A Range Condition for Polyconvex Variational Regularization

Abstract In the context of convex variational regularization, it is a known result that, under suitable differentiability assumptions, source conditions in the form of variational inequalities imply range conditions, while the converse implication only holds under an additional restriction on the operator. In this article, we prove the analogous result for polyconvex regularization. More precisely, we show that the variational inequality derived by the authors in 2017 implies that the derivative of the regularization functional must lie in the range of the dual-adjoint of the derivative of the operator. In addition, we show how to adapt the restriction on the operator in order to obtain the converse implication.


Consider a nonlinear operator equation with inexact data
where K : U ! V acts between Banach spaces, v † ; v d 2 V are exact and noisy data, respectively, and d > 0 is the noise level. A common method for the stable inversion of K is variational regularization which consists in computing regularized solutions u d a as minimizers of functionals of the form u 7 ! T a ðu; v d Þ ¼ jjKðuÞ À v d jj q þ aRðuÞ: Here, R is a typically convex regularization functional, a > 0 and q ! 1: A natural requirement for such methods is that regularized solutions converge, in some sense, to an exact solution as the noise level tends to zero. Convergence rates additionally provide bounds on the discrepancy between regularized and exact solutions in terms of the noise level. In a Banach space setting, the most common measure of discrepancy is the Bregman distance associated to R [1].
In order to guarantee convergence rates, one has to impose a source condition of some sort. Traditionally, in a linear Hilbert space setting with quadratic Tikhonov regularization, this was done by assuming that the minimum norm solution lies in the range of an operator closely related to the adjoint of K. See [2,Ch. 5] for example. Generalizing this range condition to the nonlinear Banach space setting outlined in the previous paragraph yields R 0 ðu † Þ 2 ran K 0 ðu † Þ # ; (1.2) where u † is an R-minimizing solution and K 0 ðu † Þ # is the dual-adjoint of the Gâteaux derivative of K at u † . More recently, it was shown by Hofmann et al. [3] that convergence rates can also be obtained by assuming that a variational inequality like holds for all u in a certain neighborhood of u † . Here u Ã is a subgradient of R at u † and D u Ã ðu; u † Þ denotes the corresponding Bregman distance between u and u † . Note that (1.3) does not require K or R to be differentiable. If they are, however, then the variational inequality (1.3) implies the range condition (1.2). The converse implication only holds under an additional assumption on the nonlinearity of the operator K. For a more detailed discussion of the relations between the various types of source conditions, we refer to [4, pp. 70-73].
For certain inverse problems on W 1;p ðX; R N Þ, such as image or shape registration models inspired by nonlinear elasticity [5,6], convex regularization is too restrictive, while the weaker notion of polyconvexity is more appropriate. Indeed, nonconvex regularization functionals R with polyconvex integrands are well-suited for deriving stable and convergent regularization schemes. However, since such functionals are not subdifferentiable in general, the question is how to obtain convergence rates. According to Kirisits and Scherzer [7], we addressed this issue by following Grasmair's approach of generalized Bregman distances [8]. First, we introduced the weaker concept of W poly -subdifferentiability, specifically designed for functionals with polyconvex integrands, and gave conditions for existence of W poly -subgradients. By means of the corresponding W poly -Bregman distance, we were then able to translate the convergence rates result by Hofmann et al. [3] to the polyconvex setting. The source condition derived by Kirisits and Scherzer [7] reads where w is a W poly -subgradient of R at u † and D poly w ðu; u † Þ is the corresponding generalized Bregman distance.
The main results of this article are Theorems 3.1 and 3.2 in Section 3. Theorem 3.1 states that the variational inequality (1.4) implies the range condition (1.2), given that K and R are differentiable and R satisfies the conditions guaranteeing existence of a W poly -subgradient. A major part of the proof consists in showing that R 0 ðu † Þ ¼ w 0 ðu † Þ in this case. Conversely, Theorem 3.2 states that w 0 ðu † Þ 2 ran K 0 ðu † Þ # implies (1.4), if the nonlinearities of K and w satisfy a certain inequality around u † .

Polyconvex functions and generalized Bregman distances
This section is a brief summary of the most important prerequisites by Kirisits and Scherzer [7]. For N; n 2 N we will frequently identify matrices in R NÂn with vectors in R Nn . Now, a function f :

Polyconvex functions
Every convex function is polyconvex. The converse statement only holds, if NÙn ¼ 1: The importance of polyconvex functions in the calculus of variations is due to the fact that they render functionals of the form For more details on polyconvex functions, see [9,10].

The set W poly
For the remainder of this article, unless stated otherwise, we let X & R n be an open set, p ! NÙn, and set U ¼ W 1;p ðX; R N Þ.
The following variant of the map T will prove useful. Set s 2 :¼ X N Ù n s¼2 rðsÞ and define T 2 ðAÞ :¼ ðadj 2 A; :::; adj N Ù n AÞ 2 R s2 : If u 2 U, then adj s ru consists of sums of products of s L p ðXÞ functions, and therefore, by H€ older's inequality, adj s ru 2 L p=s ðX; R rðsÞ Þ. This motivates the following two definitions: We define W poly to be the set of all functions w : Thus, the dual U Ã can be regarded a subset of W poly in a natural way.

Generalized subgradients
We denote the effective domain of R by dom R ¼ fu 2 U : RðuÞ< þ 1g. Following [8,7,11] we define the W poly -subdifferential of R at u 2 dom R as @ poly RðuÞ ¼ fw 2 W poly : RðvÞ ! RðuÞ þ wðvÞ À wðuÞ for all v 2 Ug; If RðuÞ ¼ þ1, we set @ poly RðuÞ ¼ 1. The identification of U Ã with a subset of W poly mentioned in the previous paragraph implies that @RðuÞ & @ poly RðuÞ, that is, the classical subdifferential can be regarded a subset of the W poly -subdifferential. Elements of @ poly RðuÞ are called W poly -subgradients of R at u. Concerning existence of W poly -subgradients we have shown the following result [7].
be a Carath eodory function. Assume that, for almost every x 2 X, the map ðu; nÞ7 !Fðx; u; nÞ is convex and differentiable throughout its effective domain and denote its derivative by F 0 u;n . Let p 2 ½1; 1Þ and define the following functional on U ¼ W 1;p ðX; R N Þ RðuÞ ¼ ð X Fðx; uðxÞ; TðruðxÞÞÞ dx: If Rð vÞ 2 R and the function x7 !F 0 u;n ðx; vðxÞ; Tðr vðxÞÞÞ lies in L p Ã ðX; R N Þ Â S Ã , where p Ã denotes the H€ older conjugate of p, then this function is a W poly -subgradient of R at v.

1.
If F 0 u;n ðÁ; vðÁÞ; Tðr vðÁÞÞÞ is a W poly -subgradients w 2 @ poly Rð vÞ & W poly , as postulated by Lemma 2.1, then it must be possible to write its action on u 2 U in terms of a pair ðu Ã ; v Ã Þ 2 U Ã Â S Ã 2 as in (2.5). In order to do so recall that TðAÞ ¼ ðA; T 2 ðAÞÞ. We can split the variable n 2 R s accordingly into ðn 1 ; n 2 Þ 2 R Nn Â R s 2 : Similarly, we can The integral in the bottom line corresponds to the dual pairing hv Ã ; T 2 ðruÞi S Ã 2 ;S 2 in (2.5), while the previous two terms correspond to hu Ã ; ui U Ã ;U . Therefore, u Ã is given by ðF 0 u ; F 0 n 1 Þ and v Ã by F 0 n 2 . Also, note that all integrals are well defined and finite because of the integrability conditions on the derivative of F in Lemma 2.1.

Generalized Bregman distances
Whenever R has a W poly -subgradient w 2 @ poly RðuÞ we can define the associated W poly -Bregman distance between v 2 U and u as D poly w ðv; uÞ ¼ RðvÞ À RðuÞ À wðvÞ þ wðuÞ: Note that, just like the classical Bregman distance, the W poly -Bregman distance is nonnegative, satisfies D poly w ðu; uÞ ¼ 0 whenever defined, and is not symmetric with respect to u and v. In addition, if w ¼ ðu Ã ; 0Þ 2 R poly ðuÞ, then u Ã 2 @RðuÞ and the classical and W poly -Bregman distances coincide, that is, See [8,4] for more details on (generalized) Bregman distances. In order to be able to quote the source condition by Kirisits and Scherzer [7], we need one more definition: We call u † 2 U an R-minimizing solution, if it solves the exact operator equation and minimizes R among all other exact solutions, that is, u † 2 arg minfRðuÞ : u 2 U; KðuÞ ¼ v † g: Assumption 2.1. Assume that R has a W poly -subgradient w at an R-minimizing solution u † and that there are constants b 1 2 ½0; 1Þ; b 2 ; a > 0 and q > aRðu † Þ such that wðu † Þ À wðuÞ b 1 D poly w ðu; u † Þ þ b 2 jjKðuÞ À v † jj (2. 6) holds for all u with T a ðu; v † Þ q.

A range condition
At the end of this section, we prove our main results, Theorems 3.1 and 3.2. Before that, we have to state a few preliminary results. First, we recall the definition of the dual-adjoint operator together with a characterization of its range (Lemma 3.2). Next, we compute the Gâteaux derivative of for all u 2 U and v Ã 2 V Ã : See, for instance, Section VII.1 of Ref. [12]. The operator A # is called the dual-adjoint of A. Proof. See Lemma 8.21 in Ref. [4].
Let K : DðKÞ & U ! V be a map acting between normed spaces and let u 2 DðKÞ; h 2 U: If the limit holds there for p ! 1 and some a 2 L p Ã ðXÞ and b; c ! 0. Then, the functional defined by is Gâteaux differentiable in the interior of its effective domain. Its Gâteaux derivative at u 2 int dom R is given by uðxÞ; ruðxÞÞ Á rûðxÞ dx;û 2 U: Proof. Fix u 2 int dom R andû 2 U. Assuming we can differentiate under the integral sign we have which is just Equation (3.2). It remains to show that differentiation and integration are interchangeable. For > 0 sufficiently small (see below) we define g : ðÀ; Þ Â X ! R !0 [ fþ1g, gðt; xÞ ¼ f ðx; uðxÞ þ tûðxÞ; ruðxÞ þ trûðxÞÞ: The identity @ t Ð X gðt; xÞ dx ¼ Ð X @ t gðt; xÞ dx holds true, if the following three conditions are satisfied. 3. Uniform upper bound: There is a function h 2 L 1 ðXÞ such that j@ t gðt; xÞj hðxÞ for almost every x 2 X and all t 2 ðÀ; Þ.
Proof. Identify w 2 W poly with ðu Ã ; v Ã Þ 2 U Ã Â S Ã 2 and let u;û 2 U: First, we separate the linear and nonlinear parts of w.
Assuming we can differentiate under the integral sign, the remaining limit equals As in the proof of Lemma 3.3, we have to check the conditions for interchanging integration and differentiation. Define the function gðt; xÞ ¼ v Ã ðxÞ Á T 2 ðruðxÞ þ trûðxÞÞ on ðÀ; Þ Â X. It is integrable for all t, since T 2 maps L p ðX; R NÂn Þ into S 2 and v Ã lies in S Ã 2 . It is also differentiable with respect to t, since the entries of T 2 ðruðxÞ þ trûðxÞÞ are polynomials in t. Finally, @ t g can be bounded in the following way where v Ã s denotes the L ð p s Þ Ã ðX; R rðsÞ Þ-component of v Ã : The derivative adj 0 s ðru þ trûÞ consists of sums of products of s À 1 terms of the form @ x i u j þ t@ x iû j . After expanding, every such product can be bounded by k¼0 k X m jg km j; (3.4) where each g km is a product of s À 1 L p functions and therefore lies in L p sÀ1 . Combining (3.3) with (3.4) gives an upper bound for @ t g which is independent of t. Using H€ older's inequality, it is now straightforward to verify that this bound is indeed an L 1 function.
Theorem 3.1. Let R satisfy the requirements of Lemma 2.1 at an R-minimizing solution u † 2 int dom R and let w be the W poly -subgradient thus provided. Suppose Assumption 2.1 holds for this u † and w. Moreover, assume that the integrand f of R satisfies inequality (3.1) and that K is Gâteaux differentiable at u † . Then R is Gâteaux differentiable at u † and R 0 ðu † Þ ¼ w 0 ðu † Þ 2 ran K 0 ðu † Þ # : On the other hand, recall Remark 1 to see that the W poly -subgradient w 2 @ poly Rðu † Þ provided by Lemma 2.1 is given by