On general convergence behaviours of finite-dimensional approximants for abstract linear inverse problems
TTRUNCATION AND CONVERGENCE ISSUES FOR BOUNDEDLINEAR INVERSE PROBLEMS IN HILBERT SPACE
NOE CARUSO, ALESSANDRO MICHELANGELI, AND PAOLO NOVATI
Abstract.
We present a general discussion of the main features and issuesthat (bounded) inverse linear problems in Hilbert space exhibit when the di-mension of the space is infinite . This includes the set-up of a consistent nota-tion for inverse problems that are genuinely infinite-dimensional, the analysisof the finite-dimensional truncations, a discussion of the mechanisms why theerror or the residual generically fail to vanish in norm, and the identificationof practically plausible sufficient conditions for such indicators to be small insome weaker sense. The presentation is based on theoretical results togetherwith a series of model examples and numerical tests. Introduction and outlook
In this note we discuss a number of features that are typical of bounded inverselinear problems set on infinite-dimensional
Hilbert spaces, the infinite dimension-ality being the source of phenomena that become most relevant in the numericaltreatment, and are absent when the considered space instead has finite dimension.More precisely, we shall focus on typical issues and behaviours of the sequenceof truncated, finite-dimensional problems that arise from the discretisation of theoriginal, infinite-dimensional one.As we shall explain in a moment, for specific classes of infinite-dimensional in-verse problems an already well-established insight is available in the literature con-cerning the solvability of the truncated problems and the convergence of the finite-dimensional solutions. However, for generic inverse problems the control of suchissues is surely less developed and a systematic discussion is missing.In this respect, we do not aim here at a comprehensive classification of infinite-dimensional inverse problems and we rather keep the point of view of presenting generic features and difficulties that look ‘unavoidable’ at the considered level ofgenerality. In our intentions this should provide the setting for a future thoroughanalysis of classes of infinite-dimensional inverse problems.For this reason, besides stating and proving our main results, the material willalso be presented through several model examples (and counter-examples).To fix the nomenclature and the notation, by an inverse linear problem in Hilbertspace we shall mean the problem, given a Hilbert space H , a linear operator A actingon H , and a vector g ∈ H , to determine the solution(s) f ∈ H to the linear equation(1.1) Af = g . We shall say that: (1.1) is solvable if a solution f exists, namely if g ∈ ran A ; (1.1)is well-defined if additionally the solution f is unique, i.e., if A is also injective(in which case one refers to f ‘ exact ’ solution); (1.1) is well-posed if there exists a Date : November 21, 2018.
Key words and phrases. inverse linear problems, infinite-dimensional Hilbert space, ill-posedproblems, orthonormal basis discretisation, bounded linear operators, Krylov subspaces, Krylovsolution, GMRES, conjugate gradient, LSQR. a r X i v : . [ m a t h . NA ] N ov N. CARUSO, A. MICHELANGELI, AND P. NOVATI unique solution that depends continuously (i.e., in the norm of H ) on the datum g ,equivalently, that g ∈ ran A and A has bounded inverse on its range.In applications, the linear law A that associates an input f to an output g is prescribed by some physical model, and hence within that model such a lawis exactly known. Experimental measurements produce a possibly approximateknowledge of the output g , from which one wants to obtain information on theinput f , which is the final object of interest.Of course what is ‘exactly known’ of A is its domain and action as an operatoracting on H . Other relevant features of A might not be explicitly accessible, andonly computable within some approximation: for example, if A : H → H is a (every-where defined) Hilbert-Schmidt operator, one may know its integral kernel, basedon the theoretical framework within which the problem is modelled, however itmight not be possible to write explicitly (exactly) its singular value decomposition.Although well-defined inverse linear problems are in a sense trivial theoretically,as the existence and uniqueness of the solution is not of concern, it is clear thatthere at least two main issues arising when one aims at solving them numerically.The first, which is typical already at the finite-dimensional level, namely when A is a matrix, is the fact that the measurement of g is in practice plagued by somenoise, or error of sort: as a consequence, numerically one has to deal with thepossibly ill-posed problem(1.2) Af = g + ν , where the ‘true’ physical output is some g ∈ ran A , however the actually measuredoutput is g + ν , with some small noise-like perturbation ν ∈ H for which possibly g + ν / ∈ ran A .The second issue is actually typical of the infinite-dimensional setting, on whichin fact we are going to focus most of our discussion, namely when dim H = ∞ and A is a genuine infinite-dimensional operator on H . By this we mean, as customary[21, Sect. 1.4], that A is not reduced to A = A ⊕ A by an orthogonal direct sumdecomposition H = H ⊕ H with dim H < ∞ , dim H = ∞ , and A = O .Clearly, non-trivial inverse linear problems in the sense just described are to be truncated to a finite-dimensional Hilbert space, in order to be treated numerically.This poses the questions on how close the solution(s) to the truncated problem arewith respect to the exact solution, let alone on whether the truncated problem issolvable itself.All this is very familiar and already under control for relevant classes of boundaryvalue problems on L (Ω) for some domain Ω ⊂ R d , the typical playground forGalerkin and Petrov-Galerkin finite element methods [8, 19]. In these cases A isan unbounded operator, say, of elliptic type [8, Chapter 3], [19, Chapter 4], ofFriedrichs type [8, Sect. 5.2], [9, 1, 2], of parabolic type [8, Chapter 6], [19, Chapter5], of ‘mixed’ (i.e., inducing saddle-point problems) type [8, Sect. 2.4 and Chapter4], etc. Such A ’s are assumed to satisfy (and so they do in applications) somekind of coercivity, or more generally one among the various classical conditionsthat ensure the corresponding problem (1.1) to be well-posed, such as the Banach-Ne˘cas-Babuˇska Theorem or the Lax-Milgram Lemma [8, Chapter 2].For the above-mentioned classes of inverse linear problems, the finite-dimensionaltruncation and the infinite-dimensional error analysis are widely studied and wellunderstood, as we shall comment further in due time. In that context, in orderfor the finite-dimensional solutions to converge strongly, one requires stringent yetoften plausible conditions [8, Sect. 2.2-2.4], [19, Sect. 4.2] both on the truncationspaces, that need to approximate suitably well the ambient space H (‘ approxima-bility ’, thus the interpolation capability of finite elements), and on the behaviour ofthe reduced problems, that need admit solutions that are uniformly controlled by RUNCATION AND CONVERGENCE INFINITE-DIM. INVERSE PROBLEM 3 the data (‘ uniform stability ’), and that are suitably good approximate solutions ofthe original problem (‘ asymptotic consistency ’), together with some suitable bound-edness of the problem in appropriate topologies (‘ uniform continuity ’).As plausible as the above conditions are, they are not matched by several othertypes of inverse problem of applied interest. Mathematically this is the case when-ever A does not have a ‘good’ inverse, for instance when A is a compact operatoron H with arbitrarily small singular values, or when the exact solution of the in-verse problem does not belong to the corresponding Krylov space used for thefinite-dimensional truncations.For such an abstract level of generality, for compact and generic bounded inverselinear problems, in this work we set up the theoretical formalism and settle theanalysis of the above-mentioned questions specifically when the dimension of theunderlying Hilbert space is infinite.As declared already, the purpose is to highlight non-trivial features typical ofinfinite dimensionality and discuss them through an amount of model examplesthat challenge the common intuition.In particular, we carry on the point of view that error and residual may becontrolled in a still informative way in some weaker sense than the expected normtopology of the Hilbert space. In this respect, we identify practically plausiblesufficient conditions for the error or the residual to be small in such generalisedsenses and we discuss the mechanisms why the same indicators may actually failto vanish in norm.In the concluding part of the work, we investigate the main features discussedtheoretically through a series of numerical tests, focusing on the truncation ofinfinite-dimensional inverse problems when the dimension of the truncation spaceincreases. General notation.
Besides further notation that will be declared in due time,we shall keep the following convention. H denotes a complex Hilbert space, thatwill be separable throughout this note, with norm (cid:107) · (cid:107) H and scalar product (cid:104)· , ·(cid:105) ,anti-linear in the first entry and linear in the second. Bounded operators on H are tacitly understood to be linear and everywhere defined. (cid:107) · (cid:107) op denotes thecorresponding operator norm. The space of bounded operators on H is denotedwith B ( H ). The spectrum of an operator A is denoted by σ ( A ). and O are,respectively, the identity and the zero operator, meant as finite matrices or infinite-dimensional operators depending on the context. An upper bar denotes the complexconjugate z when z ∈ C , and the norm closure V of the span of the vectors in V when V is a subset of H . For ψ, ϕ ∈ H , by | ψ (cid:105)(cid:104) ψ | and | ψ (cid:105)(cid:104) ϕ | we shall denote the H → H rank-one maps acting respectively as f (cid:55)→ (cid:104) ψ, f (cid:105) ψ and f (cid:55)→ (cid:104) ϕ, f (cid:105) ψ ongeneric f ∈ H . For identities such as ψ ( x ) = ϕ ( x ) in L -spaces we will tacitlyunderstand the ‘for almost every x ’ specification in the equality.2. Finite-dimensional truncation
Set up and notation.
Let us start with setting up a convenient formalism for the treatment of finite-dimensional truncations of linear inverse problems in infinite-dimensional Hilbertspace. In the framework of Galerkin and Petrov-Galerkin methods this is custom-arily referred to as the ‘ approximation setting ’ [8, Sect. 2.2.1].Let ( u n ) n ∈ N and ( v n ) n ∈ N be two orthonormal systems of the considered Hilbertspace H . They need not be orthonormal bases , although their completeness iscrucial for the goodness of the approximation. N. CARUSO, A. MICHELANGELI, AND P. NOVATI
In practice these are two explicitly known sets of orthonormal vectors (unlike,for instance, the possibly non-explicit orthonormal bases expressing the singularvalue decomposition of a given compact operator) that are going to be used in anumerical algorithm. In the Petrov-Galerkin nomenclature [8, 19] the u n ’s and the v n ’s span respectively the so-called ‘ solution space ’ (or ‘ trial space ’) and the ‘ testspace ’ of the problem.The choice of ( u n ) n ∈ N and ( v n ) n ∈ N depends on the specific approach. In theframework of finite element methods they can be taken to be the global shapefunctions of the interpolation scheme [8, Chapter 1]. For Krylov subspace methodsthey are just the spanning vectors of the associated Krylov subspace [16, Chapter2]. Correspondingly, for each N ∈ N , the orthonormal projections in H respectivelyonto span { u , . . . , u N } and span { v , . . . , v N } shall be(2.1) P N := N (cid:88) n =1 | u n (cid:105)(cid:104) u n | , Q N := N (cid:88) n =1 | v n (cid:105)(cid:104) v n | . Associated to a given well-defined linear inverse problem Af = g in H as (1.1),one considers the finite-dimensional truncations induced by P N and Q N , hence, foreach N , the problem to find solutions (cid:100) f ( N ) ∈ P N H to the equation(2.2) ( Q N AP N ) (cid:100) f ( N ) = Q N g . In (2.2) Q N g = (cid:80) Nn =1 (cid:104) v n , g (cid:105) v n is the datum and (cid:100) f ( N ) = (cid:80) Nn =1 (cid:104) u n , (cid:100) f ( N ) (cid:105) u n is theunknown, and the compression Q N AP N is only non-trivial as a map from P N H to Q N H , its kernel containing at least the subspace ( − P N ) H .Clearly, (2.2) (and more precisely (2.5) below) is nothing but the truncatedproblem arising from the oblique projection of the Petrov-Galerkin scheme. Whenthe special choice ( u n ) n ∈ N = ( v n ) n ∈ N is made, and hence P N = Q N for all N ’s, thisis the orthogonal projection approach of the ordinary Galerkin scheme.There is an obvious and non-relevant degeneracy (which is infinite when dim H = ∞ ) in (2.2) when it is regarded as a problem on the whole H . The actual interesttowards (2.2) is the problem resulting from the identification P N H ∼ = C N ∼ = Q N H ,in terms of which P N f ∈ H and Q N g ∈ H are canonically identified with thevectors(2.3) f N = (cid:104) u , f (cid:105) ... (cid:104) u N , f (cid:105) ∈ C N , g N = (cid:104) v , g (cid:105) ... (cid:104) v N , g (cid:105) ∈ C N , and Q N AP N with a C N → C N linear map represented by the N × N matrix A N = ( A N ; ij ) i,j ∈{ ,...,N } (2.4) A N ; ij = (cid:104) v i , Q N AP N u j (cid:105) . The matrix A N is what in the framework of finite element methods for partialdifferential equations is customarily referred to as the ‘ stiffness matrix ’.We shall call the inverse linear problem(2.5) A N f ( N ) = g N with datum g N ∈ C N and unknown f ( N ) ∈ C N , and matrix A N defined by (2.4),the N -dimensional truncation of the original problem Af = g .Let us stress the meaning of the present notation. • Q N AP N , P N f , and Q N g are objects (one operator and two vectors) referredto the whole Hilbert space H , whereas A N , f ( N ) , f N , and g N are theanalogues referred now to the space C N . RUNCATION AND CONVERGENCE INFINITE-DIM. INVERSE PROBLEM 5 • Moreover, the subscript in A N , f N , and g N indicates that the componentsof such objects are precisely the corresponding components, up to order N , respectively of A , f , and g , with respect to the tacitly declared bases( u n ) n ∈ N and ( v n ) n ∈ N , through formulas (2.3)-(2.4). • As opposite, the superscript in f ( N ) indicates that the components of the C N -vector f ( N ) are not necessarily to be understood as the first N compo-nents of the H -vector f with respect to the basis ( u n ) n ∈ N , and in particularfor N < N the components of f ( N ) are not a priori equal to the first N components of f ( N ) . In fact, if f ∈ H is a solution to Af = g , it is evidentfrom obvious counterexamples that in general the truncations A N , f N , g N do not satisfy the identity A N f N = g N , whence the notation f ( N ) for theunknown in (2.5). • Last, for a C N -vector f ( N ) the notation (cid:100) f ( N ) indicates a vector in H whosefirst N components, with respect to the basis ( u n ) n ∈ N , are precisely thoseof f ( N ) , all others being zero. Thus, as pedantic as it looks, f ( N ) = ( (cid:100) f ( N ) ) N and f N = ( (cid:99) f N ) N , and of course in general f (cid:54) = (cid:99) f N .With A , g , ( u n ) n ∈ N , and ( v n ) n ∈ N explicitly known, the truncated problem (2.5)is explicitly formulated and, being finite-dimensional, it is suited for numericalalgorithms.This poses the general question on whether the truncated problem itself is solv-able, and whether its exact or approximate solution f ( N ) is close to the exact solu-tion f and in which (possibly quantitative) sense .Let us elaborate more on these two issues in the following two subsections.2.2. Singularity of the truncated problem.
It is clear, first of all, that the question of the singularity of the truncated problem(2.5) makes sense here eventually in N , meaning for all N ’s that are large enough.For a fixed value of N the truncation might drastically alter the problem so as tomake it manifestly non-informative as compared to Af = g , such alteration thendisappearing for larger values.Yet, even when the solvability of A N f ( N ) = g N is inquired eventually in N , it isno surprise that the answer is generically negative. Example 2.1.
That the matrix A N may remain singular for arbitrary N evenwhen the operator A is injective can be seen, for example, with the truncationof the weighted (compact) right-shift operator R = (cid:80) ∞ n =1 σ n | e n +1 (cid:105)(cid:104) e n | on (cid:96) ( N )(Sect. A.3) with respect to the basis ( e n ) n ∈ N itself: indeed,(2.6) R N = · · · · · · · · · σ · · · · · · σ · · · · · · σ N − is singular irrespectively of N , with ker R N = span { e N } . (See Lemma 2.3 belowfor a more general perspective on such an example.)It is not difficult to cook up variations of the above example where the matrix A N is alternatingly singular and non-singular as N → ∞ . Example 2.2.
Of course, on the other hand, it may also well happen that the trun-cated matrix is always non-singular: the truncation of the multiplication operator
N. CARUSO, A. MICHELANGELI, AND P. NOVATI (see Sect. A.1) M = ∞ (cid:88) n =1 n | e n (cid:105)(cid:104) e n | on (cid:96) ( N ) with respect to ( e n ) n ∈ N yields the matrix M N = diag(1 , , . . . , N ), whichis a C N → C N bijection for every N .In fact, ‘bad’ truncations are always possible, as the following mechanism shows. Lemma 2.3.
Let H be a separable Hilbert space with dim H = ∞ , and let A ∈B ( H ) . There always exist two orthonormal bases ( u n ) n ∈ N and ( v n ) n ∈ N of H suchthat the corresponding truncated matrix A N defined as in (2.4) is singular for every N ∈ N .Proof. Let us pick an arbitrary orthonormal basis ( u n ) n ∈ N and construct the otherbasis ( v n ) n ∈ N inductively. When N = 1, it suffices to choose v such chat v ⊥ Au and (cid:107) v (cid:107) H = 1. Let now ( v n ) n ∈{ ,...,N − } be an orthonormal system in H satisfyingthe thesis up to the order N − v N so that ( v n ) n ∈{ ,...,N } satisfies the thesis up to order N . To this aim, let us show that a choice of v N isalways possible so that the final row in the matrix A N has all zero entries. In fact,( A N ) ij = ( Q N AP N ) ij = (cid:104) v i , Au j (cid:105) for i ∈ { , . . . , N − } and j ∈ { , . . . , N } and inorder for (cid:104) v N , Au j (cid:105) = 0 for j ∈ { , · · · , N } it suffices to take v N ⊥ ran( AP N ) , v N ⊥ ran Q N − , (cid:107) v N (cid:107) H = 1 , where P N and Q N − are the orthogonal projections defined in 2.1. Since ran( AP N )and ran Q N − are finite-dimensional subspaces of H , there is surely a vector v N ∈ H with the above properties. (cid:3) The occurrence described by Lemma 2.3 may happen both with an orthogonaland with an oblique projection scheme, namely both when P N = Q N and when P N (cid:54) = Q N eventually in N . In the standard framework of (Petrov-)Galerkin meth-ods such an occurrence is prevented by suitable assumptions on A , a typical examplebeing coercivity [8, Sect. 2.2], [19, Sect. 4.1].As in our discussion we do not exclude a priori such an occurrence. We arecompelled to regard f ( N ) as an a approximate solution to the truncated problem,in the sense that(2.7) A N f ( N ) = g N + ε ( N ) for some ε ( N ) ∈ C N . (We write ε ( N ) and not ε N because there is no reason to claim that the residual ε ( N ) in the N -dimensional problem is the actual truncation for every N of the sameinfinite-dimensional vector ε ∈ H .)It would be desirable to assume that ε ( N ) is indeed small and asymptoticallyvanishing with N , or even that ε ( N ) = 0 for N large enough, as is case in some ap-plications. Morally (up to passing to the weak formulation of the inverse problem),this is the assumption of asymptotic consistency naturally made for approximationsby Galerkin methods [8, Definition 2.15 and Theorem 2.24]. We shall make thisassumption here too, observing that in the present abstract context it is motivatedby the following property, whose proof is postponed to Section 4. Lemma 2.4.
Let A ∈ B ( H ) and g ∈ ran A . Let A N and g N be defined as in (2.3) - (2.4) above. Then there always exists a sequence ( f ( N ) ) N ∈ N such that f ( N ) ∈ C N and lim N →∞ (cid:107) A N f ( N ) − g N (cid:107) C N = 0 . In other words, there do exist approximate solutions f ( N ) to (2.5) actually sat-isfying (2.7) with (cid:107) ε ( N ) (cid:107) C N → N → ∞ . RUNCATION AND CONVERGENCE INFINITE-DIM. INVERSE PROBLEM 7
Convergence of the truncated problem: error and residual.
For an infinite-dimensional inverse problem the other major question is the van-ishing, as N → ∞ , of the two natural indicators of the displacement between theinfinite-dimensional inverse linear problem and its finite-dimensional truncation,namely the infinite-dimensional error E N and the infinite-dimensional residual R N ,defined respectively as E N := f − (cid:100) f ( N ) R N := g − A (cid:100) f ( N ) . (2.8)We qualify them as ‘infinite-dimensional’, although we shall drop this extra nomen-clature when no confusion arises, in order to distinguish them from the error andresidual at fixed N , which may be indexed by the number of steps in an iterativealgorithm.A first evident obstruction to the actual vanishing of E N when when dim H = ∞ is the use of of a non-complete orthonormal system ( u n ) n ∈ N , that is, such thatspan { u n | n ∈ N } is not dense in H . Example 2.5.
If the weighted (compact) right-shift operator R (Sect. A.3) istruncated with respect to( u n ) n ∈ N = ( e n ) n ∈ N ,n (cid:62) , ( v n ) n ∈ N = ( e n ) n ∈ N and the initial inverse problem is R f = g = e , then the exact solution is f = σ e ,yet the truncated problem can only produce approximate solutions (cid:100) f ( N ) ∈ span { e , e , . . . } , whence (cid:100) f ( N ) ⊥ f and (cid:107) (cid:100) f ( N ) − f (cid:107) H (cid:62) σ .Truncations with respect to a potentially non-complete orthonormal systemmight appear unwise, but in certain contexts are natural. One is the vast frameworkof the Krylov subspace methods [16], where one searches for approximate solutionsamong the linear combinations of the vectors g, Ag, A g, . . . and hence to performthe truncation with respect to an orthonormal basis of the Krylov subspace (2.9) K ( A, g ) := span { A k g | k ∈ N } associated to A ∈ B ( H ) and g ∈ H . Obviously, when dim K ( A, g ) = ∞ the subspace K ( A, g ) is open in H . Its closure can be the whole H , but also just a proper closedsubspace of H . Example 2.6. (i) For the right-shift operator R on (cid:96) ( N ) (Sect. A.2) and the vector g = e m +1 (one of the canonical basis vectors), K ( R, e m +1 ) = span { e , . . . , e m } ⊥ ,which is a proper subspace of (cid:96) ( N ) if m (cid:62)
1, and instead is the whole (cid:96) ( N )if g = e . Therefore the exact solution f = e m to Rf = g does not belongto K ( R, e m +1 ).(ii) For the Volterra integral operator V on L [0 ,
1] (Sect. A.5) and the func-tion g = (the constant function with value 1), it follows from (A.10) or(A.15) that the functions V g, V g, V g, . . . are (multiples of) the polyno-mials x, x , x , . . . , therefore K ( V, g ) is the space of polynomials on [0 , L [0 , u n ) n ∈ N = ( e n ) n ∈ N ,n (cid:62) spans the Krylovsubspace relative to R and e .In standard (Petrov-)Galerkin methods an occurrence as in Example 2.5 or(2.6)(i) is ruled out by an ad hoc ‘ approximability’ assumption [8, Definition 2.14 N. CARUSO, A. MICHELANGELI, AND P. NOVATI and Theorem 2.24] that can be rephrased as the request that ( u n ) n ∈ N is indeed anorthonormal basis of H .The approximability property is known to fail in situations of engineering in-terest, as is the case for the failure of the Lagrange finite elements in differentialproblems for electromagnetism [8, Sect. 2.3.3].Even when (complete) orthonormal bases of H are employed for the truncation,another feature of the infinite dimensionality must be taken into account, namelythe possibility that error and residual are asymptotically small only in some weakersense than the customary norm topology of H .There are indeed at least three meaningful senses in which the vanishing of E N or R N , as N → ∞ , can be monitored in an informative way. I. Strong ( H -norm) convergence. This is the vanishing (cid:107) R N (cid:107) H →
0, resp., (cid:107) E N (cid:107) H → (cid:107) R N (cid:107) H (cid:54) (cid:107) A (cid:107) op (cid:107) E N (cid:107) H . II. Weak convergence.
This is the vanishing R N (cid:42) E N (cid:42)
0: recall thata sequence ( ξ N ) N ∈ N in H converges weakly to ξ ∈ H as N → ∞ , ξ N (cid:42) ξ , when (cid:104) η, ξ N (cid:105) → (cid:104) η, ξ (cid:105) for any η ∈ H . III. Component-wise convergence.
This is the vanishing of each componentof the vector R N or E N with respect to the considered basis. Recall that a sequence( ξ N ) N ∈ N in H converges component-wise to ξ ∈ H as N → ∞ with respect to theorthonormal basis ( e n ) n ∈ N of H , and we write ξ N (cid:32) ξ , when (cid:104) e n , η N (cid:105) N →∞ −−−−→ (cid:104) e n , η (cid:105)∀ n ∈ N . Thus, E N (cid:32) n -th component (cid:104) u n , f − (cid:100) f ( N ) (cid:105) of E N vanishes as N → ∞ and R N (cid:32) n -th component (cid:104) v n , g − A (cid:100) f ( N ) (cid:105) of R N vanishes as N → ∞ , possibly with different vanishing rate depending on n .Clearly,(2.11) strong ⇒ weak ⇒ component-wise , and these notions are all inequivalent when dim H = ∞ (whereas they are allequivalent when dim H < ∞ ). In fact, it is standard to check that(2.12) η N (cid:107) (cid:107) H −−−−→ η as N → ∞ ⇔ (cid:40) η N (cid:42) η (cid:107) η N (cid:107) H → (cid:107) η (cid:107) H , and(2.13) η N (cid:42) η as N → ∞ ⇔ (cid:104) e n , η N (cid:105) N →∞ −−−−−→ (cid:104) e n , η (cid:105) ∀ n ∈ N sup N ∈ N (cid:107) η N (cid:107) H < + ∞ , where ( e n ) n ∈ N is an orthonormal basis of H .Despite (2.11), a mere component-wise vanishing E N (cid:32) (cid:100) f ( N ) (withrespect to the basis ( u n ) n ∈ N ) approximates the corresponding component of theexact solution f .As a matter of fact, a strong control such as (cid:107) R N (cid:107) H → (cid:107) E N (cid:107) H → not generic and only holds under specific a priori conditions on the inverse linearproblem.Thus, as already recalled in the Introduction, for elliptic boundary value prob-lems the standard Galerkin finite element method produces a strong vanishingof the error, provided that two crucial conditions are satisfied, namely a careful choice of the truncation space and the coercivity of the differential operator [19,Sect. 4.2.3]: when this is the case, the vanishing rate depends on the truncationbasis and the regularity of the solution. More generally [8, Sect. 2.3.1], standard RUNCATION AND CONVERGENCE INFINITE-DIM. INVERSE PROBLEM 9
Petrov-Galerkin methods give rise to a strong convergence of the approximate so-lution under the simultaneous validity of uniform stability, uniform boundednessand asymptotic consistency of the linear problem, and approximability by meansof the chosen truncation spaces. When the differential operator is non-coercive,additional sufficient conditions have been studied for the stability of the truncatedproblem and for the quasi optimality of the discretization scheme [5, 4, 3].On a related scenario, special classes of linear ill-conditioned problems (rank-deficient and discrete ill-posed problems) can be treated with regularisation meth-ods in which the solution is stabilised [22, 13]. The most notable regularisationmethods, namely the Tikhonov-Phillips method, the Landweber-Fridman iterationmethod, and the truncated singular value decomposition, produce indeed a stronglyvanishing error [11, 17]. Yet, when the inverse linear problem Af = g is governedby an infinite-rank compact operator A , it can be seen that the conjugate gradi-ent method, as well as α -processes (in particular, the method of steepest descent)may have strongly divergent error and residual in the presence of noise [6] and oneis forced to consider weaker forms of convergence. In fact, in [6] the presence ofcomponent-wise convergence is also alluded to.3. The compact linear inverse problem
Let us now examine, within the framework elaborated in the previous Section, theabstract truncation and convergence scheme for compact linear inverse problems.When the operator A on the given Hilbert space H is compact, it admits a‘ canonical ’ decomposition, the ‘ singular value decomposition ’ [20, Theorem VI.17](3.1) A = (cid:88) n σ n | ψ n (cid:105)(cid:104) ϕ n | , where n runs in a finite or infinite subset of N , ( ϕ n ) n and ( ψ n ) n are two orthonormalsystems of H , and 0 < σ n +1 < σ n for all n , and the above series converges inoperator norm. In the following we shall reserve the above notation for the singularvalue decomposition of the considered compact operator.The injectivity of A is tantamount as ( ϕ n ) n ∈ N being an orthonormal basis. A isnot necessarily surjective, but ran A = H if an only if ( ψ n ) n ∈ N is an orthonormalbasis.The inverse problem (1.1) for compact and injective A and g ∈ ran A is well-defined: there exists a unique f ∈ H such that Af = g .The compactness of A has two noticeable consequences here. First, since dim H = ∞ , A is invertible on its range only, and cannot have an everywhere defined boundedinverse: ran A can be dense in H , as in the case of the Volterra operator on L [0 , H , as for the weightedright-shift on (cid:96) ( N ) (Sect. A.3).Furthermore, A and its compression (in the usual meaning of Sect. 2.1) are closein a robust sense, as the following standard Lemma shows. Lemma 3.1.
With respect to an infinite-dimensional separable Hilbert space H , let A : H → H be a compact operator and let ( u n ) n ∈ N and ( v n ) n ∈ N be two orthonormalbases of H . Then (3.2) (cid:107) A − Q N AP N (cid:107) op N →∞ −−−−−→ ,P N and Q N being as usual the orthogonal projections (2.1) .Proof. Upon splitting A − Q N AP N = ( A − Q N A ) + Q N ( A − AP N ) it suffices to prove that (cid:107) A − AP N (cid:107) op N →∞ −−−−→ (cid:107) A − Q N A (cid:107) op N →∞ −−−−→
0. Let usprove the first limit (the second being completely analogous).Clearly, it is enough to prove that (cid:107) A − AP N (cid:107) op vanishes assuming further that A has finite rank. Indeed, the difference ( A − AP N ) − ( (cid:101) A − (cid:101) AP N ), where (cid:101) A is afinite-rank approximant of the compact operator A , is controlled in operator normby 2 (cid:107) A − (cid:101) A (cid:107) op and hence can be made arbitrarily small.Thus, we consider non-restrictively A = (cid:80) Mk =1 σ k | ψ k (cid:105)(cid:104) ϕ k | for some integer M ,where ( ϕ k ) Mk =1 and ( ψ k ) Mk =1 are two orthonormal systems, and 0 < σ M < · · · < σ .Now, for a generic ξ = (cid:80) ∞ n =1 ξ n v n ∈ H one has (cid:13)(cid:13) ( A − AP N ) ξ (cid:13)(cid:13) H = (cid:13)(cid:13)(cid:13) M (cid:88) k =1 σ k (cid:16) ∞ (cid:88) n = N +1 ξ n (cid:104) ϕ k , v n (cid:105) (cid:17) ψ k (cid:13)(cid:13)(cid:13) H = M (cid:88) k =1 σ k (cid:12)(cid:12)(cid:12) ∞ (cid:88) n = N +1 ξ n (cid:104) ϕ k , v n (cid:105) (cid:12)(cid:12)(cid:12) (cid:54) (cid:107) ξ (cid:107) H M (cid:88) k =1 σ k (cid:13)(cid:13) ( − P N ) ϕ k (cid:13)(cid:13) H , therefore (cid:13)(cid:13) A − AP N (cid:13)(cid:13) (cid:54) M σ · max k ∈{ ,...,M } (cid:13)(cid:13) ( − P N ) ϕ k (cid:13)(cid:13) H N →∞ −−−−−→ , since the above maximum is taken over M (hence, finitely many) quantities, eachof which vanishes as N → ∞ . (cid:3) In the following Theorem we describe the generic behaviour of well-defined com-pact inverse problem.
Theorem 3.2.
Consider • the linear inverse problem Af = g in a separable Hilbert space H for somecompact and injective A : H → H and some g ∈ ran A ; • the finite-dimensional truncation A N obtained by compression with respectto the orthonormal bases ( u n ) n ∈ N and ( v n ) n ∈ N of H .Let ( f ( N ) ) N ∈ N be a sequence of approximate solutions to the truncated problems inthe quantitative sense A N f ( N ) = g N + ε ( N ) , f ( N ) , ε ( N ) ∈ C N , (cid:107) ε ( N ) (cid:107) C N N →∞ −−−−−→ for every (sufficiently large) N . If (cid:100) f ( N ) is H -norm bounded uniformly in N , then (cid:107) R N (cid:107) H → and E N (cid:42) as N → ∞ . Proof.
We split A (cid:100) f ( N ) − g = ( A − Q N AP N ) (cid:100) f ( N ) + Q N AP N (cid:100) f ( N ) − Q N g + Q N g − g . (*)By assumption, (cid:107) Q N g − g (cid:107) H N →∞ −−−−−→ (cid:107) Q N AP N (cid:100) f ( N ) − Q N g (cid:107) H = (cid:107) A N f ( N ) − g N (cid:107) C N = (cid:107) ε ( N ) (cid:107) C N N →∞ −−−−−→ . Moreover, Lemma 3.1 and the uniform boundedness of (cid:100) f ( N ) imply (cid:107) ( A − Q N AP N ) (cid:100) f ( N ) (cid:107) H (cid:54) (cid:107) A − Q N AP N (cid:107) op (cid:107) (cid:100) f ( N ) (cid:107) H N →∞ −−−−−→ (cid:107) R N (cid:107) H → RUNCATION AND CONVERGENCE INFINITE-DIM. INVERSE PROBLEM 11
Next, in terms of the singular value decomposition (3.1) of A , where now ( ϕ n ) n ∈ N is an orthonormal basis of H , ( ψ n ) n ∈ N is an orthonormal system, and 0 < σ n +1 < σ n ∀ n ∈ N , we write (cid:100) f ( N ) = (cid:88) n ∈ N f ( N ) n ϕ n , (cid:98) f = (cid:88) n ∈ N f n ϕ n , whence 0 = lim N →∞ (cid:107) A (cid:100) f ( N ) − g (cid:107) H = lim N →∞ (cid:88) n ∈ N σ n (cid:12)(cid:12) f ( N ) n − f n (cid:12)(cid:12) . Then necessarily (cid:100) f ( N ) converges to f component-wise ( E N (cid:32) (cid:100) f ( N ) is uniformly bounded in H , thus, owing to (2.13), (cid:100) f ( N ) converges to f weakly ( E N (cid:42) (cid:3) Theorem 3.2 provides sufficient conditions for some form of vanishing of the errorand the residual. The key assumptions are: • injectivity of A , • asymptotic solvability of the truncated problems , i.e., asymptotic smallnessof the finite-dimensional residual A N f ( N ) − g N , • uniform boundedness of the approximate solutions f ( N ) .In fact, injectivity was only used in the analysis of the error in order to conclude E N (cid:42)
0; instead, the conclusion (cid:107) R N (cid:107) H → Remark 3.3 (Genericity) . Under the conditions of Theorem 3.2, the occurrenceof the strong vanishing of the residual ( (cid:107) R N (cid:107) H →
0) and the weak vanishing of theerror ( E N (cid:42)
0) as N → ∞ is a generic behaviour . For example, the compact inverseproblem R f = 0 in (cid:96) ( N ) associated with the weighted right-shift R (Sect. A.3)has exact solution f = 0. The truncated problem R N f ( N ) = 0 with respect tothe same basis ( e n ) n ∈ N , R N being the matrix (2.6), is solved by the C N -vectorswhose first N − (cid:100) f ( N ) = e N . The sequence ( (cid:100) f ( N ) ) N ∈ N ≡ ( e N ) N ∈ N converges weakly to zero in (cid:96) ( N ), whence indeed E N (cid:42)
0, and also, bycompactness, (cid:107) R N (cid:107) H →
0. However, (cid:107) E N (cid:107) H = 1 for every N , thus the error cannotvanish in the H -norm. Remark 3.4 (‘Bad’ approximate solutions) . The example considered in Remark3.3 is also instructive to understand that generically one may happen to select‘bad’ approximate solutions (cid:100) f ( N ) such that, despite the ‘good’ property (cid:107) A N f ( N ) − g N (cid:107) C N →
0, have the unsatisfactory feature (cid:107) f ( N ) (cid:107) C N = (cid:107) (cid:100) f ( N ) (cid:107) H → + ∞ : this isthe case if one chooses, for instance, (cid:100) f ( N ) = N e N . Thus, the uniform boundednessof (cid:100) f ( N ) in H required in Theorem 3.2 is not redundant. (This also shows, inview of the proof of Theorem 3.2, that whereas by compactness (cid:100) f ( N ) (cid:42) f implies (cid:107) A (cid:100) f ( N ) − Af (cid:107) →
0, the opposite implication is not true in general.)
Remark 3.5 (The density of ran A does not help) . Even if the genericity discussedin Remarks 3.3 and 3.4 is referred to compact injective operators with non-denserange, requiring ran A = H does not improve the convergence in general. Forinstance, the compact inverse problem associated with the weighted right-shift R in (cid:96) ( Z ) (Sect. A.4) involves an operator that is compact, injective, and with denserange, but its compression with Q N := P N := (cid:80) Nn = − N | e N (cid:105)(cid:104) e N | produces for every N a (2 N + 1) × (2 N + 1) square matrix that is singular and for which, therefore,all the considerations of Remarks 3.3 and 3.4 can be repeated verbatim. Remark 3.6 (‘Bad’ truncations and ‘good’ truncations) . We saw in Lemma 2.3that ‘bad’ truncations (i.e., leading to matrices A N that are, eventually in N ,all singular) are always possible. On the other hand, there always exists a “good”choice for the truncation – although such a choice might not be identifiable explicitly– which makes the infinite-dimensional residual and error vanish in a stronger sensethan what stated in Theorem 3.2, and without the extra assumption of uniformboundedness on the approximate solutions. For instance, in terms of the singularvalue decomposition (3.1) of A , it is enough to choose( u n ) n ∈ N = ( ϕ n ) n ∈ N , ( v n ) n ∈ N = ( ψ n ) n ∈ N , in which case Q N AP N = (cid:80) Nn =1 σ n | ψ n (cid:105)(cid:104) ϕ n | and A N = diag( σ , . . . , σ N ), and forgiven g = (cid:80) n ∈ N g n ψ n one has (cid:100) f ( N ) = (cid:80) Nn =1 g n σ n ϕ n , where the sequence ( g n σ n ) n ∈ N belongs to (cid:96) ( N ) owing to the assumption g ∈ ran A , whence (cid:107) f − (cid:100) f ( N ) (cid:107) H = ∞ (cid:88) n = N +1 (cid:12)(cid:12)(cid:12) g n σ n (cid:12)(cid:12)(cid:12) N →∞ −−−−−→ . The bounded linear inverse problem
It is instructive to compare the findings of the previous Section with the moregeneral case of a bounded (not necessarily compact) inverse linear problem.When dim H = ∞ and a generic bounded linear operator A : H → H is com-pressed (in the usual sense of Sect. 2) between the spans of the first N vectors ofthe orthonormal bases ( u n ) n ∈ N and ( v n ) n ∈ N , then surely Q N AP N → A as N → ∞ in the strong operator topology, that is, (cid:107) Q N AP N ψ − Aψ (cid:107) H N →∞ −−−−−→ ∀ ψ ∈ H ,yet the convergence may fail to occur in the operator norm .The first statement is an obvious consequence of the inequality (cid:107) ( A − Q N AP N ) ψ (cid:107) H (cid:54) (cid:107) ( − Q N ) Aψ (cid:107) H + (cid:107) A (cid:107) op (cid:107) ψ − P N ψ (cid:107) H valid for any ψ ∈ H . The lack of operator norm convergence is clear, for instance,when one compresses the identity operator (or any bounded, non-compact opera-tor): the operator norm limit of finite-rank operators can only be compact.For this reason, the control of the infinite-dimensional inverse problem in termsof its finite-dimensional truncated versions is in general less strong.As a counterpart of Theorem 3.2 above, let us discuss the following genericbehaviour of well-posed bounded linear inverse problems. Theorem 4.1.
Consider • the linear inverse problem Af = g in a Hilbert space H for some boundedand injective A : H → H and some g ∈ H ; • the finite-dimensional truncation A N obtained by compression with respectto the orthonormal bases ( u n ) n ∈ N and ( v n ) n ∈ N of H .Let ( f ( N ) ) N ∈ N be a sequence of approximate solutions to the truncated problems inthe quantitative sense A N f ( N ) = g N + ε ( N ) , f ( N ) , ε ( N ) ∈ C N , (cid:107) ε ( N ) (cid:107) C N N →∞ −−−−−→ for every (sufficiently large) N . Assume further that (cid:100) f ( N ) converges strongly in H ,equivalently, that (cid:107) f ( N ) − f ( M ) (cid:107) C max { N,M } N,M →∞ −−−−−−−→ . Then (cid:107) E N (cid:107) H → and (cid:107) R N (cid:107) H → as N → ∞ . RUNCATION AND CONVERGENCE INFINITE-DIM. INVERSE PROBLEM 13
Proof.
Since A (cid:100) f ( N ) − g = ( A − Q N AP N ) (cid:100) f ( N ) + Q N AP N (cid:100) f ( N ) − Q N g + Q N g − g , (*)and since by assumption (cid:107) Q N g − g (cid:107) H N →∞ −−−−−→ (cid:107) Q N AP N (cid:100) f ( N ) − Q N g (cid:107) H = (cid:107) A N f ( N ) − g N (cid:107) C N = (cid:107) ε ( N ) (cid:107) C N N →∞ −−−−−→ , then the strong vanishing of A (cid:100) f ( N ) − g is tantamount as the strong vanishing of( A − Q N AP N ) (cid:100) f ( N ) .Since in addition (cid:107) (cid:100) f ( N ) − (cid:101) f (cid:107) H N →∞ −−−−−→ (cid:101) f ∈ H , then (cid:107) ( A − Q N AP N ) (cid:100) f ( N ) (cid:107) H (cid:54) (cid:107) ( A − Q N AP N ) (cid:101) f (cid:107) H + 2 (cid:107) A (cid:107) op (cid:107) (cid:101) f − (cid:100) f ( N ) (cid:107) H N →∞ −−−−−→ Q N AP N → A ), and (*) thus implies (cid:107) R N (cid:107) H = (cid:107) A (cid:100) f ( N ) − g (cid:107) H N →∞ −−−−−→ A (cid:100) f ( N ) → g (as proved right now) and A (cid:100) f ( N ) → A (cid:101) f (by continuity),whence A (cid:101) f = g = Af and also (by injectivity) f = (cid:101) f . This shows that (cid:107) E N (cid:107) H = (cid:107) f − (cid:100) f ( N ) (cid:107) H = (cid:107) (cid:101) f − (cid:100) f ( N ) (cid:107) H → (cid:3) We observe that also here injectivity was only used in the analysis of the error,whereas it is not needed to conclude that (cid:107) R N (cid:107) H → • injectivity of A , • asymptotic solvability of the truncated problems , • convergence of the approximate solutions f ( N ) .The first two assumptions are the same as in the compact case: the first guaran-tees the existence of a unique solution and the second is a natural working hypoth-esis, by virtue of Lemma 2.4. Under such assumptions, we thus see that, in passingfrom a (well-defined) compact to a generic (well-defined) bounded inverse problem,one has to strengthen the hypothesis of uniform boundedness of the (cid:100) f ( N ) ’s to theiractual strong convergence, in order for the residual R N to vanish strongly (in whichcase, as a by-product, also the error E N vanishes strongly).Moreover, the proof of Theorem 4.1 shows that, under injectivity of A and as-ymptotic solvability of the truncated problems, the residual R N vanishes strongly,or weakly or component-wise, if and only if so does ( A − Q N AP N ) (cid:100) f ( N ) . In the com-pact case, A − Q N AP N → O in operator norm (Lemma 3.1), and it suffices thatthe (cid:100) f ( N ) ’s are uniformly bounded (or, in principle, have increasing norm (cid:107) (cid:100) f ( N ) (cid:107) H compensated by the vanishing of (cid:107) A − Q N AP N (cid:107) op ), in order for (cid:107) R N (cid:107) H →
0. Inthe general bounded case we controlled the vanishing of (cid:107) ( A − Q N AP N ) (cid:100) f ( N ) (cid:107) H byrequiring additionally that the (cid:100) f ( N ) ’s converge strongly.If instead the sequence of the (cid:100) f ( N ) ’s does not converge strongly, Theorem 4.1is not applicable, and in general one has to expect only weak vanishing of theresidual, R N (cid:42)
0, which in turn prevents the error to vanish strongly – for otherwise (cid:107) E N (cid:107) H → (cid:107) R N (cid:107) H →
0, owing to (2.10). The following exampleshows such a possibility.
Example 4.2.
For the right-shift R on (cid:96) ( N ) (Sect. A.2), an actual injective oper-ator, the inverse problem Rf = g with g = 0 admits the unique solution f = 0. Thetruncated finite-dimensional problems induced by the bases ( u n ) n ∈ N = ( v n ) n ∈ N =( e n ) n ∈ N , where ( e n ) n ∈ N is the canonical basis of (cid:96) ( N ), is governed by the sub-diagonal matrix R N = · · · · · · · · ·
01 0 · · · · · ·
00 1 0 · · · · · · . Let us consider the sequence ( (cid:100) f ( N ) ) N ∈ N with (cid:100) f ( N ) := e N for each N . Then: • R N f ( N ) = 0 = g N (the truncated problems are solved exactly), • (cid:100) f ( N ) (cid:42) • R N = g − R (cid:100) f ( N ) = − e N +1 (cid:42) A is an algebraic operator ,namely p ( A ) = O for some polynomial p (which includes finite-rank A ’s) and onetreats the inverse problem with the generalised minimal residual method (GMRES)[10].In retrospect, the arguments developed in this Section allow us to prove Lemma2.4. Proof of Lemma 2.4.
Let f solve Af = g . The sequence ( f ( N ) ) N ∈ N defined by f ( N ) := ( P N f ) N = f N (that is, (cid:100) f ( N ) = P N f )does the job, and that is a straightforward consequence of the fact that, as arguedalready at the beginning of this Section, Q N AP N → A strongly in the operatortopology. Indeed, one has by adding and subtracting Af (cid:107) A N f ( N ) − g N (cid:107) C N = (cid:107) Q N AP N (cid:100) f ( N ) − Q N g (cid:107) H (cid:54) (cid:107) ( Q N AP N − A ) f (cid:107) H + (cid:107) (1 − Q N ) Af (cid:107) H . The strong limit yields the conclusion. (cid:3) Comparison to conjugate gradient schemes
In this Section we further discuss the scope of Theorems 3.2 (compact case) and4.1 (bounded case) in application to conjugate gradient schemes for bounded, self-adjoint, positive semi-definite inverse linear problems [7, Chapt. 7], [8, Sect. 9.3.2],[19, Sect. 7.2.2]. Thus, throughout this Section A = A ∗ ∈ B ( H ) with (cid:104) h, Ah (cid:105) (cid:62) ∀ h ∈ H .In particular, we provide an additional insight on the key role played by theassumption of uniform boundedness (or even strong convergence ) of the finite-dimensional approximants.Under the above assumption on A , and with g ∈ ran A , the problem Af = g admits solution(s) in H , which form the (non-empty) manifold(5.1) S ( A, g ) := { f ∈ H | Af = g } . RUNCATION AND CONVERGENCE INFINITE-DIM. INVERSE PROBLEM 15
Clearly, if A is injective, which in this case amounts to A being positive definite,then S ( A, g ) only consists of the unique solution to the inverse problem. Moreover,any f in the solution manifold S ( A, g ) can be variationally characterised as(5.2) Φ[ f ] = min h ∈H Φ[ h ] , Φ[ h ] := (cid:104) h, Ah (cid:105) − (cid:104) h, g (cid:105) , that is, f is the minimiser of the functional Φ[ h ] (which, in specific contexts, isreferred to as the ‘energy functional’ of the problem).Based on such properties, in the framework of conjugate gradient schemes onebuilds a sequence ( f [ N ] ) N ∈ N , the so-called ‘ conjugate gradient iterates ’, by taking f [0] to be an arbitrary vector in H , and f [ N ] , for N (cid:62)
1, to be the minimiser of theproblem(5.3) min h ∈Q N Φ[ h ] , Q N := { f [0] } + span { r , Ar , . . . , A N − r } r := Af [0] − g . Here ‘iterates’ refers to the fact that the f [ N ] ’s can be equivalently obtained bymeans of certain iterative procedures [14, 18].The notation for the superscript in f [ N ] is chosen to avoid confusion with thespecial meaning already reserved to f ( N ) and (cid:100) f ( N ) in the general setting of Sect. 2.1,although it is clear that the f [ N ] ’s here are to be considered on the same conceptualfooting as the (cid:100) f ( N ) ’s, that is, they can be naturally regarded as approximate solu-tions, expected to satisfy Af [ N ] ≈ g in a suitable sense. This is suggested by thevery construction (5.3) and the variational characterisation (5.2) of the solution(s) f . That the above expectation is correct is expressed in rigorous terms by Theorem5.1 below, a classical result by Nemirovskiy and Polyak [18] (with a precursorversion by Kammerer and Nashed [15]), and discussed in more recent terms in[7, Sect. 7.2] and [12, Sect. 3.2]. In order to state it, let us introduce the map P S : H → S ( A, g ) that associates to a point h ∈ H the nearest point P S h of thesolution manifold. Then one has the following. Theorem 5.1. (Nemirovskiy and Polyak [18, Theorem 7].)
Let A = A ∗ ∈ B ( H ) with (cid:104) h, Ah (cid:105) (cid:62) ∀ h ∈ H , and let the sequence ( f [ N ] ) N ∈ N in H be defined by (5.3) above. Then (5.4) lim N →∞ (cid:107) f [ N ] − P S f [ N ] (cid:107) H = 0 , and moreover, for every γ > , (5.5) (cid:107) f [ N ] − P S f [ N ] (cid:107) H (cid:54) (cid:16) C f [0] ,γ N + 1 (cid:17) γ for some constant C f [0] ,γ > depending on f [0] and γ , provided that the problem A γ/ u = f [0] − P S f [0] admits a solution u ∈ H . When A is injective and hence S ( A, g ) only consists of the unique solution f to Af = g , (5.4) reads (cid:107) f [ N ] − f (cid:107) H → N → ∞ . In the analogy with the analysisof Theorem 4.1, the sequence of approximate solutions is convergent and the error E N indeed vanishes strongly, and so does, necessarily, the residual R N . We canthus understand Theorem 5.1 in view of our Theorem 4.1.Equally instructive is the case when A is not injective and hence the solutionmanifold S ( A, g ) contains infinitely many vectors. Again, (5.4) indicates that theapproximate solutions f [ N ] ’s are asymptotically close, in the H -norm topology, tosolutions of the considered inverse problem. However, now this does not necessarilyimply the actual convergence to a fixed solution : both the f [ N ] ’s and the corre-sponding P S f [ N ] ’s might in principle have arbitrarily large norm – in complete analogy to what one would have in Example 4.2 if one considered approximate so-lutions (cid:100) f ( N ) = N e N , instead of just (cid:100) f ( N ) = e N . In order to deduce from (5.4) that f [ N ] → f for some solution f , an additional information is needed, for examplethe property that the f [ N ] ’s are uniformly norm bounded. This sheds further lighton the requirement of strong convergence of the approximate solutions made inTheorem 4.1 needed to deduce the strong vanishing of the error.6. Counterpart remarks on linear inverse problems with noise
Let us reconsider the typical occurrence, mentioned in the Introduction, when • within the modelling of the phenomenon under investigation, the linearinverse problem Af = g is well-defined (or even well-posed), and thus, thereis a unique ‘input’ f for given ‘output’ g and with an explicitly known law f A (cid:55)−→ g ; • however, the knowledge of g obtained from measurements is disturbed byvarious forms of uncertainty.In view of the general discussion developed so far, we can make here a few remarkson such an occurrence.Now the problem Af = g cannot be studied directly, and instead one deals withthe inverse problem(6.1) A (cid:101) f = (cid:101) g in the new unknown (cid:101) f for some given (measured) (cid:101) g := g + ν ∈ H , where the ‘noise’vector ν is present albeit not known explicitly, but is typically small – for instancea small bound on (cid:107) ν (cid:107) H may be known a priori.If ν (and g ) belongs to ran A , so does (cid:101) g , and there exist an actual (possibly non-unique) solution (cid:101) f to (6.1). Theorems 3.2 and 4.1 are then applicable, replacing g with g + ν , and with analogous notation we may speak of an approximate solution f ( N ) ∈ C N such that(6.2) A N f ( N ) = g N + ν N + ε ( N ) , (cid:107) ε ( N ) (cid:107) C N N →∞ −−−−−→ . This way, Theorems 3.2 and 4.1 produce a control on the “residual with noise”( g + ν ) − A (cid:100) f ( N ) and one the “error with noise” (cid:101) f − (cid:100) f ( N ) . This only determinesthe “solution with noise”, namely (cid:101) f , and not the exact solution f , but that can bestill informative if ν is sufficiently small. For example, if A is bounded and witheverywhere defined bounded inverse, then (cid:101) f = A − ( g + ν ), whence (cid:107) (cid:101) f − f (cid:107) H (cid:54) (cid:107) A − (cid:107) op (cid:107) ν (cid:107) H , and the smallness of (cid:107) ν (cid:107) H , in terms of (cid:107) A − (cid:107) op , provides an estimateon how close f and (cid:101) f are.If, on the other hand, ν / ∈ ran A , then the problem with noise loses solvability:there is no exact solution to (6.1) and one can only think of an approximate solution (cid:101) f satisfying A (cid:101) f ≈ (cid:101) g in some sense (whence also A (cid:101) f ≈ g , since ν is convenientlysmall).Let us comment on the typical behaviour of the residual R N and the error E N associated with f , (cid:100) f ( N ) , g , for simplicity in the case where A is compact andinjective, with g ∈ ran A (thus, with f unique solution to Af = g ).6.1. Typical behaviour of R N with noise. When the truncated problem with noise is solved in the approximate sense (6.2),and the (cid:100) f ( N ) ’s are uniformly bounded in H , then necessarily(6.3) (cid:107) R N (cid:107) H = (cid:13)(cid:13) A (cid:100) f ( N ) − g (cid:13)(cid:13) H N →∞ −−−−−→ (cid:107) ν (cid:107) H . RUNCATION AND CONVERGENCE INFINITE-DIM. INVERSE PROBLEM 17
This is seen by splitting as usual R N = ( Q N AP N − A ) (cid:100) f ( N ) + (cid:0) Q N g − Q N AP N (cid:100) f ( N ) (cid:1) + (cid:0) g − Q N g (cid:1) , and observing that (cid:107) ( Q N AP N − A ) (cid:100) f ( N ) (cid:107) H (cid:54) (cid:107) Q N AP N − A (cid:107) op (cid:107) (cid:100) f ( N ) (cid:107) H → (cid:107) g − Q N g (cid:107) H →
0, and (cid:107) Q N g − Q N AP N (cid:100) f ( N ) (cid:107) H = (cid:107) A N f ( N ) − g N (cid:107) C N = (cid:107) ν N + ε ( N ) (cid:107) C N → (cid:107) ν (cid:107) H . Clearly, based on the above argument, one actually has(6.4) (cid:107) R N − ν (cid:107) H N →∞ −−−−−→ , which is in fact stronger than (6.3). Thus, ‘ the residual vanishes up to the noisethreshold ’.6.2. Typical behaviour of E N with noise. In the presence of noise one cannot expect that (cid:100) f ( N ) , even just component-wise,converges to f ; in particular, the possibility that (cid:107) E N (cid:107) H → E N (cid:42) (cid:107) E N (cid:107) H stays strictly above zero, uniformly in N , in fact with a typicalbehaviour that (cid:107) E N (cid:107) H initially decreases for not to large N , reaches a minimum,then for larger N eventually increases, possibly blowing up . (This differs fromthe behaviour of (cid:107) R N (cid:107) H , which typically decreases monotonically to (cid:107) ν (cid:107) H .) Theminimum for (cid:107) E N (cid:107) H , say, when N = N , provides the best approximant of f in H ,namely (cid:91) f ( N ) .For concreteness, let us consider the case in which the Petrov-Galerkin projectionto (6.2) is performed with the same bases ( ϕ n ) n ∈ N and ( ψ n ) n ∈ N of the canonicalsingular value decomposition (3.1) of A . Let us also assume that ν ∈ ran A (thegeneralisation of what follows to the case ν / ∈ ran A is straightforward). Thesesimplifications guarantee that for all N the matrix A N = diag( σ , . . . , σ N ) is non-singular on C N , because now Q N AP N = (cid:80) Nn =1 σ n | ψ n (cid:105)(cid:104) ϕ n | , and that (6.2) is exactlysolved by (cid:100) f ( N ) = N (cid:88) n =1 g n + ν n σ n ϕ n , having decomposed ν = ∞ (cid:88) n =1 ν n ψ n , g = ∞ (cid:88) n =1 g n ψ n , f = ∞ (cid:88) n =1 f n ϕ n , g n = σ n f n . Thus, A N f ( N ) = g N + ν N ( ε ( N ) = 0). Then (cid:107) R N (cid:107) H = (cid:13)(cid:13) g − A (cid:100) f ( N ) (cid:13)(cid:13) H = N (cid:88) n =1 | ν n | + ∞ (cid:88) n = N +1 | g n | N →∞ −−−−−→ (cid:107) ν (cid:107) H , (cid:107) E N (cid:107) H = (cid:13)(cid:13) f − (cid:100) f ( N ) (cid:13)(cid:13) H = N (cid:88) n =1 | ν n | σ n + ∞ (cid:88) n = N +1 | f n | =: α ( N ) + β ( N ) . It is clear that β ( N ) decreases monotonically to zero as N → ∞ , whereas α ( N )is monotone increasing with N . This can produce the typical initial decrease of (cid:107) E N (cid:107) H , driven by a substantial decrease of β ( N ) as opposite to a mild increaseof α ( N ), which is the case when f is mainly supported on low modes ϕ n ’s and ν instead has a substantial tail on high modes ψ n ’s. For N sufficiently large, α ( N )then becomes leading, which would produce the typical inversion of the curve of (cid:107) E N (cid:107) H versus N . Having assumed ν ∈ ran A , necessarily α ( N ) → (cid:107) A − ν (cid:107) H , thus
10 20 30 40 50 60 70 80 90 N
10 20 30 40 50 60 70 80 90 N Figure 1.
Typical behaviour of the residual (cid:107) R N (cid:107) H (left) and ofthe error (cid:107) E N (cid:107) H (right) for increasing size of the finite-dimensionaltruncation, relative to the problem Af = g considered in Example6.1, with the choice σ n = n , g n = n , ν n = . n / .with no blow-up of (cid:107) E N (cid:107) H . Reasoning as above with ν / ∈ A one would concludeinstead that the series defining α ( N ) diverges. Example 6.1.
Take, ∀ n ∈ N , σ n = n − , g n = n − , ν n = n − . Thus, A is an injective Hilbert-Schmidt operator, (cid:107) ν (cid:107) H = ζ (3) (cid:39) .
20 (where ζ ( x )denotes the Riemann zeta function), and ν / ∈ ran A . Then f n = n − , (cid:107) f (cid:107) H = β (0) = π , and β ( N ) (cid:54) ( N + 1) − → , α ( N ) ∼ ln N → + ∞ . Figure 1 displays the behaviour of residual and error in this case.7.
Numerical tests: effects of changing the truncation basis
In this final Section we examine some of the features discussed theoretically sofar through a few numerical tests concerning different choices of the truncationbases. We employed a Legendre, complex Fourier, and a Krylov basis to truncatethe problems.The two model operators that we considered are the Volterra operator V in L [0 ,
1] (Sect. A.5) and the self-adjoint multiplication operator M : L [1 , → L [1 , ψ (cid:55)→ xψ . We examined the following two inverse problems. First problem:
V f = g , with g ( x ) = x .The problem has unique solution(7.1) f ( x ) = x , (cid:107) f (cid:107) L [0 , = 1 √ (cid:39) . f is a Krylov solution, i.e., f ∈ K ( V, g ), although f / ∈ K ( V, g ). To prove thefirst fact, let us observe that K ( V, g ) is spanned by the monomials x , x , x , . . . ,i.e., K ( V, g ) = { x p | p is a polynomial on [0 , } ; therefore, if h ∈ K ( V, g ) ⊥ , then0 = (cid:82) h ( x ) x p ( x ) d x for any polynomial p ; the L -density of polynomials on [0 , x h = 0, whence also h = 0; this proves that K ( V, g ) ⊥ = { } and hence K ( V, g ) = L [0 , f / ∈ K ( V, g ) follows from f ( x ) = x · x and x / ∈ L [0 , Second problem:
M f = g , with g ( x ) = x .The problem has unique solution(7.2) f ( x ) = x , (cid:107) f (cid:107) L [1 , = (cid:114) (cid:39) . RUNCATION AND CONVERGENCE INFINITE-DIM. INVERSE PROBLEM 19 N -14 -13 F i na l E rr o r N o r m N F i na l E x a c t R e s i dua l no r m -16 N || f ( N ) || (a) Legendre basis truncation N F i na l E rr o r N o r m N -5 -4 F i na l E x a c t R e s i dua l no r m N || f ( N ) || (b) Complex Fourier basis truncation N -3 -2 -1 F i na l E rr o r N o r m N -8 -6 -4 -2 F i na l E x a c t R e s i dua l no r m N || f ( N ) || (c) Krylov basis truncation
Figure 2.
Norm of the infinite-dimensional error and residual,and of the approximated solution for the Volterra inverse problemtruncated with the Legendre, complex Fourier, and Krylov bases.and f is a Krylov solution. Indeed, K ( M, g ) = { x p | p is a polynomial on [1 , } and K ( M, g ) = { x h ( x ) | h ∈ L [1 , } = L [1 , f ∈ K ( M, g ) and f / ∈K ( M, g ).We treated both problems with three different orthonormal bases: the Legendrepolynomials and the complex Fourier modes (on the intervals [0 ,
1] or [1 , N = 100 when considering the Legendre basis, but N = 500when considering the complex Fourier basis. It is expected that there is no signif-icant numerical error from the computation of the Legendre basis, as the L [0 , L [1 ,
2] norms of the basis polynomials have less than 1% error compared totheir exact unit value.For each problem and each choice of the basis, we monitored the norm of theinfinite-dimensional error (cid:107) E N (cid:107) L = (cid:107) f − (cid:100) f ( N ) (cid:107) L ( f = f or f ), of the infinite-dimensional residual (cid:107) R N (cid:107) L = (cid:107) g − A (cid:100) f ( N ) (cid:107) L ( g = g or g ; A = V or M ), andof the approximated solution (cid:107) (cid:100) f ( N ) (cid:107) L = (cid:107) f ( N ) (cid:107) C N .Figures 2 and 4 highlight the difference between the computation in the threebases for the Volterra operator. N F i na l E rr o r N o r m -15 N F i na l E x a c t R e s i dua l no r m -15 N || f ( N ) || (a) Legendre basis truncation N F i na l E rr o r N o r m N F i na l E x a c t R e s i dua l no r m N || f ( N ) || (b) Complex Fourier basis truncation N -10 -8 -6 -4 -2 F i na l E rr o r N o r m N -10 -8 -6 -4 -2 F i na l E x a c t R e s i dua l no r m N || f ( N ) || (c) Krylov basis truncation
Figure 3.
Norm of the infinite-dimensional error, residual, andapproximated solution for the M -multiplication inverse problemtruncated with the Legendre, complex Fourier, and Krylov bases. • In the Legendre basis, (cid:107) E N (cid:107) L and (cid:107) R N (cid:107) L are almost zero. (cid:107) (cid:100) f ( N ) (cid:107) L staysbounded and constant with N and matches the expected value (7.1). Theapproximated solutions reconstruct the exact solution f at any truncationnumber. • In the complex Fourier basis, both (cid:107) E N (cid:107) L and (cid:107) R N (cid:107) L are some orders ofmagnitude larger than in the Legendre basis and decrease monotonicallywith N ; in fact, (cid:107) E N (cid:107) L and (cid:107) R N (cid:107) L display an evident convergence tozero, however attaining values that are more than ten orders of magnitudelarger than the corresponding error and residual norms for the same N inthe Legendre case. (cid:107) (cid:100) f ( N ) (cid:107) L , on the other hand, increases monotonicallyand appears to approach the theoretical value (7.1). These quite strin-gent differences in the error and residual may be attributable to the Gibbsphenomenon. In fact, reconstructing f using the Krylov approximatedsolutions produces a vector that shows a highly oscillatory behaviour nearthe end points, confirming the presence of the Gibbs phenomenon. • In the Krylov basis (cid:107) E N (cid:107) L and (cid:107) R N (cid:107) L decrease monotonically, relativelyfast for small N ’s, then rather slowly with N . Such quantities are smallerthan in the Fourier basis. (cid:107) (cid:100) f ( N ) (cid:107) L displays some initial highly oscillatorybehaviour, but quickly approaches the theoretical value (7.1). On the otherhand, the reconstruction appears to be quite good with some noticeableoscillations at the end points. RUNCATION AND CONVERGENCE INFINITE-DIM. INVERSE PROBLEM 21 x f ( x ) ActualN = 10N = 50N = 100 (a)
Legendre basis x -0.200.20.40.60.811.2 f ( x ) ActualN = 50N = 100N = 150 (b)
Complex Fourier basis x -0.200.20.40.60.811.2 f ( x ) ActualN = 10N = 50N = 100 (c)
Krylov basis
Figure 4.
Reconstruction of the exact solution f ( x ) = x fromthe solutions for the problem V f = g . The Fourier basis producesan inaccurate reconstruction due to high oscillations, resulting inhigher errors.Thus, among the considered truncations the Legendre basis yields the most ac-curate reconstruction and the complex Fourier basis yields the least accurate re-construction of the exact solution.In contrast, Figures 3 and 5 highlight the difference between the computation inthe three bases for the M -multiplication operator. • In the Legendre basis, (cid:107) E N (cid:107) L and (cid:107) R N (cid:107) L are again almost zero. (cid:107) (cid:100) f ( N ) (cid:107) L is constant with N at the expected value (7.2). The approximated solutionsreconstruct the exact solution f at any truncation number. • In the Fourier basis the behaviour of the above indicators is again qual-itatively the same, and again with a much milder convergence rate in N to the asymptotic values as compared with the Legendre case. (cid:107) E N (cid:107) L and (cid:107) R N (cid:107) L still display an evident convergence to zero. Again the highererror compared to the Legendre case is likely due to the nature of the ap-proximation of the exact solution f by oscillatory functions and the Gibbsphenomenon. • The Krylov basis displays a fast initial decrease of both (cid:107) E N (cid:107) L and (cid:107) R N (cid:107) L to the tolerance level of 10 − that was set for the residual. (cid:107) (cid:100) f ( N ) (cid:107) L alsoincreases rapidly and remains constant at the expected value (7.2). Thereconstruction of the solution is excellent, but still not quite as good as theLegendre case. x f ( x ) ActualN = 10N = 50N = 100 (a)
Legendre basis x f ( x ) ActualN = 50N = 100N = 150 (b)
Complex Fourier basis x f ( x ) ActualN = 10N = 50N = 100 (c)
Krylov basis
Figure 5.
Reconstruction of the exact solution f ( x ) = x fromthe solutions for the problem M f = g .All this gives numerical evidence that the choice of the truncation basis does affect the sequence of solutions. The Legendre basis is best suited to these problemsas f , f , g and g are perfectly representable by the first few basis vectors. Appendix A. Some prototypical example operators
Let us describe in this Appendix a few operators in Hilbert space that wereuseful in the course of our discussion, both as a source of examples or counter-examples, and as a playground to understand certain mechanisms typical of theinfinite dimensionality.A.1.
The multiplication operator on (cid:96) ( N ) . Let us denote with ( e n ) n ∈ N the canonical orthonormal basis of (cid:96) ( N ). For agiven bounded sequence a ≡ ( a n ) n ∈ N in C , the multiplication by a is the operator M ( a ) : (cid:96) ( N ) → (cid:96) ( N ) defined by M ( a ) e n = a n e n ∀ n ∈ N and then extended bylinearity and density, in other words the operator given by the series(A.1) M ( a ) = ∞ (cid:88) n =1 a n | e n (cid:105)(cid:104) e n | (that converges strongly in the operator sense). M ( a ) is bounded with norm (cid:107) M ( a ) (cid:107) op = sup n | a n | and spectrum σ ( M ( a ) ) givenby the closure in C of the set { a , a , a . . . } . Its adjoint is the multiplication by a ∗ . Thus, M ( a ) is normal. M ( a ) is self-adjoint whenever a is real and it is compactif lim n →∞ a n = 0. RUNCATION AND CONVERGENCE INFINITE-DIM. INVERSE PROBLEM 23
A.2.
The right-shift operator on (cid:96) ( N ) . The operator R : (cid:96) ( N ) → (cid:96) ( N ) defined by Re n = e n +1 ∀ n ∈ N and thenextended by linearity and density, in other words the operator given by the series(A.2) R = ∞ (cid:88) n =1 | e n +1 (cid:105)(cid:104) e n | (that converges strongly in the operator sense), is called the right-shift operator. R is an isometry (i.e., it is norm-preserving) with closed range ran R = { e } ⊥ .In particular, it is bounded with (cid:107) R (cid:107) op = 1, yet not compact, it is injective, andinvertible on its range, with bounded inverse(A.3) R − : ran R → H , R − = ∞ (cid:88) n =1 | e n (cid:105)(cid:104) e n +1 | . The adjoint of R on H is the so-called left-shift operator, namely the everywheredefined and bounded operator L : H → H defined by the (strongly convergent, inthe operator sense) series(A.4) L = ∞ (cid:88) n =1 | e n (cid:105)(cid:104) e n +1 | , L = R ∗ . Thus, L inverts R on ran R , i.e., LR = , yet RL = − | e (cid:105)(cid:104) e | . One has ker R ∗ =span { e } . R and L have the same spectrum σ ( R ) = σ ( L ) = { z ∈ C | | z | (cid:54) } , but R has noeigenvalue, whereas the eigenvalue of L form the open unit ball { z ∈ C | | z | < } .A.3. The compact (weighted) right-shift operator on (cid:96) ( N ) . This is the operator R : (cid:96) ( N ) → (cid:96) ( N ) defined by the operator-norm convergentseries(A.5) R = ∞ (cid:88) n =1 σ n | e n +1 (cid:105)(cid:104) e n | , where σ ≡ ( σ n ) n ∈ N is a given bounded sequence with 0 < σ n +1 < σ n ∀ n ∈ N andlim n →∞ σ n = 0. Thus, R e n = σ n e n +1 . R is injective and compact, and (A.5) is its singular value decomposition, withnorm (cid:107)R(cid:107) op = σ , ran R = { e } ⊥ , and adjoint(A.6) R ∗ = L = ∞ (cid:88) n =1 σ n | e n (cid:105)(cid:104) e n +1 | . Thus, LR = M ( σ ) , the operator of multiplication by ( σ n ) n ∈ N , whereas RL = M ( σ ) − σ | e (cid:105)(cid:104) e | .A.4. The compact (weighted) right-shift operator on (cid:96) ( Z ) . This is the operator R : (cid:96) ( Z ) → (cid:96) ( Z ) defined by the operator-norm convergentseries(A.7) R = (cid:88) n ∈ Z σ | n | | e n +1 (cid:105)(cid:104) e n | , where σ ≡ ( σ n ) n ∈ N is a given bounded sequence with 0 < σ n +1 < σ n ∀ n ∈ N andlim n →∞ σ n = 0. Thus, R e n = σ | n | e n +1 . R is injective and compact, with ran R dense in H and norm (cid:107)R(cid:107) op = σ . (A.7)gives the singular value decomposition. The adjoint of R is(A.8) R ∗ = L = (cid:88) n ∈ Z σ | n | | e n (cid:105)(cid:104) e n +1 | . Thus, LR = M ( σ ) = RL .The ‘inverse of R on its range’ is the densely defined, surjective, unboundedoperator R − : ran R → H acting as(A.9) R − = (cid:88) n ∈ Z σ | n | | e n (cid:105)(cid:104) e n +1 | as a series that converges on ran R in the strong operator sense.A.5. The Volterra operator on L [0 , . This is the operator V : L [0 , → L [0 ,
1] defined by(A.10) (
V f )( x ) = (cid:90) x f ( y ) d y , x ∈ [0 , .V is compact and injective with spectrum σ ( V ) = { } (thus, the spectral point0 is not an eigenvalue) and norm (cid:107) V (cid:107) op = π . It’s adjoint V ∗ acts as(A.11) ( V ∗ f )( x ) = (cid:90) x f ( y ) d y , x ∈ [0 , , therefore V + V ∗ is the rank-one orthogonal projection(A.12) V + V ∗ = | (cid:105)(cid:104) | onto the function ( x ) = 1.The singular value decomposition of V is(A.13) V = ∞ (cid:88) n =0 σ n | ψ n (cid:105)(cid:104) ϕ n | , σ n = n +1) π ϕ n ( x ) = √ (2 n +1) π xψ n ( x ) = √ (2 n +1) π x , where both ( ϕ n ) n ∈ N and ( ψ n ) n ∈ N are orthonormal bases of L [0 , V is dense, but strictly contained in H : for example, / ∈ ran V .In fact, V is invertible on its range, but does not have (everywhere defined)bounded inverse; yet V − z does, for any z ∈ C \ { } (recall that σ ( V ) = { } ),and(A.14) ( z − V ) − ψ = z − ψ + z − (cid:90) x e x − yz ψ ( y ) d y ∀ ψ ∈ H , z ∈ C \ { } . The explicit action of the powers of V is(A.15) ( V n f )( x ) = 1( n − (cid:90) x ( x − y ) n − f ( y ) d y , n ∈ N . A.6.
The multiplication operator on an annulus in L (Ω) . This is the operator M z : L (Ω r ) → L (Ω r ), f (cid:55)→ zf , where(A.16) Ω r := { z ∈ C | r < | z | < } , r ∈ (0 , .M z is a normal bounded bijection with norm (cid:107) M z (cid:107) op = 1, spectrum σ ( M z ) = Ω r ,and adjoint given by M ∗ z f = zf . References [1]
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International School for Advanced Studies – SISSA, via Bonomea 265,34136 Trieste (Italy).
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