Some applications of weighted norm inequalities to the error analysis of PDE constrained optimization problems††thanks: HA has been supported in part by the NSF grant DMS-1521590. EO has been supported in part by CONICYT through FONDECYT project 3160201. AJS has been supported in part by NSF grant DMS-1418784.
The purpose of this work is to illustrate how the theory of Muckenhoupt weights, Muckenhoupt weighted Sobolev spaces and the corresponding weighted norm inequalities can be used in the analysis and discretization of PDE constrained optimization problems. We consider: a linear quadratic constrained optimization problem where the state solves a nonuniformly elliptic equation; a problem where the cost involves pointwise observations of the state and one where the state has singular sources, e.g. point masses. For all three examples we propose and analyze numerical schemes and provide error estimates in two and three dimensions. While some of these problems might have been considered before in the literature, our approach allows for a simpler, Hilbert space-based, analysis and discretization and further generalizations.
Key words. PDE constrained optimization, Muckenhoupt weights, weighted Sobolev spaces, finite elements, polynomial interpolation in weighted spaces, nonuniform ellipticity, point observations, singular sources.
AMS subject classifications. 35J15, 35J75, 35J70, 49J20, 49M25, 65D05, 65M12, 65M15, 65M60.
The purpose of this work is to show how the theory of Muckenhoupt weights, Muckenhoupt weighted Sobolev spaces and weighted norm inequalities can be applied to analyze PDE constrained optimization problems, and their discretizations. These tools have already been shown to be essential in the analysis and discretization of problems constrained by equations involving fractional derivatives both in space and time [AO, AOS], and here we extend their use to a new class of problems.
We consider three illustrative examples. While some of them have been considered before, the techniques that we present are new and we believe they provide simpler arguments and allow for further generalizations. To describe them let be an open and bounded polytopal domain of () with Lipschitz boundary . We will be concerned with the following problems:
Optimization with nonuniformly elliptic equations. Let be a weight, that is, a positive and locally integrable function and . Given a regularization parameter , we define the cost functional
We are then interested in finding subject to the nonuniformly elliptic problem
and the control constraints
where is a nonempty, closed and convex subset of . The main source of difficulty and originality here is that the matrix is not uniformly elliptic, but rather satisfies
for almost every . Since we allow the weight to vanish or blow up, this nonstandard ellipticity condition must be treated with the right functional setting.
Problems such as (LABEL:eq:defofPDEA) arise when applying the so-called Caffarelli-Silvestre extension for fractional diffusion [AO, AOS, CS:07, NOS, NOS2], when dealing with boundary controllability of parabolic and hyperbolic degenerate equations [MR2373460, MR3171770, MR3227458] and in the numerical approximation of elliptic problems involving measures [MR3264365, NOS2]. In addition, invoking Rubio de Francia’s extrapolation theorem [MR1800316, Theorem 7.8] one can argue that this is a quite general PDE constrained optimization problem with an elliptic equation as state constraint, since there is no , only with weights.
Optimization with point observations. Let with . Given a set of prescribed values , a regularization parameter , and the cost functional
the problem under consideration reads as follows: Find subject to
and the control constraints
where is a nonempty, closed and convex subset of . In contrast to standard elliptic PDE constrained optimization problems, the cost functional (LABEL:eq:defofJp) involves point evaluations of the state. Note that these evaluations are not required for the state equation (LABEL:eq:defofPDEp) to be well posed. Additional assumptions must be made for this to make sense and, as will be seen below, the point evaluations of the state in (LABEL:eq:defofJp) lead to a subtle formulation of the adjoint problem.
Problem (LABEL:eq:defofJp)–(LABEL:eq:cc2) finds relevance in numerous applications where the observations are carried out at specific locations. For instance, in the so-called calibration problem with American options [MR2137495], in the optimal control of selective cooling of steel [MR1844451], in the active control of sound [MR2086168, ACS] and in the active control of vibrations [ACV, MR2525606]; see also [BrettElliott, BEHL, MR3150173, MR2536481, MR2193509] for other applications.
The point observation terms in the cost (LABEL:eq:defofJp), tend to enforce the state to have the fixed value at the point . Consequently, (LABEL:eq:defofJp)–(LABEL:eq:cc2) can be understood as a penalty version of a PDE constrained optimization problem where the state is constrained at a collection of points. We refer the reader to [BrettElliott, Section 3.1] for a precise description of this connection and to [MR3071172] for the analysis and discretization of an optimal control problem with state constraints at a finite number of points.
Despite its practical importance, to the best of our knowledge, there are only two references where the approximation of (LABEL:eq:defofJp)–(LABEL:eq:cc2) is addressed: [BrettElliott] and [MR3449612]. In both works the key observation, and main source of difficulty, is that the adjoint state for this problem is only in with . With this functional setting, the authors of [BrettElliott] propose a fully discrete scheme which discretizes the control explicitly using piecewise linear elements. For , the authors obtain a rate of convergence for the optimal control in the -norm provided the control and the state are discretized using meshes of size and , respectively; see [BrettElliott, Theorem 5.1]. This condition immediately poses two challenges for implementation: First, it requires to keep track of the state and control on different meshes. Second, some sort of interpolation and projection between these meshes needs to be realized. In addition, the number of unknowns for the control is significantly higher, thus leading to a slow optimization solver. The authors of [BrettElliott] were unable to extend these results to . Using the so-called variational discretization approach [Hinze:05], the control is implicitly discretized and the authors were able to prove that the control converges with rates for and for . In a similar fashion, the authors of [MR3449612] use the variational discretization concept to obtain an implicit discretization of the control and deduce rates of convergence of and for and , respectively. A residual-type a posteriori error estimator is introduced, and its reliability is proven. However, there is no analysis of the efficiency of the estimator.
In Section LABEL:sec:points below we introduce a fully discrete scheme where we discretize the control with piecewise constants; this leads to a smaller number of degrees of freedom for the control in comparison to the approach of [BrettElliott]. We circumvent the difficulties associated with the adjoint state by working in a weighted -space and prove near optimal rates of convergence for the optimal control: for and for , respectively. In addition, we provide pointwise error estimates for the approximation of the state: for and for .
Optimization with singular sources. Let be linearly ordered and with cardinality . Given a desired state and a regularization parameter , we define the cost functional
We shall be concerned with the following problem: Find subject to
where is the Dirac delta at the point and
where with , again, nonempty, closed and convex. Notice that since, for , , the solution to (LABEL:eq:defofPDEd) does not belong to . Consequently, the analysis of the finite element method applied to such a problem is not standard [Casas:85, NOS2, Scott:73]. We rely on the weighted Sobolev space setting described and analyzed in [NOS2, Section 7.2].
The state (LABEL:eq:defofPDEd), in a sense, is dual to the adjoint equation for (LABEL:eq:defofJp)–(LABEL:eq:defofPDEp), where the adjoint equation has Dirac deltas on the right hand side. The optimization problem (LABEL:eq:defofJd)–(LABEL:eq:defofPDEd) is of relevance in applications where one can specify a control at a finitely many pre-specified points. For instance, references [MR2086168, ACS] discuss applications within the context of the active control of sound and [ACV, MR2718176, MR2525606] in the active control of vibrations; see also [MR3268059, MR3150173, MR3116646].
An analysis of problem (LABEL:eq:defofJd)–(LABEL:eq:cc3) is presented in [MR3225501], where the authors use the variational discretization concept to derive error estimates. They show that the control converges with a rate of and in two and three dimensions, respectively. Their technique is based on the fact that the state belongs to with . In addition, under the assumption that they improve their results and obtain, up to logarithmic factors, rates of and . Finally, we mention that [MR2974716, MR3072225] study a PDE constrained optimization problem without control constraints, but where the controls is a regular Borel measure.
In Section LABEL:sec:delta we present a fully discrete scheme for which we provide rates of convergence for the optimal control: in two dimensions and in three dimensions, where . We also present rates of convergence for the approximation error in the state variable.
Before we embark in further discussions, we must remark that while the introduction of a weight as a technical instrument does not seem to be completely new, the techniques that we use and the range of problems that we can tackle is. For instance, for integro-differential equations where the kernel is weakly singular, the authors of [MR1306580] study the well-posedness of the problem in the weighted space. Numerical approximations for this problem with the same functional setting were considered in [MR1135762] where convergence is shown but no rates are obtained. These ideas were extended to neutral delay-differential equations in [MR3064275, MR2018123] where a weight is introduced in order to renorm the state space and obtain dissipativity of the underlying operator. In all these works, however, the weight is essentially assumed to be smooth and monotone except at the origin where it has an integrable singularity [MR1306580, MR1135762] or at a finite number of points where it is allowed to have jump discontinuities [MR3064275, MR2018123]. All these properties are used to obtain the aforementioned results. In contrast, our approach hinges only on the fact that the introduced weights belong to the Muckenhoupt class (see Definition LABEL:def:defofAp below) and the pertinent facts from real and harmonic analysis and approximation theory that follow from this definition. Additionally we obtain convergence rates which, up to logarithmic factors, are optimal with respect to regularity. Finally we must point out that the class of problems we study is quite different from those considered in the references given above.
Our presentation will be organized as follows. Notation and general considerations will be introduced in Section LABEL:sec:notation. Section LABEL:sec:A presents the analysis and discretization of problem (LABEL:eq:defofJA)–(LABEL:eq:cc). Problem (LABEL:eq:defofJp)–(LABEL:eq:cc2) is studied in Section LABEL:sec:points. The analysis of problem (LABEL:eq:defofJd)–(LABEL:eq:cc3) is presented in Section LABEL:sec:delta. Finally, in Section LABEL:sec:NumExp, we illustrate our theoretical developments with a series of numerical examples.
2 Notation and preliminaries
Let us fix notation and the setting in which we will operate. In what follows is a convex, open and bounded domain of () with polytopal boundary. The handling of curved boundaries is somewhat standard, but leads to additional technicalities that will only obscure the main ideas we are trying to advance. By we mean that there is a nonessential constant such that . The value of this constant might change at each occurrence.
2.1 Weights and weighted spaces
Throughout our discussion we call a weight a function such that for a.e. . In particular we are interested in the so-called Muckenhoupt weights [MR1800316, Turesson].
Definition 2.1 (Muckenhoupt class)
Let and be a weight. We say that if
where the supremum is taken over all balls .
From the fact that many fundamental consequences for analysis follow. For instance, the induced measure is not only doubling, but also strong doubling (cf. [NOS2, Proposition 2.2]). We introduce the weighted Lebesgue spaces
and note that [NOS2, Proposition 2.3] shows that their elements are distributions, therefore we can define weighted Sobolev spaces
which are complete, separable and smooth functions are dense in them (cf. [Turesson, Proposition 2.1.2, Corollary 2.1.6]). We denote .
We define as the closure of in and set . On these spaces, the following Poincaré inequality holds
where the hidden constant is independent of , depends on the diameter of and depends on only through .
The literature on the theory of Muckenhoupt weighted spaces is rather vast so we only refer the reader to [MR1800316, NOS2, Turesson] for further results.
2.2 Finite element approximation of weighted spaces
Since, the spaces are separable for , and smooth functions are dense, it is possible to develop a complete approximation theory using functions that are piecewise polynomial. This is essential, for instance, to analyze the numerical approximation of (LABEL:eq:defofPDEA) with finite element techniques. Let us then recall the main results from [NOS2] concerning this scenario.
Let be a conforming triangulation (into simplices or -rectangles) of . We denote by a family of triangulations, which for simplicity we assume quasiuniform. The mesh size of is denoted by . Given we define the finite element space
where, if is a simplex, — the space of polynomials of degree at most one. In the case that is an -rectangle — the space of polynomials of degree at most one in each variable. Notice that, by construction, for any and .
The results of [NOS2] show that there exists a quasi-interpolation operator , which is based on local averages over stars and thus well defined for functions in . This operator satisfies the following stability and approximation properties:
Finally, to approximate the PDE constrained optimization problems described in Section LABEL:sec:introduccion we define the space of piecewise constants by
2.3 Optimality conditions
To unify the analysis and discretization of the PDE constrained optimization problems introduced and motivated in Section LABEL:sec:introduccion and thoroughly studied in subsequent sections, we introduce a general framework following the guidelines presented in [GH:09, MR2516528, MR2441683, MR0271512, JCarlos, Tbook]. Let and be Hilbert spaces denoting the so-called control and observation spaces, respectively. We introduce the state trial and test spaces and , and the corresponding adjoint trial and test spaces and , which we assume to be Hilbert. In addition, we introduce:
A bilinear form which, when restricted to either or , satisfies the conditions of the BNB theorem; see [Guermond-Ern, Theorem 2.6].
A bilinear form which, when restricted to either or is bounded. The bilinear forms and will be used to describe the state and adjoint equations.
An observation map , which we assume linear.
A desired state .
A regularization parameter and a cost functional
All our problems of interest can be cast as follows. Find subject to:
and the constraints
where is nonempty, bounded, closed and convex. We introduce the control to state map which to a given control, , associates a unique state, , that solves the state equation (LABEL:eq:abstate). As a consequence of (LABEL:a) and (LABEL:b), the map is a bounded and linear operator. With this operator at hand we can eliminate the state variable from (LABEL:eq:absJ) and introduce the reduced cost functional
Then, our problem can be cast as: Find over . As described in (LABEL:e) we have so that is strictly convex. In addition, is weakly sequentially compact in . Consequently, standard arguments yield existence and uniqueness of a minimizer [Tbook, Theorem 2.14]. In addition, the optimal control can be characterized by the variational inequality
where denotes the Gâteaux derivative of at [Tbook, Lemma 2.21]. This variational inequality can be equivalently written as
where denotes the optimal adjoint state and solves
The optimal state is the solution to (LABEL:eq:abstate) with .
2.4 Discretization of PDE constrained optimization problems
Let us now, in the abstract setting of Section LABEL:sub:optim, study the discretization of problem (LABEL:eq:absJ)–(LABEL:eq:ccabs). Since our ultimate objective is to approximate the problems described in Section LABEL:sec:introduccion with finite element methods, we will study the discretization of (LABEL:eq:absJ)–(LABEL:eq:ccabs) with Galerkin-like techniques.
Let be a parameter and assume that, for every , we have at hand finite dimensional spaces , , , and . We define , which we assume nonempty. About the pairs , for , we assume that they are such that satisfies a BNB condition uniformly in ; see [Guermond-Ern, §2.2.3]. In this setting, the discrete counterpart of (LABEL:eq:absJ)–(LABEL:eq:ccabs) reads: Find
subject to the discrete state equation
and the discrete constraints
As in the continuous case, we introduce the discrete control to state operator , which to a discrete control, , associates a unique discrete state, , that solves (LABEL:eq:abstateh). is a bounded and linear operator.
The pair is optimal for (LABEL:eq:Jdis)–(LABEL:eq:ccdis) if solves (LABEL:eq:abstateh) and the discrete control satisfies the variational inequality
where the discrete adjoint variable solves
To develop an error analysis for the discrete problem described above, we introduce the -orthogonal projection onto . We assume that . In addition, we introduce two auxiliary states that will play an important role in the discussion that follows. We define
i.e., is defined as the solution to (LABEL:eq:abstateh) with replaced by . We also define
this is, is the solution to (LABEL:eq:absadjh) with replaced by .
The main error estimate with this level of abstraction reads as follows.
Lemma 2.2 (abstract error estimate)
Let and be the continuous and discrete optimal pairs that solve (LABEL:eq:absJ)–(LABEL:eq:ccabs) and (LABEL:eq:Jdis)–(LABEL:eq:ccdis), respectively. If
then, for any , we have the estimate
where the constant depends on and but does not depend on .
Proof. Since by definition and by assumption , we set and in (LABEL:eq:VIabs) and (LABEL:eq:VIabsh) respectively. Adding the ensuing inequalities we obtain
Denote . In order to estimate this term we add and subtract to obtain
Since is the unique solution to (LABEL:eq:hatph), we have that
Similarly, since solves (LABEL:eq:hatyh), we derive
Set and which, by assumption (LABEL:eq:intersection), are admissible test functions to obtain
This, and the continuity of the bilinear form allow us to bound (LABEL:I) as follows:
where denotes the norm of the bilinear form .
Let us now analyze the remaining terms in (LABEL:eq:aux), which we denote by II. To do this, we rewrite II as follows:
Now, notice that
Next, since the bilinear form is continuous, we arrive at
The remaining term, which we will denote by III, is treated by using, again, that the bilinear form is continuous. This implies that for any
From (LABEL:eq:phMphu) and the fact that the discrete spaces satisfy a discrete BNB condition uniformly in we conclude
Collecting these derived estimates we bound the term II.
The estimates we obtained for I, II and III readily yield the claimed result.
The use of this simple result will be illustrated in the following sections.
Remark 2.3 (on the choice of )
In (LABEL:eq:errest) the choice of can be arbitrary. A judicious choice will be fundamental in the error analysis of the optimization problem with point observations (LABEL:eq:defofJp)–(LABEL:eq:cc2).
3 Optimization with nonuniformly elliptic equations
In this section we study the problem (LABEL:eq:defofJA)–(LABEL:eq:cc) under the abstract framework developed in Section LABEL:sub:optim. Let be a convex polytope and where the -Muchenkhoupt class is given by Definition LABEL:def:defofAp. In addition, we assume that is symmetric and satisfies the nonuniform ellipticity condition (LABEL:eq:nonunifellip).
Owing to the fact that the diffusion matrix satisfies (LABEL:eq:nonunifellip) with , as shown in [FKS:82], the state equation (LABEL:eq:defofPDEA) is well posed in , whenever . For this reason, we set:
which, as a consequence of (LABEL:eq:nonunifellip) with and the Poincaré inequality (LABEL:eq:Poincare), is bounded, symmetric and coercive in .
. Notice that, if and then
where we have used the Poincaré inequality (LABEL:eq:Poincare).
The cost functional as in (LABEL:eq:defofJA).
For , we define the set of admissible controls by
which is closed, bounded and convex in . In addition, since the functional (LABEL:eq:defofJA) is strictly convex. Consequently, the optimization problem with nonuniformly elliptic state equation (LABEL:eq:defofJA)–(LABEL:eq:cc) has a unique optimal pair [Tbook, Theorem 2.14]. In this setting, the first order necessary and sufficient optimality condition (LABEL:eq:VIabs) reads
where the optimal state solves
and the optimal adjoint state solves
The results of [FKS:82], again, yield that the adjoint problem is well posed.
Let us now propose a discretization for problem (LABEL:eq:defofJA)–(LABEL:eq:cc), and derive a priori error estimates based on the results of Section LABEL:sub:absdisc. Given a family of quasi-uniform triangulations of we set:
, where the discrete space is defined in (LABEL:eq:Upc).
, where the set of admissible controls is defined in (LABEL:eq:UA).
is the -orthogonal projection onto , which we denote by and is defined by
The definition of yields that .
, where the discrete space is defined in (LABEL:eq:V).
Notice that, since , assumption (LABEL:eq:intersection) is trivially satisfied. We obtain the following a priori error estimate.
Corollary 3.1 (a priori error estimate)
Let and be the continuous and discrete optimal controls, respectively. If then
where the hidden constant is independent of .
Proof. We invoke Lemma LABEL:lem:abserror and bound each one of the terms in (LABEL:eq:errest). First, since , the results of [NOS2] imply that
Indeed, since solves (LABEL:eq:adjA) and solves (LABEL:eq:hatph), the term satisfies
Adding and subtracting the terms and appropriately, where denotes the interpolation operator described in §LABEL:sub:FEM, and using the coercivity of we arrive at
Using the regularity of and we obtain the claimed bound.
We now handle the second term involving the derivative of the reduced cost . Since it can be equivalently written using (LABEL:eq:VIabs), invoking the definition of given by (LABEL:eq:proj-1), we obtain
The Poincaré inequality (LABEL:eq:Poincare), in conjunction with the stability of the discrete state equation (LABEL:eq:abstateh), yield
for all . This yields control of the last term in (LABEL:eq:errest).
These bounds yield the result.
Remark 3.2 (regularity of and )
The results of Corollary LABEL:col:errA rely on the fact that . Reference [MR2780884] provides sufficient conditions for this to hold.
Theorem 3.3 (rate of convergence)
In the setting of Corollary LABEL:col:errA, if we additionally assume that then, we have the optimal error estimate
where the hidden constant is independent of .
Proof. We bound and . Using that and a Poincaré-type inequality [NOS2, Theorem 6.2], we obtain
Now, to estimate the term , it is essential to understand the regularity properties of . From [Tbook, Section 3.6.3], solves (LABEL:eq:VIA) if and only if
The assumption immediately yields [KSbook, Theorem A.1], which allows us to derive the estimate
Collecting the derived results we arrive at the desired estimate.
4 Optimization with point observations
Here we consider problem (LABEL:eq:defofJp)–(LABEL:eq:cc2). Let be a convex polytope, with . We recall that denotes the set of observable points with .
To analyze problem (LABEL:eq:defofJp)–(LABEL:eq:cc2) using the framework of weighted spaces we must begin by defining a suitable weight. Since , we know that . For each we then define
and the weight
As [NOS2, Lemma 7.5] shows, with this definition we have that . With this