Robust preconditioners for PDEconstrained optimization with limited observations
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Abstract
Regularization robust preconditioners for PDEconstrained optimization problems have been successfully developed. These methods, however, typically assume observation data and control throughout the entire domain of the state equation. For many inverse problems, this is an unrealistic assumption. In this paper we propose and analyze preconditioners for PDEconstrained optimization problems with limited observation data, e.g. observations are only available at the boundary of the solution domain. Our methods are robust with respect to both the regularization parameter and the mesh size. That is, the condition number of the preconditioned optimality system is uniformly bounded, independently of the size of these two parameters. The method does, however, require extra regularity. We first consider a prototypical elliptic control problem and thereafter more general PDEconstrained optimization problems. Our theoretical findings are illuminated by several numerical results.
Keywords
PDEconstrained optimization Preconditioning Minimum residual methodMathematics Subject Classification
65F08 65N21 65K101 Introduction
In practice, observations are rarely available throughout the entire domain of the state equation. On the contrary, the purpose of solving an inverse problem is typically to use data recorded at the surface of an object to compute internal properties of that object: Impedance tomography, the inverse problem of electrocardiography (ECG), computerized tomography (CT), etc. This fact, combined with the discussion above, motivate the need for further improving numerical methods for solving KKT systems arising in connection with PDEconstrained optimization.
This paper is organized as follows. In the next section we derive the KKT system associated with the model problem (1)–(3). Our \(\alpha \) robust preconditioner is presented in Sect. 3, along with a number of numerical experiments. Sections 4 and 5 contain our analysis, and the method is generalized in Sects. 6 and 7. In Sect. 8 we discuss the preconditioner when applied to a standard finite element approximation of the problem. Section 9 provides a discussion of our findings, including their limitations.
2 KKT system
Consider the PDE (2) with the boundary condition (3). A solution u to this elliptic PDE, with source term \(f\in {L^2({\varOmega })}\), is known to have improved regularity, i.e. \(u\in H^{s}({\varOmega })\), for some \(s\in [1, 2]\), with s depending on the domain \({\varOmega }\). In the remainder of this paper we assume that the solution u is in \({H^2({\varOmega })}\) for any source term \(f\in {L^2({\varOmega })}\). This assumption is known to hold if \({\varOmega }\) is convex or if \(\partial {\varOmega }\) is \(C^2\), see e.g. [5, 7].
We will see below that, in order to design a regularization robust preconditioner for (1)–(3), it is convenient to express the state equation in the form (6), instead of employing integration by parts/Green’s formula to write it on the standard selfadjoint form.
2.1 Optimality system
3 Numerical experiments
In (13), we have implicitly used the same discretization for the control variable and the Lagrange multiplier. In (10)–(12), both variables belong to \({L^2({\varOmega })}\), so it seems natural to preserve this correspondence in the discretization. In fact, we can see from (10) that \(f = \alpha ^{1} w\), so the control could be eliminated from the system prior to the discretization. This would result in a \(2\times 2\) block system in place of (13). While solving the smaller system is more practical in terms of computational costs, we find that the analysis is more clearly presented for the \(3\times 3\) system (13).
In the current numerical experiments, we employ the Bogner–Fox–Schmit (BFS) rectangle for discretizing the state variable \(u \in {\bar{H}^2({\varOmega })}\). That is, the finite element field consists of bicubic polynomials that are continuous, have continuous first order derivatives and mixed second order derivatives at each vertex of the mesh. BFS elements are \(C^1\) on rectangles and therefore \(H^2\)conforming. The control f and Lagrange multiplier w are discretized with discontinuous bicubic elements.
Remark
3.1 Eigenvalues
Let us first consider the exact preconditioner \(\mathcal {B}_{\alpha }\) defined in (14). If \(\mathcal {B}_{\alpha }\) is a good preconditioner for the discrete optimality system (13), then the spectral condition number of \(\mathcal {B}_{\alpha }\mathcal {A}_{\alpha }\) should be small and bounded, independently of the size of both the regularization parameter \(\alpha \) and the discretization parameter h.
3.2 Multilevel preconditioning
In practice, the action of \(\mathcal {B}_{\alpha }\) is replaced with a less computationally expensive operation \({\widehat{\mathcal {B}_{\alpha }}}\). Note that \(\mathcal {B}_{\alpha }\) has a block structure, and that computationally efficient approximations can be constructed for the individual blocks. The only challenging block of the preconditioner is the biharmonic operator \(\alpha R+M_\partial \). Order optimal multilevel algorithms for forth order operators discretized with the Bogner–Fox–Schmit was developed in [16]. Specifically, it was shown that a multigrid Vcycle using a symmetric \(4\times 4\) block Gauss–Seidel smoother, where the blocks contain the matrix entries corresponding to all degrees of freedom associate with a vertex in the mesh, results in an order optimal approximation. The remaing blocks of the preconditioners are weighted mass matrices which are efficiently handled by two symmetric GaussSeidel iterations for the (1,1) and (3,3) blocks.
We estimated condition numbers of the individual blocks of \(\mathcal {B}_{\alpha }^{1}\) preconditioned with their respective approximations. The results are reported in Tables 1 and 2. A slight deterioration in the performance of the multigrid cycle can be seen for very small values of \(\alpha > 0\).
3.3 Iteration numbers
Condition numbers of M preconditioned with symmetric Gauss–Seidel iterations
Iterations  1  2  3 

(\(h=2^{8}\))  1.931  1.303  1.126 
Estimated condition numbers of \(\alpha R + M_\partial \) preconditioned with one Vcycle multigrid iteration
\(\alpha \)\h  \(2^{4}\)  \(2^{6}\)  \(2^{8}\) 

1  1.130  1.136  1.140 
\(10^{4}\)  1.129  1.135  1.139 
\(10^{8}\)  1.237  1.150  1.149 
\(10^{12}\)  1.252  1.259  1.253 
Number of preconditioned Minres iterations needed to solve the optimality system to a relative error tolerance \(\varepsilon = 10^{12}\)
\(\alpha \)\h  \( 2^{4}\)  \(2^{5}\)  \(2^{6}\)  \(2^{7}\) 

1  53( 4.33)  53 (4.36)  53 (4.36)  53 (4.36) 
\(10^{1}\)  57 (4.31)  57 (4.34)  57 (4.35)  57 (4.35) 
\(10^{2}\)  75 (4.31)  72 (4.34)  70 (4.35)  68 (4.35) 
\(10^{3}\)  79 (4.31)  79 (4.34)  77 (4.35)  73 (4.35) 
\(10^{4}\)  81 (4.30)  81 (4.33)  79 (4.35)  77 (4.35) 
\(10^{5}\)  82 (4.33)  81 (4.33)  79 (4.35)  79 (4.35) 
\(10^{6}\)  81 (4.35)  79 (4.36)  79 (4.35)  81 (4.35) 
\(10^{7}\)  70 (4.35)  81 (4.37)  81 (4.36)  79 (4.35) 
\(10^{8}\)  62 (4.36)  70 (4.36)  79 (4.36)  81 (4.36) 
\(10^{9}\)  62 (4.36)  64 (4.37)  68 (4.37)  78 (4.36) 
\(10^{10}\)  62 (4.36)  63 (4.36)  64 (4.37)  67 (4.37) 
4 Analysis of the KKT system
By using standard techniques for saddle point problems, one can show that the system (20) satisfies the Brezzi conditions [1], provided that \(\alpha >0\). Therefore, for every \(\alpha > 0\), this set of equations has a unique solution. Nevertheless, if the standard norms of \({L^2({\varOmega })}\) and \({H^2({\varOmega })}\) are employed in the analysis, then the constants in the Brezzi conditions will depend on \(\alpha \). More specifically, the constant in the coercivity condition will be of order \(O(\alpha )\), and thus becomes very small for \(0 < \alpha \ll 1\). This property is consistent with the ill posed nature of (1)–(3) for \(\alpha =0\), and makes it difficult to design \(\alpha \) robust preconditioners for the algebraic system associated with (20).
Similar to the approach used in [9, 10, 14], we will now introduce weighted Hilbert spaces. The weights are constructed such that the constants appearing in the Brezzi conditions are independent of \(\alpha \). Thereafter, in Sect. 5, we will show how these scaled Hilbert spaces can be combined with simple maps to design \(\alpha \) robust preconditioners for our model problem.
4.1 Weighted norms
4.2 Brezzi conditions
Lemma 1
Proof
Lemma 2
Proof
4.3 Boundedness
Having established that the Brezzi conditions hold, with constants that are independent of \(\alpha \), we next explore the boundedness of \(\mathcal {A}_{\alpha }\).
Lemma 3
Proof
Lemma 4
Proof
4.4 Isomorphism
Theorem 1
4.5 Estimates for the discretized problem
The stability properties (32) are not necessarily inherited by discretizations. However, the structure used to prove the socalled “infsup condition” in Lemma 1 is preserved in the discrete system provided that the same discretization is employed for the control and the Lagrange multiplier. Furthermore, the boundedness properties, Lemmas 3 and 4, certainly also hold for conforming discretizations.
It remains to adress the coercivity condition, Lemma 2, for the discretized problem. We consider finite dimensional subspaces \(U_h\subset U = {\bar{H}^2({\varOmega })}\) and \(W_h\subset W = {L^2({\varOmega })}\). For certain choices of \(U_h\) and \(W_h\), the estimate of Lemma 2 carries over to the finitedimensional setting.
Lemma 5
Proof
Assume that \((1{\varDelta }) U_h \subset W_h\), and that (34) holds for \((f_h,u_h)\in W_h\times U_h\). Then \(f_h +(1{\varDelta }) u_h \in W_h\), and (34) implies \(f_h +(1{\varDelta }) u_h = 0\). Therefore, \((f_h, u_h)\) satisfies (29) and the estimate (33) follows from Lemma 2. \(\square \)
If the state is discretized with \(C^1\)conforming bicubic Bogner–Fox–Schmit rectangles, as in Sect. 3, then Lemma 5 is satisfied if the control and Lagrange multiplier is discretized with discontinuous bicubic elements on the same mesh. For triangular meshes, one could choose Argyris triangles for the state variable and piecewise quintic polynomials for the control and Lagrange multiplier variables.
We remark that Lemma 5 provides a sufficient, but not necessary criterion for stability of the discrete problem, and usually may imply far more degrees of freedom in the discrete space \(W_h\subset W\) than is actually needed. The usefulness of Lemma 5 is that the estimates (35) can, in principle, always be obtained by choosing a sufficiently large space for the control and Lagrange multiplier.
5 Preconditioning
Lemma 6
Proof
We say that a preconditioner \(\mathcal {B}_{\alpha }\) for \(\mathcal {A}_{\alpha }\) is robust with respect to the parameter \(\alpha \) if \(\kappa (\mathcal {B}_{\alpha }\mathcal {A}_{\alpha })\) is bounded uniformly in \(\alpha \). The significance of Lemma 6 is that such a robust preconditioner can be found by identifying (parameterdependent) norms in which \(\mathcal {A}_{\alpha }\) and \(\mathcal {A}_{\alpha }^{1}\) are both uniformly bounded.
5.1 Parameterrobust minimum residual method
Theorem 2
Proof
Remark
In this paper we only consider the minimum residual method, and we therefore require that the preconditioner is selfadjoint and positive definite. More generally, if other Krylov subspace methods are to be applied to (20), then preconditioners lacking symmetry or definiteness may be considered.
We mention in particular that a preconditioned conjugate gradient method for problems similar to (20) was proposed in [14], based on a clever choice of inner product.
6 Generalization
Is our technique applicable to other problems than (1)–(3)? We will now briefly explore this issue, and show that the preconditioning scheme derived above yields \(\alpha \) robust methods for a class of problems.
The scaling (25)–(27) was also investigated in [10], but for a family of abstract problems posed in terms of Hilbert spaces. More specifically, for general PDEconstrained optimization problems, subject to Tikhonov regularization, and with linear state equations. But in [10] no assumptions about the control, state or observation spaces were made, except that they were Hilbert spaces. Under these circumstances, it was proved that the coercivity and the boundedness, of the operator associated with the KKT system, hold with \(\alpha \)independent constants. Nevertheless, in this general setting, the infsup condition involved an \(\alpha \)dependent constant, which, eventually, yielded theoretical iteration bounds of order \(O([\log \left( \alpha ^{1} \right) ]^2)\) for Minres.
In the present paper we were able to prove an \(\alpha \)robust infsup condition for the model problem (1)–(3). This is possible because both the control f and the dual/Lagrangemultiplier w belong to \({L^2({\varOmega })}\). From a more general perspective, it turns out that this is the property that must be fulfilled in order for our approach to be successful: The control space and the dual space, associated with the state equation, must coincide. This will usually lead to additional regularity requirements for the state space.
 (A1)
 \(A:U \rightarrow W'\) is a continuous linear operator with closed range. In particular, there is a constant \(c_1\) such that for all \(u \in U\),$$\begin{aligned} \left\ u \right\ _{U/ {\text {Ker}}A} = \inf _{{\tilde{u}} \in {\text {Ker}}A} \left\ u{\tilde{u}} \right\ _U \le c_1\left\ A u \right\ _{W'}. \end{aligned}$$
 (A2)
 \(T:U \rightarrow O\) is linear and bounded, and invertible on the kernel of A. That is, there is a constant \(c_2\) such that for all \(u\in {\text {Ker}}A\),$$\begin{aligned} \left\ u \right\ _U \le c_2 \left\ T u \right\ _O. \end{aligned}$$
We set \(\mathcal {V}= W_\alpha \times U_{\alpha } \times W_{\alpha ^{1}}\). As in Sect. 4, \(\mathcal {A}_{\alpha }:\mathcal {V}\rightarrow \mathcal {V}'\) can be shown to be an isomorphism, with parameterindependent estimates obtained in the weighted norms.
Theorem 3
Example 1
Example 2
Example 3
Note that we here consider the case in which observation data is assumed to be available throughout the entire domain \({\varOmega }\) of the state equation.
We remark that in [13, 14], parameterrobust preconditioners were proposed for the “prototype” problem, using the standard variational formulation (52) of the PDE. Those methods do not require improved regularity for the state space. Instead, they require that observations are available throughout the computational domain.
7 Eigenvalue analysis
In Sect. 6 it was shown that the condition number of \(\mathcal {B}_{\alpha }\mathcal {A}_{\alpha }\), with \(\mathcal {A}_{\alpha }\) defined in (44) and \(\mathcal {B}_{\alpha }\) defined in (51), can be bounded independently of \(\alpha \), as well as independently of the operators appearing in (42)–(43). Moreover, the numerical experiments indicate that the eigenvalues are contained in three intervals, independently of the regularization parameter \(\alpha \), see Fig. 2. In this section we detail the structure of the spectrum of the preconditioned system considered in Sect. 6, and we obtain sharp estimates for the constants appearing in Theorem 3.
 (B1)

M is a selfadjoint and positive definite,
 (B2)

\( K + R\) is positive definite,
 (B3)

K is selfadjoint and positive semidefinite.
Theorem 4
Proof
The estimate (57) follows from (56), noting that \(\vert {\text {sp}}(\mathcal {B}_\alpha \mathcal {A}_{\alpha }) \vert \subset [r_2, r_3]\). From (63) it can be seen that the roots of r are eigenvalues of \(\mathcal {B}_\alpha \mathcal {A}_{\alpha }\) if \({\text {Ker}}K \) is nontrivial. \(\square \)
Remark
8 Discretization with \(H^1\) conforming finite elements
The theory outlined in Sect. 4 provides a robust preconditioning technique for the optimality system (20) assuming additional regularity and making use of \(H^2\) conforming elements. However, this additional regularity only appears relevant to the discretization of the (2, 2) block of the ideal preconditioner (51), since the coefficient matrix in (44) only involves second order operators. It therefore seems reasonable that the use of sophisticated \(H^2\) conforming elements could be avoided in favour of standard \(H^1\) conforming elements, provided that we can implement an approximate inverse to the fourth operator appearing in the preconditioner.
The operator \(K_h + A_h M_h^{1} A_h\) in the (2,2) block of (65) coincides with Schur complement of a CiarletRaviart mixed finite element formulation of the fourth order problem (16)–(18), and can be thought of as nonlocal fourth order operator. Multigrid techniques for a similar operator was studied in [8], where a multigrid Wcycle applied to a local operator approximating the Schur complement was shown to be an efficient preconditioner.
Table 4 presents iteration numbers and estimated condition numbers for a simplistic scheme where we replace the (2,2) block in (65) with \(K_h + A_h {\tilde{M}}_h^{1} A_h\), where \({\tilde{M}}_h\) is a lumped mass matrix. For the appxroximate inversion of (65), we applied an algebraic multigrid Wcycle for the (2,2) block and two symmetric Gauss–Seidel iterations to the remaining two diagonal blocks. The experiment was carried out on a unit square domain and an Lshaped domain, with both domains triangularized with structured meshes. For the Lshaped domain, the \(H^2\)regularity discussed in the beginning of Sect. 2 is known not to hold.
The iteration numbers reported in Table 4 appears bounded, although we observe an increase in the estimated condition number for the Lshaped domain as the mesh is refined. Although the condition number with an exact inverse (65) is bounded in accordance with the analysis in Sect. 7, this appears not to be the case when the exact inverse of the (2,2) block is replaced with an AMG cycle.
\(\alpha \backslash h\)  \(2^{6}\)  \(2^{7}\)  \(2^{8}\)  \(2^{9}\) 

Square domain  
\(10^{10}\)  81 (6.80)  82 (6.80)  88 (6.87)  93 (6.90) 
\(10^{8}\)  90 (6.65)  93 (6.82)  91 (6.63)  89 (6.63) 
\(10^{6}\)  95 (7.10)  90 (6.63)  89 (6.66)  88 (6.71) 
\(10^{4}\)  89 (6.63)  89 (6.68)  88 (6.72)  86 (6.73) 
\(10^{2}\)  79 (6.63)  79 (6.70)  78 (6.73)  78 (6.73) 
1  69 (6.68)  68 (6.74)  67 (6.75)  67 (6.75) 
Lshaped domain  
\(10^{10}\)  80 (6.80)  82 (6.76)  88 (6.81)  93 (6.87) 
\(10^{8}\)  90 (6.65)  93 (6.73)  91 (6.63)  89 (6.63) 
\(10^{6}\)  93 (6.92)  90 (6.63)  89 (6.64)  88 (6.71) 
\(10^{4}\)  89 (6.64)  89 (6.68)  92 (8.16)  92 (10.2) 
\(10^{2}\)  85 (8.25)  87 (10.2)  89 (13.1)  90 (17.4) 
1  90 (16.4)  93 (22.5)  94 (32.0)  86 (46.9) 
9 Discussion
Previously, parameter robust preconditioners for PDEconstrained optimization problems have been successfully developed, provided that observation data is available throughout the entire domain of the state equation. For many important inverse problems, arising in industry and science, this is an unrealistic requirement. On the contrary, observation data will typically only be available in subregions, of the domain of the state variable, or at the boundary of this domain. We have therefore explored the possibility for also constructing robust preconditioners for PDEconstrained optimization problems with limited observation data.
For an elliptic control problem, with boundary observations only, we have developed a regularization robust preconditioner for the associated KKT system. Consequently, the number of Minres iterations required to solve the problem is bounded independently of both regularization parameter \(\alpha \) and the mesh size h. In order to achieve this, it was necessary to write the elliptic state equation on a nonstandard, and nonselfadjoint, variational form. If this approach is employed, then the control and the Lagrange multiplier will belong to the same Hilbert space, which leads to extra regularity requirements for the state. This fact makes it possible to construct parameter weighted metrics such that the constants appearing in the Brezzi conditions, as well as the constants in the inequalities expressing the boundedness of the KKT system, are independent of \(\alpha \) and h. Consequently, the spectrum of the preconditioned KKT system is uniformly bounded with respect to \(\alpha \) and h, which is ideal for the Minres scheme. These properties were illuminated through a series of numerical experiments, and the preconditioned Minres scheme handled our model problem excellently.
The use of a nonselfadjoint form of the elliptic state equation leads to additional challenges for conforming discretization schemes and in multigrid implementations. For the numerical experiments, we employed a \(C^1\) finite element discretization that is \(H^2\)conforming, where the rectangular elements are tensor products of Hermite intervals. This discretization is limited to structured meshes. While there are other, more flexible \(C^1\) finite element discretizations available in two dimensions (e.g. Argyris and Bell triangles), all of the methods suffer from high computational cost due the smoothness requirements imposed on the nodal basis functions. In three dimensions, the situation is even worse, and \(C^1\) discretizations with tetrahedrons become nearly intractable, see e.g. [15].
Some of the difficulties with traditional \(C^1\) finite element discretizations can be avoided with Galerkin methods making use of basis functions that naturally fulfill the smoothness requirements. Examples of such methods include discretization with spline basis functions, such as isogeometric analysis [3]. Another approach is the virtual element method [2]. However, the development of multilevel methods for the fourth order operator in the preconditioner (51) would remain a challenging problem.
We have also demonstrated that the technique is applicable also outside of \(H^2\)conforming discretizations.
Our findings for the simple elliptic control problem were generalized to a broader class of KKT systems. It turns out that the methodology is applicable whenever the control and the Lagrange multiplier belong to the same space, and extra regularity properties are fulfilled by the state equation  these are the key issues. From a theoretical perspective, this is in many cases not a severe restriction, but it gives rise to new challenges for the discrete problems. This is even the case for the elliptic state equation considered in this text. Also, our approach will not yield \(\alpha \) independent bounds if the control is only defined on a subdomain of the domain of the state equation. In such cases, the spaces for the control and the Lagrange multiplier will not coincide. How to design efficient parameterrobust preconditioners for such problems, is, as far as the authors know, still an open problem.
Footnotes
 1.In [10, 11] it is proved that the number of needed preconditioned Minres iterations cannot grow faster thanFurthermore, in [11] it is explained why iterations counts of the kind (5) often will occur in practice.$$\begin{aligned} a + b \left[ \log _{10} \left( \alpha ^{1} \right) \right] ^2. \end{aligned}$$
Notes
Acknowledgements
The first author is grateful to Adrian Hope for conducting numerical experiments concerning Hermite elements.
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