Parameter estimation for Euler equations with uncertain inputs

Transcription

1 Parameter estimation for Euler equations wit uncertain inputs Sergiy Zuk Abstract Te paper presents a new state estimation algoritm for 2D incompressible Euler equations wit periodic boundary conditions and uncertain but bounded inputs and initial conditions. Te algoritm converges (in L 2 -sense) to a least squares estimator given incomplete and noisy observations. It is also sown experimentally tat te proposed algoritm applies to several conservation laws developing sock discontinuities. Te results are illustrated by numerical examples. I. INTRODUCTION Data assimilation algoritms represent a backbone of modern cyber-pysical systems. Tese algoritms allow one to optimally combine a priori knowledge encoded in equations of matematical pysics wit a posteriori information in te form of sensor readings. Weater forecasting is one of many examples were data assimilation is applied to improve predictions generated by ydrodynamical models. Te divergence-free Euler equations provide te most basic model of incompressible flows of omogeneous inviscid fluids [7], yet tis model may provide insigts in studies of turbulence. We refer te reader to [2] for a detailed overview of various matematical questions related to Euler equations. In tis 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 te Euler equations. Te resulting vorticity equation describes te rotation of te vorticity of te fluid velocity field (see Section II). We stress tat te omogenous vorticity equation possesses a nice property: te L 2 -norm of te vorticity is conserved over time. Te latter, in fact, allows one to prove existence and uniqueness IBM Researc, Dublin, Ireland, {sergiy.zuk,tigran}@ie.ibm.com Tigran Tcrakian of te solution of 2D Euler equations globally (in time) [7, Cor.3.3, p.6]. Assuming periodic boundary conditions, we apply Fourier-Galerkin (FG) approximation: we project te vorticity equation onto a subspace generated by {e inx } n N and obtain an ODE for te projection coefficients, a FG model. Note tat Fourier-Galerkin approximation possesses a spectral convergence rate provided te solution of te Euler equation is smoot []. Altoug te main results of tis work may be derived for te case of bounded domains wit non-penetration boundary conditions or unbounded domains (R 2 ), we focus on smoot periodic flows to simplify te convergence analysis. We refer te reader to Section II, were oter types of boundary conditions and weak solutions of Euler equations are discussed. Given an ODE representing projection coefficients, we design a discrete-in-time FG model suc tat te L 2 -norm of te discrete vorticity is conserved over time (see Section III-B). Tis property allows us to prove te convergence of te approximations provided by te discrete FG model to a solution of te continuous FG model. We ten derive te state estimator (in te form of a minimax filter) for te discrete FG model and note tat te combination of te mentioned convergence proof wit results of [6] may be used to derive a continuous formulation of te minimax filter (see Section III- C). To te best of our knowledge, tis result is new. A similar approac as been used to design data assimilation algoritms for scalar macroscopic traffic flow models [9] and flood models (StVenant equations) []. We stress tat te proposed metod sows very good performance not only for smoot periodic flows but also for conservation laws wit sock discontinuities and non-periodic boundary conditions (see Section IV). Data assimilation for systems of yperbolic conservation laws based on te calculus of variations

2 was proposed in [3], were te autors adopt te strategy: optimize, ten discretize so tat te estimate of te initial density does not depend on a discretization metod. In contrast, te present paper solves te filtering problem; tat is, te state estimate at time instant, t, depends on te observation at te same time instant and te previous estimate only. A comparison of classical filtering algoritms (extended Kalman filter and ensemble Kalman filter) for scalar conservation laws wit quadratic non-linearity may be found in [4]. Adaptive parameter estimators for yperbolic equations were considered for instance in [5]. Tis paper is organized as follows. Te next section presents a brief overview of Euler equations. Section II-A presents discrete in time FG model for Euler equations; te state estimation algoritm is sown in section III-C. Numerical experiments are given in section IV. Section V contains concluding remarks. Notation. Let Ω denote a subset of R 2 wit boundary Ω, and n is a normal vector for Ω poining outside, Ω T := Ω [, T ]. C s (Ω) denotes a space of continuously differentiable functions on Ω (up to order s), L 2 (Ω) is te space of square-integrable functions on Ω, H (Ω) is a Sobolev space of functions wit weak derivatives of L 2 (Ω)-class. div( u) = x u + x2 u 2, curl( u) = x u 2 x2 u, u = ( x u, x2 u), u = ( x2 u, x u) u v denotes te canonical inner product of R 2, H 2 := H H denotes te cartesian product of H wit itself. L 2 (t, t, H) := {f : f(t) H and T f(t) 2 H < + }. We write u = v a.e. on Ω if u(x) = v(x) for almost all x Ω. II. EULER EQUATIONS Assume tat ω verifies te vorticity-stream formulation of te Euler equation: t ω + u ω = f, ψ = ω, u = ū + ψ, ω() = curl( u ), (x, t) Ω T := [ r, r] 2 [, T ], () were ū R 2 is a given vector, u C 2 (Ω) 2 is a smoot -periodic vector-function and f C (Ω T ) as zero mean, f(x, t)dx =. It is not ard to Ω prove (see [7, Prop 2.4,p.5]) tat u = (u, u 2 ) C 2 (Ω T ) 2 is a smoot -periodic solution of te incompressible Euler equation on Ω T : d u + ( u ) u + p = f, div( u) =, u() = u + ū, (2) were f C 2 (Ω T ) and curl( f) = f, and te pressure p is a function of u and f. We note tat Euler equation (2) as te unique smoot solution u for Ω = R 2 provided f = and u is a smoot function suc tat div( u ) = and curl( u ) L (R 2 ) (see [7, L.3.2, p.93 and Cor.3.3, p.6]). For te case of bounded domains wit smoot boundary one may also derive existence of smoot solutions for (2) provided u and f are smoot and u n = on Ω (see [, p.356]). Weak solutions (or Sobolev space solutions) for (2) were constructed in [2] provided curl( u ) is bounded, and f C(, T, L p (Ω)) is so tat curl( f) is bounded. Solutions of L 2 (, T, L 2 (R 2 ) 2 )-class corresponding to so called vortex seets were discussed in [7, p.33]. A. Fourier-Galerkin approximation Define b( u, w, v) := ( u w)vdx. Ω Assume tat div( u) = and u, w, v are smoot -periodic functions on Ω. We find integrating by parts: b( u, w, v) = b( u, v, w). (3) Now, we apply Fourier spectral metod to construct a finite dimensional approximation of (). Let s Z 2 and define φ s (x) := eiπr s x 2r. It is known tat {φ s } s Z 2 is a total ortonormal system in L 2 (Ω) and (φ s, φ k ) L2 (Ω) = δ s,k δ s2,k 2 were s i, k i are components of s, k. Now, we multiply () by φ s

3 and integrate over Ω. We obtain a weak vorticity formulation: d (ω, φ s) L2 (Ω) + b( u, ω, φ s ) = (f, φ s ) L2 (Ω). Let us now set ω N (x, t) := k ω i N k(t)φ k (x) wit ω k (t) := (ω(t), φ k ) L 2 (Ω) and plug ω N into te above formulation. We get te following ODE for te coefficients: ω s + b( u, φ k, φ s )ω k (t) = (f, φ s ) L 2 (Ω). k i N Define ω(t) := {ω s (t)} s,2 N, f(t) := {(f(t), φ s ) L2 (Ω)} s,2 N and set B( ω) := {b( u, φ k, φ s )} s,2, k,2 N. Now we use tis notation to rewrite te finite dimensional vorticity formulation in te vector form: d ω + B( ω) ω = f wit initial condition ω() = ω were ω := {ω s ()} s,2 N. We stress tat B( ω) = B ( ω) as b( u, φ k, φ s ) = b( u, φ s, φ k ) by (3) so tat B( w) is skew-symmetric. A. Problem statement Let ω solve III. MAIN RESULTS d ω + B( ω) ω = f, ω() = ω, (4) and assume tat a vector-function y is observed in te following form ( k,2 M): y k (t) = (H k, φ s ) L 2 (Ω)ω s (t) + η k (t), (5) s,2 N were H k L 2 (Ω) and η = {η k } k,2 M is a measurable vector-function modelling noise. Our aim is to construct a state estimate ˆω(T ) for ω(t ) given data y and assuming tat ω S ω + T f Q f + η R η (6) provided S, Q, R are symmetric positive definite matrices of appropriate dimensions. Let us stress tat if ω is te solution of (4), ω N corresponds to ω and ω solves () ten according to [, T.5.] we ave: ω N (, t) ω(, t) L2 (Ω) e g(t) N 2l ω(, ) 2 H l (Ω) + e g(t) N 2 l max τ t ω(, τ) H l (Ω), g(t). (7) Tis so called spectral convergence rate justifies our coice of te state equation. Namely, (4) is a standard finite dimensional Fourier-Galerkin model wit uncertain input f. Now, following te idea of [4], [5] we may incorporate te effect of te unresolved modes (te projection error) by simply adding anoter model error term e and introduce an additional algebraic equation to filter out inadmissible e (see [8], [3]) so tat te exact projection coefficients of ω will be among te solutions of (4). On te oter and, te above convergence rate estimate suggests tat te effect of e is negligible for reasonably large N and l (so for smoot ω) and it may be terefore absorbed by f (by increasing te size of te ellipsoid (6)). B. Discrete Fourier-Galerkin model Assume f =. If we multiply () by ω and integrate over Ω we get: 2 t ω 2 L 2 (Ω) = (f, ω) L 2 (Ω) as b( u, ω, ω) = by (3). Hence, L 2 (Ω)-norm of ω is conserved. We stress tat (4) as te same property, namely ω 2 R is conserved as B( ω) is skewsymmetric. In wat follows we propose a metod 2N+ wic approximates ω(j) by w j := w(j), j =, m, := T m, m N and te norm of w j is conserved. By Newton-Leibniz formula we get: ω((j + )) = ω(j) (j+) j B( ω(τ)) ω(τ)dτ Define B j := B( ω(j)). Approximating te integral by te trapezoidal rule one gets: (I+ 2 B j+) ω((j+)) = (I 2 B j) ω(j)+o( 3 ). ω N (x, t) := k i N ω k(t)φ k (x) were te coefficients ω k are components of ω

4 By noting tat B j+ = B j +B( d ω (j))+o(2 ) we can simplify te above equation compromising te order of te approximation: specifically, approximating B j+ by B j + B( d ω (j)) we reduce te order down to O( 2 ) (locally); if we simply take B j we get O()-approximation. In wat follows we stick to te latter and define w j as a solution of te linear system: (I + 2 B j) w j+ = (I 2 B j) w j, w = ω. (8) Note tat (I + 2 B j) is invertible as B j is skewsymmetric and K j := (I + 2 B j) (I 2 B j) is te Caley transform of te skew-symmetric matrix B j. Hence, K j is an ortogonal matrix and so w j 2 R 2N+ = w 2 R 2N+. For j =,..., m and j t < (j + ) we define te following functions: U (m) wj++ wj (t) := 2, V (m) (t) = w j and W (m) (t) = w j (t j)b( w j )U (m) (t). Since w j 2 R = w 2N+ 2 R it follows tat 2N+ V (m) L2 (,T ), U (m) L2 (,T ) C < + and so W (m) L2 (,T ) C 2 < +. Terefore, te sequences of piecewise constant functions, {V (m) } m N, {U (m) } m N contain weakly convergent subsequences in L 2 (, T ). Te same olds true for te sequence {W (m) } m N. We denote te convergent subsequences by {V (mi) }, {U (mi) } and {W (mi) } respectively and let V, U and W be teir limiting functions. We claim tat all te tree mentioned sequences converge strongly and V = U = W. Indeed, dw (mi) dw (mi) = B(V (mi) (t))u (mi) (t) and so is bounded in L 2 (, T ). Hence, {W (mi) } weakly converges in H (, T ) tat implies strong convergence in L 2 (, T ). Now, V (mi) U (mi) L2 (,T ) 3 2 m 2 w 2 C 2 R 2N+ 2 were C := 2N+ j= B(e j ) 2, e j is j-t canonical basis vector. Hence, by taking weak limit (m ) and using weak lowersemicontinuity of L 2 -norm we get: V = U. On te oter and W (mi) V (mi) 2 L 2 (,T ) 3 3 m j= B( w j)u (mi) (t) 2 R 2N+ 3 3 C w 4 R 2N+. By te same argument we get: W = V. Taking weak limits in dw (mi) = B(V (mi) (t))u (mi) (t) we find tat: dw = B(W )W (t). Now, recalling te spectral convergence rate estimate (7) and noting tat tere exists unique smoot solution of () we deduce tat any convergent subsequence of U (m), V (m) and W (m) as te same limit. Te latter proves tat te entire sequences U (m), V (m) and W (m) are weakly convergent and sare te same limiting function W wic is te unique solution of (4). Let us summarize te above results. Proposition : If f = ten (i) (4) as a unique solution ω suc tat ω() R 2N+ = ω(t) R 2N+, (ii) a sequence of piecewise constant functions V (m) (t) = w(j), j t < (j + ) converges to ω in L 2 (, T ) provided w(j) solves (8) and := T m, m N. We note tat it is not ard to generalize point (ii) of te latter proposition to te case f. We omit tis generalization ere for te sake of space. C. State estimator Following [6] we introduce te following system of linear Hamiltonian equations: ( I 2 B j 2 H RH )( Uj+ 2 Q I+ 2 Bj 2 H RH ) ( I+ )( V j+ = 2 B j IPj ) 2 Q I, 2 Bj (9) were P j := U j V j for j > and P = S, and B j := B(ˆω j ), and ˆω j solves te following system

5 of linear equations: (I + 2 B j + 2 P j+h RH)ˆω j+ = (I 2 B j 2 P jh RH)ˆω j () + 2 P j+ H R y j+ + P j H R y j 2 were H := {(H k, φ s ) L 2 (Ω)} k,2, s,2 N. By combining te idea of te proof of Proposition wit a well-known stabilization effect brougt by P j, te solution of approximated Riccati equation (see [6] for te details), one may prove te following proposition. Proposition 2: Let U j, V j, P j and ˆω j be defined as above and := T m, m N. A sequence of piecewise constant functions V (m) (t) = V j, U (m) (t) = U j, ˆω (m) (t) = ˆω j for j t < (j+) converges to U, V and ˆω in L 2 (, T ). Moreover, U, V and ˆω represent te unique solution of te following system of equations: dˆω = B(ˆω)ˆω + P H RH( y H ˆω), ˆω() =, du = B (ˆω)U + H RHV, U() = I, dv = Q U B(ˆω)V, V () = S, P = V U. () In fact, te latter proposition presents a continuous version of te minimax filter (equation ()) and equations (9)-() define an approximation for te filter. In te following section we illustrate te convergence properties of tis approximation on numerical examples. IV. NUMERICAL EXPERIMENTS A. Euler equations For te divergence-free Euler system, we look for solutions in te space, span{φ k (x) := φ k (x)φ k2 (y)} k=...n x,k 2=...N y, were ψ k (z) = sin(k πz/l), L = 2r is te lengt of eac side of te spatial domain, and N x and N y are te numbers of basis functions in eac of te spatial, (a) at t =.5 (b) at t =.5 (c) at t =. (d) at t =. (e) at t = 3. (f) at t = 3. Fig. : s and trut for different times dimensions. Te coice of basis satisfies te boundary conditions, ω = on Ω. Te corresponding Fourier-Galerkin model (4) is subject to te perturbation vector f wic as only two non-zero entries, meaning tat in effect, we perturb only two modes: one low, and one ig frequency mode. For te filter, we first generate observations y by running (4) te Euler system forward in timewit initial condition represented by a translation and scaling of Gaussian distributions. Te filter is initialized to zero, meaning we assume no knowledge of te initial condition. Observations are taken on an evenly spaced 5 5 grid away from te boundary, were te state is fixed at zero. For bot te generation of observations and te filtering, we coose N x = N y = 6. Figure sows te estimate and trut at different times. Te sparsity of te observations is apparent from Figure a,

6 u u.4.2 n=, m=.4.2 n=2, m=2.2 time =. Spectral coefficient Spectral coefficient Observations Perturbed observations.2 Time Time.6 (a) Projection coefficients for k = k 2 = (b) Projection coefficients for k = k 2 = n=3, m=3.4.2 Spectral coefficient Relative L 2 norm of estimation error x Fig. 3: True traffic state versus estimate at t= Time Time time = 3. (c) Projection coefficients for(d) Relative L 2 -error of te k = k 2 = 3 estimate.2 Observations Perturbed observations.8 wic is from an early point in te estimation. In Figure c, we see tat by t =., te estimate does still not fully mimic te trut, but tat te flow is being captured. In Figure e, te estimate appears to capture te trut quite well. Figures 2a - 2c sow te estimated and true projection coefficients for some of te low wave numbers. B. Ligtill-Witam-Ricards (LWR) model In tis section we apply te above metod to scalar conservation laws. Recall from [9] tat te standard equilibrium traffic-flow model, LWR model consists of a scalar conservation law, t u(x, t) + x f(u(x, t)) =, (2) wit initial data u (x) = u(x, ) were u : R R + R is te traffic density, x R and t R + are te independent variables, space and time respectively, and f : R R is te flux function. ) A typical flux function is f(u) = uv m ( u u m were te constants, V m and u m, are te maximum speed and te maximum density respectively. We impose periodic boundary conditions on te interval (, 2π). Unlike te Euler equation presented above, x Fig. 4: True traffic state versus estimate at t=3 tis model develops sock discontinuities even subject to periodic boundary conditions and smoot initial condition. Te filter as been applied to te Fourier-Galerkin model 2 wic as been used to generate observations. Te filter as been initialized to. Te estimation results are presented on te figures 3-5. We refer te reader to [9] for te furter details. C. StVenant equations Tis final example sows tat te proposed state estimator may andle systems of conservation laws 2 Te model ad artificial viscosity term activated on iger order modes to allow for correct sock tracking

7 u time = 6..2 Observations Perturbed observations x Fig. 7:, u and te estimate after 2 time steps Fig. 5: True traffic state versus estimate at t=6 Fig. 8:, u and te estimate after 2 time steps Fig. 6: Initial conditions for, u and te estimate wit non-periodic boundary conditions. Te standard equilibrium flood model consists of a system of scalar conservation laws: t + x (u) =, t (u) + x (u 2 + g2 2 ) = (3) wit boundary conditions u(, t) = u l (t) and (, t) = l (t) on (, ), were is te fluid dept, u is te averaged velocity and g is te gravitational constant. Te Discountinuos Galerkin metod as been applied to te above equation to generate observations. Tere was no perturbation, so f =. Te initial conditions for te model and filter are presented at Figure 6. Te estimation results are presented on te figures 7-8. We refer te reader to [] for te furter details. V. CONCLUSION Te paper presents te discrete and continuous versions of te minimax state estimator for Euler equations. Te estimator is derived for a Fourier- Galerkin model wic approximates smoot solutions of te Euler equation wit spectral convergence rate. A very curious topic for te future researc is to develop te idea of [9] for weak solutions of Euler equations, tat is to combine te presented state estimation approac and vanising viscosity metod [] to design a state estimator for L solutions of Euler equations in vorticitystream formulation. We stress tat te presented metod and convergence results apply to Navier- Stokes equations in dimension 2 witout major modifications.

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