On the Numerical Evaluation of Fractional Sobolev Norms. Carsten Burstedde. no. 268

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1 On the Numerical Evaluation of Fractional Sobolev Norms Carsten Burstedde no. 268 Diese Arbeit ist mit Unterstützung des von der Deutschen Forschungsgemeinschaft getragenen Sonderforschungsbereiches 611 an der Universität Bonn entstanden und als Manuskript vervielfältigt worden. Bonn, April 2006

2 On the Numerical Evaluation of Fractional Sobolev Norms Carsten Burstedde February 24, 2006 Abstract In several important and active fields of modern applied mathematics, such as the numerical solution of PDE-constrained control problems or various applications in image processing and data fitting, the evaluation of (integer and real) Sobolev norms constitutes a crucial ingredient. Different approaches exist for varying ranges of smoothness indices and with varying properties concerning exactness, equivalence and the computing time for the numerical evaluation. These can usually be expressed in terms of discrete Riesz operators. We propose a collection of criteria which allow to compare different constructions. Then we develop a unified approach which is valid for non-negative real smoothness indices for standard finite elements, and for positive and negative real smoothness for biorthogonal wavelet bases. This construction delivers a wider range of exactness than the currently known constructions and is computable in linear time. Keywords: fractional Sobolev norm, elliptic PDE, control problem, data fitting, finite elements, biorthogonal wavelets MSC (2000) classification: 65K10, 65J10 1 Introduction Sobolev norms are a fundamental mathematical concept which is widely used for modern research in the areas among many others of applied mathematics and numerical simulation. They arise in a natural way in the weak formulation of partial differential and integral equations, as regularization terms in image processing and data fitting, and in the objective functional of PDE-constrained optimal control problems. There exists an abundant collection of literature on these subjects, see for example [2, 10, 17, 39, 44, 50, 55]. However, the role of equivalences between Sobolev norms in the functional analytic setting and discrete norms on coefficient spaces as a central building block for the theory and practice of fast solvers has matured only during the last 10 to 15 years. Such norm equivalences relate to the approximate evaluation of norms and the development of optimal preconditioners for partial differential equations, such as multigrid [11, 38, 55], BPX-type [12, 58, 59], and wavelet schemes [18, 24, 30, 31], see also [33, 35, 48]. For these applications, mostly the order of smoothness is relevant, and not the actual value of a norm. Estimations of equivalence up to uniformly bounded constant factors are generally sufficient to establish theoretical results. Therefore, computations up to spectral equivalence 1

3 are also common in practice. These can be performed via the application of Riesz matrices and multilevel scaling operations, which are approximate discrete representations of operators in function space. Recently, the focus of interest is somewhat shifting from the solution of a single partial differential equation towards more complex problem formulations. Motivated by industrial applications, the subject of optimization subject to PDE constraints is quickly gaining importance, see e.g. [6,8,37,53,54]. For such control problems, a penalty functional which usually contains weighted sums of Sobolev norms has to be minimized, where the physical model represented by a set of PDEs must be respected as a constraint. To compare different models and simulations, and to satisfy the objectives often given by applications in engineering, it is important that the numerical modeling of the norms under consideration obeys a minimal set of criteria. These applications require in the first place that norms of integral smoothness, i.e., on the spaces L 2 or H 1, are computed exactly up to discretization error. To preserve the optimal runtime complexity of state-of-the-art numerical methods, also the evaluation of norms must be performed in a time which is proportional to the number of unknowns. Thus, the fast computation is a crucial requirement from the practical point of view. There are also applications which motivate to consider negative or positive real norms. This is also interesting from a purely mathematical point of view. For example, control problems with boundary control naturally lead to norms on H 1 2 or its dual, and the weak formulation of elliptic PDEs and integral equations motivates to consider functions in (H 1 ). As there exist different definitions of non-integer Sobolev norms, it is only reasonable to demand evaluation up to equivalence here. Thus, a consistent modeling implies that equivalence for all occuring smoothness indices is a necessity. Moreover, we propose as a further criterion that the norm of constant functions which reduces to the L 2 norm shall be evaluated exactly. We have included this seemingly obvious criterion since it is by no means respected by all existing approaches. Having motivated this set of criteria, it remains to state that there are no constructions with finite elements or wavelets we know of which satisfy all of them (this statement will become more clear later in this document). The numerical evaluation of Sobolev norms in this more general context has to our knowledge not yet been systematically studied. (We exclude spectral elements [9] from our considerations, as the compact support of all basis functions and nonperiodic boundary conditions are important for the applications we have in mind.) In this paper, we present a framework which realizes the complete set of criteria for finite elements and non-negative real order of smoothness, and for biorthogonal wavelets and arbitrary smoothness. It is derived from the concept of additive multilevel preconditioners in the spirit of BPX-type and wavelet methods. A first sketch of the general idea restricted to the wavelet context was given in [13]. We realize here a suitable construction for finite elements, and improve the original wavelet-specific construction (which up to now suffered from an artificial correction to guarantee the exactness for constant functions). This new approach is largely similar in nature for the realms of finite elements and wavelets, which justifies in our opinion to call it a unified scheme for the numerical evaluation of general, that is, integer and real Sobolev norms. This paper is organized as follows. In Section 2, we cover the theoretical basis for the subjects of multiresolution analysis and biorthogonality, which are essential for the development of our construction of Riesz operators. Section 3 is dedicated to the development of our scheme in the finite element context. This also serves the purpose of motivation and introduction of the main 2

4 idea. The general construction for primal and dual norms is realized in the wavelet setting in 4, using the full theory of biorthogonal space decompositions. The last Section 5 will deal with issues of extension and generalisation and practical concerns. Let us now comment on some notational conventions. To emphasize the role of duality, we will write v, ṽ for the dual pairing of two functions v H, ṽ H, where H denotes a general Hilbert space with dual H. For the special case H = H = L 2, this expression reduces to the standard inner product (v, ṽ) L2. For collections of functions Φ = {φ} H, which we consistently interpret as (possibly infinite) column vectors, and some single function f H, the term Φ, f is to be understood as a column vector, and f, Φ as a row vector, according to Φ, f := ( φ, f ) φ Φ, f, Φ := Φ, f T. (1.1) The relations for the dual functions are defined analogously. Consequently, by-element dual pairings of two function sequences Φ H and Φ = { φ} H are written as (not necessarily square and possibly infinite) matrices Φ, Φ with entries Φ, Φ := ( φ, φ ). (1.2) φ Φ, φ Φ It follows that linear transformations of two function sequences Φ, Φ by (not necessarily square) matrices A and B satisfy AΦ, B Φ = A Φ, Φ B T. (1.3) Vectors and matrices are written in bold face. The identity matrix is denoted by I N, where the dimension N is omitted when it is clear from the context. Recall that we specify by H s the Sobolev space of (possibly fractional) index of smoothness s R over a domain Ω R n, and abbreviate its dual as H s := (H s ) [1]. The relation a b means that ca b Ca with constants c, C which are independent of any parameters on which a or b may depend. Likewise, we introduce the notation a < b for a Cb, and a > b is naturally defined as b < a. 2 General Considerations This section covers the essential theory. We have aimed to keep it as lean and non-redundant as possible. Most of it can be found in literature on wavelet methods for PDEs, such as [24]. 2.1 Multiresolution The concept of multiresolution analysis is central for (multilevel) finite elements and (biorthogonal B-spline) wavelet bases alike, see e.g. [29, 35, 45, 46]. We collect some basic facts here and introduce our notation and conventions. Definition 2.1. Let H be a Hilbert space over the domain Ω R n with inner product (, ) H and norm H := (, ) 1/2 H. A multiresolution sequence S = {S j} j j0 is a set of nested closed subspaces S j H with the following properties, S j0 S j H, clos H = H, (2.1) where j 0 Z denotes the coarsest level of resolution. 3 j j 0 S j

5 Introducing the notation S(Φ) := clos H (span Φ) (2.2) for any countable collection of functions Φ H, the subspaces S j typically have the form S j = S(Φ j ) with Φ j := {φ j,k : k j }, (2.3) where φ j,k are suitable basis functions for S j over corresponding (possibly infinite) index sets j with cardinality N j := # j (2.4) and accumulated cardinality N J := J j=j 0 # j. (2.5) The generator bases or single-scale bases {Φ j } j j0 defined in (2.3) are chosen to be uniformly stable in the sense that c l2 ( j ) c T Φ j H (2.6) for any coefficient vector c l 2 ( j ). As a special case it follows that the basis functions are normed in H, φ j,k H 1. (2.7) In the following, we will drop the subscript on norms over the sequence spaces l 2, simply writing, unless we wish to emphasize the choice of the index set. This applies equally to vectors and matrices. Furthermore, we consider general ρ-adic refinement, that is, the number of basis functions grows geometrically between two levels, N j, N j ρ nj. (2.8) The practically most relevant case is of course given by ρ = 2. For numerical purposes, it is important that the basis functions satisfy the locality condition diam supp(φ j,k ) ρ j. (2.9) As the spaces S j are nested, the generator functions of adjacent scales obey the two-scale relations Φ T j = Φ T j+1g j,0, (2.10) which define certain N j+1 N j refinement matrices G j,0 : l 2 ( j ) l 2 ( j+1 ). As a consequence of locality (2.9), the number of entries per row and column is uniformly bounded by a constant. It follows that the matrices G j,0 are uniformly sparse, that is, the number of nonzero entries is proportional to the number of unknowns. Generator functions on coarse levels j can be expressed by functions on any finer level J by the iterated use of (2.10), Φ T j = Φ T J T J,j with T J,j := G J 1,0 G J 2,0 G j,0, T J,J := I. (2.11) Collecting the basis functions from all levels results in a redundant set or frame Θ J, Θ T J := (ΦT j 0,..., Φ T J 1, Φ T J ) = Φ T J (T J,j0,..., T J,J 1, T J,J ) =: Φ T J H J, (2.12) 4

6 with the rectangular N J N J transformation matrix H J. We now introduce the notion of complement spaces, which play an important role in the context of multilevel preconditioning and norm equivalences. To this end, we infer from (2.1) that there exist sequences Ψ j := {ψ j,k : k j } such that S(Φ j+1 ) = S(Φ j ) S(Ψ j ), (2.13) where the direct sum is not necessarily meant to be orthogonal. This relation trivially implies # j+1 = (# j ) + (# j ). With W j := S(Ψ j ) (2.14) and (2.3), the relation (2.13) can be abbreviated as S j+1 = S j W j. (2.15) When an explicit specification of the complement basis Ψ j is needed, it is usually selected such that the collections Φ j Ψ j are uniformly stable in the sense of (2.6) for all j j 0. Then it follows further that there exist matrices G j,1 : l 2 ( j ) l 2 ( j+1 ) satisfying Ψ T j = ΦT j+1 G j,1. (2.16) It is important for us that the basis Ψ j also has compact support analogous to (2.9), which causes G j,1 to be uniformly sparse as well. The complete two-scale relations now read (Φ T j, Ψ T j ) = Φ T j+1g j with G j := (G j,0, G j,1 ) : l 2 ( j+1 ) l 2 ( j+1 ). (2.17) From (2.13) it follows that G j is invertible. Moreover, Φ j Ψ j is uniformly stable if and only if G j 1, G 1 j 1. (2.18) 2.2 Mass, Stiffness and Riesz Matrices Suppose now that the general multiresolution analysis outlined in the previous section is defined on the particular Hilbert space H = L 2 (Ω). The generator bases Φ j may for example be variants of the nodal basis, spline bases or constructions based on polynomials, or multivariate extensions thereof, provided that uniform stability holds, i.e., (2.6) is satisfied. This section is concerned with the computation of inner products. Let v 1, v 2 S j L 2 be two functions with expansion coefficients c 1, c 2 l 2, that is, v i = c T i Φ j. Clearly, their L 2 inner product can be evaluated according to (v 1, v 2 ) L2 = (c T 1 Φ j, c T 2 Φ j ) L2 = c T 1 (Φ j, Φ j ) L2 c 2 =: c T 1 M j c 2 (2.19) with the mass matrix M j = (Φ j, Φ j ) L2. It is symmetric and positive definite. By (2.6), its spectral condition number κ(m j ) is uniformly bounded, c T M j c = (c T Φ j, c T Φ j ) L2 = c T Φ j 2 L 2 c 2 = c T c = κ(m j ) 1. (2.20) Thus, the L 2 norm of any function v = c T Φ j S j can be computed via the mass matrix, v 2 L 2 = c T M j c. (2.21) 5

7 The computation of higher order integral Sobolev norms is analogous. Consider for example the H 1 seminorm, v 2 H 1 = ( v, v) L2 = ( c T Φ j, c T Φ j ) L2 = c T ( Φ j, Φ j )c =: c T A j c, (2.22) where A j = ( Φ j, Φ j ) L2 is the standard finite element stiffness matrix for the Laplace operator. Above definitions of mass and stiffness matrices are special cases of Riesz matrices R Z for spaces Z, which fulfil the equality v 2 Z = ct R Z c. (2.23) Some examples for functions v S j Z are R L2 = M j, R H 1 = M j + A j. (2.24) Extensions to Sobolev norms of higher integral order are straightforward, provided that the basis functions Φ j are sufficiently smooth. In the following, it will be at times useful to represent operators over a redundant set of functions. Using (2.12), we denote the resulting (positive semidefinite) matrices by M j := (Θ j, Θ j ) = H T j M jh j, Ā j := ( Θ j, Θ j ) = H T j A jh j, etc. (2.25) The goal of this paper is to construct Riesz matrices for Sobolev spaces of arbitrary real smoothness H s, s R, for a possibly large range of s ( γ, γ). Here we understand by H s := (H s ) the dual space of H s, s 0. As we will see, the possible range of smoothness indices s depends on the existence of appropriate norm equivalences. For finite element bases, we cover the case s 0, while for biorthogonal wavelet bases, the whole range s R is in principle accessible. However, it is an important point that fractional Sobolev norms, s Z, are only defined up to equivalence [1]. On the one hand, this implies that unique Riesz matrices do not exist for such s. On the other hand, this guarantees some freedom in the actual construction, since equivalence is all that can reasonably be expected. After these preparations, we postulate the following criteria for constructions of Riesz matrices R H s over a range s [0, γ) (for finite elements) or s ( γ, γ) (for wavelets), which compute Sobolev norms numerically according to (2.23). These have before been formulated in [13]. Conclusion 2.2. We aim to construct Riesz operators with the following properties. 1. Norms of integral smoothness must be computed exactly. 2. Norms of fractional smoothness must be computed up to equivalence. 3. The norm of constant functions must be computed exactly for any s. 4. The computations up to any fixed level j must require only O(N j ) operations. The first two criteria are important for applications in optimization and control: Exactness for the integral case ensures comparability between different implementations or problems, while the postulate of equivalence is fundamental for the treatment of fractional smoothnesses. The third criterion implements a certain degree of numerical consistency, while the fourth is clearly indispensable for algorithms of optimal computational complexity. We will in the following provide constructions which satisfy all of the above requirements. 6

8 2.3 Norm Equivalences The existence of norm equivalences between Sobolev spaces H s and weighted multilevel decompositions provides among many other uses the basis for optimal preconditioners of elliptic PDEs and the construction of biorthogonal wavelets. They also play a fundamental role in the construction of Riesz operators discussed here. Therefore, we shortly comment on the main mathematical ideas, which can be expressed in the general framework of biorthogonal space decompositions, see e.g. [23, 24, 28]. Theorem 2.3. Let S j0 S j and S j0 S j be two sequences of nested subspaces of H = L 2 (Ω). (a) Suppose that inf 0 v S j sup 0 ṽ S j v, ṽ v L2 ṽ L2 > 1, (2.26) and the analogous condition (2.26) with interchanged roles of S j and S j hold. Then there exists a sequence {Q j } j of uniformly bounded projectors Q j : L 2 L 2 with Im(Q j ) = S j and Im(I Q j ) = ( S j ) L 2. For the adjoint projectors ( Q j ) j holds analogously Im( Q j ) = S j and Im(I Q j ) = (S j ) L 2. Additionally, it follows that Q j Q j+1 = Q j and Q j Qj+1 = Q j, respectively. (b) Under the extra assumptions that there exists 0 < γ < d such that inf v w L2 < w S j ρ js v H s (2.27) for v H s, 0 s d (which is usually called a direct or Jackson estimate) and v H s < ρ js v L2 (2.28) for v S j, 0 s γ (which is usually referred to as inverse or Bernstein estimate), and that analogous assumptions (2.27), (2.28) with constants 0 < γ < d hold for { S j } j, it follows (with the definitions Q j0 1 := Q j0 1 := 0) that and j=j 0 v j 2 < H s j=j 0 ρ 2js v j 2 L 2 for v j Im(Q j Q j 1 ), s ( d, γ), (2.29) j=j 0 ρ 2js (Q j Q j 1 )v 2 L 2 < v 2 H s for v Hs, s ( γ, d). (2.30) Corollary 2.4. In particular, for s ( γ, γ), v ((Q j Q j 1 )v) j is a bounded mapping from H s into l 2,s (Q), where ( l 2,s (Q) := {(v j ) j : v j Im(Q j Q j 1 ), (v j ) j l2,s (Q) := ) 1 ρ 2js v j 2 2 L 2 j=0 } <, (2.31) with bounded inverse (v j ) j j=0 v j. Therefore the symbols < in (2.29) and (2.30) can be replaced by symbols in this case. This leads to the norm equivalence v 2 H s j=j 0 ρ 2js (Q j Q j 1 )v 2 L 2, v H s, s ( γ, γ). (2.32) Analogous results hold for interchanged roles of (γ, d) and ( γ, d), with Q j replaced by Q j. 7

9 It is convenient to express these relations in terms of operators [24], Σ s v := which act as a shift in the Sobolev scale, With these definitions, it follows that j=j 0 ρ (j j0)s (Q j Q j 1 )v, (2.33) Σ t v H s v H s+t, s, s + t ( γ, γ). (2.34) Σ 0 = id and Σ 1 s = Σ s. (2.35) To ensure a consistent notation, we will in the following use the convention X j0 1 : = 0 for any operator, vector or matrix X unless stated otherwise. Above statements about projectors are connected to the multiresolution sequences introduced previously, with H = L 2 (Ω), by the following remark. Remark 2.5. We can identify W j = Im(Q j+1 Q j ), cf. (2.15), and its dual counterpart Wj = Im( Q j+1 Q j ). Consequently, W j = ( S j ) L 2 Sj+1 and Wj = (S j ) L 2 Sj+1. (2.36) It follows trivially that the W j and W j are biorthogonal between levels, W j1 W j2 for j 1 j 2. (2.37) Therefore, the collection of spaces W j, Wj is called a biorthogonal decomposition of L 2. This includes the spaces W j0 1 := S j0 and W j0 1 := S j0. The prerequisites for Theorem 2.3 can be verified by the following considerations. When S j and S j are chosen as finite element spaces, direct and inverse estimates (2.27), (2.28) are standard facts, see e.g. [10, 28, 48]. For functions which are dyadically refined (2.8) and local (2.9), the inverse estimates (2.28) are satisfied with γ = t and γ = t + 3 2, provided that the functions from the spaces S j and S j are piecewise smooth and globally t, respectively t, times continuously differentiable. For the case of discontinuous finite elements, this holds with t = t = 1. The direct estimates (2.27) are usually enforced by the requirement that, relative to their meshes, S j and S j contain all t or t times continuously differentiable piecewise polynomials of degree d 1 and d 1, respectively. Theorem 2.3 is quite general in the sense that it only refers to properties of the subspaces S j and S j. To develop the theory in the remaining parts of this paper, we additionally assume the existence of biorthogonal bases as follows. Definition 2.6. The collections of functions Φ j S j and Φ j S j are called biorthogonal if The collection Φ j is also called the dual basis of Φ j. Φ, Φ = I. (2.38) Now that such bases are specified, the projectors Q j and Q j can be written down explicitly [15]. 8

10 Theorem 2.7. Let Φ j and Φ j be biorthogonal, uniformly stable bases of the nested spaces S j and S j, respectively. Then, (2.26) is satisfied, and the projectors defined by fulfil the conditions from Theorem 2.3. Q j v := v, Φ j Φ j, Qj ṽ := ṽ, Φ j Φ j (2.39) Summarising these results, the norm equivalence (2.32) can be ensured by the choice of biorthogonal bases for the spaces S j and S j which are uniformly stable and yield polynomial exactness of orders d and d, respectively. In the remainder of this document, we will employ this formalism for constructions of Riesz operators which satisfy our requirements motivated in the introduction. This will be done in two different settings, namely for standard finite elements and biorthogonal B-spline wavelets. 3 Evaluation with Finite Elements The formalism of biorthogonal space decompositions covers standard finite elements as a special case. For example, the optimality of BPX-type preconditioners [12, 58, 59] can be derived using Corollary 2.4 for s [0, γ) [25]. We will use a similar reasoning in this section to make use of the norm equivalences in the finite element setting. However, the crucial part here is exactness for integral order of smoothness, which is not important for preconditioning. In this sense, the following derivations are by necessity sharper than the cited results. Here we obtain a construction of Riesz operators which fulfills the requirements from Conclusion 2.2 for nonnegative orders of smoothness. 3.1 Adapting the Framework to Finite Elements Let us assume that Φ j is a piecewise polynomial, uniformly stable and local finite element basis of order d, which implies that γ = d 1 2. This fits into the biorthogonal framework by setting S j := S j, Φj := M 1 j Φ j. (3.1) It follows that Φ j and Φ j are biorthogonal (however, this definition of Φ j does not yield local functions), and that W j = W j. The projectors have the form Q j = Q j = (, Φ j ) L2 M 1 j Φ j. (3.2) The uniform stability of Φ j implies that also Φ j is uniformly stable and (2.26) holds. Thus, all prerequisites for Theorem 2.3 are satisfied, and the norm equivalence (2.32) is valid for s ( γ, γ). A basic ingredient of the norm equivalence is the projection of a function v S J onto a complement space, (Q j Q j 1 )v, with j J. For any given expansion v = c T Φ J, the result can be computed by use of (2.10), (2.11), (2.19) and (3.2) as (Q j Q j 1 )v = (c T Φ J, Φ j ) L2 M 1 j Φ j (c T Φ J, Φ j 1 ) L2 M 1 ( ) = c T M J T J,j M 1 j T J,j 1 M 1 j 1 GT j 1 Φ j =: c T F J,j Φ j =: d T J,j Φ j, 9 j 1 Φ j 1 (3.3)

11 where we have abbreviated the projection matrix onto the complement space as F J,j and the projected coefficent vector as d J,j. With (2.32), this representation enables us to calculate the H s norm of v up to equivalence, v 2 H s J j=j 0 ρ 2js (Q j Q j 1 )v 2 L 2 = J j=j 0 ρ 2js d T J,j Φ j 2 L 2 = Because of the orthogonality of the complement spaces (2.37), it follows that J j=j 0 ρ 2js d T J,j M jd J,j. (3.4) (d T J,j 1 Φ j1, d T J,j 2 Φ j2 ) L2 = 0 for j 1 j 2. (3.5) Defining the diagonal scaling matrix D j for the basis Θ j by and the cumulative scaled coefficient vector e J,s by D j := diag(ρ 0 I j0, ρ 1 I j0 +1,..., ρj j 0 I j ) (3.6) e T J,s := (ρ0 d T J,j 0, ρ s d T J,j 0 +1,..., ρ(j j 0)s d T J,J ) = (dt J,j 0,..., d T J,J ) D s J, (3.7) we have collected all ingredients for our first result. Theorem 3.1. The norm v 0,s for v = c T Φ J, defined by v 2 0,s := et J,s M J e J,s, (3.8) is equal to the L 2 norm for s = 0 and for constant functions, and equivalent to the norm on H s for s ( γ, γ). It can be computed up to discretization error accuracy in O(N J ) operations. Proof. The uniform scaling of e J,s with ρ j 0s preserves the equivalence from (3.4). Exactness for s = 0 is ensured by (2.35). The norm for constant functions is always computed correctly since in this case only the first component ρ 0 d T J,j 0 is nonzero, independent of s. All matrices have a special, uniformly sparse structure and can therefore be applied in O(N J ) operations. Since T J,J = I, the mass matrices on the highest level J cancel out. Yet, the inversion of the lower level mass matrices M j, j < J, during the evaluation of F J,j needs to be done numerically. Because these are uniformly well-conditioned and uniformly sparse, an iterative solver such as the method of conjugate gradients, combined with a nested iteration strategy, delivers the result up to discretization error accuracy ρ jd in at most O(N j ) operations [14,41]. A geometric series argument ensures this bound for the cumulative sum of all F J,j over the range of levels j. 3.2 A Unified Scheme for Norms of Higher Order While Theorem 3.1 yields exactness for s = 0, it only guarantees equivalence for integer s N. Therefore, we develop an analogous construction which is exact for s = 1. In principle, this can be repeated for all s = 2, 3,..., resulting in a scale of exact integer Sobolev norms. To this end, let v = c T Φ J be an arbitrary function in S j. Using the operator Σ s from (2.33), (2.34), we obtain v 2 H s Σ s 1v 2 H 1. (3.9) 10

12 The term on the right hand side can be computed via Σ s 1 v = J j=j 0 ρ (j j0)(s 1) (Q j Q j 1 )v = where we have used the definitions from (3.3) and (3.7). Theorem 3.2. The norm v 1,s for v = c T Φ J, defined by J j=j 0 ρ (j j0)(s 1) d T J,jΦ j = e T J,s 1Θ J, (3.10) v 2 1,s := et J,s 1( M J + ĀJ)e J,s 1, (3.11) is equal to the H 1 norm for s = 1 and for constant functions, and equivalent to the norm on H s for s ( γ, γ). It can be computed up to discretization error accuracy in O(N J ) operations. Proof. Equivalence to the H s norm is assured by construction (3.9). The expression above is obtained using (3.10) and the definition of the extended mass and stiffness matrices (2.25). Exactness for general functions and s = 1 follows from (2.35). For constant functions, the contribution of Ā vanishes and the situation is identical as in Theorem 3.1. Also, the reasoning for a computation time in O(N J ) is analogous. Based on these results, we obtain a unified formulation of a norm for [0, 1] which is exact for s = 0 and s = 1 and equivalent in between by a convex combination of 0,s and 1,s. We generalize this ansatz for integer s as follows. Proposition 3.3. Suppose that there exist norms i,s for i {0, 1,..., S}, with S < γ, which fulfil i,i = H i and i,s H s for s i 1, 0 s < γ. Moreover, they shall be exact for constant functions and computable with an effort of O(N J ). With the following definition of the hat functions h i (x) centered at i, x (i 1) for i 1 x < i, h i (x) := (i + 1) x for i x i + 1, (3.12) 0 else, it follows that the norm S 2 s := h i (s) 2 i,s (3.13) i=0 is equal to Sobolev norms for integer smoothness s = 0, 1,..., S and equivalent for 0 s S. Furthormore, it is exact for constant functions and computable with an effort of O(N J ). To illuminate some characteristics of this construction, we rewrite it in a slightly different form. To this end, define the scaled Riesz matrices on the redundant set, R 0 := M J, R1 := 1 D J ( M J + 1 ĀJ) D J, etc. (3.14) using the notation from (2.23), (2.25). Then, using the definitions (3.8) and (3.11), equation (3.13) takes the form ( S ) v 2 s = et R J,s J,s e J,s := e T J,s h i (s) R i e J,s (3.15) with a newly defined unified Riesz operator R J,s. 11 i=0

13 Conclusion 3.4. We close this section with the following considerations, which apply to the numerical scheme for the evaluation of Sobolev norms just presented above. As can be inferred from (3.15), the vectors d J,j need only be computed once for each j and can be reused for the scaling operations (3.7). Therefore, single applications of the matrices M J and (T J,j M 1 j ) T, j 0 j < J, suffice. The single scaled Riesz matrices from (3.14) are uniformly well-conditioned (when zero eigenvalues are ignored). This is a consequence of the integer norm equivalences, and can also be seen from a classical BPX preconditioning argument. An analogous statement for the convex combination R J,s (3.15) does not follow in an obvious way. The reason is that the eigensystems of the Riesz operators R i are different, and the number of zero eigenvalues is not preserved. In summary, we have established a family of Sobolev norms which fulfil the criteria from Conclusion 2.2. Explicit representations for s = 0 and s = 1 have been provided in (3.8) and (3.11), while the extension to higher orders of smoothness is possible with the techniques described above. 3.3 Numerical Examples To illustrate the evaluation scheme described above, we have conducted several computations for a number of sample functions and smoothness indices. We have chosen the functions f i (x) := cos(2 i πx), i = 0,..., 3, g 1 (x) as an asymmetric hat function with peak at 1 3 and g 2(x) := x. These functions are plotted below. f i f 0 f 3 f 2 f x x g i g 2 = x For each function, we have created a table where the rows correspond to increasing levels of resolution j, and the columns to the five smoothness indices 0, 1 3, 1 2, 3 4 and 1. The exact values for the L 2 and H 1 norms are provided in the bottom row. g 1 j\s 0 1/3 1/2 3/ /2 f 0 2 H s 5.44 j\s 0 1/3 1/2 3/ /2 f 1 2 H s

14 j\s 0 1/3 1/2 3/ /2 f 2 2 H s 79.5 j\s 0 1/3 1/2 3/ /2 f 3 2 H s 316. First of all, it can be seen that the computed L 2 and H 1 norms converge to the exact values in all cases. Convergence starts later for the more oscillatory functions, which makes sense since their periods are multiples of the grid size for low resolutions. The norms for non-integer smoothness interpolate between the extreme cases s = 0 and s = 1. We notice also that the change of the norm between s = 0 and s = 1 3 is the most extreme. This is allowed, since only equivalence is required for non-integer s. Yet, it indicates that there may be ways to improve the dependence of the construction on the smoothness parameter s. j\s 0 1/3 1/2 3/ /3 g 1 2 H s 29/6 j\s 0 1/3 1/2 3/ /2 g 2 2 H s The results for g 1 and g 2 are similar to the results for the cosine functions f i. For the square root function g 2, where the H 1 norm is just not finite, it can be seen that the corresponding values grow with the resolution. In summary, we may say that the numerical studies confirm the predicted behavior in all cases. 4 Evaluation with Wavelets In this section, we will cover the construction of Riesz operators for biorthogonal wavelets, where both the primal and the dual basis have compact support. Compared to standard finite elements, the process of biorthogonalization introduces additional complexity in the construction of the wavelet bases. However, once this is done, the resulting framework is perfectly symmetric between the primal and the dual side, which means that Sobolev norms of negative smoothness indices can be evaluated in much the same way as for positive smoothness. In particular, the mass matrices which are present in the finite element context drop out completely, which leads to a simpler structure of the projectors and the relations between different levels of resolution. 13

15 4.1 Introduction to Biorthogonal Wavelets Biorthogonal wavelets are based on a multiresolution analysis just as described in Section 2. We have already introduced the sequence of nested spaces S j0 S j H and motivated the two-scale relations (Φ T j, ΨT j ) = ΦT j+1 G j with G j := (G j,0, G j,1 ) : l 2 ( j+1 ) l 2 ( j+1 ) (4.1) between the single-scale bases Φ j and the complement bases Ψ j = {ψ j,k : k j }, which span the associated subspaces S j = S(Φ j ), W j = S(Ψ j ), S j+1 = S j W j. (4.2) It will be convenient to abbreviate Ψ j0 1 := Φ j, j0 1 := j, and to apply analogous definitions for the dual basis. Then the formalism allows to specify primal and dual multi-scale bases Ψ, Ψ of infinite cardinality, Ψ := Ψ j, Ψ := Ψ j. (4.3) j=j 0 1 j=j 0 1 For any fixed maximum level J, the transformation (4.1) can be iterated, which leads to a matrix W j : l 2 ( J ) l 2 ( J ) operating between Φ J and the multi-scale basis Ψ (J) up to that level, Ψ T (J) := (ΦT j 0, Ψ T j 0,..., Ψ T J 1) = Φ T J W J. (4.4) This multi-scale transformation W J is composed of two-scale operators, ( ) Gj 0 W J := W J,J 1 W J,j0 with W J,j :=. (4.5) 0 I It can be used to switch between single-scale and multi-scale representations of functions. Let for example the expansion coefficients of a function v S J in the generator basis be denoted by c, and the coefficients in the multi-scale basis by d, i.e., v = c T Φ J = d T Ψ (J). (4.6) The coefficient vector d T =: (c T j 0, d T j 0,..., d T J 1 ) contains contributions from all scales. The first part holds the coefficients of the generator basis on the coarsest level which provides a rough approximation of v. The following coefficients are ordered by their level and add successive layers of detail information with zero mean to the approximation. Inserting (4.4) into (4.6), the transformation from the detail coefficients to the single-scale coefficients emerges as c = W J d. (4.7) Since all sub-transformations W J,j are uniformly sparse, by a geometric series argument also W J can be applied in O(N J ) arithmetic operations. Therefore, it is called the fast wavelet transform. Already for the finite element setting, we made use of the central Theorem 2.3 which guarantees the existence of biorthogonal projectors and a range of norm equivalences. However, its full generality is only exploited by biorthogonal wavelets which are based on two distinct sets of nested spaces, namely the primal and dual collections S j0 S j H and Sj0 S j H. (4.8) 14

16 We assume in the following that H = H = L 2 (Ω), ρ = 2 and that there exist bases Φ j, Ψ j and Φ j, Ψ j which are uniformly stable, local and of polynomial exactness of the order d and d, respectively, and that these bases are biorthogonal, These relations in turn imply (2.26). Φ j, Φ j = I, Ψ j, Ψ j = I, j = j 0, j 0 + 1,.... (4.9) Existence of such bases on the real line has been shown in [22], where B-splines are used as primal functions and the dual functions are implicitly specified by recursion formula. These have later been modified to construct wavelets on the unit interval [27]. Wavelets where both primal and dual functions are piecewise polynomials have also been constructed [28, 51]. For practical purposes, several extensions and optimisations have been developed, see e.g. [4, 13, 49, 57]. Because of the nestedness of the spaces, there exist two-scale and multi-scale transformations G j, W j resp. Gj, Wj for both collections, see (2.10), (2.16) and (4.5). Biorthogonality (4.9) implies that G T 1 j = G j and Wj T 1 = W j. A representation of the projectors from Theorem 2.3 is given by Q j =, Φ j Φ j, Qj =, Φ j Φ j. (4.10) This form can be used to substitute the differences (Q j Q j 1 )v in (2.32), which will in turn yield the Riesz basis property of Ψ and Ψ, defined as follows. Definition 4.1. A collection of functions Σ H is called a Riesz system for H if v H = d T Σ H d, v = d T Σ H. (4.11) A Riesz system is called a Riesz basis if it is also a basis of H. In other words, the norm of a function v = d T Σ H can be estimated from below and above by the sequence norm of its expansion coefficients d. This form of equivalence has already been encountered in (2.6), meaning that Φ j forms a Riesz basis for S j. Here, the Riesz basis property is extended to the full infinite multi-scale basis Ψ. Using (4.10) in Theorem 2.3 leads to the following discrete norm equivalence [27]. Theorem 4.2. Let Ψ, Ψ be a pair of uniformly stable, biorthogonal multi-scale bases in the sense of (2.18) and (4.9), and let the direct and inverse inequalities (2.27) and (2.28) hold. Then it follows that v 2 H s j=j js (v, Ψ j ) L2 2, v H s, s ( γ, γ), (4.12) and the dual relation, obtained by switching the roles of Ψ j and Ψ j, holds for s ( γ, γ). Thus, setting s = 0 implies that Ψ and Ψ are Riesz bases for L 2, and the expansion of a function v L 2 can be written as v = (v, Ψ) L2 Ψ = (v, Ψ)L2 Ψ, (4.13) with v L2 (v, Ψ) L2 (v, Ψ) L2. (4.14) 15

17 For general s, the formulation (4.12) suggests to absorb the scaling factors 2 js into the wavelet basis. To this end, we define Ψ s := D s Ψ := {2 (j (j 0 1))s ψ j,k } j j0 1,k j, (4.15) where D is the diagonal matrix consisting of entries 2 j, j being the level of each function. The dual basis is defined by Ψ s := D s Ψ. (4.16) In view of (4.9), these definitions imply Ψ s, Ψ s = I. (4.17) In summary, the main theorem characterising Sobolev spaces by wavelet expansions emerges as follows. Corollary 4.3. Under the assumptions of Theorem 4.2 and for any s ( γ, γ), Ψ s is a Riesz basis for H s, and Ψ s is a Riesz basis for H s, namely, H s ( Ψ s, ) L2 and H s (Ψ s, ) L2. (4.18) Thus, wavelets can be used to characterize function spaces in terms of expansion coefficients. The transformation between Sobolev spaces of different smoothness can be achieved through the shift operator Σ s, which has been introduced in (2.33) and can be rewritten as Σ t : H s+t H s, v = d T Ψ s+t Σ t v = (D t d) T Ψ s+t = d T Ψ s. (4.19) To transform between the single-scale basis of L 2 and a certain wavelet basis of smoothness s, the scaling in terms of the diagonal matrix D can also be incorporated into the multi-scale transformation (4.4) according to (Ψ s (J) )T = Φ T J Ws J with W s J := W JD s. (4.20) The following result relates the conditioning of WJ s, which can be understood as a generalisation of (2.18), to the Riesz basis property for Ψ s [24]. Theorem 4.4. The transformations W s J if and only if Ψ s is a Riesz basis of H s. are well-conditioned in the sense that W s J 1, (Ws J ) 1 1, (4.21) As an orthogonal basis is by definition a self-dual Riesz basis, it fits into the framework described here. However, orthogonality can be considered as a requirement which is too strong. Although compactly supported orthogonal wavelet bases for the Sobolev spaces considered here have been constructed [29], they have some disadvantages which make them harder to handle in numerical simulations. In particular, they are not piecewise polynomials, and they are only implicitly defined. The framework of biorthogonal bases offers much more flexibility to construct multi-scale bases which are explicitly defined, well-conditioned in the sense of (4.21), and lead to multi-scale transformations which can be applied with linear computational complexity. In summary, biorthogonality is as far as one can deviate from orthogonality. Essential features of orthogonality are preserved: Biorthogonal wavelets satisfy the Riesz basis property for the Sobolev spaces in question, and the expansion coefficients of a function can be computed by inner products with (dual) basis functions. 16

18 4.2 Primal and Dual Integral Sobolev Norms Let a function be expanded in an unscaled wavelet basis according to L 2 v = d T Ψ. The L 2 norm can be evaluated by v 2 L 2 = (d T Ψ, d T Ψ) L2 = d T (Ψ, Ψ) L2 d =: d T M Ψ d, (4.22) with the (infinite-dimensional) mass matrix M Ψ in wavelet coordinates. computed according to The norm in H 1 is v 2 H 1 = v 2 L 2 + v 2 H 1 = d T M Ψ d + ( d T Ψ, d T Ψ) L2 = d T (M Ψ + A Ψ )d, (4.23) with the definition of the stiffness matrix in wavelet coordinates, A Ψ := ( Ψ, Ψ) L2. When we expand v in the natural wavelet basis for H 1 according to (4.15), H 1 v = d T Ψ 1 = d T D 1 Ψ, we can write the H 1 norm as v 2 1 = dt D 1 (M Ψ + A Ψ )D 1 d =: d T R 1 d, (4.24) where we have hidden the definition of R 1, the Riesz matrix in the natural wavelet basis. Note that due to Corollary 4.3, R 1 is uniformly well-conditioned. Furthermore, M Ψ, A Ψ and R 1 are symmetric and positive definite. Norms of higher integral order N 0 i < γ are computed analogously via Riesz matrices R i with κ(r i ) 1. For practical applications, M Ψ and A Ψ are not applied directly. transform from (4.4), (4.5) can be used via the identities Instead, the fast wavelet M Ψ(j) = W T j M jw j, A Ψ(j) = W T j A jw j. (4.25) In the framework of biorthogonal wavelets, the evaluation of dual norms is done with relative ease. As an example of the main technique, we first assure that L 2 can be identified with its dual. To this end, we apply the general definition of dual norms, v H := v, w sup, (4.26) 0 w H w H to the expansion L 2 ṽ = d T Ψ =: d T Ψ. For any L 2 w = w T Ψ, we may write ṽ L2 = ṽ, w d T Ψ, w T Ψ sup = sup = sup 0 w L 2 w L2 0 w l 2 (w T M Ψ w) w l 2 dt w (w T M Ψ w) 1 2. (4.27) Here we have used (4.22) and the biorthogonality condition (4.9). After all, the expression is formulated in vectors and matrices on l 2 and can be reduced further by the substitution g := M 1/2 Ψ w as follows, 1/2 Ψ d) T g (M ṽ L2 = sup 0 g l 2 (g T g) 1 2 = ( d T M 1 Ψ d) 1 2 = (d T M Ψ d) 1 2 = ṽ L2. (4.28) From this identity, we conclude that biorthogonal wavelet expansions on L 2 are consistent with the fact that L 2 can be identified with its dual, and the norms are identical. Even when the expansion v = d T Ψ is not available, the dual norm can be efficiently computed, as M Ψ is 17

19 uniformly well-conditioned, and its inverse can be applied in O(N J ) operations for any given level J. After these preparations, we discuss the norm of H 1 = (H 1 ). We expand the function H 1 ṽ = d T Ψ1, and the trial function H 1 w = w T Ψ 1. The numerator of (4.26) is derived analogously to (4.27), ṽ, w = d T Ψ1, w T Ψ 1 = d T Ψ 1, Ψ 1 w = d T w. (4.29) We have used here the general biorthogonality relation (4.17). With a similar substitution as in the derivation of (4.28), we arrive at v 2 1 = d T R 1 1 d. (4.30) Remark 4.5. Due to the uniformly bounded condition number of R 1, the norm v H 1 can be evaluated efficiently in the wavelet setting. In contrast, a standard finite element ansatz would necessitate the multilevel projections detailed in Section 3, and an additional mass matrix would have need to be applied in (4.29). The framework of biorthogonal wavelets is thus particularly suited for the evaluation of positive and negative integral Sobolev norms. Norms of higher order can be processed in the same fashion as exercised here for H 1, L 2 and H 1, provided that the ansatz functions are sufficiently regular. 4.3 A Unified Scheme to Evaluate Arbitrary Real Norms In this section, we finally put all ingredients together to establish a construction of Riesz matrices which fulfills the criteria from Conclusion 2.2. In order to compare our construction to existing approaches, we first give a very short review of the state of the art of the evaluation of general Sobolev norms in the wavelet framework, and comment on the deficits. To this end, let us consider the expansion of a function v in the unscaled wavelet basis, v = d T Ψ. In [26], the norm used is v (1) H := (d T D 2s d) 1 s 2. (4.31) This norm is always equivalent to H s, but never exact for s Z. simplest approximation R s := I. The choice in [14] is It corresponds to the v (2) H s := (d T D s M Ψ D s d) 1 2, (4.32) which is always equivalent, but only exact for s = 0 (the L 2 case). This conforms with the approximation R s := R 0 and can be seen as a qualitative improvement compared to (4.31). Anyway, both variants permit the evaluation of the corresponding dual norms up to equivalence, and they satisfy the fundamental requirement of computational efficiency. To obtain a construction which is exact for both L 2 and H 1, one might consider a multiplicative interpolation as follows, v (3) H := ( d T M 1 s s 2 Ψ (M Ψ + A Ψ ) s M 1 s 2 Ψ d) 1 2. (4.33) This expression is motivated by the analogous definition for spectral elements, where it yields the norm in the sense of a Fourier decomposition for all s R. In the context of wavelets, it is exact for s = 0 and s = 1, and equivalent in between. However, the fractional powers of matrices 18

20 require a singular value decomposition, which is prohibitively expensive in practice. Moreover, constant functions are not preserved. Thus, it seems that this ansatz is again not appropriate. In summary, none of the above constructions based on wavelet discretizations meet the criteria from Conclusion 2.2. However, it turns out that the interpolation by convex combination which we devised in Section 3 can be generalized to the wavelet setting. To this end, let d s be the coefficient vector of a function v = d T s Ψs in a natural wavelet basis Ψ s for H s, and suppose that there exist representations R i for i {0, 1,..., S}, with S < γ, which fulfil H i = d T i R id i and H s d T s R id s for s i 1, 0 s < γ. Thus, H i v = d T i Ψi, v 2 i = dt i R id i. (4.34) Examples are R 0 = M Ψ and R 1 as defined in (4.24). Higher orders can be derived from the standard definition of Sobolev norms. We know that all R i are symmetric positive definite and spectrally equivalent to the identity matrix. In the unscaled representation, these Riesz matrices transform according to H i v = d T Ψ, v 2 i = d T D i R i D i d. (4.35) With the definition of the hat function h i (x) from (3.12), we formulate our proposal for a unified Riesz matrix within the wavelet framework. Theorem 4.6. For s R with 0 s S, the norm defined by H s v = d T Ψ, S v 2 s := h i (s) d T D s R i D s d, (4.36) i=0 or alternatively written in the scaled wavelet basis H s v = d T s Ψ s as S v 2 s = h i (s) d T s R id s = d T s i=0 ( S ) h i (s)r i d s =: d T s R sd s, (4.37) i=0 is equal to the standard Sobolev norms for integral s and equivalent for fractional s. It can be computed in linear time. Proof. For s N 0, only one summand survives with i = s, and the above formula reduce to the expression given in (4.34) or (4.35), respectively. For fractional s, the formulation on the right of (4.37) is a convex combination of two summands which are both spectrally equivalent to the identity matrix. Consequently, the whole sum is spectrally equivalent to the identity matrix, and the expression yields a norm which is equivalent to the corresponding fractional Sobolev norm. Likewise, the computational efficiency and the exactness for constant functions is inherited as a feature of the matrices R i. Thus, the general Riesz matrix R s has a uniformly bounded condition number. Therefore, its inverse can be computed numerically with an effort of O(N J ), where J is any given maximum level of resolution, by conjugate gradient iterations combined with a nested iteration ansatz [13]. This allows to evaluate dual norms by a generalization of the considerations in Section

21 Corollary 4.7. For an expansion in the natural dual basis Ψ s of H s, 0 s < min{ γ, γ}, the dual norm can be computed by inverting R s, i.e., H s ṽ = d T s Ψ s, ṽ 2 s = d T s R 1 s d s. (4.38) This expression is exact for integer s, and equivalent for s R. Its numerical computation up to discretization error accuracy requires an effort linear in the amount of degrees of freedom. Indeed, compared to the finite element setting, wavelets permit the evaluation of primal and dual norms within an elegant formalism. This construction satisfies all criteria from Conclusion Numerical Examples In this section we collect results for the evaluation of primal and dual results using a biorthogonal spline wavelet basis. The wavelets are constructed as in [27] with some corrections as outlined in [13]. The lowest level for this basis is j 0 = 3. We have used the functions f 0, f 3, g 1 and g 2 as introduced in Section 3.3. The left table of each row contains the values for the primal norms, and the right table those for the dual norms. j\s 0 1/3 1/2 3/ /2 f 0 2 H s 5.44 j\s 0 1/3 1/2 3/ /2 f 3 2 H s 316. j\s 0 1/3 1/2 3/ /3 g 1 2 H s 29/6 j\s 0 1/3 1/2 3/ /2 g 2 2 H s j\s 0 1/3 1/2 3/ /2 f 0 2 (H s ) j\s 0 1/3 1/2 3/ /2 f 3 2 (H s ) j\s 0 1/3 1/2 3/ /3 g 1 2 (H s ) j\s 0 1/3 1/2 3/ /2 g 2 2 (H s ) 20