ON THE DEVELOPMENT OF PARAMETER-ROBUST PRECONDITIONERS AND COMMUTATOR ARGUMENTS FOR SOLVING STOKES CONTROL PROBLEMS

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1 Electronic Transactions on Numerical Analysis. Volume 44, , 25. Coyright c 25,. ISSN ETNA ON THE DEVELOPMENT OF PARAMETER-ROBUST PRECONDITIONERS AND COMMUTATOR ARGUMENTS FOR SOLVING STOKES CONTROL PROBLEMS JOHN W. PEARSON Abstract. The develoment of reconditioners for PDE-constrained otimization roblems is a field of numerical analysis which has recently generated much interest. One class of roblems which has been investigated in articular is that of Stokes control roblems, that is, the roblem of minimizing a functional with the Stokes or Navier-Stokes) equations as constraints. In this manuscrit, we resent an aroach for reconditioning Stokes control roblems using reconditioners for the Poisson control roblem and, crucially, the alication of a commutator argument. This methodology leads to two block diagonal reconditioners for the roblem, one of which was reviously derived by W. Zulehner in 2 [SIAM J. Matrix Anal. Al., 32 2), ] using a nonstandard norm argument for this saddle oint roblem, and the other of which we believe to be new. We also derive two related block triangular reconditioners using the same methodology and resent numerical results to demonstrate the erformance of the four reconditioners in ractice. Key words. PDE-constrained otimization, Stokes control, saddle oint system, reconditioning, Schur comlement, commutator AMS subject classifications. 65F8, 65F, 65F5, 76D7, 76D55, 93C2. Introduction. Decades ago, a significant area of research in numerical analysis was the numerical solution of the Stokes and Navier-Stokes equations, two artial differential equations PDEs) that are crucial to the field of fluid dynamics. Preconditioned Krylov subsace methods for the solutions of the saddle oint systems relating to each of these roblems are given in [24] and [], resectively, for instance. Since then, a further area of numerical analysis has become rominent: that of PDE-constrained otimization, which involves minimizing a functional with one or more PDEs as constraints. Consequently, the develoment of solvers for Stokes control roblems, one of the most fundamental such roblems, has itself become a well researched area. There has been much success in this field: iterative solvers for a class of these roblems that are indeendent of the mesh-size h have been devised for the time-indeendent roblem in [9] and the time-deendent roblem in [23]. Further, a multigrid solver constructed in [8] is shown to be itself indeendent of h. However, generating Krylov subsace solvers that are robust with resect to the regularization arameter as well as the mesh-size has roved to be a more difficult task one notable excetion is the reconditioned MINRES aroach derived in [26] using a nonstandard norm argument, which does exhibit this indeendence. In this manuscrit, we consider the time-indeendent Stokes control roblem where the velocity and the control variable are included in the cost functional but the ressure is not. We consider these roblems using fundamental saddle oint theory and exlain how it is ossible to use this to construct reconditioners for the Stokes control roblem using a Poisson control reconditioner along with a commutator argument, the concet of which we shall describe. There are many reasons why we believe such an investigation is of considerable interest. Firstly, it enables us to re-derive the reconditioner of Zulehner [26] within a ure saddle oint framework. We are also able to derive a new block diagonal reconditioner for this roblem that is robust with resect to mesh-size and the control regularization coefficient, as well as two block triangular reconditioners which aear to have the same roerty. Finally, and erhas most intriguingly, we believe that the theory outlined in this aer can be alied to the much Received July 3, 23. Acceted October 28, 24. Published online on February 6, 25. Recommended by M. Benzi. School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, The King s Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, UK j.earson@ed.ac.uk). 53

2 54 J. W. PEARSON harder Navier-Stokes control roblem, which we will address in a future manuscrit [6]. We are also able to use the methodology resented here to exlain why the choice of whether or not to regularize the ressure is crucial from a reconditioning oint of view. This manuscrit is structured as follows. In Section 2, we detail two areas of background which we will make use of: those of saddle oint theory and reconditioners for Poisson control roblems. In Section 3, we combine these with the theory of commutator arguments to derive the four aforementioned reconditioners for Stokes control roblems two block diagonal and two block triangular). We also state the dominant comutational oerations required to aly our reconditioners and discuss the imortance of the inclusion or omission of a ressure term in the cost functional. In Section 4, we rovide numerical results to demonstrate how the reconditioners erform in ractice, and in Section 5, we make some concluding remarks. 2. Background. In this section, we introduce two fundamental subject areas which we utilize in the remainder of this manuscrit. Firstly, in Section 2., we outline some basic roerties of saddle oint systems which we make use of. Secondly, in Section 2.2, we detail the theory of solving Poisson control roblems, which we also exloit. 2.. Saddle oint systems. The matrix systems that we consider the iterative solution of in the remainder of this manuscrit are all of saddle oint form, that is, of the form [ ] [ ] [ ] Φ Ψ T x b 2.) =, Ψ Θ x 2 b 2 }{{} Λ where Φ R m m, Ψ R m, m, have full row rank and Θ R. In the roblems that we study, Φ and Θ are also symmetric. A comrehensive review of systems of saddle oint tye is given in []. We are seeking reconditioners for equations of the form 2.). Therefore we utilize the observations that if we recondition Λ with P, P 2, or P 3, where [ ] [ ] Φ Φ P = Θ + ΨΦ Ψ T, P2 = Ψ Θ + ΨΦ Ψ T, [ ] Φ P 3 = Ψ Θ ΨΦ Ψ T, then the non-zero eigenvalues of λ P 2 P 2 Λ and P Λ are given by 3 Λ) = {±}, λ P Λ) = {} for any choice of Θ, and the non-zero eigenvalues of 3 P λ P {, Λ) = 2 ± 5) Λ are given by }, rovided Θ =. The above results are given in [2, 3] in the case Θ = ; the eigenvalue results for P 2 Λ and P 3 Λ in the case Θ are shown in [9]. Now, the matrices P Λ and P 2 Λ are diagonalizable but P 3 Λ is not, so consequently a Krylov subsace method for the solution of 2.) with ideal reconditioners P, P 2, and P 3 should converge in 3, 2, and 2 iterations, resectively [3], in the relevant cases rovided that Λ is non-singular. Naturally, we are not exlicitly constructing P, P 2, and P 3 as this would be In the roblem we will consider, the matrix is singular, however, the reconditioners described are often effective for such systems also.

3 PARAMETER-ROBUST SOLVERS FOR STOKES CONTROL 55 comutationally wasteful but instead construct aroximations P, P 2, and P 3 such that the actions of the inverses of our reconditioners may be alied efficiently. Having develoed these reconditioners, one may consider the MINRES algorithm [4] with reconditioners of the form P and reconditioners of the form P 2 and P 3 used in conjunction with solvers such as GMRES [2] and the Bramble-Pasciak Conjugate Gradient method [3]. We note that the quantity S := Θ + ΨΦ Ψ T is imortant in all three of the above reconditioners; this term is commonly known as the negative) Schur comlement of Λ, and much emhasis will be laced on aroximating this quantity of the matrix systems we consider Otimal control of Poisson s equation. In literature including [8, 2, 26], the iterative solution of the matrix system resulting from the distributed Poisson control roblem min y,u 2 y ŷ 2 L 2Ω) + 2 u 2 L 2Ω) s.t. y = u, in Ω, y = g, on Ω, is considered. Here, the domain on which we are working is denoted as Ω R d, d {2, 3}, with boundary Ω. Moreover, y denotes the state variable with ŷ some desired state), u the control variable, and a regularization arameter or Tikhonov arameter). The symbol denotes the Lalacian oerator. Discretizing the above roblem using equal-order finite element basis functions for y, u, and leads to the 2 2 matrix system [8] [ M K 2.2) K ] [ ] y M = [ ] Mŷ + c, d where y and ŷ are the discretized versions of y and ŷ, resectively, c and d are vectors corresonding to the boundary conditions, and is the discretized version of the adjoint variable or Lagrange multilier), which is related to u by u =. Here M denotes a finite element mass matrix and K a finite element stiffness matrix, two frequently used tyes of matrices, both of which are ositive definite. Two reconditioners which are robust for all values of the mesh-size h and the regularization arameter, and which we denote P P and P2 P here, have been develoed and tested for the matrix system 2.2) in [26] and [8], resectively: [ ] M + K P P = M +, K) [ ] M P2 P ) ) = K + M M K +. M These two reconditioners have been derived in very different ways: P P was obtained using a nonstandard norm argument in [26] and P2 P using the saddle oint theory described in Section 2.. Each of these reconditioners for the Poisson control roblem may be extended to an effective reconditioner for Stokes control roblems as we demonstrate in Section 3. The crucial ste in constructing P2 P is that of aroximating the Schur comlement of 2.), KM K + ) ) M. In [8], it is shown that if we aroximate this by K + M M K + M, then the eigenvalues of the reconditioned Schur comlement are all contained within the interval [ 2, ].

4 56 J. W. PEARSON 3. Otimal control of the Stokes equations. The roblem that we consider for the majority of this section is the following distributed Stokes control roblem: 2 v v 2 L 2Ω) + 2 u 2 L 2Ω) s.t. v + = u, in Ω, min v,u v =, in Ω, v = g, on Ω. Again we work on a domain Ω R d, d {2, 3}, with boundary Ω and with a regularization arameter. Here, v denotes the velocity in d dimensions and the ressure term, both of which are state variables in this roblem. u is the control variable in d dimensions. We also introduce at this oint the adjoint variables λ which is equal to u) and µ. Discretizing this roblem results in the matrix system [26] 3.) M K B T B K B T M B v λ = µ M v + c d f, where M and K here denote d d block matrices with mass and stiffness matrices from the velocity sace on the block diagonals, and B reresents the negative of the divergence oerator on the finite element sace in matrix form. The vector v corresonds to the target function v, λ and µ are related to the adjoint variables λ and µ, and the vectors c, d, and f take account of boundary conditions. We note at this oint that this matrix is in general singular, as it is well known that the vector of ones is a member of the nullsace of B T see [6, Chater 5] for instance) the matrix in 3.) therefore has two zero eigenvalues one corresonding to each aearance of B T ). 2 However, this may be avoided by restricting the ressure sace to the orthogonal comlement of the nullsace as in this case the matrix B T will clearly no longer have a nullsace. We also note that in the construction of the functional being minimized in this otimal control roblem, we have not regularized the ressure term the roblem where ressure is regularized was considered in [9, 23], for instance. This is extremely imortant from a reconditioning oint of view, and in Section 3.4 we exlain why this makes a major difference. We consider discretizing this roblem using the well-studied inf-su stable) Taylor-Hood finite element basis functions, that is, discretizing the velocity v using Q2-basis functions and the ressure using Q-basis functions. We discretize the control u and adjoint variable λ using Q2-functions and the adjoint variable µ using Q-functions. It is not immediately obvious how the reconditioners derived for the Poisson control roblem in the revious section can be alied to the more difficult Stokes control roblem. In this section, we exlain how this may be achieved. Note that this definition of M and K is slightly different from the definition used in Section 2.2 where these terms simly denoted a single mass or stiffness matrix. 2 On the continuous level, the zero eigenvalues arise from the fact that an arbitrary constant may be added to the solution of the ressure or the adjoint variable µ yielding another solution.

5 PARAMETER-ROBUST SOLVERS FOR STOKES CONTROL Derivation of block diagonal reconditioners. To commence our derivation, we reorder the matrix system 3.) so that we are dealing with the system M K B T v M v + c K M BT λ B µ = d f. } B {{ } A This is a saddle oint system of the form 2.) with [ ] M K Φ = K M, Ψ = [ B B ], Θ = [ ]. Note that the, )-block Φ is of the form of the matrix system 2.2) relating to the Poisson control roblem. We will use this to motivate two block diagonal reconditioners related to two reconditioners for Poisson control detailed in Section 2.2. These reconditioners will be of the form [ ] Φ 3.3) P = ΨΦ Ψ T. ) arox Such a strategy also leads to block triangular reconditioners of the form [ ] [ Φ Φ 3.4) P = Ψ ΨΦ Ψ T or ) arox Ψ ΨΦ Ψ T ) arox We will derive two such block triangular reconditioners in Section First reconditioner. We motivate our first reconditioner for the Stokes control system 3.2) using the reconditioner P P for the Poisson control roblem of Section 2.2. We first note that the, )-block of the Stokes control roblem 3.2) is of the form of the matrix involved in the Poisson control roblem, so we write, in the notation of 3.3), [ ] [ ] M K M + K Φ = K M M + =: K) Φ. Here, the notation Φ Φ indicates that Φ has been constructed with the aim that the singular values of Φ Φ are bounded within a fixed small) interval. The next ste is to find a good aroximation to the Schur comlement ΨΦ Ψ T of the matrix system 3.3); we justify a otential aroximation by writing [ ] [ ] [ ] ΨΦ Ψ T B M K B T = B K M B T [ ] [ ] [ ] B M + K B M + B T K) B T =: Ψ Φ Ψ T [ ] BM + K) = B T BM + K) B T. We highlight the fact that, in general, the aroximate identity Φ Φ does not necessarily tell us that Ψ Φ Ψ T ΨΦ Ψ T unless Ψ is a square and invertible matrix, which is not ].

6 58 J. W. PEARSON the case here). However, this seems to be a reasonable motivation for an aroximation which is otentially effective, and we indeed find that this strategy does lead to a good aroximation of ΨΦ Ψ T for this roblem. Furthermore, an eigenvalue analysis carried out in [] for this reconditioner verifies its otency for the Stokes control roblem. At this oint, as it is done in [26], we may aroximate BM + K) B T by M + K ) in the above exression, where M and K denote finite element mass and stiffness matrices, resectively, of the ressure sace. Hence, we may write that [ ] M ΨΦ Ψ T + K ) M + K ) =: ΨΦ Ψ T ) arox. Therefore, utting all of the above working together, we ostulate that M + K P = M + K) M + K ) M + K is an effective reconditioner for A. This is exactly the reconditioner roosed by Zulehner in [26] using a nonstandard norm argument. We will demonstrate the effectiveness of this reconditioner by dislaying numerical results in Section Second reconditioner. We are also able to derive a new block diagonal reconditioner for the Stokes control system 3.2) using the reconditioner P2 P for the Poisson control roblem. We treat the, )-block of the Stokes control system by using the reconditioner for Poisson control, writing in the notation of 3.3)) [ ] [ ] M K M Φ = K M KM K + [ M ] M ) ) K + M M K + =: M Φ. We now again search for a good aroximation to the Schur comlement we roceed as follows: [ ] [ ] [ ] ΨΦ Ψ T B M B T B KM K + M B T [ BM B T ] = ) B KM K + M B T. Once more, we have assumed in the above working that Φ being a good aroximation to Φ leads to Ψ Φ Ψ T aroximating ΨΦ Ψ T well; for this roblem we find that this heuristic does indeed lead to an effective aroximation. We do not yet have a feasible reconditioner as the matrices BM B T and ) B KM K + M B T cannot be inverted without comuting the inverses of M or KM K + M. However, it is well known that BM B T may be well aroximated ) This may be done by alying the commutator argument of Section 3..2 with L := + I. This is carried out in a very similar fashion in [23] for matrices of this form for time-deendent Stokes control roblems.

7 PARAMETER-ROBUST SOLVERS FOR STOKES CONTROL 59 by K see [6, Chater 8]), so we use this for the first block of our Schur comlement aroximation. ) We therefore now seek an idea for aroximating Σ := B KM K + M B T so that we obtain a chea and invertible aroximation to the Schur comlement. We do this by using a commutator argument, a tye of which is described in [6] for the Navier-Stokes equations, for instance. We examine the commutator E = L) L), where L = 2 + I. This is an oerator carefully chosen to give us a matrix that we can use to aroximate Σ. Now, discretizing this commutator using finite elements gives E h = M L)M B T M B T M L ), where L = KM K + M. Pre-multilying by BL M and ost-multilying by L M, where L = K M K + M, then gives BM B T L M BL B T, where, crucially, we assume that the commutator E h is small. We may now use the fact that BM B T K and substitute in the exression for L to obtain that 2 Σ = B KM K + ) M B T K L M, and therefore that Σ M L K = M K M K + M ) K = M K M + K. We note that such an argument has been used a number of times before we give a brief summary of some alications in Section 5. Thus, a second ossible reconditioner for A is M ) ) K + P 2 = M M K + M K K M M + K which we ostulate being an effective reconditioner. We again verify that this is the case by numerical results resented in Section 4. We note at this oint that this reconditioner is a more flexible one as we find that a reconditioner of this form may be alied to the more difficult and general linearizations The aroximation BM B T K may be justified by the observations that = on the continuous level, and that the matrices K, B, M, and B T relate to the continuous oerators,, I, and, resectively. 2 An aroximation of the form BL B T K L M was first introduced by Cahouet and Chabard in [4] for the forward Stokes roblem. Such arguments have since been used to develo iterative solvers for a variety of fluid dynamics roblems. ),

8 6 J. W. PEARSON of the Navier-Stokes control roblem [6]. In more detail, when a Picard-tye iteration is alied to this roblem, we may rearrange the matrix system obtained so that we have as the, )-block a matrix corresonding to the convection-diffusion control roblem as oosed to a Poisson control roblem here. Using a reconditioner derived for the convection-diffusion control roblem in [7], we may aly a similar commutator argument to aroximate the Schur comlement of the matrix systems for Navier-Stokes control for this roblem, we find ) that we need to aroximate BM B T and B F M F T + M B T, where F arises from the differential oerator relating to the Navier-Stokes equations. By doing this we arrive at iterative solvers for the Navier- Stokes control roblem. It is likely that such strategies could also be alied to the linear systems obtained when Newton iteration is alied to the roblem Block triangular reconditioners. A useful asect of our aroach is that we may consider develoing robust reconditioners for the Stokes control roblem that are not of the block diagonal form of P and P 2. We do this by considering various block triangular reconditioners of the Poisson control matrix system. [ ] Φ Firstly, we may consider a reconditioner of the form Ψ ΨΦ Ψ T stated ) arox in 3.4) that is in some sense analogous to P as derived in Section 3... We could in fact consider the same aroximations Φ and ΨΦ Ψ T ) arox as we did to construct P, Φ = ΨΦ Ψ T ) arox = [ M + K [ M ] M +, K) + K ) M + K ) ], to develo the following block triangular reconditioner for A: M + K P 3 = M + K) B M + K ) B M + K ), which may be alied within the GMRES algorithm. In addition to this reconditioner, we may form a block lower triangular reconditioner for the Stokes control roblem that is based on the following block triangular reconditioner P P 3 for the Poisson control roblem: P P 3 = [ M K K + M ) M K + M ] ), which was shown to be effective for that roblem in [8]. We may, once again, use this as an aroximation to the, )-block of the Stokes control matrix A.

9 PARAMETER-ROBUST SOLVERS FOR STOKES CONTROL 6 Let us consider how we may recondition the Schur comlement of A while using this aroximation of the, )-block. We write, in the notation of 3.4), [ ] [ ] [ ] ΨΦ Ψ T B M K B T = B K M B T [ ] [ ] [ ] B M B T B K ŜP B T [ ] [ B M ] [ ] B T = B Ŝ P KM Ŝ B T =: Ψ Φ Ψ T P [ BM B T ] = BŜ P KM B T BŜ P [ BT ] K ) BŜ P KM B T M K M + K + 2 M =: ΨΦ Ψ T ) arox, where Ŝ P = K + ) M M K + ) M. In the working above, we have again used the aroximation BM B T K. To aroximate the matrix BŜ P BT, we have used the same commutator argument as in Section 3..2 excet with L = ŜP = KM K + M + 2 K and L = K M K + M + 2 K. Therefore, alying the block triangular) saddle oint theory of Section 2., we arrive at a block triangular reconditioner for A, namely, M K ŜP P 4 = B K ). B BŜ P KM B T M K M + K + 2 M Of course, we would not be able to aly the MINRES algorithm with the reconditioners P 3 or P 4 ; instead we would use the GMRES algorithm of [2]. However, numerical tests indicate that P 3 and P 4 are effective reconditioners for A nevertheless we refer to Section 4 for a demonstration of this assertion Further comments. We now wish to make some further observations about the reconditioners which we have roosed. Firstly, it is natural to consider the effectiveness of the new commutator arguments we have introduced as such arguments are heuristic in nature. We therefore carry out numerical tests on our aroximations; in articular we look for the maximum and minimum non-zero) eigenvalues of 3.5) 3.6) [ [ M K M + ] K B M K M + K + 2 ] M KM K + M ) B T, B KM K + M + 2 K) B T, which relate to the two new commutator arguments introduced in this aer, and which are utilized in the reconditioners P 2 and P 4, resectively. In Table 3., we rovide eigenvalues

10 62 J. W. PEARSON TABLE 3. Maximum and minimum non-zero) eigenvalues for the commutator aroximation 3.5) used in the block diagonal reconditioner for different values of h and. h λ 2 λ max λ 2 λ max λ 2 λ max λ 2 λ max TABLE 3.2 Maximum and minimum non-zero) eigenvalues for the commutator aroximation 3.6) used in the block triangular reconditioner for different values of h and. h λ 2 λ max λ 2 λ max λ 2 λ max λ 2 λ max of the matrix 3.5) for a range of mesh-sizes and values of, and in Table 3.2, we resent the same results for 3.6). For the results in both tables, an evenly saced grid with Taylor- Hood elements was used with the values of h stated corresonding to the distance between Q2-nodes. We can see that the aroximations are effective ones for a range of arameter values, esecially for smaller values of. We note a benign deendence of the effectiveness of the aroximations on h, but the results obtained are still very reasonable. Another ertinent question is how chea it is to aly our roosed reconditioners. We therefore detail the main comutational oerations required to aroximate P, P 2, P 3, and P4 excluding matrix multilications, which are comaratively chea). For the uroses of these descritions, we view the reconditioners as 4 4 block matrices and refer to each block in this way. Oerations needed to aly P : -, ): multigrid oeration for M + K, -2, 2): multigrid oeration for M + K, -3, 3): Chebyshev semi-iteration for M and multigrid oeration for K, -4, 4): Chebyshev semi-iteration for M and multigrid oeration for K, -total: 2 Chebyshev semi-iterations and 4 multigrids. Oerations needed to aly P2 : -, ): Chebyshev semi-iteration for M, -2, 2): 2 multigrid oerations for K + M, -3, 3): multigrid oeration for K, -4, 4): 2 Chebyshev semi-iterations for M and multigrid oeration for K, -total: 3 Chebyshev semi-iterations and 4 multigrids.

11 PARAMETER-ROBUST SOLVERS FOR STOKES CONTROL 63 Oerations needed to aly P3 : these are the same as for P and hence in total: -total: 2 Chebyshev semi-iterations and 4 multigrids. Oerations needed to aly P4 : -, ): Chebyshev semi-iteration for M, -2, 2): 2 multigrid oerations for K + M, -3, 3): multigrid oeration for K, -4, 3): Chebyshev semi-iteration for M and 2 multigrid oerations for K + M, -4, 4): 2 Chebyshev semi-iterations for M and multigrid oeration for K, -total: 4 Chebyshev semi-iterations and 6 multigrids. We can see from this list of oerations that the alication of each reconditioner esecially P, P 2, and P 3 ) is fairly chea, and therefore that our methods should be comutationally effective ones. Finally, an imortant question arising from this work relates to whether the methodology can be alied to other roblems of Stokes control tye. In more detail, rather than considering distributed control roblems of the form described in this manuscrit, one could examine formulations where the control is only alied on the boundary or within some subdomain. It is likely that much of the methodology within this aer could be alied to these more diverse roblems, however, two major issues will inevitably arise: A robust aroximation Φ of the, )-block Φ will become harder to construct. In articular deriving an aroximation to the Schur comlement of the Poisson control roblem, which is involved in the construction of Φ, will become heuristic in nature when subdomain roblems are considered [5, Chater 4], as oosed to the rigorous nature of the reconditioners for the Poisson control roblem on the whole domain [8, 26] used in Sections 3.. and The alication of commutator arguments to build Schur comlement aroximations Ŝ becomes more troublesome as such arguments have not been so widely tested on subdomain roblems. The Schur comlement aroximations will also be imacted by the less robust, )-block aroximation Φ. In summary, whereas we believe this work has the otential to be extended to more comlex roblems of Stokes and Navier-Stokes control tye, it is clear that significant investigation will need to be carried out in relation to the validity of the, )-block and Schur comlement aroximations before such an aroach could be reliably alied Penalty term alied to ressure. In this section we briefly consider the Stokes control roblem 2 v v 2 L 2Ω) + α 2 2 L 2Ω) + 2 u 2 L 2Ω) s.t. v + = u, in Ω, min v,,u v =, in Ω, v = g, on Ω, which is identical to the roblem we have studied in the revious sections, excet that we imose an additional term in the cost functional relating to the ressure with α being the corresonding enalty arameter).

12 64 J. W. PEARSON It is useful to consider reconditioning of the resulting matrix system [9] 3.7) M K B T v K M BT λ B µ = } B {{ αm } B M v + c d f αm in light of the framework discussed in this aer, in articular, whether it is ossible to recondition the roblem arising from the Stokes control roblem with a ressure enalty term in the same way as it is done to recondition the system arising without this ressure term. For brevity, we simly consider develoing a reconditioner of the form P for the matrix system 3.7) we find that the same issues arise when trying to construct reconditioners of the form P 2, P 3, and P 4 ). We may construct an aroximation of the, )-block of B exactly as we did for the matrix system A in Section 3.. as the, )-blocks of A and B are the same). When we attemt to construct an aroximation of the Schur comlement of B in a similar way for A, in the derivation of P, we obtain the following: [ S B = αm ] + [ = αm ] + [ ] [ ] [ ] B M K B T B K M B T [ ] [ ] [ ] B M + K B M + B T K) B T ]. [ BM + K) B T αm + BM + K) B T At this oint we face a roblem the 2, 2)-block of our roosed Schur comlement aroximation could be ositive definite, negative definite, or indefinite, deending on the values of α,, and h used, thus creating major issues when attemting to construct a ositive definite reconditioner which we require for use with MINRES). Even if the values of α,, and h were such that αm + BM + K) B T is ositive definite, it is far from clear how we may efficiently aroximate this matrix in a similar way as BM + K) B T was aroximated in Section 3... We are therefore unable to derive a arameter-robust reconditioner using our aroach. We therefore conclude that the Stokes control roblem involving a enalty term for the ressure is a harder roblem to solve robustly than the roblem without, at least if the methodology discussed in this manuscrit is considered. We oint the reader to [9] for a solver for the time-indeendent roblem with ressure enalty term that is indeendent of the mesh-size h but not the enalty arameter ) and to [23] for an extension of this solver to the time-deendent case. 4. Numerical exeriments. Having motivated the theoretical otential of our aroach, we now seek to demonstrate how our reconditioners erform in ractice. To do this, we consider two test roblems. The first roblem we look at is an otimal control analogue of the

13 PARAMETER-ROBUST SOLVERS FOR STOKES CONTROL x 2 x x.5.5 x a) = b) = 4 FIG. 4.. Plots of the comuted velocity solution to the first test roblem for different x 2 x 5 x 2 x a) = 2 b) = 6 FIG Plots of the comuted ressure solution to the first test roblem for different. lid-driven cavity roblem on the domain Ω = [, ] 2 : min v,u 2 v 2 L + 2Ω) 2 u 2 L 2Ω) s.t. v + = u, in Ω, v =, { in Ω, v = [, ] T on [, ] {}, [, ] T on Ω\ [, ] {}). We wish to observe how well the four reconditioners resented in this aer erform when solving the matrix system relating to this roblem. In Table 4., we dislay the number of MINRES iterations and CPU times for solving this roblem with reconditioner P to a tolerance of 6 for a variety of h and. In Table 4.2, the number of iterations and CPU times for solving the same roblem using MINRES with reconditioner P 2 to the same tolerance is given. Finally in Tables 4.3 and 4.4, we reort the iteration count and CPU times for solving The CPU times include the time taken to construct the matrices M and K involved in the reconditioner. We construct these matrices in the same way as in the Incomressible Flow & Iterative Solver Software IFISS) ackage [5, 22]. Where aroriate, we follow the recie detailed in [6, Chater 8] of imosing a Dirichlet boundary condition in the matrix K at the node on the velocity sace corresonding to the inflow boundary condition.

14 66 J. W. PEARSON TABLE 4. Number of iterations and CPU times in seconds) when alying MINRES to the first test roblem with reconditioner P for a variety of h and. h 2 3, , , , , 478 SIZE ) 26) ).232) ) 26) ).38) ) ) ) ) TABLE 4.2 Number of iterations and CPU times in seconds) when alying MINRES to the first test roblem with reconditioner P 2 for a variety of h and. h 2 3, , , , , 478 SIZE ) 25) ).249) ) 25) ).52) ) ) ) ) the roblem to the same tolerance with the GMRES algorithm used in the Incomressible Flow & Iterative Solver Software IFISS) ackage 2 [5, 22], reconditioned with the matrices P 3 and P 4. In Figures 4. and 4.2, we dislay solutions to the test roblem for velocity and ressure for different values of. In each of the tables and figures, the value of h indicated corresonds to the sacing between Q2-nodes. When generating these results, we use 2 stes of Chebyshev semi-iteration to aroximate the inverses of mass matrices; see [25] for more details. To aroximate the inverses of K, M + K, and K + M in our reconditioners note that the last two matrices are the same u to a multilicative factor), we emloy the algebraic multigrid AMG) routine HSL MI2 from the Harwell Subroutine Library HSL) [2], using 2 V-cycles with 2 re- and ost- relaxed Jacobi) smoothing stes to aroximate each matrix inverse. In all tables in this 2 All results in Tables are obtained using a tri-core 2.5 GHz workstation.

15 PARAMETER-ROBUST SOLVERS FOR STOKES CONTROL 67 TABLE 4.3 Number of iterations and CPU times in seconds) when alying GMRES to the first test roblem with reconditioner P 3 for a variety of h and. h 2 3, , , , , 478 SIZE ) 28) ).263) ) 3) ).65) ) ) ) ) TABLE 4.4 Number of iterations and CPU times in seconds) when alying GMRES to the first test roblem with reconditioner P 4 for a variety of h and. h 2 3, , , , , 478 SIZE ) 4) ).26) ) 5) ).63) ) ) ) ) section, the symbol denotes that the coarsening of the AMG routine failed when alied to M + K or K + M this occurs in the secific and imractical arameter regime where h is large and is small and is caused by the resence of ositive off-diagonal entries. In these cases, we resent results obtained using direct solves rather than AMG. To test our methods further, we also consider the following second test roblem on Ω = [, ] 2 : 2 v v 2 L 2Ω) + 2 u 2 L 2Ω) s.t. v + = u, in Ω, min v,u v =, in Ω, v = v, on Ω,

16 68 J. W. PEARSON.5 x x.5 x x.5 a) Velocity v b) Pressure x 2.5 µ x.5 x x.5 c) Adjoint velocity λ d) Adjoint ressure µ FIG Plots of the comuted solution to the second test roblem with = 4. where v = [ 2 x ) 2 x + x ), 2 + x ) x2 x 2 ) ] T [ 2 x ) 2 x x ), 2 x ) x2 x 2 ) ] T [ 2 + x ) 2 x + x ), 2 + x ) x2 + x 2 ) ] T [ 2 + x ) 2 x x ), 2 x ) x2 + x 2 ) ] T in [, ] [, ], in [, ] [, ], in [, ] [, ], in [, ] [, ], and x = [x, x 2 ] T denotes the satial coordinates. The target state v within this roblem setu corresonds to a recirculating flow with symmetry built into the roblem. In Figure 4.3, we dislay solution lots for this roblem, and in Tables 4.5 and 4.6, we resent numerical results for solving this roblem using MINRES reconditioned with P and P 2. Although we do not resent results for our GMRES-based solvers for this roblem, we note that the numerical features of these solvers are similar to those when tested on the first test roblem. The results shown in Tables indicate that the four reconditioners discussed in this manuscrit are robust with resect to mesh-size and regularization arameter. The iteration count is low for all four solvers considering the comlexity of the roblems. In many ractical roblems, the value of is within the range [ 6, ]; all methods erform well in this regime. We note that the block diagonal reconditioner P introduced in [26]) and the block triangular reconditioner P 3 based on it solve the roblem in the shortest time in all cases The only arameter regime where we do not observe comlete robustness is that of very small, when we observe some degradation in the erformance of the AMG routine used.

17 PARAMETER-ROBUST SOLVERS FOR STOKES CONTROL 69 TABLE 4.5 Number of iterations and CPU times in seconds) when alying MINRES to the second test roblem with reconditioner P for a variety of h and. h 2 3, , , , , 478 SIZE ) 26) ).24) ) 26) ).6) ) ) ) ) TABLE 4.6 Number of iterations and CPU times in seconds) when alying MINRES to the second test roblem with reconditioner P 2 for a variety of h and. h 2 3, , , , , 478 SIZE ) 25) ).276) ) 25) ).54) ) ) ) ) considered and with the lowest iteration count in most cases. However, the strategy involved in constructing these reconditioners is highly secific to this roblem. We believe that the flexibility in the methodology used to construct P 2 and P 4 would enable us to consider the more general and much harder Navier-Stokes control roblem, and therefore it is imortant to note that these reconditioners also seem to achieve robustness, albeit with larger iteration counts and CPU times than P and P 3. Of the two reconditioners P 2 and P 4, we note that the reconditioner P 4 solves the roblem in fewer iterations than P 2 but greater CPU time due to the added comlexity of the GMRES algorithm though this could artially be offset by using restarts within the GMRES method). We find that in the Navier-Stokes control case, using reconditioners of the form P 2 and P 4 would result in convergence to the solution of the matrix systems involved in similar CPU times [6] because a non-symmetric solver such as GMRES has to be used in both cases as both equivalent reconditioners would be non-symmetric in the Navier-Stokes case. We also note that in the arameter regime of small, the iteration count when the reconditioner P 4 is

18 7 J. W. PEARSON TABLE 4.7 Comarison of the H -norms of the iterative solution v l) and the direct solution v l,dir) for the state v and the L 2 -norms of the iterative solution u l) and the direct solution u l,dir) for the control u when alying MINRES to the first test roblem with the reconditioner P 2. Results are given for a variety of mesh levels l which corresond to h = 2 l ) and values of. v l) H v l,dir) Ω) H Ω) v l,dir) H Ω) u l) L2 Ω) ul,dir) L2 Ω) u l,dir) L2 Ω) Level l TABLE 4.8 Comarison of the H -norms of the iterative solution v l) and the direct solution v l,dir) for the state v and the L 2 -norms of the iterative solution u l) and the direct solution u l,dir) for the control u when alying GMRES to the first test roblem with the reconditioner P 4. Results are given for a variety of mesh levels l which corresond to h = 2 l ) and values of. v l) H v l,dir) Ω) H Ω) v l,dir) H Ω) u l) L2 Ω) ul,dir) L2 Ω) u l,dir) L2 Ω) Level l used is even smaller than that when P or indeed P 2 ) is alied. We believe that to extend this methodology to obtain an effective solver for the analogous Navier-Stokes control roblem, a reconditioner of the form of either P 2 or P 4 can therefore be considered. When testing our new methods, it is also desirable to ascertain whether the solutions obtained are accurate reflections of the true solutions and are reasonably unaffected by the stoing criteria within MINRES and GMRES which by definition are related to the reconditioners used). In Tables 4.7 and 4.8 we therefore comare the values of v l) H Ω) and u l) L2Ω) for the iterative solution of v and u on each mesh level l with the values v l,dir) H Ω) and u l,dir) L2Ω) obtained using a direct method for the mesh levels where

19 PARAMETER-ROBUST SOLVERS FOR STOKES CONTROL 7 we find using a direct method to be feasible and comutationally non-rohibitive). We resent these results for the first test roblem, using the new reconditioners P 2 with MINRES) and P 4 with GMRES) for a range of mesh levels and values of. In these tables we find that the scaled norms are largely around 6 as exected and deend very little on the mesh level and value of and hence the changing reconditioner). This gives a good indication that our iterative schemes are solving the roblem well and are resenting accurate solutions to the linear systems tested. 5. Concluding remarks. The use of commutator arguments has been an extremely valuable tool when develoing iterative methods for roblems in fluid dynamics. In [] for instance, such an argument was alied in order to develo a solver for the Navier-Stokes equations which erformed well for a wide range of values of mesh-size and viscosity. Since then, such arguments have also been alied to good effect when deriving iterative schemes for PDE-constrained otimization roblems, for examle in [23] to obtain mesh-indeendent solvers for time-deendent Stokes control and in [26] to arrive at a mesh- and regularizationrobust solver for a class of Stokes control roblems. Also, in [7], commutator arguments for the Navier-Stokes equations are analyzed for a range of boundary conditions. In this manuscrit, we have used new commutator arguments to derive further mesh- and regularization-robust solvers for these roblems: block diagonal and block triangular. We have also exlained the role of saddle oint theory and that of reconditioners for the Poisson control roblem in generating solvers for the more difficult Stokes control roblem. We rovided numerical results to justify the otency of this aroach and exlained the imortance of the ressure regularization term or lack of it) from an iterative solver oint of view. We believe that the arguments we have introduced in this manuscrit may be extended to generate robust solvers for a class of the harder Navier-Stokes control roblems we will discuss this in a future aer [6]. In addition, future research in this area could include the alication of these techniques to roblems with state or control constraints, boundary control roblems, or time-deendent Stokes-tye roblems, as well as tackling otimal control roblems derived secifically from real-world data. Acknowledgment. The author thanks an anonymous referee for his/her careful reading of the manuscrit and helful comments. He would also like to thank Andy Wathen for his invaluable hel and advice while this manuscrit was being reared as well as Martin Stoll and Walter Zulehner for useful discussions about this work. The author was suorted for this work by the Engineering and Physical Sciences Research Council UK), Grant EP/P5526/. REFERENCES [] M. BENZI, G. H. GOLUB, AND J. LIESEN, Numerical solution of saddle oint roblems, Acta Numer., 4 25),. 37. [2] J. BOYLE, M. D. MIHAJLOVIC, AND J. A. SCOTT, HSL_MI2: an efficient AMG reconditioner for finite element roblems in 3D, Internat. J. Numer. Methods Engrg., 82 2), [3] J. H. BRAMBLE AND J. E. PASCIAK, A reconditioning technique for indefinite systems resulting from mixed aroximations of ellitic roblems, Math. Com., 5 988),. 7. [4] J. CAHOUET AND J.-P. CHABARD, Some fast 3D finite element solvers for the generalized Stokes roblem, Internat. J. Numer. Methods Fluids, 8 988), [5] H. C. ELMAN, A. RAMAGE, AND D. SILVESTER, Algorithm 866: IFISS, a Matlab Toolbox for modelling incomressible flow, ACM Trans. Math. Software, 33 27), Art. 4 8 ages). [6] H. C. ELMAN, D. J. SILVESTER, AND A. J. WATHEN, Finite Elements and Fast Iterative Solvers: With Alications in Incomressible Fluid Dynamics, Oxford University Press, New York, 25. [7] H. C. ELMAN AND R. S. TUMINARO, Boundary conditions in aroximate commutator reconditioners for the Navier-Stokes equations, Electron. Trans. Numer. Anal., 35 29), htt://etna.mcs.kent.edu/vol.35.29/ dir

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