there exists a unique solution due to Lax-Milgram and therefore s = 0. In
−Lau (y, p, u)h = 0, s3 just depends on s1 and we can restrict ourselves to solving
the first two equations. From −Lap (y, p, u)v = 0 it follows
(s1 , v)H01 = β(s2 , (I − Nyp (y, p, u))v)H01
and therefore
(s1 , (I − Gy (y, u))w)H01 = β(s2 , (I − Nyp (y, p, u))(I − Gy (y, u))w)H01
= β((I − Nyp (y, p, u))∗ s2 , (I − Gy (y, u))w)H01
= β((I − Gy (y, u))s2 , (I − Gy (y, u))w)H01 .
We plug this into −Lay (y, p, u)w = 0 and obtain the bilinear form
a(w, v) = −(v, w)H01 + αβ((I − Gy (y, u))v, (I − Gy (y, u))w)H01
− β(v, Nyy (y, p, u)w)H01 = 0.
For coercivity we need a(w, w) ≥ C&w&2H 1 . With
0
a(w, w) = −&w&2H 1 + αβ&(I − Gy (y, u))w&2H 1 − β(w, Nyy (y, p, u)w)H01
0
0
≥ −&w&2H 1 + αβ&(I − Gy (y, u))w&2H 1 − β&Nyy (y, p, u)&&w&2H 1
0
0
0
and
&(I − Gy (y, u))w&H01 ≥ &w&H01 − &Gy (y, u)w&H01 ≥ (1 − ρ)&w&H01
it follows
a(w, w) ≥ −&w&2H 1 + αβ(1 − ρ)2 &w&2H 1 − β&Nyy (y, p, u)&&w&2H 1
0
0
0
= (αβ(1 − ρ)2 − 1 − β&Nyy (y, p, u)&)&w&2H 1 .
0
10
Therefore, from the assumption
αβ(1 − ρ)2 − 1 − β&Nyy (y, p, u)& > 0
we obtain positivity.
We still need to show, that the stationary point is a local minimizer of La .
Since (y ∗ , u∗ ) is a minimizer of (P1) and we assume that the sufficient second
order condition is fulfilled, there exist constants C > 0 and ε > 0 such that for
(y, u) ∈ H01 × L2 with &y − y ∗ &2H 1 + &u − u∗ &2L2 ≤ ε it holds
0
&u − u∗ &2L2 ≤ C(f (y, u) − f (y ∗ , u∗ ))
= C(L(y, p, u) + (p, G(y, u) − y)H01 − L(y ∗ , p∗ , u∗ ))
= C(La (y, p, u) − La (y ∗ , p∗ , u∗ ) + (p, G(y, u) − y)H01
−
α
β
&G(y, u) − y&2H 1 − &Ny (y, p, u) − p&2H 1 )
0
0
2
2
Under the assumption
(p, G(y, u) − y)H01 ≤
β
α
&G(y, u) − y&2H 1 + &Ny (y, p, u) − p&2H 1
0
0
2
2
(26)
this results in
&u − u∗ &2L2 ≤ C(La (y, p, u) − La (y ∗ , p∗ , u∗ )).
(27)
La (y ∗ , p∗ , u∗ ) ≤ La (y, p, u),
(28)
Therefore
for any (y, p, u) in a neighborhood of (y ∗ , p∗ , u∗ ). α and β must be big enough
to fulfill (26).
So far we know that any minimizer of problem (P1) is also a minimizer of the
augmented Lagrangian. In the next step we show that the One Shot iterate is
a descent direction for La and therefore the One Shot Method converges to the
right minimum.
Lemma 4. If there exist constants α > 0 and β > 0 such that the following
conditions are fulfilled
α(1 − ρ) −
α2
β
&Gu &2 > 1 + &Nyy &,
2µ
2
β
β(1 − ρ) > 1 + &Nyy &,
2
µ
γ> ,
2
(29a)
(29b)
(29c)
then the One Shot iterate yields descent on La .
Proof. We follow the proof of Lemma with the difference that for the descent
direction s
(−∇La , s) ≥ C ! s!2 ,
11
with a constant C > 0 and the norm !s!2 := &s1 &2H 1 + &s2 &2H 1 + &s3 &2L2 needs
0
0
to be fulfilled. Hence for the bilinear form
B(w, v, h) := −2(w, v)H01 + α(w, w − Gy w)H01 − β(v, Nyy w)H01 + β(v, v − Gy v)H01
+ γ(h, h)L2 − α(w, Gu h)H01
with (w, v, h) ∈ H01 × H01 × L2 , we need to verify the existence of a constant
C > 0 such that
B(w, v, h) ≥ C(&w&2H 1 + &v&2H 1 + &h&2L2 ).
0
0
After applying Young’s inequality
ab ≤
a2
νb2
+
2ν
2
with the parameter ν chosen as 1, ρ, &Nyy & and
combination with (5) it follows that
α&Gu &2
µ
respectively and in
α1 2
β
α
ρ&w&2H 1 −
ρ &w&2H 1 − &Nyy &&v&2H 1
0
0
0
2
2ρ
2
1
β
β
+ β&v&2H 1 − ρ&v&2H 1 −
ρ2 &v&2H 1
0
0
0
2
2ρ
B(w, v, h) ≥ −&w&2H 1 − &v&2H 1 + α&w&2H 1 −
0
−
0
0
β 1
&Nyy &2 &w&2H 1
0
2 &Nyy &
µ
α
α α&Gu &2
&w&2H 1 −
&Gu &2 &h&2L2
0
2
µ
2 α&Gu &2
β
α2
= &w&2H 1 (α(1 − ρ) −
&Gu &2 − 1 − &Nyy &)
0
2µ
2
β
+ &v&2H 1 (β(1 − ρ) − 1 − &Nyy &)
0
2
µ
2
+ &h&L2 (γ − ).
2
+ γ&h&2L2 −
Therefore, from the assumption
α(1 − ρ) −
β
α2
&Gu &2 − 1 − &Nyy & > 0,
2µ
2
β
β(1 − ρ) − 1 − &Nyy & > 0,
2
µ
γ − > 0,
2
we obtain positivity.
Remark 3. From (29a) and (29b) we get estimates for α and β. From (29a)
for example
β
α
α[(1 − ρ) −
&Gu &2 ] > 1 + &Nyy & > 1 > 0
(31)
2µ
2
it follows with α, β > 0 that
(1 − ρ) −
2µ
α
&Gu &2 > 0 ⇒ α <
(1 − ρ).
2µ
&Gu &2
12
Additionally we get a lower bound, because by (31) it follows
(1 − ρ) −
α
1
&Gu &2 > .
2µ
α
2ρµ
α
The case α < 1 yields (1 − ρ) − 2µ
&Gu &2 > 1 and therefore α < − &G
2 , which
u&
results in a contradiction. Therefore we have α > 1.
By the following lemma we even obtain a result stating the sufficiently decrease
in a neighborhood of the minimizer.
Lemma 5. Let the slightly stronger conditions than (29)
β
α2
&Gu &2 > 1 + CNyy
2µ
2
β
β(1 − ρ) > 1 + CNyy
2
µ
γ>
2
α(1 − ρ) −
(32)
and (26) be satisfied for a constant CNyy with
&Nyy (y, p, u)& < CNyy ,
(33)
that is independent of any iterates (yk , pk , uk ). Then the angle condition holds
in a neighborhood U(y ∗ , u∗ ) of the optimum (y ∗ , u∗ ), i.e.
(−∇La , s)
≥ C > 0.
&∇La & ! s!
(34)
Proof. The proof is based on the inequality (29a) and (29b) that in general may
depend on the iterates. Since the constant in the angle condition needs to be
independent of the iterates, it is necessary to find suitable approximations of
the critical terms &Gu (y, u)& and &Nyy (y, p, u)&. By definition this is fulfilled
for the norm
&Gu (y, u)& :=
&Gu (y, u)[h]&H01 =
sup
&h&L2 =1
sup
&h&L2 =1
&(−∆)−1 h&H01 .
(35)
For
&Nyy (y, p, u)& :=
&Nyy (y, p, u)[v, w]&
sup
&v&H 1 =&w&H 1 =1
0
=
0
|
sup
&v&H 1 =&w&H 1 =1
0
≤ |1 +
!
0
!
vw dx +
Ω
!
pey vw dx|
Ω
pey dx|
Ω
it follows with a generic constant C that
&Nyy (y, p, u)& ≤ C&p&L2 &ey &L2 .
The term &ey &L2 is bounded due to (3) and with (6c) we get
&Nyy (y, p, u)& ≤ C&u&L2 .
13
(36)
For a given starting iterate (y0 , p0 , u0 ) we consider the level set
N0 := {(y, p, u) ∈ H01 × H01 × L2 : La (y, p, u) ≤ La (y0 , p0 , u0 )}.
(37)
From (27) it follows that
&u − u∗ &2L2 ≤ C(La (y0 , p0 , u0 ) − La (y ∗ , p∗ , u∗ ))
for (y, p, u) ∈ N0 , particularly
&u&2L2 ≤ C(La (y0 , p0 , u0 ) − La (y ∗ , p∗ , u∗ ) + &u∗ &2L2 )
and finally since the right hand side is independent of any iterates
&Nyy (y, p, u)& ≤ CNyy .
For the angle condition we get with the same arguments as in the proof of
Lemma 4 and (33) that
α2
β
&Gu &2 − 1 − CNyy )
0
2µ
2
β
µ
+ &s2 &2H 1 (β(1 − ρ) − 1 − CNyy ) + (γ − )&s3 &2L2 ,
0
2
2
(−∇La , s) ≥ &s1 &2H 1 (α(1 − ρ) −
i.e. with (32) we receive
(−∇La , s) ≥ δ∗ ! s!2 ,
but here the constant δ∗ is independent of any iterates. Additionally we get by
definition
&Lay & =
sup
&w&H 1 =1
|Lay w| =
0
≤
sup
&w&H 1 =1
0
sup
&w&H 1 =1
|(s2 , w)H01 + α(s1 , (Gy − I)w)H01 + β(s2 , Nyy w)H01 |
0
'
(
&s2 &H01 &w&H01 + α(1 + ρ)&s1 &H01 &w&H01 + β&s2 &H01 &Nyy &&w&H01
= &s2 &H01 + α(1 + ρ)&s1 &H01 + β&Nyy &&s2 &H01
&Lap & =
sup
&v&H 1 =1
|Lap v| =
0
sup
&v&H 1 =1
|(s1 , v)H01 + β(s2 , (Gy − I)v)H01 |
0
≤ &s1 &H01 + β(1 + ρ)&s2 &H01
&Lau & =
sup
&h&L2 =1
|Lau h| =
sup
&h&H 1 =1
| − γ(s3 , h)L2 + α(s1 , Gu h)H01 |
0
≤ γ&s3 &L2 + α&Gu &&s1 &H01
Therefore with (33) it follows
*
)
&∇La & ≤ &s1 &H01 [α(1 + ρ) + 1 + α&Gu &] + &s2 &H01 1 + βCNyy + β(1 + ρ) + γ&s3 &L2
≤ C ! s!,
with a constant C > 0. Finally we get
δ∗ ! s!2
(−∇La , s)
≥
> 0.
&∇La & ! s!
C ! s ! !s!
14
4.2
The stationary viscous Burgers equation
Theoretically it would be possible to do the same analysis for the Burgers equation as for the solid fuel ignition model by using F (y) = −∆. But this choice
does not describe a numerical stable method and therefore we choose the One
Shot iteration with F (y) = (−∆ + y · ∇) as follows:
yk+1 = (−∆ + yk · ∇)−1 (uk )
(38a)
−1
pk+1 = (−∆ − yk · ∇ − div(yk ))
1
uk+1 = uk − (µuk + pk ).
γ
T
(yk − zd − (∇yk ) pk )
(38b)
(38c)
For the convergence analysis we define the Lagrangian as in the general setting
to be
L(y, p, u) := f (y, u)− < ϕ, y − G(y, u) >H −1 ,H01
(39)
with the operator
ϕ := −∆p − (y · ∇)p − div(y)p
for the adjoint p ∈
H01 .
Remark 4. One can obtain the optimality system in a standard way by differentiating the Lagrangian defined above. With the following relationships
!
!
((v · ∇)y)p dx = (∇y)T p v dx,
Ω
!
((y · ∇)v)p dx =
Ω
Ω
! "
yi ∂i vj pj dx = −
! "
yi ∂i (pj )vj dx −
!
! "
∂i (yi )pj vj dx
Ω i,j
Ω i,j
=−
∂i (yi pj )vj dx
Ω i,j
Ω i,j
=−
! "
(y · ∇)pv + div(y)pv dx
Ω
and (14) we obtain
Ly (y, p, u)w = (y − zd , w)L2 − < ϕy w, y − G(y, u) >H −1 ,H01 − < ϕ, w − Gy (y, u)w >H −1 ,H01
!
= (y − zd , w)L2 − (−(w · ∇) − div(w))p(y − G(y, u)) dx
!
− (−∆ − (y · ∇) − div(y))p(w − (−∆ + y · ∇)−1 (−(w · ∇)G(y, u))) dx
!
= (y − zd , w)L2 − (w · ∇)y p − (w · ∇)G(y, u) p dx
!
− p(−∆ + y · ∇)(w − (−∆ + y · ∇)−1 (−(w · ∇)G(y, u))) dx
!
= (y − zd , w)L2 + (w · ∇)G(y, u) p − (∇y)T pw dx
!
− p(−∆w + y · ∇w + (w · ∇)G(y, u))
!
= (y − zd , w)L2 − (−∆ − (y · ∇) − div(y))pw + (∇y)T pw dx
15
Finally
Ly (y, p, u)w =
!
(−∆ − (y · ∇) − div(y))[(−∆ − (y · ∇) − div(y))−1 (y − zd − (∇y)T p) − p]w dx
Similarly we can compute the derivative with respect to the adjoint and the
control.
!
Lp (y, p, u)v = − (−∆ − (y · ∇) − div(y))v (y − G(y, u)) dx
!
= − v (−∆ + y · ∇)(y − G(y, u)) dx
!
= − v(−∆y + (y · ∇)y − u) dx.
This yields the state equation. And finally
!
Lu (y, p, u)h = µ(u, h)L2 + (−∆ − (y · ∇) − div(y))p Gu (y, u)h dx
!
= µ(u, h)L2 + p(−∆ + y · ∇)Gu (y, u)h dx
!
= (µu + p)h dx.
The analysis is more complex in this case, because the operator F (y) depends
on y. Again we can define the augmented Lagrangian as
La (y, p, u) := L(y, p, u) +
β
α
&G(y, u) − y&2H 1 + &Ny (y, p, u) − p&2H 1
0
0
2
2
For the derivative we obtain with the help of

  
G(y, u) − y
s1
s = s2  = Ny (y, p, u) − p
− γ1 Lu (y, p, u)
s3
Lay (y, p, u)w = Ly (y, p, u)w + α(G(y, u) − y, (Gy (y, u) − I)w)H01
+ β(Ny (y, p, u) − p, Nyy (y, p, u)w)H01
!
= (−∆ − (y · ∇) − div(y))s2 w dx
+ α(s1 , (Gy (y, u) − I)w)H01 + β(s2 , Nyy (y, p, u)w)H01
Lap (y, p, u)v = Lp (y, p, u)v + β(Ny (y, p, u) − p, (Nyp (y, p, u) − I)v)H01
!
= v (−∆ + y · ∇)s1 dx + β(s2 , (Nyp (y, p, u) − I)v)H01
Lau (y, p, u)h = Lu (y, p, u)h + α(G(y, u) − y, Gu (y, u)h)H01
= −γ(s3 , h)L2 + α(s1 , Gu (y, u)h)H01 .
16
We proceed in the same way as in subsection 4.1. For s = 0 it follows ∇La = 0.
In addition if the related bilinear form to −∇La = 0 is coercive, we obtain
s = 0. The bilinear form reads
!
B(w, v, h) := −2 v(−∆ + y · ∇)w dx + α(w, w − Gy w)H01
− β(v, Nyy w)H01 + β(v, v − Nyp v)H01 + γ(h, h)L2 − α(w, Gu h)H01 .
For the subsequent approximations we need to assume that &Nyp & ≤ ρ < 1,
since the contraction factor of the adjoint equation is not given as for the solid
fuel ignition model, see Remark 1. Compared to the bilinear form in subsection
4.1 there is the additional term
!
−2 (y · ∇)wv dx.
Because of the continuous injection H01 +→ L4 we obtain
!
−2 (y · ∇)wv dx ≥ −&y&2H 1 &w&2H 1 − C∗2 &v&2H 1
0
0
0
with a constant C∗ . Therefore
β
α2
&Gu &2 − 1 − &Nyy & − &y&2H 1 )
0
0
2µ
2
β
+ &v&2H 1 (β(1 − ρ) − 1 − &Nyy & − C∗2 )
0
2
µ
2
+ &h&L2 (γ − )
2
B(w, v, h) ≥ &w&2H 1 (α(1 − ρ) −
and finally
Lemma 6. If there exist constants α > 0 and β > 0 such that the following
conditions are fulfilled
α(1 − ρ) −
α2
β
&Gu &2 > 1 + &Nyy & + &y&2H 1 ,
0
2µ
2
β
β(1 − ρ) > 1 + &Nyy & + C∗2 ,
2
µ
γ> ,
2
(40a)
(40b)
(40c)
and
< F (y)∗ p, G(y, u)−y >H −1 ,H01 ≤
β
α
&G(y, u)−y&2H 1 + &Ny (y, p, u)−p&2H 1 (41)
0
0
2
2
then the One Shot Method converges to a local minimum of problem (P2).
(41) is the analogue to (27) for the general setting. You need this condition to
obtain a local minimum of the augmented Lagrangian.
17
5
Numerical Results
In this section we test the One Shot Method for the two example problems, the
solid fuel ignition model and the viscous Burgers equation.
5.1
5.1.1
The solid fuel ignition model
Discretization
Let Ω = [0, 1]2 be subdivided into shape-regular meshes Th consisting of elements
K. The optimization problem is solved by discretizing the One Shot Method
(22) with continuous piecewise linear finite elements. Let
V h := {v h ∈ C0 (Ω)| v h |K ∈ P1 (K), K ∈ Th },
then we consider y h , ph , uh ∈ V h as the finite element approximations to the
primal y, the dual p and the control u. Therefore in every iteration step of the
One Shot Method we solve the finite element problem
h
h
a(yk+1
, v h ) = (δ eyk + uhk , v h ), v h ∈ V h
a(phk+1 , v h )
=
h
(δ phk eyk
uhk+1 = uhk −
+
ykh
h
(42a)
h
− zd , v ), v ∈ V
h
1
(µuhk + phk ).
γ
Here a(·, ·) denotes the bilinear form
!
a(u, v) =
∇u · ∇v dx,
Ω
(42b)
(42c)
u, v ∈ H01
and (·, ·) the standard L2 scalar product.
5.1.2
Optimization
We update y h , ph , uh according to (42). As a stopping criteria we verify that
the residual of the coupled iteration in the L2 -norm is less than some given
tolerance tol, i.e.
h
res = &yk+1
− ykh &L2 + &phk+1 − phk &L2 + &uhk+1 − uhk &L2 ≤ tol.
All the subsequent results are obtained with the help of the finite element library
deal.II. The linear problem in each iteration step is solved with a conjugated
gradient method.
We first set u0 ≡ 1, y0 ≡ 1, p0 ≡ 0, tol = 10−3 and zd (x1 , x2 ) = x1 x2 . The
numerical results for µ = γ = 0.005 and δ = 1 can be seen in Figure 1 on page
19.
The choice of δ has a huge effect on the solutions. The control for various δ and
µ = γ = 0.01 is shown in Figure 2 on page 20.
18
1
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0.8
0.6
0.4
0.2
0
1
1
0.8
0
0.8
0.6
0.2
0.4
0
0.4
0.6
0.8
0.6
0.2
0.2
0.4
0.4
0.6
1 0
(a) Desired state
0.8
0.2
1 0
(b) State
3
0
2.5
-0.002
-0.004
2
-0.006
1.5
-0.008
1
-0.01
0.5
-0.012
0
-0.014
1
1
0.8
0
0.8
0.6
0.2
0.4
0
0.4
0.6
0.8
0.2
0.6
0.2
0.4
1 0
(c) Control
0.4
0.6
0.8
0.2
1 0
(d) Dual
Figure 1: Numerical Solution for µ = γ = 0.005, tol = 10−3 and δ = 1
Additionally the method seems not to be robust with respect to the parameters
δ, µ and the desired state zd . The biggest regularization parameter that we
obtain results for is µ = 0.005. In [2] the authors experienced similar problems.
They solved the discretized optimality system with a multigrid One Shot approach.
The number of iterations for a complete One Shot optimization are displayed
in Tables 1 to 3 for various regularization parameters µ, preconditioners γ and
degrees of freedom. Here δ is fixed to 1. ∞ indicates divergence of the One Shot
Method or that the maximal number of iterations was reached.
# Dofs
81
289
1089
4225
γ = 0.04
∞
∞
∞
∞
γ = 0.05
∞
∞
∞
∞
γ = 0.06
19
19
19
19
γ = 0.08
9
9
9
9
γ = 0.1
8
8
8
8
γ = 0.2
10
10
10
10
γ = 0.5
24
24
24
24
Table 1: Number of iterations for Solid Fuel Ignition Model with µ = 0.1, δ = 1
and tol = 10−3
19
γ=1
44
44
44
44
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
1
1
0.8
0
0.8
0.6
0.2
0.4
0
0.4
0.6
0.8
0.6
0.2
0.2
0.4
0.4
0.6
1 0
(a) δ = 1
0.8
0.2
1 0
(b) δ = 2
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
1
0.8
0.6
0.4
0.2
0
-0.2
1
1
0.8
0
0.8
0.6
0.2
0.4
0
0.4
0.6
0.8
0.6
0.2
0.2
0.4
0.4
0.6
1 0
(c) δ = 3
0.8
0.2
1 0
(d) δ = 4
0.5
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1.2
-1.4
0
-0.5
-1
-1.5
-2
-2.5
1
1
0.8
0
0.8
0.6
0.2
0.4
0
0.4
0.6
0.8
0.2
1 0
(e) δ = 5
0.6
0.2
0.4
0.4
0.6
0.8
0.2
1 0
(f) δ = 6
Figure 2: Control for µ = γ = 0.01 and tol = 10−3
20
#Dofs
81
289
1089
4225
γ = 0.004
∞
∞
∞
∞
γ = 0.005
∞
∞
∞
∞
γ = 0.007
56
57
58
58
γ = 0.008
29
30
30
30
γ = 0.01
21
21
21
21
γ = 0.02
16
16
16
16
Table 2: Number of iterations for Solid Fuel Ignition Model with µ = 0.01, δ = 1
and tol = 10−3
#Dofs
81
289
1089
4225
γ = 0.0025
∞
∞
∞
∞
γ = 0.004
96
102
104
104
γ = 0.005
40
40
40
40
γ = 0.01
27
27
27
27
γ = 0.1
70
70
70
70
γ=1
271
271
271
271
Table 3: Number of iterations for Solid Fuel Ignition Model with µ = 0.005,
δ = 1 and tol = 10−3
For all choices of µ and γ the number of iterations seems to be independent
of the number of grid points. The best results are obtained for γ close to µ,
more precisely for µ = 0.1 the best results are obtained for γ = µ, but for
µ = 0.01 and µ = 0.005 they are obtained for γ = 2µ. In all three cases the
numerical results reflect the condition (40c) in Lemma 4. The method diverges
if the condition is violated, i.e. γ ≤ µ/2.
5.1.3
Adaptive scheme
The numerical mesh independence results above encourage the idea of an adaptive One Shot strategy. Therefore we extend (42) by an additional adaptive
step, i.e. we refine or coarsen the grid according to an error estimator and interpolate the solutions to the new mesh. For the refinement process we chose a
residual type error estimator for the cells and the faces with respect to the state
and adjoint state.
1 "
hE & [n · ∇y]E &2E
2
E∈∂K
1 "
:= h2K &∆p + δ pey + y − zd &2K +
hE & [n · ∇p]E &2E
2
2
ηK,y
:= h2K &∆y + δ ey + u&2K +
2
ηK,p
E∈∂K
2
2
2
ηK
:= ηK,y
+ ηK,p
Overall the Adapted One Shot Method is stated in Algorithm 2 on page 23.
The 40% of the cells with the largest error indicator are refined, those 30% with
the smallest are coarsened. This error estimater may not be optimal for the
problem but serves as a good test. The final adaptively generated grids are
plotted for µ = γ = 0.1, δ = 1 in Figure 3 on page 22 and δ = 2 in Figure 4 on
page 22 respectively.
21
γ = 0.1
40
40
40
40
γ=1
181
180
180
179
Figure 3: Adaptively generated grid for µ = γ = 0.1 and δ = 1
Figure 4: Adaptively generated grid for µ = γ = 0.1 and δ = 2
22
Algorithm 2 Adapted One Shot Method
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
Choose u0 , y0 , p0 , tol > 0
for k = 0, 1, 2, . . . do
h
calculate yk+1
, phk+1 and uhk+1 using (42)
if res < tol1 then
calculate ηK for all K and refine or coarsen the mesh
interpolate the solution vectors to the new mesh
end if
if res < tol2 then
STOP
end if
end for
5.2
The stationary viscous Burgers equation
For the discretization of the Burgers equation a SUPG scheme was implemented
using deal.II. For the 1D case we solve in every step of the One Shot Method
the following problem
!!
!
−νyk+1
+ yk yk+1
= uk
−νp!!k+1 − yk p!k+1 = yk − zd
1
uk+1 = uk − (µuk + pk ).
γ
(43)
As a first example we choose zd = 1, µ = 0.1, ν = 0.1 and ν = 0.01. The
solutions for the second case can be seen in Figure 5 on page 24.
Preconditioning the linear system in every iteration step, that is solved using the
GMRES method, seems to be very important for the Burgers equation. While
for the first example with µ = 0.1 and ν = 0.1 the SSOR preconditioner in
deal.II worked fine, the preconditioner had to be improved for the solution with
ν = 0.01. deal.II offers a wrap around to the PETSc library, which itself offers
various solvers and preconditioners. Therefore whenever possible we solved the
system using deal.II, but for comparison in this case we also computed the solutions with PETSc using an ILU preconditioner. Otherwise in case the deal.II
solver failed, we switched to PETSc completely.
Tables 4 and 5 show the number of iterations for various degrees of freedom and
preconditioners γ with ν = 0.1. In the first case the deal.II solver was used and
in the second case a PETSc solver was used.
#Dofs
33
65
129
257
γ = 0.1
∞
∞
∞
∞
γ = 0.5
∞
∞
∞
∞
γ = 0.8
90
90
90
90
γ=1
65
65
65
65
γ = 1.2
65
65
65
65
γ=2
98
98
98
98
γ=3
137
136
136
136
γ=5
205
205
205
205
Table 4: Number of iterations for 1D Burgers equation with µ = 0.1, ν = 0.1,
zd = 1, without PETSc
23
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
(a) State
2.5
0
2
-0.05
1.5
-0.1
1
-0.15
0.5
-0.2
0
-0.25
0
0.2
0.4
0.6
0.8
1
0
0.2
(b) Control
0.4
0.6
0.8
1
(c) Dual
Figure 5: Solution to 1D Burgers equation for µ = 0.1 and ν = 0.01
#Dofs
33
65
129
257
γ = 0.1
∞
∞
∞
∞
γ = 0.5
∞
∞
∞
∞
γ = 0.8
85
88
91
95
γ=1
62
64
66
69
γ = 1.2
61
63
66
70
γ=2
91
96
101
107
γ=3
126
133
141
150
γ=5
186
199
212
227
Table 5: Number of iterations for 1D Burgers equation with µ = 0.1, ν = 0.1,
zd = 1, with PETSc
The mesh independence can be clearly seen in the case of the deal.II solver. On
the contrary PETSc seems to destroy the mesh independent behaviour.
For ν = 0.01 the results can be seen in Table 6.
A small increase in the number of iterations can be observed, which again might
be the influence of PETSc.
As a second example we chose µ = 0.1, ν = 0.1 and zd = sin(13x). The solutions
are displayed in Figure 6 on page 25, the number of iterations in Tables 7 and
8.
Again the difference in the mesh independent behaviour between deal.II and
PETSc solver is clear.
24
#Dofs
33
65
129
257
γ=1
∞
∞
∞
∞
γ=2
162
159
161
164
γ = 2.5
121
123
127
134
γ=3
128
134
142
150
γ=4
158
166
176
187
γ=5
187
197
209
224
Table 6: Number of iterations for 1D Burgers equation with µ = 0.1, ν = 0.01,
zd = 1, with PETSc
1
0.04
0.8
0.03
0.6
0.02
0.01
0.4
0
0.2
-0.01
0
-0.02
-0.2
-0.03
-0.4
-0.04
-0.6
-0.05
-0.8
-0.06
-1
0
0.2
0.4
0.6
0.8
1
-0.07
0
0.2
(a) Desired state
0.4
0.6
0.8
1
0.6
0.8
1
(b) State
0.6
0.08
0.4
0.06
0.2
0.04
0
0.02
-0.2
0
-0.4
-0.02
-0.6
-0.04
-0.8
-0.06
0
0.2
0.4
0.6
0.8
1
0
0.2
(c) Control
0.4
(d) Dual
Figure 6: Solution to 1D Burgers equation for µ = 0.1, ν = 0.1 and zd = sin(13x)
#Dofs
33
65
129
257
γ=1
∞
∞
∞
∞
γ=2
205
210
210
210
γ = 2.5
134
134
134
134
γ=3
155
156
156
156
γ=4
197
197
197
197
γ=5
235
235
235
235
Table 7: Number of iterations for 1D Burgers equation with µ = 0.1, ν = 0.1,
zd = sin(13x), without PETSc
25
#Dofs
33
65
129
257
γ=1
∞
∞
∞
∞
γ=2
190
200
215
225
γ = 2.5
122
129
137
145
γ=3
141
150
159
169
γ=4
177
189
202
215
γ=5
211
226
242
258
Table 8: Number of iterations for 1D Burgers equation with µ = 0.1, ν = 0.1,
zd = sin(13x), with PETSc
As a next step the Burgers equation in two space dimensions is solved. In every
iteration step the following linear system needs to be solved:
(−∆ + yk · ∇)yk+1 = uk
(−∆ − yk · ∇ − div(yk ))pk+1 = yk − zd − (∇yk )T pk
1
uk+1 = uk − (µuk + pk ).
γ
The results for the 2D Burgers equation with µ = 0.1 and ν = 0.1 are shown
in Tables 9 and 10. Here the solver is more sensitive to changes in degrees of
freedom compared to the one-dimensional case. But at least for the best value
of γ = 1.2 the deal.II solver seems to behave mesh independent.
#Dofs
882
1922
3362
5202
γ = 0.6
∞
∞
∞
∞
γ = 0.7
∞
170
142
130
γ = 0.8
129
98
88
84
γ=1
69
59
56
55
γ = 1.2
55
52
52
51
γ=2
74
73
73
73
Table 9: Number of iterations for the 2D Burgers equation with µ = 0.1 and
ν = 0.1, without PETSc
#Dofs
882
1922
3362
5202
γ = 0.6
∞
∞
∞
∞
γ = 0.7
335
224
199
189
γ = 0.8
166
136
128
125
γ=1
94
86
85
86
γ = 1.2
87
90
93
95
γ=2
134
141
146
150
Table 10: Number of iterations for the 2D Burgers equation with µ = 0.1 and
ν = 0.1, with PETSc
We add adaptation steps as in Algorithm 2. After 3 adaptive steps with 40% of
the cells with the largest gradient jumps being refined and 8% of the cells with
the smallest being coarsened, the grid exhibits a fine region near the boundary.
The solution on this adaptively generated grid is illustrated in Figure 7.
26
1
1
v1
0.4
0.2
0
0.2
0.4
x
0.6
0.8
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0.8
0.6
y
y
0.6
0
v2
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0.8
0.4
0.2
0
1
(a) First component of Primal
0
0.2
0.4
x
0.6
0.8
1
(b) Second component of Primal
1
1
v1
v2
2.4
2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
y
0.6
0.4
0.6
0.4
0.2
0
2.4
2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0.8
y
0.8
0.2
0
0.2
0.4
x
0.6
0.8
0
1
(c) First component of Control
0
0.2
0.4
x
0.6
0.8
1
(d) Second component of Control
1
1
v1
v2
-0.02
-0.04
-0.06
-0.08
-0.1
-0.12
-0.14
-0.16
-0.18
-0.2
-0.22
-0.24
y
0.6
0.4
0.6
0.4
0.2
0
-0.02
-0.04
-0.06
-0.08
-0.1
-0.12
-0.14
-0.16
-0.18
-0.2
-0.22
-0.24
0.8
y
0.8
0.2
0
0.2
0.4
x
0.6
0.8
0
1
(e) First component of Dual
0
0.2
0.4
x
0.6
0.8
1
(f) Second component of Dual
Figure 7: Solution to 2D Burgers equation on adaptively refined grid for µ = 0.1,
ν = 0.1 and zd = 1
27
Additionally the One Shot Method is tested for µ = 0.01 and ν = 0.1. The
results are displayed in Table 11. Again the grid is adapted in the same way as
above and plotted in Figure 8.
#Dofs
882
1922
3362
5202
γ = 0.6
∞
∞
∞
∞
γ = 0.7
∞
431
438
446
γ = 0.75
455
452
461
471
γ = 0.8
473
475
486
497
γ=1
564
571
586
599
Table 11: Number of iterations for the 2D Burgers equation with µ = 0.01 and
ν = 0.1, with PETSc
Figure 8: Adaptively refined grid for 2D Burgers equation with µ = 0.01 and
ν = 0.1
6
Conclusions
In this paper we studied the One Shot Method in a Hilbert space setting. For two
example problems, the solid fuel ignition Model and the Burgers equation, we
set up a convergence analysis. In detail, we showed that an augmented Lagrange
function is an exact penalty function and that the One Shot iteration serves as a
descent direction to the augmented Lagrangian. The numerical results show the
convergence for a variety of preconditioners. Additionally it can be seen, that
the method is mesh independent. Finally we extended the One Shot Method
by an additional adaptive step.
28
References
[1] J. M. Burgers, Application of a model system to illustrate some points
of the statistical theory of free turbulence, Proc. Acad. Sci. Amsterdam,
43:2-12, 1940.
[2] A. Borzi and K. Kunisch, The numerical solution of the steady state solid
fuel ignition model and its optimal control, SIAM J. SCI. COMPUT.,
Vol.22, No. 1, pp. 263-284.
[3] E. Casas, O. Kavian and J.-P. Puel, Optimal control of an ill-posed elliptic semilinear equation with an exponential nonlinearity, ESAIM Control,
Optim. Calc. Var., Vol. 3, 1998, pp. 361-380.
[4] J.C. De Los Reyes, Constrained Optimal control of Stationary Viscous
Incompressible Fluids by Primal-Dual Active Set Methods, Dissertation
Uni Graz, 2003 .
[5] A. Hamdi and A. Griewank, Properties of an augmented Lagrangian for
design optimization, Preprint-Number SPP1253-11-01.
[6] A. Hamdi and A. Griewank, Reduced Quasi-Newton Method for simultaneous design and optimization, Preprint-Number SPP1253-11-02.
[7] E. Hopf, The partial differential equation ut + uu! = µu!! , Communications
an Pure and Applied Mathematics, pages 201-230, 1950.
[8] K. Ito and K. Kunisch, Optimal control of the solid fuel ignition model
with H 1 -cost, SIAM J. CONTROL OPTIM., Vol.40, No. 5, pp. 1455-1472.
[9] A. Kauffmann, Optimal control of the solid fuel ignition model, Dissertation
TU Berlin, 1998 .
[10] A. Kauffmann and K. Kunisch, Optimal control of the solid fuel ignition
model, ESAIM:PROCEEDINGS, Vol.8, pp.65-76, 2000.
[11] E. Özkaya and N. Gauger, Single-step One-Shot aerodynamic Shape Optimization, Preprint-Number SPP1253-10-04.
[12] E. Özkaya and N. Gauger, Automatic transition from simulation to oneshot shape optimization with Navier-Stokes equation , GAMM-Mitt. 33,
No. 2, 133-147.
[13] S. Ta’asan, One-shot methods for optimal control of distributed parameter
systems I:Finite dimensional control, ICASE 91-2, 1991.
[14] F. Tröltzsch,
Vieweg, 2005.
Optimale Steuerung partieller Differentialgleichungen,
[15] S. Volkwein, Mesh independence of an augmented Lagrangian-SQP method
in Hilbert spaces and control problems for the Burgers equation, Dissertation, 1997.
29