Parameter estimation for Euler equations with uncertain inputs

Parameter estimation for Euler equations with uncertain inputs
Sergiy Zhuk
Tigran Tchrakian
Abstract— The paper presents a new state estimation
algorithm for 2D incompressible Euler equations
with periodic boundary conditions and uncertain but
bounded inputs and initial conditions. The algorithm
converges (in L2 -sense) to a least squares estimator
given incomplete and noisy observations. It is also
shown experimentally that the proposed algorithm
applies to several conservation laws developing shock
discontinuities. The results are illustrated by numerical examples.
I. I NTRODUCTION
Data assimilation algorithms represent a backbone
of modern cyber-physical systems. These algorithms allow one to optimally combine a priori
knowledge encoded in equations of mathematical
physics with a posteriori information in the form of
sensor readings. Weather forecasting is one of many
examples where data assimilation is applied to
improve predictions generated by hydrodynamical
models. The divergence-free Euler equations provide the most basic model of incompressible flows
of homogeneous inviscid fluids [7], yet this model
may provide insights in studies of turbulence. We
refer the reader to [2] for a detailed overview
of various mathematical questions related to Euler
equations.
of the solution of 2D Euler equations globally (in
time) [7, Cor.3.3, p.116]. Assuming periodic boundary conditions, we apply Fourier-Galerkin (FG) approximation: we project the vorticity equation onto
a subspace generated by {einx }|n|≤N and obtain an
ODE for the projection coefficients, a FG model.
Note that Fourier-Galerkin approximation possesses
a spectral convergence rate provided the solution of
the Euler equation is smooth [1]. Although the main
results of this work may be derived for the case
of bounded domains with non-penetration boundary
conditions or unbounded domains (R2 ), we focus
on smooth periodic flows to simplify the convergence analysis. We refer the reader to Section II,
where other types of boundary conditions and weak
solutions of Euler equations are discussed.
In this paper we design a new state estimator for 2D
Euler equations subject to uncertain but bounded
input and unknown initial conditions. We rely on a
vorticity-stream formulation of the Euler equations.
The resulting vorticity equation describes the rotation of the vorticity of the fluid velocity field (see
Section II). We stress that the homogenous vorticity
equation possesses a nice property: the L2 -norm of
the vorticity is conserved over time. The latter, in
fact, allows one to prove existence and uniqueness
Given an ODE representing projection coefficients,
we design a discrete-in-time FG model such that
the L2 -norm of the discrete vorticity is conserved
over time (see Section III-B). This property allows
us to prove the convergence of the approximations
provided by the discrete FG model to a solution
of the continuous FG model. We then derive the
state estimator (in the form of a minimax filter)
for the discrete FG model and note that the combination of the mentioned convergence proof with
results of [6] may be used to derive a continuous
formulation of the minimax filter (see Section IIIC). To the best of our knowledge, this result is
new. A similar approach has been used to design
data assimilation algorithms for scalar macroscopic
traffic flow models [9] and flood models (StVenant
equations) [11]. We stress that the proposed method
shows very good performance not only for smooth
periodic flows but also for conservation laws with
shock discontinuities and non-periodic boundary
conditions (see Section IV).
IBM
Research,
Dublin,
{sergiy.zhuk,tigran}@ie.ibm.com
Data assimilation for systems of hyperbolic conservation laws based on the calculus of variations
Ireland,
was proposed in [3], where the authors adopt the
strategy: “optimize”, then “discretize” so that the
estimate of the initial density does not depend on
a discretization method. In contrast, the present
paper solves the filtering problem; that is, the state
estimate at time instant, t, depends on the observation at the same time instant and the previous
estimate only. A comparison of classical filtering
algorithms (extended Kalman filter and ensemble
Kalman filter) for scalar conservation laws with
quadratic non-linearity may be found in [4]. Adaptive parameter estimators for hyperbolic equations
were considered for instance in [5].
where ū ∈ R2 is a given vector, ~u0 ∈ C 2 (Ω)2 is a
1
smooth 1-periodic
R vector-function and f ∈ C (ΩT )
has zero mean, Ω f (x, t)dx = 0. It is not hard to
prove (see [7, Prop 2.4,p.50]) that ~u = (u1 , u2 )> ∈
C 2 (ΩT )2 is a smooth 1-periodic solution of the
incompressible Euler equation on ΩT :
This paper is organized as follows. The next section
presents a brief overview of Euler equations. Section II-A presents discrete in time FG model for
Euler equations; the state estimation algorithm is
shown in section III-C. Numerical experiments are
given in section IV. Section V contains concluding
remarks.
We note that Euler equation (2) has the unique
smooth solution ~u for Ω = R2 provided f~ = 0 and
~u0 is a smooth function such that div(~u0 ) = 0 and
curl(~u0 ) ∈ L1 (R2 ) (see [7, L.3.2, p.93 and Cor.3.3,
p.116]). For the case of bounded domains with
smooth boundary one may also derive existence
of smooth solutions for (2) provided ~u0 and f~
are smooth and ~u0 · ~n = 0 on ∂Ω (see [10,
p.356]). Weak solutions (or Sobolev space solutions) for (2) were constructed in [12] provided
| curl(~u0 )| is bounded, and f~ ∈ C(0, T, Lp (Ω))
is so that | curl(f~)| is bounded. Solutions of
L2 (0, T, L2 (R2 )2 )-class corresponding to so called
vortex sheets were discussed in [7, p.303].
Notation. Let Ω denote a subset of R2 with boundary ∂Ω, and ~n is a normal vector for ∂Ω poining
outside, ΩT := Ω × [0, T ]. C s (Ω) denotes a space
of continuously differentiable functions on Ω (up
to order s), L2 (Ω) is the space of square-integrable
functions on Ω, H 1 (Ω) is a Sobolev space of
functions with weak derivatives of L2 (Ω)-class.
div(~u) = ∂x1 u1 +∂x2 u2 , curl(~u) = ∂x1 u2 −∂x2 u1 ,
∇u = (∂x1 u, ∂x2 u)> , ∇⊥ u = (−∂x2 u, ∂x1 u)>
~u · ~v denotes the canonical inner product of R2 ,
H 2 := H × H denotes the cartesian product of
H withR itself. L2 (t0 , t1 , H) := {f : f (t) ∈
T
H and 0 kf (t)k2H dt < +∞}. We write u = v
a.e. on Ω if u(x) = v(x) for almost all x ∈
Ω.
d~u
+ (~u · ∇)~u + ∇p = f~ ,
dt
div(~u) = 0 , ~u(0) = ~u0 + ū ,
(2)
where f~ ∈ C 2 (ΩT ) and curl(f~) = f , and the
pressure p is a function of ~u and f~.
A. Fourier-Galerkin approximation
Define
Z
(~u · ∇w)vdx .
b(~u, w, v) :=
Ω
Assume that div(~u) = 0 and ~u, w, v are smooth
1-periodic functions on Ω. We find integrating by
parts:
II. E ULER EQUATIONS
b(~u, w, v) = −b(~u, v, w) .
Assume that ω verifies the vorticity-stream formulation of the Euler equation:
Now, we apply Fourier spectral method to construct
a finite dimensional approximation of (1). Let s ∈
iπr −1 s·x
Z2 and define φs (x) := e 2r . It is known that
{φs }s∈Z2 is a total orthonormal system in L2 (Ω)
and (φs , φk )L2 (Ω) = δs1 ,k1 δs2 ,k2 where si , ki are
components of s, k. Now, we multiply (1) by φs
∂t ω + ~u · ∇ω = f ,
−∆ψ = ω ,
⊥
~u = ū + ∇ ψ , ω(0) = curl(~u0 ) ,
2
(x, t) ∈ ΩT := [−r, r] × [0, T ] ,
(1)
(3)
and integrate over Ω. We obtain a weak vorticity
formulation:
d
(ω, φs )L2 (Ω) + b(~u, ω, φs ) = (f, φs )L2 (Ω) .
dt
P
Let us now set ω N (x, t) := |ki |≤N ωk (t)φk (x)
with ωk (t) := (ω(t), φk )L2 (Ω) and plug ω N into
the above formulation. We get the following ODE
for the coefficients:
X
ω̇s +
b(~u, φk , φs )ωk (t) = (f, φs )L2 (Ω) .
|ki |≤N
Define ω
~ (t) := {ωs (t)}|s1,2 |≤N , f~(t)
{(f (t), φs )L2 (Ω) }|s1,2 |≤N and set
:=
B(~
ω ) := {b(~u, φk , φs )}|s1,2 |,|k1,2 |≤N .
Now we use this notation to rewrite the finite dimensional vorticity formulation in the vector form:
d~
ω
+ B(~
ω )~
ω = f~ with initial condition ω
~ (0) =
dt
ω
~ 0 where ω
~ 0 := {ωs (0)}|s1,2 |≤N . We stress that
B(~
ω ) = −B > (~
ω ) as b(~u, φk , φs ) = −b(~u, φs , φk )
by (3) so that B(w)
~ is skew-symmetric.
A. Problem statement
Let ω
~ solve
ω
~ (0) = ω
~0 ,
kω N (·, t) − ω(·, t)kL2 (Ω) ≤ eg(t) N −2l kω(·, 0)k2H l (Ω)
+ eg(t) N 2−l max kω(·, τ )kH l (Ω) , g(t) ≥ 0 .
0≤τ ≤t
(7)
This so called spectral convergence rate justifies
our choice of the state equation. Namely, (4) is a
standard finite dimensional Fourier-Galerkin model
with uncertain input f~. Now, following the idea
of [14], [15] we may incorporate the effect of the
unresolved modes (the projection error) by simply
adding another model error term e and introduce an
additional algebraic equation to filter out inadmissible e (see [8], [13]) so that the exact projection
coefficients of ω will be among the solutions of (4).
On the other hand, the above convergence rate
estimate suggests that the effect of e is negligible
for reasonably large N and l (so for smooth ω) and
it may be therefore absorbed by f~ (by increasing
the size of the ellipsoid (6)).
B. Discrete Fourier-Galerkin model
III. M AIN RESULTS
d~
ω
+ B(~
ω )~
ω = f~ ,
dt
Let us stress that if ω
~ is the solution of (4), ω N
1
corresponds to ω
~ and ω solves (1) then according
to [1, T.5.1] we have:
(4)
and assume that a vector-function ~y is observed in
the following form (|k1,2 | ≤ M ):
X
~yk (t) =
(Hk , φs )L2 (Ω) ωs (t) + ηk (t) , (5)
|s1,2 |≤N
where Hk ∈ L2 (Ω) and ~η = {ηk }|k1,2 |≤M is a
measurable vector-function modelling noise. Our
aim is to construct a state estimate ω̂(T ) for ω
~ (T )
given data ~y and assuming that
Z T
ω
~ 0> S~
ω0 +
f~> Qf~ + ~η > R~η dt ≤ 1
(6)
0
provided S, Q, R are symmetric positive definite
matrices of appropriate dimensions.
Assume f = 0. If we multiply (1) by ω and integrate over Ω we get: 21 ∂t kωk2L2 (Ω) = (f, ω)L2 (Ω) as
b(~u, ω, ω) = 0 by (3). Hence, L2 (Ω)-norm of ω is
conserved. We stress that (4) has the same property,
namely k~
ω k2R2N +1 is conserved as B(~
ω ) is skewsymmetric. In what follows we propose a method
which approximates ω
~ (jh) by w̃j := w̃(jh), j =
T
0, m, h := m
, m ∈ N and the norm of w̃j is
conserved.
By Newton-Leibniz formula we get:
Z (j+1)h
ω
~ ((j + 1)h) = ω
~ (jh) −
B(~
ω (τ ))~
ω (τ )dτ
jh
Define Bj := B(~
ω (jh)). Approximating the integral by the trapezoidal rule one gets:
h
h
ω ((j+1)h) = (I− Bj )~
ω (jh)+O(h3 ) .
(I+ Bj+1 )~
2
2
1 ω N (x, t)
P
:= |ki |≤N ωk (t)φk (x) where the coefficients
ωk are components of ω
~
d~
ω
By noting that Bj+1 = Bj + hB( (jh)) + O(h2 )
dt
we can simplify the above equation compromising
the order of the approximation: specifically, approxd~
ω
imating Bj+1 by Bj + hB( (jh)) we reduce the
dt
order down to O(h2 ) (locally); if we simply take
Bj we get O(h)-approximation. In what follows we
stick to the latter and define w̃j as a solution of the
linear system:
(I +
h
h
Bj )w̃j+1 = (I − Bj )w̃j , w̃0 = ω
~ 0 . (8)
2
2
Note that (I +
symmetric and
h
2 Bj )
Kj := (I +
is invertible as Bj is skewh
h
Bj )−1 (I − Bj )
2
2
is the Caley transform of the skew-symmetric matrix Bj . Hence, Kj is an orthogonal matrix and so
kw̃j k2R2N +1 = kw̃0 k2R2N +1 .
For j = 1, . . . , m − 1 and jh ≤ t <
(j + 1)h we define the following functions:
w̃j+1 +w̃j
U (m) (t) :=
, V (m) (t) = w̃j and
2
(m)
W
(t) = w̃j − (t − jh)B(w̃j )U (m) (t).
Since kw̃j k2R2N +1 = kw̃0 k2R2N +1 it follows that
kV (m) kL2 (0,T ) , kU (m) kL2 (0,T ) ≤ C1 < +∞ and
so kW (m) kL2 (0,T ) ≤ C2 < +∞. Therefore,
the sequences of piecewise constant functions,
{V (m) }m∈N , {U (m) }m∈N contain weakly convergent subsequences in L2 (0, T ). The same holds
true for the sequence {W (m) }m∈N . We denote the
convergent subsequences by {V (mi ) }, {U (mi ) } and
{W (mi ) } respectively and let V ∗ , U ∗ and W ∗ be
their limiting functions.
We claim that all the three mentioned sequences
converge strongly and V ∗ = U ∗ = W ∗ .
dW (mi )
Indeed,
= −B(V (mi ) (t))U (mi ) (t) and
dt
dW (mi )
so
is bounded in L2 (0, T ). Hence,
dt
{W (mi ) } weakly converges in H 1 (0, T ) that
implies strong convergence in L2 (0, T ). Now,
(mi )
(mi )
3
2
1
2
1
2
k
≤ h m kw̃0 k2R2N +1 C2
P2N +1
2
j=1 kB(ej )k , ej is j-th
kV
−U
where C :=
canonical basis
L2 (0,T )
vector.
Hence,
by
taking
weak limit (m → ∞) and using weak lowersemicontinuity of L2 -norm we get: V ∗ = U ∗ . On
the other hand kW (mi ) − V (mi ) k2L2 (0,T ) ≤
Pm
h3
(mi )
(t)k2R2N +1
≤
j=1 kB(w̃j )U
3
3
h
4
3 Ckw̃0 kR2N +1 . By the same argument we
get: W ∗ = V ∗ . Taking weak limits in
dW (mi )
= −B(V (mi ) (t))U (mi ) (t) we find
dt
dW ∗
that:
= −B(W ∗ )W ∗ (t).
dt
Now, recalling the spectral convergence rate estimate (7) and noting that there exists unique smooth
solution of (1) we deduce that any convergent subsequence of U (m) , V (m) and W (m) has the same
limit. The latter proves that the entire sequences
U (m) , V (m) and W (m) are weakly convergent and
share the same limiting function W ∗ which is the
unique solution of (4).
Let us summarize the above results.
Proposition 1: If f = 0 then
(i) (4) has a unique solution ω
~ such that
ω (t)kR2N +1 ,
k~
ω (0)kR2N +1 = k~
(ii) a sequence of piecewise constant functions
V (m) (t) = w̃(jh), jh ≤ t < (j + 1)h
converges to ω
~ in L2 (0, T ) provided w̃(jh)
T
solves (8) and h := m
, m ∈ N.
We note that it is not hard to generalize point
(ii) of the latter proposition to the case f 6= 0.
We omit this generalization here for the sake of
space.
C. State estimator
Following [6] we introduce the following system of
linear Hamiltonian equations:
>
I− h
2 Bj
h −1
2Q
>
h
2 H RH
I+ h
2 Bj
Uj+1
Vj+1
=
>
I+ h
2 Bj
h −1
2Q
>
h
2 H RH
I− h
2 Bj
(9)
where Pj := Uj−1 Vj for j > 0 and P0 = S −1 , and
Bj := B(ω̂j ), and ω̂j solves the following system
I
Pj
,
of linear equations:
(I +
h
h
Bj + Pj+1 H > RH)ω̂j+1
2
2
h
h
(10)
= (I − Bj − Pj H > RH)ω̂j
2
2
h Pj+1 H > R~yj+1 + Pj H > R~yj
+
,
2
2
(a) Estimate at t = 0.05
(b) Truth at t = 0.05
(c) Estimate at t = 1.00
(d) Truth at t = 1.00
(e) Estimate at t = 3.00
(f) Truth at t = 3.00
where H := {(Hk , φs )L2 (Ω) }|k1,2 |,|s1,2 |≤N . By
combining the idea of the proof of Proposition 1
with a well-known stabilization effect brought by
Pj , the solution of approximated Riccati equation
(see [6] for the details), one may prove the following proposition.
Proposition 2: Let Uj , Vj , Pj and ω̂j be defined
T
, m ∈ N. A sequence
as above and h := m
of piecewise constant functions V (m) (t) = Vj ,
U (m) (t) = Uj , ω̂ (m) (t) = ω̂j for jh ≤ t < (j +1)h
converges to U , V and ω̂ in L2 (0, T ). Moreover,
U , V and ω̂ represent the unique solution of the
following system of equations:
dω̂
= −B(ω̂)ω̂ + P H > RH(~y − H ω̂) , ω̂(0) = 0 ,
dt
dU
= B > (ω̂)U + H > RHV , U (0) = I ,
dt
dV
= Q−1 U − B(ω̂)V , V (0) = S −1 , P = V U −1 .
dt
(11)
In fact, the latter proposition presents a continuous
version of the minimax filter (equation (11)) and
equations (9)-(10) define an approximation for the
filter. In the following section we illustrate the
convergence properties of this approximation on
numerical examples.
IV. N UMERICAL EXPERIMENTS
A. Euler equations
For the divergence-free Euler system, we look
for solutions in the space, span{φk (x) :=
φk1 (x)φk2 (y)}k1 =0...Nx ,k2 =0...Ny , where ψk1 (z) =
sin(k1 πz/L), L = 2r is the length of each side
of the spatial domain, and Nx and Ny are the
numbers of basis functions in each of the spatial
Fig. 1: Estimates and truth for different times
dimensions. The choice of basis satisfies the boundary conditions, ω = 0 on ∂Ω. The corresponding
Fourier-Galerkin model (4) is subject to the perturbation vector f~ which has only two non-zero
entries, meaning that in effect, we perturb only two
modes: one low, and one high frequency mode.
For the filter, we first generate observations ~y by
running (4) the Euler system forward in timewith
initial condition represented by a translation and
scaling of Gaussian distributions. The filter is initialized to zero, meaning we assume no knowledge
of the initial condition. Observations are taken on
an evenly spaced 15 × 15 grid away from the
boundary, where the state is fixed at zero. For both
the generation of observations and the filtering,
we choose Nx = Ny = 60. Figure 1 shows the
estimate and truth at different times. The sparsity
of the observations is apparent from Figure 1a,
n=1, m=1
n=2, m=2
1.4
1.4
Truth
Estimate
time = 1.00
Truth
Estimate
1.2
1.2
1.2
Estimate
Truth
1
Observations
Spectral coefficient
Spectral coefficient
1
0.8
0.6
0.8
Perturbed observations
1
0.6
0.4
0.8
0.4
0.2
0.2
0
0
0.5
1
1.5
2
2.5
Time
3
3.5
4
4.5
−0.2
5
u
0.6
0
0
0.5
1
1.5
2
2.5
Time
3
3.5
4
4.5
5
0.4
(a) Projection coefficients for (b) Projection coefficients for
k1 = k2 = 1
k1 = k2 = 2
n=3, m=3
0.2
1.4
0.6
Truth
Estimate
0
1.2
Relative L2−norm of estimation error
0.4
Spectral coefficient
0.2
0
−0.2
−0.4
0
1
2
3
4
5
6
x
0.8
0.6
Fig. 3: True traffic state versus estimate at t=1
0.4
0.2
−0.6
−0.8
1
0
0
0.5
1
1.5
2
2.5
Time
3
3.5
4
4.5
5
0
0.5
1
1.5
Time
2
2.5
3
time = 3.00
(c) Projection coefficients for(d) Relative L2 -error of the
k1 = k2 = 3
estimate
1.2
Estimate
Truth
Observations
1
Perturbed observations
0.8
B. Lighthill-Whitham-Richards
model
(LWR)
In this section we apply the above method to scalar
conservation laws. Recall from [9] that the standard equilibrium traffic-flow model, LWR model
consists of a scalar conservation law,
∂t u(x, t) + ∂x f (u(x, t)) = 0,
(12)
with initial data u0 (x) = u(x, 0) where u : R ×
R+ → R is the traffic density, x ∈ R and t ∈
R+ are the independent variables, space and time
respectively, and f : R → R is the flux function.
A typical flux function is f (u) = uVm 1 − uum
where the constants, Vm and um , are the maximum
speed and the maximum density respectively. We
impose periodic boundary conditions on the interval
(0, 2π). Unlike the Euler equation presented above,
0.6
u
which is from an early point in the estimation. In
Figure 1c, we see that by t = 1.00, the estimate
does still not fully mimic the truth, but that the flow
is being captured. In Figure 1e, the estimate appears
to capture the truth quite well. Figures 2a - 2c show
the estimated and true projection coefficients for
some of the low wave numbers.
0.4
0.2
0
0
1
2
3
4
5
6
x
Fig. 4: True traffic state versus estimate at t=3
this model develops shock discontinuities even subject to periodic boundary conditions and smooth
initial condition. The filter has been applied to the
Fourier-Galerkin model2 which has been used to
generate observations. The filter has been initialized
to 0. The estimation results are presented on the
figures 3-5. We refer the reader to [9] for the further
details.
C. StVenant equations
This final example shows that the proposed state
estimator may handle systems of conservation laws
2 The model had artificial viscosity term activated on higher
order modes to allow for correct shock tracking
time = 6.00
1.2
Estimate
Truth
Observations
1
Perturbed observations
0.8
u
0.6
0.4
0.2
0
0
1
2
3
4
5
Fig. 7: h, hu and the estimate after 20 time steps
6
x
Fig. 5: True traffic state versus estimate at t=6
Fig. 8: h, hu and the estimate after 120 time steps
Fig. 6: Initial conditions for h, hu and the estimate
V. C ONCLUSION
with non-periodic boundary conditions. The standard equilibrium flood model consists of a system
of scalar conservation laws:
∂t h + ∂x (hu) = 0 ,
∂t (hu) + ∂x (hu2 +
gh2
)=0
2
(13)
with boundary conditions u(0, t) = ul (t) and
h(0, t) = hl (t) on (0, 1), where h is the fluid depth,
u is the averaged velocity and g is the gravitational
constant. The Discountinuos Galerkin method has
been applied to the above equation to generate
observations. There was no perturbation, so f~ = 0.
The initial conditions for the model and filter are
presented at Figure 6. The estimation results are
presented on the figures 7-8. We refer the reader
to [11] for the further details.
The paper presents the discrete and continuous
versions of the minimax state estimator for Euler
equations. The estimator is derived for a FourierGalerkin model which approximates smooth solutions of the Euler equation with spectral convergence rate. A very curious topic for the future
research is to develop the idea of [9] for weak
solutions of Euler equations, that is to combine the
presented state estimation approach and vanishing
viscosity method [1] to design a state estimator
for L∞ solutions of Euler equations in vorticitystream formulation. We stress that the presented
method and convergence results apply to NavierStokes equations in dimension 2 without major
modifications.
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