Existence Theory of Abstract Approximate Deconvolution Models of Turbulence

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1 Existence Theory of Abstract Approximate Deconvolution Models of Turbulence Iuliana Stanculescu Abstract This report studies an abstract approach to modeling the motion of large eddies in a turbulent flow. If the Navier-Stokes equations NSE are averaged with a local, spatial convolution type filter, φ = g δ φ, the resulting system is not closed due to the filtered nonlinear term uu. An approximate deconvolution operator D is a bounded linear operator which is an approximate filter inverse Du = approximation of u. Using this general deconvolution operator yields the closure approximation to the filtered nonlinear term in the NSE uu DuDu. Averaging the Navier-Stokes equations using the above closure, possible including a time relaxation term to damp unresolved scales, yields the approximate deconvolution model ADM w t + Dw Dw ν w + q + χw = f and w =. Here w u, χ, and w is a generalized fluctuation. We derive conditions on the general deconvolution operator D that guarantee the existence and uniqueness of strong solutions of the model. We also derive the model s energy balance. Introduction Many approaches have been used to simulate turbulent flows. In large eddy simulation LES the evolution of averages is sought. These averages are defined through a local spatial averaging process associated with an averaging radius δ. Once an averaging radius and a filtering process is selected, an LES model can be developed and then solved numerically. One of the most interesting approaches to generate LES model is via approximate deconvolution or approximate/asymptotic inverse of the filtering operator. Approximate deconvolution models ADM are systematic rather than ad hoc. They can achieve high theoretical accuracy and shine in practical tests; they contain few or no fitting/tuning parameters. The ADM approach has thus proven itself to be very promising. However, among the very many known approximate deconvolution operators from image processing, e.g. [], so far only two have been studied for LES modeling, the van Cittert deconvolution operator and Geurts approximate inverse filter. Their success suggest that it is time to develop a general theory of LES-ADM as a guide to development of models based on other, possibly better, deconvolution operators and refinement of existing ones. Two basic requirements of an acceptable ADM are that a unique, strong solution exists and that the model s global energy balance be close in some sense to that of the NSE. Herein, we consider these two important questions. We find, in Section 3 see P, P, and P3, conditions on the approximate deconvolution operator D that guarantee that the ADM has a unique strong solution. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA,56, U.S.A.; ius@pitt.edu, ius, partially supported by NSF grant DMS 586

2 In Section 4 see Theorem 4., we derive the model s energy balance and show that under these conditions on the deconvolution operator the model correctly captures the global energy balance of the large scales. The underlying fluid velocity ux,t and pressure px,t are solutions of the Navier-Stokes equations NSE: u t + u u ν u + p = f and u = in R n,. where ν = µ/ρ is the kinematic viscosity, f is the body force, and R n n = or 3 is the flow domain. We consider L-periodic boundary conditions to separate the interior closure problem from other important problems associated with filtering through a boundary, [4] and finding boundary conditions for local averages near wall laws, [], ux + Le j, t = ux, t, j =,..., n. The Navier-Stokes equations are supplemented by the initial condition, the usual normalization condition in the periodic case of zero mean velocity and pressure, and the assumption that all data are square integrable with zero mean ux, = u x and u dx = p dx =, u x, t dx <, fx, t dx <, and fx, tdx =, for t,. where =, L n. Let overbar denote a local, spatial averaging operator, such as averaging by convolution, that is linear and commutes with differentiation. Averaging the NSE, the non-closed equations for u and p, known as the Space Filtered Navier-Stokes equations SFNSE, are u t + u u ν u + p = f and u =..3 Since in general uu u u, the closure problem is to replace uu by a tensor Su, u depending only on u, not on u. If we denote filtering by u = Gu and D is an approximate inverse of G so u = Du, then the variant of approximate deconvolution model we consider herein approximates uu = DuDu =: Su, u. The deconvolution problem is central in image processing, []. Thus many algorithms can be adapted to give a possible LES closure model. The goal of deconvolution in LES is to recover accurately the resolved scales asymptotically as δ. The resulting LES model should have a lucid energy balance and favorable properties for its approximate solution. As an example, the N th van Cittert approximate deconvolution operator D N, defined precisely in Section 5, see also [], [6], [4], is an easy to construct bounded linear operator on L satisfying φ = D N φ + Oδ N+, for smooth φ. In other words, D N is an asymptotic as δ approximate inverse of G. With the van Cittert approximate deconvolution operator, the closure problem in.3 can be solved approximately but systematically by: u u D N u D N u + Oδ N+, for smooth u. More generally and beyond the van Cittert operator, there is little analytical guidance as to the properties needed of a deconvolution operator to produce a reliable LES model. Since inverting a filter is an ill-posed problem, a deconvolution operator can also be generated by any method for solving approximately ill-posed problems. Thus, there are many possible choices of D available. Once D is selected, we define the higher order fluctuation w = w Dw, if I DG is symmetric positive semi-definite and w = I DG I DGw otherwise.

3 In general, any approximate deconvolution operator D, used as a closure approximation, leads to the approximate deconvolution LES model w t + Dw Dw ν w + q + χw = f w =.4 w t= = u qdx =. The time relaxation term χw, where χ is a model parameter, is often included in the ADM to damp marginally resolved scales, see [9] and [6], so we include it in our analysis. If D is a symmetric positive definite operator then it induces a deconvolution weighted L inner product and norm defined by u, v D := u Dvdx and v D = v, v D. For specificity, we select the filtering operation v := I δ v. In Sections 3 and 4 we show that if the deconvolution operator D : L L is a bounded, self adjoint, and positive definite operator that commutes with differentiation, then a weak solution exists for the deconvolution model.4. In Section 5 we show that the weak solution is a unique strong solution and satisfies the energy equality: wt D + δ wt D + = t f, w D dτ + t ν w D + νδ w D + χw, I δ w D dτ w D + δ w D..5 Remark.. [ An Important Difference Between Deconvolution Models ] The SFNSE can be rewritten as u t + u u ν u + p + uu u u = f. Another possible deconvolution LES model approximates uu u u = Du Du Du Du. In the simplest case D = I this is the Bardina model w t + w w ν w + q + ww w w = f..6 This common approach is NOT covered by the theory herein. In fact, we believe but cannot prove yet that this approach can be unstable, unless sufficient ad hoc eddy viscosity is added thereby destroying the high accuracy of the ADM approach. We therefore have a strong preference for the simpler, accurate, and unconditionally stable model.4 over the similarity type deconvolution model.6. Preliminaries, Notations, and Function Spaces We start by briefly reviewing the concept of averaging/filtering in LES and define the function spaces and the norms needed for the variational formulation of the scale similarity model.. Averaging operators used in LES The idea of LES is to split the velocity into u = u + u a local, spatial average u and a fluctuation about the mean u. It is widely believed that given the random and chaotic character of the fluctuations, their average effects on the mean motion can successfully be modeled and thus the mean can be predicted accurately. The mean is defined by filtering or mollification convolution with an approximate identity. The goal is to predict the mean accurately. Let g denote a filter, 3

4 such as the Gaussian filter gx = 6 π e 6 x, and let δ denote the selected averaging radius. Define g δ x := δ 3 gx/δ. Averages are defined by convolution with the kernel g. Given a velocity u, its mean u and fluctuation u are defined by u = Gu and u = u u when ux := g δ x x ux dx.. R 3 Typical choices used in LES include the top-hat filter, sharp spectral cut-off, the Gaussian filter and differential filters, defined next, which also fit into the convolution formalism. Definition.. [ Differential Filter ] Let A := δ + I and let ϕ denote the unique L-periodic solution in of: Aϕ := δ ϕ + ϕ = ϕ.. Remark.. Under periodic boundary conditions and for constant δ both. and. commute with differentiation, preserve incompressibility, and can be written as convolution with approximate identity. Differential filters are well-established in LES, starting with the work of Germano [] and continuing with [], [8]. They have many connections to regularization processes such as the Yoshida regularization of semigroups and the very interesting work of Foias, Holm, Titi [7] and others on Lagrange averaging of the Navier-Stokes equations. The mean ux is the weighted average of u about the point x. As δ, the points near x are weighted more and more heavily, so intuitively we expect that u u as δ. This is known for convolution filters. For differential filters, it is also not difficult to show. Remark.. Expanding the velocity u in Fourier series we obtain ux = k ûke ik x where k is the wave number vector and k = πn L, for n Z3. Let k = k = k + k + k 3 be its magnitude. The Fourier coefficients are ûk = L 3 uxe ik x dx. Parseval s equality leads to L 3 ux dx = k ûk = k ûk. k =k Lemma.. Let u = δ + I u. Then for any u L k =k u u in L as δ. Proof. Given ε >, we show that for δ small enough u u < ε. Indeed, using Parseval s equality, as in Remark., we obtain u u = δ k + ûk = δ k + δ k ûk k k <k M k =k k =k and for any positive integer M we can write u u = δ k + δ k ûk + δ k + δ k k>m k =k ûk. 4

5 Furthermore, we have δ k k =k + δ k ûk k =k ûk and thus, for M large enough <k M k =k δ k + δ k ûk < ε..3 For this M we have k>m k =k δ k + δ k ûk δ M + δ M u.4 < ε, for some δ = δm. From.3 and.4 the conclusion follows.. Function Spaces, Weak and Strong Solution The notation for the function spaces we are using follows Temam, [] and Layton and Lewandowski [5]. Let represent the L norm. For k, let V k # denote the space of all [, L]3 -periodic functions with restriction on the cell =, L 3 in the Sobolev space H k. Thus We denote by V k # the associated norm: ϕ V k # = V k # = { u H k loc u is L-periodic }. / j ϕx dx, for all ϕ V#. k j= For the variational formulation of the approximate deconvolution model, we consider the spaces of periodic, divergence-free functions: and Further, we define D T = D = V k # = { u V k # u = } H = { u L u is L-periodic and u = } V = { u H u = }. { ψ C ψ is L-periodic, has compact support and { ψ C [, T ψ, t is L-periodic, has compact support and } ψdx = ψdx = Definition.. Let D be a symmetric, positive definite and bounded operator on L. The D inner product and norm on L are φ, ψ D := φ Dψdx and φ D = φ, φ D, for every φ L. }. 5

6 Definition.3. Let D and G be bounded operators on L. We define { φ := I DG φ, if I DG is symmetric positive semi-definite I DG I DG φ, otherwise. The semi-inner product and semi-norm on L are φ, ψ := φ, ψ and φ = φ, φ, for every φ L. To make progress in the mathematical understanding of an LES model the key idea is the notion of weak solution. For the NSE the notion of weak solution was introduced by Leray, [3]. We now introduce the notion of weak solution for the ADM.4. Definition.4. Let f L, T ; V and w V #. A function w : [, T ] R n is a weak solution of.4 if w L, T ; V # L, T ; H.5 and wt, ϕt for all ϕ D T. = T T [ w, ϕ ] ν w, ϕ DwDw, ϕ χw, ϕ dt t f, ϕdt + w, ϕ.6 Definition.5. The pair w, q is a strong solution of the deconvolution model.4 if w is a weak solution and w V # H, for a.e. t [, T ] w V #[, T ], for a.e. x.7 q V # L, for a.e. t, T ]. Remark.3. Because the inclusion V# H is compact, the inverse of the Stokes operator is bounded, self-adjoint, and compact. This implies that there exists an orthonormal basis of H consisting of eigenfunctions of the Stokes operator. 3 Existence of Weak Solutions of Approximate Deconvolution Models Due to the nonlinearity in.4 small changes in the deconvolution operator can yield significant positive or negative changes in the solution of the induced model. The averaging/convolution operator G is bounded, self-adjoint, positive definite and commutes with differentiation. We thus postulate that deconvolution operator D, its approximate inverse, is also bounded, self-adjoint, positive definite and commutes with differentiation. We postulate the following properties of the deconvolution operator D: P. D is a bounded linear operator on L, P. D is self-adjoint and positive definite, P3. D commutes with differentiation. Note that P, P, and P3 imply in particular that the D norm and the L norm are equivalent on L, i.e. there exist two positive constants C and C such that C ϕ ϕ D C ϕ, for all ϕ L. 6

7 Remark 3.. Let {ψ j } be eigenfunctions of the Stokes operator in H. Then, due to the periodic boundary conditions, span{ψ j } j is invariant under the Stokes operator and any other constant coefficient differential operators. By P3, it is also invariant under D. Under postulates P, P, and P3 on the deconvolution operator D, we prove that the ADM model.4 admits weak solutions, in the sense of the Definition.4. We follow the exposition on the existence of the weak solutions of the NSE, in Galdi [8]. The analysis includes the case χ = of no time relaxation. Theorem 3.. Let T > and D be a deconvolution operator satisfying the properties P, P, and P3. For w V # H and f L, T ; V, there exists a weak solution w L, T ; V # L, T ; H of.4 in the sense of the Definition.4. Moreover, for any t, T ], the following stability bound holds: t wt + δ wt + ν wτ + δ wτ dτ t C fτ dτ + w + δ w. 3. Proof. Let {ψ j } D be an orthonormal basis of H consisting of eigenfunctions of the Stokes operator, as in Remark.3. We are looking for Galerkin approximate solutions of the form: w k x, t = η kr tψ r x, for k N. 3. r= Since p, ψ j = p, ψ j = for j =,..., k, the functions {w k } k satisfy the ODE system: wk t, ψ j + ν w k, ψ j + Dw k Dw k, ψ j wk, ψ j for all j =,..., k. Using 3. in 3.3 and 3.4 it follows that r= η kr t ψ r, ψ j + ν η kr ψ r, ψ j + r= r,i= = f, ψ j 3.3 = w, ψ j, 3.4 η kr η ki Dψ r Dψ i, ψ j = f, ψ j η kr ψ r, ψ j = w, ψ j for all j =,..., k. For simplification, let a rj = ψ r, ψ j, a rij = Dψ r Dψ i, ψ j, f j = f, ψj and cj = w, ψ j. With these and since ψ i, ψ j = δ ij, equations 3.5 and 3.6 become r= η kj t + ν a rj η kr + a rij η kr η ki = f j 3.7 r= r,i= η kj = c j. 3.8 for j =,..., k. Since f j L, T, for j =,..., k from the theory of ODEs we know that the problem has a unique solution η kj W,, T k, for a small enough time interval T k T. Because w V # H, there exists u H such that u = w and: w, ψ j = u, ψ j, j =,..., k

8 From Remark 3., span{ψ j } j is invariant under the Stokes and differential operators. In particular span{ψ j } j is invariant under the deconvolution operator D and δ +I. Since Dw k is a linear combination of {ψ j } we deduce that Dw k span {ψ j } j and δ + Iw k span{ψ j } j. So, we can use δ + I w k as test function in 3.9. wk, δ + Iw k = u, δ + Iw k = u, w k. 3. Thus, w k +δ w k u + w k and we have the following à priori estimate w k + δ w k u. 3. Further, in 3.3 we can replace ψ j by I δ Dw k span {ψ j } j. Since I δ is symmetric, the nonlinear term vanishes: Dw k Dw k, I δ Dw k = Dw k Dw k, Dw k = and 3.3 becomes wk t, I δ Dw k + ν w k, I δ Dw k = f, Dw k. Applying properties P, P, and P3 of D, and integrating between and t we are led to w kt D + δ w k t D + ν = t t wk τ D + δ w k τ D dτ f, Dw k τdτ + w k D + δ w k D. 3. Moreover, since the D-norm and the L -norm are equivalent on L there exists a constant C such that t w kt + δ w k t + ν wk τ + δ w k τ dτ t t C f dτ + w k τ dτ + w k + δ w k. 3.3 Gronwall s inequality in 3.3 implies that for all t, T ] w kt + δ w k t + ν Ce t t Also, for all t, T ] we have that t wk τ + δ w k τ dτ f dτ + w k + δ w k. 3.4 t T e t f dτ + w k + δ w k Ce T f dτ + w k + δ w k. Let M := e T T f dτ + w k + δ w k, so that M is a constant independent of t and k. In particular, from 3.4 we deduce w k t M /. 3.5 Using the definition of the L -norm and 3. we get w k t = w k t, w k t = k r= η kr t. Thus / η kr t ηkrt = w k t M /. r= 8

9 We already know that T k T. The à priori bound 3.5 shows that we cannot have T k < T, since it would imply η kr t is unbounded as t T k, for some r {,..., k}. This shows T k = T, for all k N. Next, we investigate the properties of convergence of {w k } as k. To begin, let j be fixed but arbitrary. Define the sequence {N k t} k, where N k t = w k t, ψ j, for t, T ]. We shall first show that the sequence so defined satisfies the properties. {N k t} k is uniformly bounded,. {N k t} k is equicontinuous. The first property follows from ψ j =, the Cauchy- Schwarz inequality, and 3.5. N k t w k t ψ j M /. To prove the second, let us note that for every t and s in, T N k t N k s = w k t w k s, ψ j = η kr t η kr s ψ r, ψ j. Integrating between and t in 3.7 we first obtain that the coefficients η kr satisfy the equation and thus N k t N k s = η kj t = η kj + t s f, ψ j dτ ν t t s f j ν r= a rj η kr r= w k, ψ j dτ n,i= t s a rij η kr η ki dτ 3.6 Dw k Dw k, ψ j dτ. 3.7 Using the Cauchy- Schwarz inequality and the orthogonality of { ψ j }, we can bound each term f, ψ j f ψ j = f, w k, ψ j w k ψ j Dw k Dw k, ψ j = Dw k Dw k, ψ j C D max ψ j w k w k j C D M / max ψ j w k. j Putting everything together and letting K = ν ψ j and K = max j ψ j we have N k t N k s t s f + K w k + K M / w k dτ. 3.8 Further more, Cauchy-Schwarz inequality in time gives: N k t N k s t s { f L s,t;l + K + K M / } w k L s,t;l. 3.9 Finally, since f L, T ; V and since w k L s,t;l w k L,T ;L for any T >, our argument is over and {N k t} k is equicontinuous. By the Arzela-Ascoli Theorem, from {N k t} we may select a subsequence, which we redenote by {N k t} uniformly convergent to a continuous function Nt, i.e: w k t, ψ j Nt uniformly in t. 3. 9

10 Since the estimate 3.5 is independent of t, we obtain that {w k } is a bounded sequence in L, T ; H. By the weak compactness of L, T ; H it follows that there exists a convergent subsequence of {w k } in L, T ; H. If we redenote the convergent subsequence by {w k } we obtain that there exists wt in H such that From 3. and 3. follows that w k t, ψ j wt, ψ j uniformly in t. 3. V t = wt, ψ j, for all t, T ]. Using 3.3, we deduce that {w k } is a bounded sequence in L, T ; V #. By the weak compactness of L, T ; V #, there exists a convergent subsequence of {w k } in L, T ; V #. Again, if we redenote the convergent subsequence by {w k }, then there exists w t V # such that w k t, vt w t, vt uniformily in, T ], for all j N 3. and for all v V #. However, V # H. Uniqueness of the limit of a sequence together with 3. and 3. yield that wt = w t. It means that w L, T ; V # L, T ; H. This implies that along a subsequence w k w in L, T, L w k w in L, T, L. 3.3 To prove 3. we replace ψ j by I δ ψ j in 3.3 and let k. The nonlinear term gives T T Dw k Dw k, I δ ψ j dτ = = T T DwDw, I δ ψ j dτ T Dwk Dw k, ψ j dτ DwDw, ψj dτ T Dwk wdw k, ψ j dτ DwDw wk, ψ j dτ C D max ψ j w k w L,T,L j w k L,T,L + C D max ψ j w k w L,T,L j w L,T,L, which tends to as k by 3.3. This concludes our proof. Proposition 3.. If w is a weak solution of the model.4, then Proof. By the Riesz Representation Theorem and density T w t = w t, φdt. φ w t L, T ; L. 3.4 sup φ D Let φ D. Multiplying the first equation of.4 by φ, applying the symmetric positive definite property of the filtering operation and the incompressibility condition we obtain w t, φ = f, φ Dw Dw, φ + ν w, φ. 3.5 We can bound each term in the right hand side. First, we note that Definition. leads to φ φ. Using the Cauchy-Schwarz inequality it follows that f, φ f φ f φ, 3.6 w, φ w φ. 3.7

11 If we also use Ladyshenskaya s and Poicaré s inequalities we obtain Dw Dw, φ C Dw / Dw / Dw φ Cδ, D w / w 3/ φ. 3.8 With inequalities and the regularity properties of w, from 3.5 we deduce w t, φ C δ, D, w, w, w φ. Let K := C δ, D, w, w, w. Then w t, φ K φ. Integration in time and Cauchy-Schwarz inequality in the right hand side lead to T w t, φdt CK, T. φ Now take sup φ and the conclusion follows. Remark 3.. An important consequence of Proposition 3. is that w t, v + Dw Dw, v + ν w, v = f, v, 3.9 for any v L, T ; L. Indeed, since D is dense in L, T ; L, there exist a sequence of functions {φ n } n in D such that φ n w. If we multiply the first equation of.4 by {φ n } n and let n we obtain 3.9, using the Proposition 3. to pull the limit into the first term and the regularity proven in the Theorem 3. for the remaining terms. 4 ADM Energy Balance and Uniqueness In this section we prove that the weak solution of the ADM.4 is a unique strong solution. We also show that the model satisfies an energy equality rather than inequality. In our proofs we include the case when I DG is symmetric positive semi-definite and thus φ := I DGφ. First we prove uniqueness. Theorem 4.. Assume that w H # H and f L, T ; V. The weak solution of.4 is unique. Proof. By contradiction, assume that there exist two solutions w, q and w, q of.4. Let φ := w w thus φ := w w and r := q q. Then, subtracting the weak formulations of w, q and w, q, it follows that φ, r is a weak solution of φ t + Dw Dw Dw Dw ν φ + r + χφ = φ = 4. φ =. subject to periodic boundary condition and zero mean. From Remark 3., we can multiply the first equation of 4. by I δ Dφ L, T ; L. After algebraic manipulation and integration by parts, the nonlinear term becomes: Dw Dw Dw Dw, I δ Dφdx = Dφ Dφ Dw dx. Since both I DG and D are symmetric, positive semi-definite and positive definite respectively, and all operators commute, the higher order fluctuation term is non-negative χ φ, I δ Dφ = χ ψ, ψ + χ ψ, ψ, where ψ = D / φ. 4.

12 The other terms are φ t, I δ Dφ = d dt φ D + δ φ D ν φ, I δ Dφ = ν φ D + δ φ D r, I δ Dφ = r, I δ D φ =. Thus, we obtain d dt φ D + δ φ D + ν φ D + δ φ D Dφ Dφ Dw dx. 4.3 Next, we apply the Cauchy-Schwarz inequality in the right hand side Dφ Dφ Dw dx Dw L 4 Dφ L 4 Dφ. The Sobolev embedding theorem gives that there exists C = C, f, w, D such that Dφ Dφ Dw dx C φ. Furthermore, using the equivalence of the D and the L norm, there is a constant C = C δ such that Putting everything together, 4.3 implies φ C φ + δ φ. d dt φ D + δ φ D C φ + δ φ 4.4 But, φ =. Gronwall s Lemma implies that φ vanishes everywhere for all t. Hence, uniqueness follows. Lemma 4.. The LES ADM model.4 has a unique strong solution. Proof. From the Definition.5, any strong solution is, in particular, a weak solution as well. From Theorems 3. and 4. a unique weak solution exists. Remark 3. shows that this weak solution is also a strong solution. Theorem 4.. Let w be the unique strong solution of.4. Then w satisfies the energy equality: +χ t for all t, T ]. t wt D + δ wt D + ν w τ, δ + IDwτ t dτ = f, wτ D dτ + wτ D + δ wτ D dτ w D + δ w D, 4.5 Proof. Multiply the first equation of.4 by the test function δ + IDw. The nonlinear term vanishes because: DwDw, I δ Dw = DwDw, Dw = Rewriting, we obtain: wt, I δ Dw ν w, I δ Dw +χ w, δ + IDw = f, I δ Dw. 4.6

13 We have: wt, I δ Dw k = d w dt D + δ w D ν w, I δ Dw k = ν w D + δ w D f, I δ Dw = f, Dw. 4.7 Replacing 4.7 in 4.6 and integrating between and t we obtain 4.5, for all t, T ]. Remark 4.. In Theorem 4., the kinetic energy of the model = total energy dissipation = t the initial kinetic energy of the model = total energy input by the body force f = wt D + δ wt D, [ ν wτ D + δ wτ D + χ w τ, δ + Iwτ ] D dτ, t w D + δ w D, f, wτ D dτ. Thus Theorem 4. means that the kinetic energy of the model + total energy dissipation = the initial kinetic energy of the model + total energy input by the body force f. 5 Examples of Deconvolution Operators and their properties The basic problem in deconvolution is: given u + noise find u approximately. In other words given u solve Gu = u for u. 5. Many averaging operators G are symmetric and positive semi-definite. If the averaging operator is smoothing, the deconvolution problem will be not stably invertible due to small divisor problems. With these constraints in mind we review a few examples of deconvolution operators and their properties. In this section we consider the filtering operation be given by... The van Cittert deconvolution operator. The van Cittert method of approximate deconvolution, see [], constructs a family D N inverses to G using N steps of fixed point iterations. of Algorithm 5.. [van Cittert Algorithm]: Choose For n =,,,..., N perform Set D N u := u N. u = u. u n+ = u n + {u Gu n }. A very detailed mathematical theory of the van Cittert deconvolution operator and resulting approximate deconvolution models are already known, see [], [3], [5], [5] and [6]. We point out the following lemma, proved in [6], concerned with properties of the approximate deconvolution operator D N. Lemma 5.. The operator D N : L L is bounded, symmetric and positive definite. Proof. For the proof see [ [6], Lemma.]. 3

14 . The Accelerated van Cittert deconvolution operator. In our second example, approximations to u are obtained via the algorithm defined below. Algorithm 5.. [Accelerated van Cittert Algorithm]: Given relaxation parameters ω n, choose For n =,,,..., N perform Set D ω N u := u N. u = u. u n+ = u n + ω n {u Gu n }. Proposition 5.. Let the averaging operator be the differential filter Gϕ := δ + I ϕ. If the relaxation parameters ω i are positive, for i =,,..., N, then the Accelerated van Cittert deconvolution operator D ω N : L L is symmetric positive definite. Proof. The proof follows from [[7], Lemma 3.]. 3. Tichonov regularization deconvolution operator. Our third example is the Tichonov regularization deconvolution operator. More precisely, since G := δ + I is symmetric positive-definite, given u and µ > small, an approximate solution to the deconvolution problem 5. can be calculated as the unique minimizer in L of the functional F µ v = Gv, v u, v + µ v, v. The resulting family of Tichonov regularization deconvolution operators is D µ = lim µ G + µi 5. and the approximate solution of 5. is u µ = lim µ G + µi u. The family of operators D µ has the following properties, see []. for any µ >, D µ is a bounded linear operator,. lim µ D µ ϕ = ϕ for all ϕ L. Proposition 5.. Let the averaging operator be the differential filter Gϕ := δ + I ϕ. Let µ > be fixed. The operator D µ : L L is a bounded self-adjoint and positive definite. Proof. Remark that G is a linear, self-adjoint positive definite operator. Since D µ is a function of G and the identity operator I, it is also linear and self-adjoint. The Spectral Mapping Theorem implies that the eigenvalues of D µ are given by λd µ = λg + µi. Since G, see [6], we have that the eigenvalues of D µ are strictly positive and < µ λd µ <, for µ > µ Thus D µ is also bounded and positive definite. 4. Geurts approximate filter inverse. One of the first studies of deconvolution as a basis for LES models was done by Geurts in [9]. Let φ be the top hat filter, φx = x+δ δ x δ φx dx. Briefly, he developed approximate inverse of the top-hat convolution filter, Dφ := x+δ x δ d x x φx dx, 4

15 based on exactness of polynomials of degree µ. If Π µ = {px : px = a + a x a µ x µ } the criteria that determined the deconvolution kernel d was DGφ = φ, for all φ Π µ. 5.4 In D in multiple dimension extension is by tensor product, the top-hat filter, g has the special property that g { polynomial of degree µ} = { polynomial of degree µ}. Using this property, for L, let dx = { d + d x d L x L, if x π, if x > π. On [ π, π] the coefficients d, d,..., d L are uniquely determined by exactness of polynomials of degree L +. d g x l = x l, l =,,..., L The deconvolution operator Dφ = d φ is self adjoint, commutes with G. The theoretical development in [9] and associated test suggests that D satisfies: DGφ φ Cφδ L+ for smooth φ D LL L Cδ, L <. The deconvolution operator D L are tabulated for L =,,, 3 in Geurts [[9], Table ]. The tophat filter, and thus the associated deconvolution operator, is important in many applications, but it is not clear if or how the theory developed herein could be extended to it. This is because ĝk = sinkδ//kδ/, both changes sign and has zeros. At this point it appears to be an interesting and important open question to extend Geurts construction to the other filters, such as the Gaussian. Extension to dynamic inverse models has been done in []. 5. A variation of the Geurts Approximate Filter Inverse, [9]. The construction of Geurts [9], can be modified so as to fit in the theory herein. Indeed, first we shall interpret the construction in wave number space. With the differential filter ĝδk = δ k + we have ĝk as k, since the filtering is smoothing. High order accuracy on the large scales means exactness on the high degree polynomials on x, i.e. Guerts condition 5.4 and, equivalently, high order contact of dk to ĝk at k =. Since the convolution should be a bounded operator and d = we can pose the problem seeking a rational deconvolution kernel Then the accuracy conditions are If, additionally, dk = + n k n l k l + b k b l k l, b l. d = satisfied by the choice of the th coefficients n and b d m dk dk m = dm k= dk m âk, m =,,..., µ. k= + b k b l k l, for all k R dk >, for all k R, then the deconvolution operator satisfies the conditions of the theory. 5

16 6. Not all deconvolution operators used in image processing satisfy P, P, and P3 above. Direct deconvolution, D = δ +I is not bounded and thus not satisfying P. The Accelerated van Cittert with negative relaxation parameters is not positive definite, violating P. The resulting deconvolution operator is not positive definite. Furthermore, there are very many iterative methods, such as steepest descent and the conjugate gradient method, that can be truncated to give deconvolution operators. However these often produce approximations to u which depend nonlinearly on u. Thus, the resulting approximate deconvolution operators are nonlinear, violating P. 6 Extension to other filters and Conclusions When developing a mathematical foundation of an LES model, the first analytical problem that arises is existence of solutions of the model. We developed a general theory about existence of solutions of deconvolution models. The averaging operator chosen was a specific differential filter. More generally, if the filter G satisfies ĝk for all k, then the exact filter inverse A can be defined as an unbounded operator with dense domain and close range. If additionally, ĝk as k with O k or faster then the existence theory developed herein can be extended to the filter G. This includes the Gaussian filter, for example, but excludes the top filter and sharp spectral cutoff. One main result of this work is finding near minimal conditions on the deconvolution operator that guarantee existence and uniqueness of the strong solution of a deconvolution model. We also proved that under P, P, and P3 the models satisfy an energy equality, which describes the evolution of the kinetic energy in a fluid s flow. It is important to remark that there are many possible deconvolution operators that don t satisfy the conditions P, P, and P3. Acknowledgement I thank Professor William Layton for suggesting the problem, all the support and help. I thank Professor Geurts for a helpful communication on inverse modeling in LES. References [] N. A. Adams and S. Stolz, Deconvolution methods for subgrid-scale approximation in large eddy simulation, Modern Simulation Strategies for Turbulent Flow, R.T. Edwards, Zurich,. [] M. Bertero and B. Boccacci, Introduction to Inverse Problems in Imaging, IOP Publishing Ltd., Bristol, 998. [3] L. C. Berselli, T. Iliescu and W. Layton, Large Eddy Simulation, Springer, Berlin, 6. [4] A. Dunca, V.John, W. Layton, The Commutation Error of the Space Averaged Navier- Stokes Equations on a Bounded Domain, Advances in Mathematical Fluid Mechanics 3, Birkhuser Verlag Basel, 4, [5] A. Dunca, Space avereged Navier-Stokes equations in the presence of walls, Phd Thesis, University of Pittsburgh. 4. [6] A. Dunca and Y. Epshteyn, On the Stolz-Adams Deconvolution Model for the Large- Eddy Simulation of Turbulent Flows, SIAM J. Math. Anal. 6, [7] C. Foias, D. D. Holm and E. S. Titi, The Navier-Stokes-alpha model of fluid turbulence, Physica D, 5-53,

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