The Bayesian Formulation and Well-Posedness of Fractional Elliptic Inverse Problems

THE BA YESIAN F ORMULA TION AND WELL-POSEDNESS OF FRA CTIONAL ELLIPTIC INVERSE PR OBLEMS NICOL ´ AS GARC ´ IA TRILLOS AND DANIEL SANZ-ALONSO Abstract. W e study the in verse problem of reco v ering the order and the diffusion coefficient of an elliptic fractional partial differen tial equation from a finite n um ber of noisy observ ations of the solution. W e work in a Bay esian framew ork and sho w conditions under which the posterior distribution is giv en by a change of measure from the prior. Moreov er, w e show w ell-posedness of the inv erse problem, in the sense that small p erturbations of the observed solution lead to small Hellinger p erturbations of the associated posterior measures. W e th us provide a mathematical foundation to the Ba y esian learning of the order —and other inputs— of fractional mo dels. Key w ords. Ba y esian in v erse problems, extension problem, fractional partial differential equations 1. In tro duction 1.1. Aim and Relev ance The great promise of nonlo cal mo dels described by fractional partial differential equations (FPDEs) has been largely confirmed in appli- cations as v aried as groundwater flow, finance, materials science, and mathematical biology . A common scenario is that fractional order mo dels are w ell established, and y et the correct order is hard to determine as it ma y dep end on sp ecific features of the problem. This is for instance the case in the mo deling of visco elastic materials [4], where in practice the order and parameters of the mo dels are determined by lab oratory exp erimen ts that often con tain non-negligible amoun ts of uncertaint y and errors [20]. The aim of this pap er is to provide a probabilistic framework for a fractional elliptic equation. This framew ork ackno wledges any uncertaint y in the order and the diffusion co efficien t of the equation, and allows to reduce said uncertaint y by the use of partial and noisy measurements of the output solution. In this w ay w e extend the Ba y esian form ulation of elliptic in verse problems prop osed in [11] to fractional order mo dels, and in tro duce —to the b est of our knowledge— the first Bay esian approach to learning the order of a FPDE. Other than their applied imp ortance in mo delling, there is a further motiv ation for the study of fractional equations, and specifically their in v ersion. Indeed, the solution to classical in teger order PDEs in ph ysical space can some times be reco v ered from a FPDE in the low er-dimensional b oundary of the physical domain. Output measuremen ts are often tak en in the boundary , and hence relate naturally to the fractional equation. With the Bay esian approach prop osed in this pap er, uncertaint y in the solution in the full ph ysical domain —after b oundary measurements are taken— could b e characterized as follo ws: i) describ e the uncertain ty in the inputs of the FPDE; ii) reduce the uncertain ty b y use of b oundary measuremen ts; iii) propagate the remaining uncertaint y in the inputs to characterize the uncertaint y in the solution to the FPDE; and iv) use the mapping from b oundary to physical domain to characterize the uncertaint y in the full solution. An example is given b y the full 3D quasi-geostrophic equations, whose streamlines can b e recov ered from the 2D surface quasi-geostrophic equations (whic h contains a fractional diffusiv e term) by solving an elliptic PDE in 3D. Data assimilation using b oundary measurements for this system has b een studied in a sequen tial context [16]. An example that is more closely related to the framework of this pap er is given by the thin obstacle problem in the introduction of [8]. 1.2. F ramew ork and Main Results In order to describ e the inv erse problem of interest, we no w introduce a working definition of the FPDE that will constitute our 1 2 The Bay esian F ormulation and W ell-Posedness of F ractional Elliptic Inv erse Problems forw ard mo del. A more detailed mathematical account will b e given in Section 3. W e w ork in a b ounded Lipschitz domain D ⊂ R d , and let L A := − div( A ( x ) ∇ x ) . F or 0 < s < 1 w e let L s A denote a fractional p ow er of the elliptic op erator L A (see equation (3.1), Section 3). W e then consider the Neumann problem ( L s A p = f , in D , ∂ A p = 0 , on ∂ D , (1.1) where ∂ A p := A ( x ) ∇ p · ν, and ν is the exterior unit normal to ∂ D. Extensions to other b oundary conditions are p ossible. The right-hand side f is assumed to b e known, and conditions on its regularity will b e given. The inputs s and A are assumed to contain non-negligible uncertain ty . W e supp ose, how ev er, that the diffusion co efficient A = A ( x ) is known to b e symmetric and strictly elliptic. The later means that there are p ositive constan ts λ A and Λ A suc h that, for almost every x ∈ D , λ A I d ≤ A ( x ) ≤ Λ A I d , (1.2) where I d is the d × d iden tit y matrix, and M 1 ≥ M 2 if M 1 − M 2 is p ositive semi-definite. The constants λ A and Λ A can b e recov ered sharply in terms of the minimum and maxim um eigen v alues of the A ( x ) ov er D . F or such optimal λ A and Λ A w e refer to Λ A /λ A as the el lipticity of A . Under mild assumptions equation (1.1) has a unique solution p = p s,A . The forwar d map is then defined as the map F : X → Z from inputs ( s, A ) ∈ X to the solution p s,A ∈ Z. W e consider differen t choices of input parameter space X and corresp onding space of outputs Z , see Settings 4.1 and 4.6 b elow. W e in vestigate the inv erse problem of learning the inputs ( s, A ) ∈ X from a finite dimensional vector y ∈ R m of partial and noisy measurements of the output solution p ∈ Z. More precisely , w e assume the additive Gaussian observ ation mo del y = G  ( s, A )  + η , (1.3) where G is the comp osition of the forward map F : X → Z with a b ounded linear func- tional O : Z → R m represen ting an observation map , and η is a vector of measurement errors that we assume to b e centered and Gaussian with known p ositive definite cov ari- ance Γ, η ∼ N (0 , Γ) . W e follow the Bay esian approac h to inv erse problems [17], [23] and put a prior distribution µ 0 on the inputs ( s, A ) ∈ X , aiming to capture b oth the uncertaint y and the a v ailable kno wledge ab out the inputs. The prior is then conditioned on the observed data y to produce —via Bay es’ rule— a p osterior distribution µ y on the space X of input parameters. Note, ho wev er, that application of Bay es’ rule in this setting requires careful justification since our space of parameters is infinite dimensional. W e will pro vide suc h justification here in tw o different settings. That is the con ten t of our first main result: Theorem 1.1. Under the c onditions of Setting 4.1 or Setting 4.6 b elow, the forwar d map F : X → Z is c ontinuous. Ther efor e, if the prior µ 0 is any me asur e with µ 0 ( X ) = 1 , then the Bayesian inverse pr oblem of r e c overing inputs u := ( s, A ) ∈ X of the FPDE (1.1) fr om data y = G ( u ) + η , η ∼ N (0 , Γ) , is wel l formulate d: the p osterior µ y is wel l define d in X and it is absolutely c ontinuous with r esp e ct to µ 0 . Mor e over, the R adon-Niko dym derivative is given by dµ y dµ 0 ( u ) = 1 Z exp  − 1 2 | y − G ( u ) | 2 Γ ,  , Z = Z X exp  − 1 2 | y − G ( u ) | 2 Γ  dµ 0 ( u ) , (1.4) N. Garc ´ ıa T rillos, and D. Sanz-Alonso 3 wher e | · | Γ = | Γ − 1 / 2 · | , and | · | is the Euclide an norm in R m . The pro of is given in Section 4. W e remark that measurability of G w ould suffice for the ab ov e result to hold. Measurability of G is implied by the shown contin uity of F and the assumed contin uity of O . Our second main result concerns well-posedness of the Bay esian inv erse problem, in the sense that small p erturbations in the data lead to small Hellinger p erturbations of the corresp onding p osterior measures. W e recall that the Hellinger distance b etw een t wo probability measures µ, µ 0 (defined in the same measurable space) is given by D Hell ( µ, µ 0 ) 2 = 1 2 Z  r dµ dν − r dµ 0 dν  2 dν, (1.5) where ν is any reference probability measure with resp ect to which b oth µ and µ 0 are absolutely con tinuous (e.g. 1 2 µ + 1 2 µ 0 ). W e then hav e: Theorem 1.2. Supp ose Setting 4.1 or Setting 4.6 b elow ar e in plac e, and that the prior µ 0 satisfies Assumption 5.1. Then ther e is C = C ( r ) such that, for al l y 1 , y 2 ∈ R m with | y 1 | , | y 2 | ≤ r , D Hell ( µ y 1 , µ y 2 ) ≤ C | y 1 − y 2 | . (1.6) The pro of can b e found in Section 5. Stabilit y of the Hellinger distance guar- an tees stability of posterior exp ectations under perturbations in the data. Precisely , (1.6) implies that for h ∈ L 2 µ y ( X ) ∩ L 2 µ y 0 ( X ) there is C , dep ending only on r and on the exp ectation of h 2 under µ y and µ y 0 , suc h that    E µ y h − E µ y 0 h    ≤ C | y − y 0 | . 1.3. Scop e and No v elt y Existence and well-posedness results similar to Theorems 1.1 and 1.2 hav e b een established for a num b er of Bay esian inv erse problems, see e.g. [23], [11]. Ho w ev er, the pro ofs of our main results require new techniques that rely on state-of-the art (F)PDE regularit y theory . W e now briefly review some of the nov el features in the scop e and analysis of the Bay esian inv erse problem studied in this pap er: 1. Bay esian learning of the order of the mo del —and p otentially of spatially- v ariable order mo dels— is b ound to find applications in finance, material sci- ence, the geophysical sciences, and b eyond. Our results build on the recently dev elop ed theory of Ba yesian inv erse problems in function space [23]. W e pro- vide a brief review in Section 2. 2. W e show (Subsection 4.1.1) con tin uit y of the forw ard map using spectral theory of self-adjoint, compact op erators. This pro of relies on the sp ectral definition of the fractional op erator L s A , describ ed in Subsection 3.1, and allows, at most, for righ t-hand side f ∈ L 2 ( D ) . 3. W e make use of the extension problem for elliptic FPDEs, reviewed in Subsec- tion 3.2. This p ow erful idea allo ws to study elliptic FPDEs by means of an asso ciated elliptic PDE in higher dimensional space with degenerate diffusion co efficien t. The extension problem for the fractional Laplacian was introduced in [7], and then generalized in [21], [22] to more general second order elliptic op erators, such as the ones considered in this pap er. W e provide a second pro of of contin uit y of the forward map using the extension problem in Subsection 4.1.2. This approach allows for source f ∈ H − s , at least for certain priors. 4 The Bay esian F ormulation and W ell-P osedness of F ractional Elliptic Inv erse Problems 4. The pro of of Theorem 1.1, and sp ecially that of Theorem 1.2, requires careful analysis of how different constants app earing in regularit y estimates dep end on the ellipticity of the base elliptic operator L A . In particular, and as part of our analysis, we study the effect of the ellipticity on different fractional Sob olev norms (Remark 4.5) and on the Cacciop oli estimates in [8]. The later is necessary in order for the theory to cov er log-normal type priors —widely used in applications— for the diffusion co efficient. 1.4. Literature W e conclude this introduction by relating our work to the literature. An extensive review of fractional dynamics, their applications, and their connection to stochastic pro- cesses is [18]. The in terplay b etw een fractional diffusion and sto chastic pro cesses sheds ligh t in to their key applied relev ance: the F eynman-Kac form ula for general α -stable L ´ evy pro cesses [5], [2] —widely used, for instance, in finance— is a fractional Laplacian diffusion [6] (with integer order for Bro wnian motion). F ractional deriv ativ es hav e b een used to model groundw ater flo w [3], and a deep analysis of the fractional p orous medium equation is given in [12]. The regularit y theory for the Neumann problem (1.1) —as w ell as the Diric hlet problem— has been thoroughly studied in [8]. Two important to ols in the analysis of elliptic FPDEs are the extension problem, by which the analysis can b e reduced to that of an elliptic PDE with degenerate diffusion co efficient [7], [22] [21], and the use of c haracterizations of the fractional Laplacian based on the heat semigroup or Poisson kernels [8]. On the computational side, finite element metho ds for fractional problems hav e also b een studied via the extension argument [19] —see also [1]. The in- v erse problem considered here could b e amenable to classical regularization approaches [14], and there has b een recent in terest in inv erse problems for related fractional mo dels [15], [28]. The Bay esian formulation that w e adopt has, how ev er, t wo main app ealing features, see e.g. [23]. First, the prior provides a natural wa y (with a clear probabilistic in terpretation) of regularizing the otherwise underdetermined inv erse problem. Second, the solution to the Ba y esian inv erse problem, i.e. the posterior measure, contains in- formation on the remaining uncertaint y in the inputs after the observ ations hav e b een assimilated. A precise understanding of the uncertain t y in the inputs is key in order to characterize the uncertain t y in the solution to (1.1). In this regard, the numerical propagation of uncertaint y through differential mo dels is an active area of research [25], [26]. A textb o ok on the Bay esian approach to inv erse problems is [17]. The formulation w as extended to function space settings —suc h as the one considered in this pap er— in [23]. The infinite dimensional elliptic in verse problem was studied in [11], and p osterior consistency was established in [24]. The deriv ation of p osterior consistency results in the fractional setting will b e the sub ject of future work, as will b e the in vestigation of spatially-v arying order mo dels [27] and priors [9]. W e also intend to study the sp ecific computational c hallenges of the Bay esian inv erse problem arising from the fractional forw ard mo del. Outline Section 2 reviews the Bay esian approach to inv erse problems in function space. Section 3 introduces the mathematical formulation of the forward mo del (1.1). In Section 4 we sho w con tin uit y of the forw ard map, thereby proving Theorem 1.1. Section 5 contains the pro of of Theorem 1.2. A simple example is given in Section 6. W e close in Section 7. The pro ofs of some auxiliary results are brought together in an app endix. Notation W e let S + ( R d ) b e the space of d × d real p ositive definite matrices. F unction spaces of zero-mean functions will be denoted with a subscript “avg”. F or instance, L 2 avg ( D ) will denote the space of functions h ∈ L 2 ( D ) with R D h ( x ) dx = 0 . N. Garc ´ ıa T rillos, and D. Sanz-Alonso 5 2. Ba y esian Inv erse Problems Let X and Z be tw o se parable Banach spaces. X will represent the space of input parameters for (1.1), and Z will represent the space of corresp onding output solutions. Let F : X → Z b e the forwar d map from inputs to outputs, and let O : Z → R m b e the observation map from outputs to data. Supp ose that F and O are Borel measurable maps and denote G := O ◦ F . W e consider the in verse problem y = G ( u ) + η , (2.1) where the aim is to reco v er the input u from data y . W e assume that η ∼ N (0 , Γ) for giv en p ositive definite Γ . W e follo w the Bay esian approach and put a prior µ 0 on the unkno wn u . The conditional law of u given y is known as the p osterior measure, and will b e denoted µ y . The follo wing t wo propositions are an immediate consequence of the theory of inv erse problems in function space introduced in [23], and further dev elop ed in [10]. W e will use them to show our main results Theorem 1.1 and 1.2. Prop osition 2.1 (Posterior definition) . Supp ose that the map G : X → R m is me asur able and that µ 0 ( X ) = 1 . Then the p osterior distribution µ y is absolutely c ontinuous with r esp e ct to µ 0 , and the R adon-Niko dym derivative is given by dµ y dµ 0 ( u ) = 1 Z exp  − 1 2 | y − G ( u ) | 2 Γ ,  , Z = Z X exp  − 1 2 | y − G ( u ) | 2 Γ ,  dµ 0 ( u ) . (2.2) Prop osition 2.2 (Hellinger con tin uity) . Assume that µ 0 ( X ) = 1 , and that G ∈ L 2 µ 0 ( X ) . Then ther e is C = C ( r ) such that, for al l y 1 , y 2 ∈ R m with | y 1 | , | y 2 | ≤ r , D Hell ( µ y 1 , µ y 2 ) ≤ C | y 1 − y 2 | . 3. F orw ard Mo del In this section we giv e the mathematical formulation of the forw ard model (1.1). It is imp ortant to note that there is no canonical w a y to define fractional pow ers of the base elliptic op erator L A = − div( A ( x ) ∇ x ) in bounded domains. In this pap er we adopt the sp ectral definition –see equation (3.1) b elow. The asso ciated fractional elliptic problem has been recently studied both analytically [8] and computationally [19]. W e refer to [1] for further discussion on differen t definitions of fractional Laplacians. 3.1. Basic F orm ulation and Sp ectral Considerations The diffusion coeffi cien t A ( x ) ∈ R d × d of L A will b e ass umed to b e symmetric, b ounded, measurable, and to satisfy a uniform elliptic condition, see (1.2). Since w e are considering the Neumann problem, we restrict the domain of L A to H 1 avg ( D ) , and note that L 2 avg ( D ) admits an orthonormal basis of eigenfunctions of L A , ψ k ∈ H 1 avg ( D ) , k ≥ 0 , with corresp onding eigenv alues 0 < λ 1 ≤ λ 2 % ∞ . This allows us to define, for 0 < s < 1 and p ( x ) = P ∞ k =1 p k ψ k ( x ) , L s A p ( x ) = ∞ X k =1 λ s k p k ψ k ( x ) . (3.1) The domain of L s A is the space H s avg ( D ) of functions p ∈ L 2 avg ( D ) with k p k 2 H s A := ∞ X k =1 λ s k h p, ψ k i 2 L 2 < ∞ . (3.2) 6 The Bay esian F ormulation and W ell-P osedness of F ractional Elliptic Inv erse Problems This space has Hilb ert structure when endow ed with the inner pro duct h p, q i H s A := ∞ X k =1 λ s k p k q k , where q = P ∞ k =1 q k ψ k ∈ H s avg ( D ) . The space H s avg ( D ) do es not dep end on A . Indeed H s avg ( D ) can b e equiv alently defined as zero-mean functions in the closure of C ∞ ( D ) with resp ect to the norm k · k 2 H s := k · k 2 L 2 + [ · ] 2 H s , where [ p ] H s := Z D Z D  p ( x ) − p ( z )  2 | x − z | d +2 s dx dz , see [8]. Indeed, the analysis in Subsection 4.1 —see Remark 4.5— shows that there is C s indep enden t of A such that [ p ] H s ≤ C s k p k H s A . This will b e used in Subsection 4.2 in to order to formulate the Bay esian inv erse problem in the case of smo oth A. An y functional f ∈ H − s avg ( D ) acting on H s avg ( D ) can b e written as f = P ∞ k =1 f k ψ k , where P k =1 λ − s k f 2 k < ∞ . F or any suc h f there is a unique solution p = p s,A ∈ H s avg ( D ) to (1.1) giv en by p s,A = ∞ X k =1 λ − s k f k ψ k . In the extreme case s = 0, if f ∈ L 2 avg ( D ) then (1.1) has a unique solution p ∈ L 2 avg ( D ). In Section 4 we study the sp ectral prop erties of the op erator L − 1 A that maps g ∈ L 2 avg ( D ) to the solution p ∈ L 2 avg ( D ) to ( L A p = g , in D, ∂ A p = 0 , on ∂ D . (3.3) The previous paragraph implies that L − 1 A is well defined. Moreov er L − 1 A is contin uous, compact, and self-adjoint with resp ect to the usual L 2 -inner pro duct. 3.2. The Extension Problem In this subsection w e introduce the extension problem that will b e used to show contin uity of the forw ard map in Subsection 4.1.2. F or uniformly elliptic A and s ∈ (0 , 1) let a := 1 − 2 s , and let B ( x ) :=  A ( x ) 0 0 1  ∈ R ( d +1) × ( d +1) . (3.4) W e denote by P s,A : D × (0 , ∞ ) → R the solution to the extension problem      div( y a B ∇ P s,A ) = 0 , in D × (0 , ∞ ) , ∂ A P s,A = 0 , on ∂ D × [0 , ∞ ) , P s,A ( x, 0) = p s,A ( x ) , on D . (3.5) In w eak form (3.5) can b e formulated as Z Ω Z ∞ 0 h B ∇ P s,A , ∇ φ i y a dy dx = c s Z Ω L s A p s,A φ ( x, 0) dx, ∀ φ ∈ H 1 avg ( D × (0 , ∞ ) , y a dy dx ) . (3.6) N. Garc ´ ıa T rillos, and D. Sanz-Alonso 7 Here c s is a constant only dep ending on s , φ ( · , 0) is interpreted as the trace of φ on D × { 0 } , and H 1 avg ( D × (0 , ∞ ) , y a dy dx ) is the space of functions in H 1 ( D × (0 , ∞ ) , y a dy dx ) satisfying Z D φ ( x, y ) dx = 0 , a.e. y ∈ (0 , ∞ ); F or conv enience we recall that the weigh ted Sob olev space H 1 ( D × (0 , ∞ ) , y a dy dx ) is defined as the completion of smo oth functions φ under the norm k φ k 2 H 1 ( D × (0 , ∞ ) ,y a dy dx ) := Z D Z ∞ 0 | φ ( x, y ) | 2 y a dy dx + Z D Z ∞ 0 |∇ φ ( x, y ) | 2 y a dy dx =: k φ k 2 L 2 ( D × (0 , ∞ ) ,y a dy dx ) + [ φ ] 2 H 1 ( D × (0 , ∞ ) ,y a dy dx ) . 4. Ba y esian F ormulation of F ractional Elliptic Inv erse Problems In this section we show c on tin uity of the forward map under tw o sets of regularity conditions on the diffusion co efficient A and the right-hand side f of the elliptic FPDE (1.1). These conditions are found in Settings 4.1 and 4.6 b elow. Con tin uit y of the forw ard map, combined with Prop osition 2.1, establishes Theorem 1.1. In Subsection 4.1 (Setting 4.1) we imp ose no regularity on the elliptic diffusion co efficient, whereas in Subsection 4.2 (Setting 4.6) we assume that it is differentiable. In the former setting solutions to (1.1) are not necessarily contin uous, while in the later setting solutions to (1.1) are contin uous [8]. 4.1. Non-smo oth Case: Measuremen ts from Bounded Linear F unctionals This subsection is devoted to the pro of of Theorem 1.1 in the following setting. Setting 4.1. The right-hand side f of (1.1) is in L 2 avg ( D ) . We let E b e the sp ac e of matrix-value d functions A ( x ) in L ∞ ( D : S + ( R d )) for which the el liptic c ondition (1.2) is in plac e. We let X := (0 , 1) × E and Z := L 2 avg ( D ) . We let F : X → Z , ( s, A ) 7→ p s,A which maps inputs ( s, A ) ∈ X into the solution p s,A ∈ Z to the FPDE (1.1) . Final ly, we let O : Z → R m b e a b ounde d line ar functional, and set G = O ◦ F . W e provide tw o different pro ofs. The first one is based on standard results on the stabilit y of the sp ectrum of compact self-adjoin t op erators. The second one relies on PDE tec hniques proposed in [7] and [8], where the fractional equation is interpreted as a Diric hlet to Neumman map of an appropriate elliptic equation on an extended domain. 4.1.1. The Sp ectral Approac h Let us start with the pro of based on sp ectral metho ds. F or a given A ∈ E , we recall that as a map b etw een L 2 avg ( D ) into itself, L − 1 A is a self-adjoint (with resp ect to the usual L 2 -inner pro duct) and compact op erator. F urthermore, its eigenfunctions coincide with those of L A and its eigen v alues are the recipro cals of those of L A . Lemma 4.2 and Prop osition 4.3 b elow are prov ed in the App endix for completeness. They are k ey in the sp ectral pro of of Theorem 1.1. Lemma 4.2. L et A, A 0 ∈ E . Then, k L − 1 A − L − 1 A 0 k op ≤ C λ A λ A 0 k A − A 0 k ∞ , (4.1) wher e k·k op is the op er ator norm for op er ators fr om L 2 avg ( D ) into itself and C is a c onstant that dep ends only on the domain D . 8 The Bay esian F ormulation and W ell-P osedness of F ractional Elliptic Inv erse Problems Prop osition 4.3. F or every fixe d N ∈ N , A ∈ E ther e exist c onstants C and δ > 0 (de- p ending on N and A ) such tha t for every A 0 ∈ E with k A − A 0 k ∞ ≤ δ we have     1 λ i − 1 λ 0 i     ≤ C k A − A 0 k ∞ , i = 1 , . . . , N , and k ψ i − ψ 0 i k L 2 ≤ C k A − A 0 k ∞ , i = 1 , . . . , N , for some orthonormal set { ψ 1 , . . . , ψ N } of eigenfunctions of L − 1 A with eigenvalues 1 λ 1 ≥ · · · ≥ 1 λ N and some orthonormal set { ψ 0 1 , . . . , ψ 0 N } of eigenfunctions of L − 1 A 0 with eigenvalues 1 λ 0 1 ≥ · · · ≥ 1 λ 0 N . Pr o of of The or em 1.1, Setting 4.1, sp e ctr al appr o ach. Fix ( s, A ) ∈ X and ε > 0. Because f ∈ L 2 ( D ), we may pick N ∈ N in such a wa y that regardless of the orthonormal basis of eigenfunctions { ψ 1 , ψ 2 , . . . } of L − 1 A (with corresp onding eigenv alues 1 λ 1 , 1 λ 2 , . . . ) we ha v e max n 1 λ 2 1 , 1 o ∞ X i = N +1 h ψ i , f i 2 L 2 < ε 2 . Let us now take A 0 ∈ E with k A 0 − A k ∞ < δ , where δ is as in Prop osition 4.3, and consider tw o bases { ψ 1 , . . . , ψ N , . . . } , { ψ 0 1 , . . . , ψ 0 N , . . . } of L 2 avg ( D ) consisting of eigenfunc- tions of L − 1 A and L − 1 A 0 for whic h the first N corresp onding eigenfunctions are related as in Prop osition 4.3. Recall that p s,A := L − s A f may b e written as p s,A = ∞ X i =1 1 λ s i h f , ψ i i L 2 ψ i . No w, for s 0 ∈ (0 , 1) with s 0 ≥ s , it is straigh tforward to chec k that p s 0 ,A := L − s 0 A f solv es the equation (1.1) with fractional p ow er s but with righ t hand side equal to f 0 := ∞ X i =1 1 λ s 0 − s i h f , ψ i i L 2 ψ i . Notice that f 0 b elongs to L 2 avg ( D ) since s 0 ≥ s and f ∈ L 2 avg ( D ). In particular, it follows that L s A ( p s,A − p s 0 ,A ) = f 0 − f . (4.2) Hence, λ 2 s 1 k p s,A − p s 0 ,A k 2 L 2 ≤ ∞ X i =1 λ 2 s i h p s,A − p s 0 ,A , ψ i i 2 L 2 = k f 0 − f k 2 L 2 , and so k p s,A − p s 0 ,A k L 2 ≤ 1 λ s 1 k f 0 − f k L 2 ≤ max  1 λ 1 , 1  k f 0 − f k L 2 . N. Garc ´ ıa T rillos, and D. Sanz-Alonso 9 The norm k f 0 − f k L 2 can b e estimated by k f 0 − f k 2 L 2 ≤ max     1 λ s 0 − s 1 − 1  2 , 1 λ s 0 − s N − 1 ! 2    k f k 2 L 2 + max ( 1 λ 2( s 0 − s ) 1 , 1 ) ∞ X i = N +1 h f , ψ i i 2 L 2 ≤ C A,N | s 0 − s | 2 + C A ∞ X i = N +1 h f , ψ i i 2 L 2 , so that in particular, k p s,A − p s 0 ,A k L 2 ≤ C A,N | s 0 − s | + C A ∞ X i = N +1 h f , ψ i i 2 L 2 ! 1 / 2 ≤ C A,N | s 0 − s | + C A ε. (4.3) Changing the roles of s 0 and s , w e can sho w in a similar fashion that even when 0 < s 0 < s inequalit y (4.3) is still v alid. Let us now introduce the op erators S N and S 0 N S N g := N X i =1 1 λ s 0 i h g , ψ i i L 2 ψ i , S 0 N g := N X i =1 1 λ 0 s 0 i h g , ψ 0 i i L 2 ψ 0 i , whic h are truncated versions of L − s 0 A and L − s 0 A 0 resp ectiv ely . It follo ws that, k p s 0 ,A − S N p s 0 ,A k 2 L 2 ≤ 1 λ 2 s 0 1 ∞ X i = N +1 h f , ψ i i 2 L 2 < ε 2 and similarly , k p s 0 ,A 0 − S 0 N p s 0 ,A 0 k 2 L 2 ≤ 1 λ 0 2 s 0 1 ∞ X i = N +1 h f , ψ 0 i i 2 L 2 ≤ 1 λ 0 2 s 0 1 k f k 2 L 2 − N X i =1 h f , ψ 0 i i 2 L 2 ! ≤ max  1 λ 0 2 1 , 1  k f k 2 L 2 − N X i =1 h f , ψ 0 i i 2 L 2 ! ≤ C A k f k 2 L 2 − N X i =1 h f , ψ i i 2 L 2 ! + C A,N k A − A 0 k 2 ∞ ≤ C A ε 2 + C A,N k A − A 0 k 2 ∞ where the fourth inequality follows from Prop osition 4.3. The ab ov e inequalities com- bined with Prop osition 4.3 imply that k S N p s 0 ,A − S 0 N p s 0 ,A k L 2 ≤ C A,N k A − A 0 k ∞ . Com bining all the previous inequalities with (4.3), we conclude that k p s,A − p s 0 ,A 0 k L 2 ≤ C A,N ( | s − s 0 | + k A − A 0 k ∞ ) + C A ε. (4.4) 10 The Bay esian F ormulation and W ell-P osedness of F ractional Elliptic Inv erse Problems Therefore, for all ( s 0 , A 0 ) satisfying | s − s 0 | + k A − A 0 k ∞ < min n ε C A,N , δ o , kF ( s, A ) − F ( s 0 , A 0 ) k L 2 < C A ε, and con tinuit y of F is prov ed. 4.1.2. The Extension Approac h Let us no w consider the second pro of of Theorem 1.1 in Setting 4.1. This proof serv es as an alternative to studying the stability of the sp ectra of L − 1 A . Pr o of of The or em 1.1, Setting 4.1, extension appr o ach. Let P s,A and P s,A 0 b e the so- lutions to (3.5) with inputs ( s, A ) and ( s, A 0 ), resp ectively . Using the test function φ := P s,A − P s,A 0 in the asso ciated weak formulations (3.6) we deduce that Z Ω Z ∞ 0 h B ∇ P s,A , ∇ ( P s,A − P s,A 0 ) i y a dy dx = c s Z Ω L s A p s,A ( p s,A − p s,A 0 ) dx and that Z Ω Z ∞ 0 h B 0 ∇ P s,A 0 , ∇ ( P s,A − P s,A 0 ) i y a dy dx = c s Z Ω L s A 0 p s,A 0 ( p s,A − p s,A 0 ) dx. Since b oth L s A p s,A and L s A 0 p s,A 0 are equal to f , we deduce that Z Ω Z ∞ 0 h B ∇ P s,A , ∇ ( P s,A − P s,A 0 ) i y a dy dx = Z Ω Z ∞ 0 h B 0 ∇ P s,A 0 , ∇ ( P s,A − P s,A 0 ) i y a dy dx. Subtracting R Ω R ∞ 0 h B ∇ P s,A 0 , ∇ ( P s,A − P s,A 0 ) i y a dy dx from both sides of the abov e equa- tion w e obtain Z Ω Z ∞ 0 h B ∇ ( P s,A − P s,A 0 ) , ∇ ( P s,A − P s,A 0 ) i y a dy dx = Z Ω Z ∞ 0 h ( B 0 − B ) ∇ P s,A 0 , ∇ ( P s,A − P s,A 0 ) i y a dy dx F rom this it follows that min { 1 , λ A } Z Ω Z ∞ 0 |∇ ( P s,A − P s,A 0 ) | 2 y a dy dx ≤ k A − A 0 k ∞  Z Ω Z ∞ 0 |∇ P s,A 0 | 2 y a dy dx  1 / 2  Z Ω Z ∞ 0 |∇ ( P s,A − P s,A 0 ) | 2 y a dy dx  1 / 2 . (4.5) Therefore, for A 0 suc h that k A − A 0 k ∞ ≤ λ A / 2 ,  Z Ω Z ∞ 0 |∇ ( P s,A − P s,A 0 ) | 2 y a dy dx  1 / 2 ≤ max  1 , 1 λ A  k A − A 0 k ∞  Z Ω Z ∞ 0 |∇ P s,A 0 | 2 y a dy dx  1 / 2 = c s max  1 , 1 λ A  k A − A 0 k ∞  Z Ω L s A 0 p s,A 0 p s,A 0 dx  1 / 2 ≤ c s max  1 , 1 λ A  k A − A 0 k ∞ 1 λ 0 s/ 2 1 k f k L 2 ≤ c s max  1 , 1 λ A  k A − A 0 k ∞ 1 λ s/ 2 A k f k L 2 (4.6) N. Garc ´ ıa T rillos, and D. Sanz-Alonso 11 where the first equality follows using the weak formulation for the equation satisfied b y P s,A 0 taking as test function φ := P s,A 0 and the second to last inequalit y follo ws from the fact that L s A 0 p s,A 0 = f , and the last inequalit y follo ws from the v ariational represen tation of the first eigenv alue of L − 1 A 0 and equation (8.2). It has been sho wn that the trace of the w eigh ted Sobolev space H 1 ( D × (0 , ∞ ) , y a dy dx ) is the space H s , see Section 7 of [8] and references within. In particular, there exists a constant C s dep ending only on s such that [ p s,A − p s,A 0 ] H s ≤ k p s,A − p s,A 0 k H s ≤ C s k P s,A − P s,A 0 k H 1 (Ω × (0 , ∞ ) ,y a dy dx ) . On the other hand, there exists a constant C D (only dep ending on the domain D ) such that k P s,A − P s,A 0 k H 1 (Ω × (0 , ∞ ) ,y a dy dx ) ≤ C D [ P s,A − P s,A 0 ] H 1 (Ω × (0 , ∞ ) ,y a dy dx ) . (4.7) Indeed, this follo ws from the following considerations. First, b y F ubini’s theorem the function P s,A ( · , y ) − P s,A 0 ( · , y ) b elongs to H 1 ( D ) for almost ev ery y ∈ (0 , ∞ ). Second, for a.e. y ∈ (0 , ∞ ) b oth P s,A ( · , y ) and P s,A 0 ( · , y ) hav e a verage (in x ) equal to zero, and hence b y Poincar ´ e inequality in D Z D | P s,A ( x, y ) − P s,A 0 ( x, y ) | 2 dx ≤ C D Z D |∇ x ( P s,A ( x, y ) − P s,A 0 ( x, y )) | 2 dx, a.e. y ∈ (0 , ∞ ) . In tegration with resp ect to y a dy yields Z ∞ 0 Z D | P s,A − P s,A 0 | 2 y a dxdy ≤ C D Z ∞ 0 Z D |∇ x ( P s,A ( x, y ) − P s,A 0 ( x, y )) | 2 y a dxdy ≤ C D Z ∞ 0 Z D |∇ ( P s,A ( x, y ) − P s,A 0 ( x, y )) | 2 y a dxdy , th us establishing (4.7). F rom (4.6) and (4.7), we deduce that [ p s,A − p s,A 0 ] H s ≤ C D,s max  1 , 1 λ A  k A − A 0 k ∞ 1 λ s 1 k f k L 2 . Finally we use the fact that both p s,A and p s,A 0 ha ve a v erage zero and Poincar ´ e inequalit y in H s (whic h follo ws for instance from a compactness argument and Theorem 7.1 in [13]) to conclude that k p s,A − p s,A 0 k H s ≤ C s [ p s,A − p s,A 0 ] H s ≤ C D,s max  1 , 1 λ A  k A − A 0 k ∞ 1 λ s 1 k f k L 2 . (4.8) The ab ov e shows that for ev ery fixed s the map A 7→ F ( s, A ) is lo cally Lipschitz con tinuous when the range is not only endow ed with the L 2 -norm, but in fact when it is endo wed with the stronger H s -norm. The contin uit y of the map F in the first coordinate (i.e. fixing A and changing s ) can b e obtained directly from the represen tation of L s A in terms of eigenv alues and eigenfunctions. Remark 4.4. If the mar ginal of the prior µ 0 on its first c o or dinate was supp orte d in [ ¯ s, 1] for ¯ s > 0 , then we c ould take the sp ac e Z to b e e qual to H ¯ s with norm k·k H ¯ s . The ab ove pr o of shows that the forwar d map F is inde e d c ontinuous. Notic e that in that c ase, the observation map O may b e c onstructe d using H − ¯ s ; the b ottom line is that the map G = O ◦ F is c ontinuous. 12 The Bay esian F ormulation and W ell-P osedness of F ractional Elliptic Inv erse Problems Remark 4.5. As in (4.6) and (4.8) , and using the tr ac e the or em for H 1 ( D × (0 , 1) , y a dy dx ) we de duc e that [ p s,A ] H s ≤ k p s,A k H s ≤ C s k P s,A k H 1 = C s [ P s,A ] H 1 + C s k P s,A k L 2 ≤ C s k p s,A k H s A . Sinc e we also have [ p s,A ] 2 H s A = ∞ X i =1 h f , ψ i i 2 L 2 λ s i ≤ 1 λ s 1 k f k 2 L 2 , we c onclude that [ p s,A ] H s ≤ C s λ s/ 2 1 k f k L 2 ≤ C s λ s/ 2 A k f k L 2 . (4.9) We wil l use this identity in the next subse ction when we discuss the c ontinuity of the map G in the p ointwise observations c ase. 4.2. Smo oth Case: P oin t wise Observ ations The presentation of this subsection is parallel to that of the previous one. Here we pro ve Theorem 1.1 under the following setting: Setting 4.6. We let f ∈ L p avg ( D ) for some p ≥ 2 , and let s p := d 2 p . We let X := ( s p , 1] ∩ C 1 ( D : S + ( R d )) , and let Z := C 0 ( D ) . As b efor e, we let F : X → Z , ( s, A ) 7→ p s,A map inputs ( s, A ) ∈ X to the solution p s,A ∈ Z to the fr actional PDE (1.1) . Final ly, we let O : Z → R m b e a b ounde d line ar functional, and set G = O ◦ F . Remark 4.7. The assumptions on f and A in Setting 4.6 guar ante e that the solu- tion p s,A is c ontinuous [8]. Pointwise observations of the solution c onstitute a natur al example of observation map O . That is, O ( p ) :=    p ( x 1 ) . . . p ( x m )    , for some fixe d p oints x 1 , . . . , x m ∈ D . T o e ase the notation we wil l often write C 1 inste ad of C 1 ( D : S + ( R d )) . No c onfusion wil l arise fr om doing so. Note that C 1 is c ontaine d in the sp ac e E of uniformly el liptic matrix-value d functions of Setting 4.1. W e are ready to prov e Theorem 1.1 in the ab ov e setting. The pro of combines the analysis of the non-smooth case in Subsection 4.1 with the regularity theory of Caffarelli and Stinga [8] to show con tinuit y of the forward map. The formulation of the inv erse problem then follows from Prop osition 2.1. Pr o of of The or em 1.1, Setting 4.6. Let A ∈ C 1 , s ∈ ( s p , 1] and let ε > 0. Consider p s,A := L − s A f . W e ha ve already sho wn that there exists δ > 0 such that if | s − s 0 | + k A − A 0 k ∞ < δ then k p s,A − p s 0 ,A 0 k L 2 ≤ ε. N. Garc ´ ıa T rillos, and D. Sanz-Alonso 13 No w, from [8] it follows that p s,A = L − s A f is an α -H¨ older con tinuous function with α = 2 s − d p if 2 s − d p < 1 , and α -H¨ older con tinuous for any α < 1 if 2 s − d p ≥ 1. Moreov er, its H¨ older seminorm is b ounded by [ p s,A ] C 0 ,α ( D ) ≤ C ( k p s,A k L 2 + [ p s,A ] H s + k f k L p ) , (4.10) for a constan t C dep ending on s , the mo dulus of contin uit y of A and the ellipticity of A (see Theorem 1.2 in [8]). Using Remark 4.5 we deduce that [ p s,A ] C 0 ,α ( D ) ≤ C k f k L p , for a constant that dep ends on s , on the mo dulus of contin uit y of A , the ellipticity of A , and the first eigenv alue of L A . Hence, for any ( s 0 , A 0 ) ∈ X [ p s 0 ,A 0 ] C 0 ,α ( D ) ≤ C ( k p s 0 ,A 0 k L 2 + [ p s 0 ,A 0 ] H s + k f k L p ) , where the constant C may be tak en to b e uniform o v er all s 0 , A 0 with | s − s 0 | + k A − A 0 k C 1 ≤ δ for some δ > 0 small enough. On the other hand, a simple argument sho ws that for arbitrary α -H¨ older con tinuous functions φ 1 , φ 2 w e hav e k φ 1 − φ 2 k ∞ ≤ C D,α max { [ φ 1 ] C 0 ,α , [ φ 2 ] C 0 ,α } d/ (2 α + d ) · k φ 1 − φ 2 k 2 α/ (2 α + d ) L 2 , where C D,α is a constant that dep ends on α and the domain D . W e can then deduce that if | s − s 0 | + k A − A 0 k C 1 < δ then, k p s,A − p s 0 ,A 0 k ∞ ≤ C k f k d/ (2 α + d ) L p ε 2 α/ (2 α + d ) . This shows the contin uity of the forw ard map F : X → Z when X = ( s p , 1) × C 1 and Z = C 0 ( D ) is endow ed with the sup norm. 5. Hellinger Con tin uous Dep endence on Observ ations W e now study the Hellinger con tin uit y of the posterior distribution µ y as y changes. In light of Prop osition 2.2 and the discussion pro ceeding it, w e imp ose some conditions on the priors µ 0 for whic h we can guarantee the stability of p osterior distributions. Assumption 5.1. In what fol lows let s − = 0 for Setting 4.1, and s − = s p for Setting 4.6. We assume that the prior µ 0 satisfies µ 0 ( X ) = 1 and that its mar ginals µ 0 , 1 and µ 0 , 2 on the first and se c ond variables satisfy 1. F or every p olynomial q in two variables Z     q  1 λ A , k A k      dµ 0 , 2 ( A ) < ∞ . 2. The supp ort of µ 0 , 1 is c ontaine d in [ s − + ε, 1 − ε ] for some smal l enough ε > 0 . Notice that with these assumptions, µ 0 , 2 is allo w ed to giv e positive mass to sets of A with large ellipticit y and large norm, provided the mass deca ys fast enough. In particular, µ 0 , 2 can be chosen to be log-Gaussian (like in Remark 5.2 below). In con trast, notice that the assumptions on µ 0 , 1 , that we mak e for simplicity , rule out any p ossible degeneracy in the estimates obtained in Section 4 as s approaches the endp oin ts of the in terv al [ s − , 1]. 14 The Bay esian F ormulation and W ell-P osedness of F ractional Elliptic Inv erse Problems Remark 5.2. One example of inter est of a prior µ 0 for which µ 0 , 2 satisfies the first c ondition in Assumption 5.1 is the fol lowing. Supp ose that we pick A ac c or ding to A ( x ) := e − v ( x ) I d , wher e we r e c al l I d is the identity matrix and wher e v is chosen ac c or ding to a Gaussian pr o c ess in the appr opriate Banach sp ac e. It fol lows that 1 λ A ≤ e k v k and k A k ≤ e k v k . We c onclude that for any p olynomial q in two variables, Z E     q  1 λ A , k A k      dµ 0 , 2 ( A ) < ∞ , thanks to pr op erties of Gaussian pr o c esses. Pr o of of The or em 1.2. Since in both Settings 4.1 and 4.6 w e assume th at the observ ation map O is a b ounded functional, the pro of reduces to sho wing that F ∈ L 2 µ 0 ( X ) by Prop osition 2.2. The pro of in Setting 4.1 is straightforw ard, since for ev ery ( s, A ) ∈ X w e hav e kF  s, A  k L 2 ≤ k f k L 2 λ s 1 ≤ max  1 , 1 λ 1  k f k L 2 ≤ max  1 , C D λ A  k f k L 2 , where we are using λ 1 to denote the first (nonzero) eigenv alue of L A ; the last inequality follo ws from the v ariational formula for the first nonzero eigenv alue of L A (see the App endix). Assumption 5.1 on µ 0 guaran tees that F b elongs to L 2 µ 0 ( X ). Notice that in this case we are not fully using Assumption 5.1. Indeed, we are only using the fact that 1 /λ A is square integrable. Let us now consider the proof in setting 4.6. W e base our analysis on equations (4.9) and (4.10). Recall that the constan t app earing in the regularit y estimates in (4.10) dep ends on the ellipticity of A , the mo dulus of con tinuit y of A and s , but their explicit dep endence is not provided in [8]. F or our purp oses, understanding the dep endence of the constan ts on the parameters of the problem is imp ortant. F ortunately , the dep en- dence of the constants in the estimates in [8] can b e track ed down b y mo difying slightly some of the crucial estimates and inequalities in [8]. W e p oint out the steps that need to b e taken to achiev e this. First, the constant in the Cacciopp oli inequality in Lemma 3.2 in [8] can b e written as C = ˜ C  Λ λ  2 , where ˜ C is universal and in particular do es not dep end on A . This follo ws directly from the pro of provided in [8], given that the “Cauch y inequality with ε > 0” can b e applied with ε = λ 4(2Λ+1) . Second, the approximation Lemma (Corollary 3.3 in [8]) is deduced using the Cac- ciopp oli estimate mentioned abov e and a compactness argumen t. It is at this stage that one loses the explicit dep endence of constants on ellipticit y . One can actually restate N. Garc ´ ıa T rillos, and D. Sanz-Alonso 15 this appro ximation lemma by modifying the definition of “normalized solution” U giv en in the mentioned corollary . Indeed, one can replace the normalization condition to Z B 1 U ( x, 0) 2 dx + Z B ∗ 1 U 2 y a dy dx ≤  λ Λ  2 . Then, the same argument in [8], shows that the n um b er δ can b e c hosen to hav e the form δ = ˜ δ ( ε ) · λ Λ . With these mo difications, one can follow the arguments in [8] to ultimately pro ve that the constant in (4.10) dep ends p olynomially on the ellipticity and the mo dulus of contin uit y of A . This dep endence on ellipticit y and mo dulus of contin uity is all we need to guarantee that for µ 0 satisfying Assumption 5.1, the forw ard map F b elongs to L 2 µ 0 ( X ). 6. Example The purp ose of this section is to illustrate some asp ects of the theory in an analytically tractable set-up. A more thorough numerical inv estigation is b ey ond the scop e of this work. Unlik e in the rest of the pap er, we assume here A ≡ 1 to b e known, so that L s A = − ∆ s . Thus w e restrict our attention to the case where the only uncertain ty arises from the order s, and we will henceforth drop A from the notation. 6.1. F ramew ork W e consider the toy mo del ( − ∆ s p = cos( bx ) − 2 b sin( bπ ) , in [ − π , π ] , p 0 ( − π ) = p 0 ( π ) = 0 . (6.1) This framew ork allows us to easily write the eigenfunctions and eigen v alues of the Neu- mann Laplacian: φ k ( x ) = cos( k x ) , λ k = k 2 , k = 1 , 2 , . . . . (6.2) Moreo ver f ( x ) := cos( bx ) − 2 b sin( bπ ) has zero mean and admits the F ourier representa- tion f ( x ) = ∞ X k =1 ( − 1) k b 2 − k 2 cos( k x ) , − π ≤ x ≤ π . The solution p = p s to (6.1) is thus given by p s ( x ) = ∞ X k =1 ( − 1) k k 2 s ( b 2 − k 2 ) cos( k x ) , − π ≤ x ≤ π . This expression giv es a tangible representation of the forward map F ( s ) := p s . Note that, for any 0 < s ≤ 1 , p s is contin uous and thus can b e ev aluated p oint wise. W e now describ e our observ ation mo del. Let m ≥ 1 . W e define a grid of m p oints in [ − π , π ] as follows. If m = 1 w e let x 1 = π . If m ≥ 2 we let h = 2 π / ( m − 1) and set x j = − π + j h, j = 0 , . . . , m − 1 . W e then define, for con tinuous p , O ( p ) := [ p ( x 1 ) , . . . , p ( x m )] T , and G = O ◦ F . With the ab ov e definitions the inv erse problem of in terest can b e written as y = G ( s ) + η , (6.3) where G ( s ) = [ p s ( x 1 ) , . . . , p s ( x m )] T ∈ R m , and η is assumed to be Gaussian distributed, η ∼ N (0 , γ 2 I m × m ) . 16 The Bay esian F ormulation and W ell-P osedness of F ractional Elliptic Inv erse Problems 6.2. Numerical Illustrations In this subsection w e represent some posterior measures on the order s of equation (6.1). W e put a uniform prior on s , s ∼ µ 0 ≡ U [0 , 1] , and generate synthetic data from the underlying parameter s ∗ = 0 . 7 , according to the mo del (6.3) with b = 1 / 2 . W e consider three different v alues for: i) the num b er of p oin twise observ ations m ; and (ii) the level of the noise in the observ ations, γ . The results are sho wn in Figure 6.1. The p osterior is exp ected to concen trate around the “true” underlying parameter s = 0 . 7 in the regimes m → ∞ or γ → 0 , and this can be observ ed in Figure 6.1. W e remark that the plotted p osteriors µ y ( ds ) depend of course of the realization of the data y , and the p lots below represent only one suc h realization. W e conducted many exp eriments all of which demonstrate this concentration phenomena. In particular, it is p erhaps surprising that even with one p oint wise observ ation ( m = 1) accurate reconstruction of the order s is p ossible provided that the observ ation noise γ is small enough. Fig. 6.1: Posterior measures on the order s. The data is generated from s ∗ = 0 . 7 . The three rows correspond, resp ectively , to m = 1 , 100 , 10000 . The three columns corresp ond to γ = 0 . 3 , 0 . 15 , 0 . 075 . 7. Conclusions W e conclude by summarizing the main outcomes of this work, and b y describing some future research directions. • W e provide the fundamental framework for Bay esian learning of the order and diffusion co efficien t of a FPDE. • W e com bine tw o thriving researc h areas: the Ba y esian formulation of in v erse problems in function space, and the analysis of regularity theory for elliptic FPDEs. • W e generalize the Bay esian formulation of (in teger order) elliptic inv erse prob- lems [11] to the fractional case. W e also generalize the existing theory by considering a matrix-v alued —rather than scalar— diffusion co efficient. • There are many researc h questions that stem from this pap er. First, it w ould b e in teresting to deriv e posterior consistency and w eak conv ergence results suc h N. Garc ´ ıa T rillos, and D. Sanz-Alonso 17 as those a v ailable for the (integer) elliptic problem ([24] and [11]). Second, we in tend to inv estigate the computational challenges that arise from the in version of nonlo cal mo dels. Finally , there is applied interest in the learning of v ariable order models, where the order of the equation is allo wed to v ary throughout the ph ysical domain. 8. App endix In this app endix we provide a pro of of Lemma 4.2 and Prop osition 4.3. Pr o of of L emma 4.2. Let g ∈ L 2 avg ( D ) , and let p := L − 1 A g , p 0 := L − 1 A 0 g . Considering the test function p − p 0 w e deduce that Z D  A ∇ p, ∇ ( p − p 0 )  dx = Z D g ( p − p 0 ) dx = Z D  A 0 ∇ p 0 , ∇ ( p − p 0 )  dx. Subtracting R D  A 0 ∇ p, ∇ ( p − p 0 )  dx from b oth sides of the ab ov e equation we obtain − Z D  ( A − A 0 ) ∇ p, ∇ ( p − p 0 )  dx = Z D  A 0 ∇ ( p − p 0 ) , ∇ ( p − p 0 )  dx. By Cauc h y-Sc hw arz inequality the left hand side of the abov e expression is b ounded ab o v e by k A − A 0 k ∞  Z D |∇ p | 2 dx  1 / 2  Z D |∇ ( p − p 0 ) | 2 dx  1 / 2 . Hence, λ A 0 Z D |∇ ( p − p 0 ) | 2 dx ≤ Z D  A 0 ∇ ( p − p 0 ) , ∇ ( p − p 0 )  dx ≤ k A − A 0 k ∞  Z D |∇ p | 2 dx  1 / 2  Z D |∇ ( p − p 0 ) | 2 dx  1 / 2 . W e conclude that,  Z D |∇ ( p − p 0 ) | 2 dx  1 / 2 ≤ 1 λ A 0 k A − A 0 k ∞  Z D |∇ p | 2 dx  1 / 2 . The ab ov e inequality combined with P oincar ´ e inequality and a standard energy argu- men t yield k p − p 0 k L 2 ≤ C  Z D |∇ ( p − p 0 ) | 2 dx  1 / 2 ≤ C λ A 0 λ A k A − A 0 k ∞ k g k L 2 ( D ) , where C is a constan t only dep ending on D . Therefore, k L − 1 A − L − 1 A 0 k op ≤ C λ A 0 λ A k A − A 0 k ∞ . Before pro ving Prop osition 4.3 we show that for arbitrary uniformly elliptic A and A 0 w e hav e that | λ A − λ A 0 | ≤ k A − A 0 k ∞ . (8.1) 18 The Bay esian F ormulation and W ell-P osedness of F ractional Elliptic Inv erse Problems F or an arbitrary d × d matrix B we denote b y | B | its spectral norm. If B is p ositive definite, i.e. B ∈ S + ( R d ) , w e denote by λ min B its smallest eigen v alue. Then, for B , B 0 ∈ S + ( R d ) , | λ min B − λ min B 0 | ≤ | B − B 0 | . (8.2) Indeed, b y Courant Fisher theorem, if x is an arbitrary element of R d with unit Eu- clidean norm, λ min B ≤ h B x, x i =  ( B − B 0 ) x, x  + h B 0 x, x i ≤ | B − B 0 | + h B 0 x, x i . Minimizing the right-hand side ov er x ∈ R d with unit norm, λ min B ≤ | B − B 0 | + λ min B 0 . A symmetry argument then gives (8.2). It then follows from (8.2) that, for a.e. x ∈ D , λ A ≤ λ min A ( x ) ≤ λ min A 0 ( x ) + | A ( x ) − A 0 ( x ) | ≤ λ min A 0 ( x ) + k A − A 0 k ∞ , and so λ A ≤ λ A 0 + k A − A 0 k ∞ . Changing the roles of A 0 and A we obtain (8.1). Pr o of of Pr op osition 4.3. The pro of is by induction on N . Base c ase N = 1 . The first eigenv alues of L − 1 A and L − 1 A 0 can b e written resp ectively as 1 λ 1 = max k p k L 2 =1 h L − 1 A p, p i L 2 1 λ 0 1 = max k p k L 2 =1 h L − 1 A 0 p, p i L 2 . No w, for an arbitrary p ∈ L 2 avg ( D ) with k p k L 2 = 1, w e hav e 1 λ 1 ≥ h L − 1 A p, p i L 2 = h ( L − 1 A − L − 1 A 0 ) p, p i L 2 + h L − 1 A 0 p, p i L 2 ≥ −k L − 1 A − L − 1 A 0 k op + h L − 1 A 0 p, p i L 2 ≥ − C λ A λ A 0 k A − A 0 k ∞ + h L − 1 A 0 p, p i L 2 , (8.3) where the last inequality follows from Lemma 4.2. Maximizing ov er all unit norm p ∈ L 2 avg ( D ) in the last line of the ab ov e expression gives 1 λ 0 1 − 1 λ 1 ≤ C λ A λ A 0 k A − A 0 k ∞ . Rev ersing the roles of A and A 0 and com bining with the ab ov e inequality     1 λ 0 1 − 1 λ 1     ≤ C λ A λ A 0 k A − A 0 k ∞ . N. Garc ´ ıa T rillos, and D. Sanz-Alonso 19 Therefore, if k A − A 0 k ∞ ≤ λ A 2 w e deduce using (8.1) that     1 λ 0 1 − 1 λ 1     ≤ C A k A − A 0 k ∞ , (8.4) for a constant C A that only dep ends on A . Let us now fo cus on pro ving the statement ab out eigenfunctions. Let ψ 0 1 b e a unit norm eigenfunction of L − 1 A 0 with eigen v alue 1 λ 0 1 . Let us denote by 1 ˜ λ 1 , 1 ˜ λ 2 , . . . the differ ent eigen v alues of L − 1 A . Also, denote by P i the pro jection on to the eigenspace of L − 1 A asso ciated to the eigenv alue 1 ˜ λ i . Then, L − 1 A ψ 0 1 = 1 λ 1 P 1 ψ 0 1 + ∞ X i =2 1 ˜ λ i P i ψ 0 1 , so that L − 1 A ψ 0 1 − 1 λ 1 ψ 0 1 = ∞ X i =2  1 ˜ λ i − 1 λ 1  P i ψ 0 1 . Th us,    L − 1 A ψ 0 1 − 1 λ 1 ψ 0 1    2 L 2 = ∞ X i =2  1 ˜ λ i − 1 λ 1  2 k P i ψ 0 1 k 2 L 2 ≥  1 λ 1 − 1 ˜ λ 2  2 ∞ X i =2 k P i ψ 0 1 k 2 L 2 =  1 λ 1 − 1 ˜ λ 2  2 k ψ 0 1 − P 1 ψ 0 1 k 2 L 2 . (8.5) Hence, k ψ 0 1 − P 1 ψ 0 1 k L 2 ≤ λ 1 ˜ λ 2 ˜ λ 2 − λ 1    L − 1 A ψ 0 1 − 1 λ 1 ψ 0 1    L 2 ≤ C A  k L − 1 A ψ 0 1 − L − 1 A 0 ψ 0 1 k L 2 +    L − 1 A 0 ψ 0 1 − 1 λ 1 ψ 0 1    L 2  ≤ C A  k L − 1 A − L − 1 A 0 k op +    1 λ 1 − 1 λ 0 1     ≤ C A k A − A 0 k ∞ , (8.6) where the last inequality follows from Lemma 4.2 and from (8.4) (assuming that k A 0 − A k ∞ ≤ λ A 2 ). Cho osing δ A := min { 1 2 C A , λ A 2 } , where C A is the constant in the last line of (8.6), w e notice that if k A − A 0 k ∞ < δ A , then ψ 1 := 1 k P 1 ψ 0 1 k P 1 ψ 0 1 20 The Bay esian F ormulation and W ell-P osedness of F ractional Elliptic Inv erse Problems is a normalized eigenfunction of L − 1 A with eigen v alue 1 λ 1 . Moreo ver, k ψ 1 − ψ 0 1 k L 2 ≤ C A k A − A 0 k ∞ for a constant C A (not necessarily equal to that in (8.6)). Inductive step. Let us now supp ose that the result is true for N − 1, so that there are constan ts C A and δ A > 0 suc h that if k A − A 0 k ∞ ≤ δ A then    1 λ i − 1 λ 0 i    ≤ C A k A − A 0 k ∞ , i = 1 , . . . , N − 1 , and k ψ i − ψ 0 i k L 2 ≤ C A k A − A 0 k ∞ , i = 1 , . . . , N − 1 , for some orthonormal eigenfunctions ψ 1 , . . . , ψ N − 1 of L − 1 A with eigenv alues 1 λ 1 , . . . , 1 λ N − 1 , and some orthonormal eigenfunctions ψ 0 1 , . . . , ψ 0 N − 1 of L − 1 A 0 with eigen v alues 1 λ 0 1 , . . . , 1 λ 0 N − 1 . Let us start with the statement ab out the eigenv alues. Indeed, 1 λ N and 1 λ 0 N can b e written resp ectiv ely as 1 λ N = max p ∈ Σ ⊥ , k p k =1 h L − 1 A p, p i L 2 1 λ 0 N = max p ∈ Σ 0 ⊥ , k p k =1 h L − 1 A 0 p, p i L 2 , where Σ := span { ψ 1 , . . . , ψ N − 1 } and Σ 0 = span { ψ 0 1 , . . . , ψ 0 N − 1 } , are as ab ov e. F or an arbi- trary p 0 ∈ Σ 0 ⊥ with k p 0 k L 2 = 1, let us consider p := p 0 − N − 1 X i =1 h p 0 , ψ i i L 2 ψ i . F rom the induction hypothesis      N − 1 X i =1 1 λ i h p 0 , ψ i i L 2 ψ i      L 2 =      N − 1 X i =1 1 λ i h p 0 , ψ i i L 2 ψ i − N − 1 X i =1 1 λ 0 i h p 0 , ψ 0 i i L 2 ψ 0 i      L 2 ≤ C N ,A k A − A 0 k ∞ , (8.7) pro vided k A 0 − A k ∞ ≤ δ A . So if k A 0 − A k ∞ ≤ δ A , then h L − 1 A p, p i L 2 = h L − 1 A p 0 , p 0 i L 2 − N − 1 X i =1 1 λ i h p 0 , ψ i i 2 L 2 ≥ h L − 1 A p 0 , p 0 i L 2 − C N ,A k A − A 0 k ∞ , (8.8) where the last inequality follows from (8.7). On the other hand, assuming k A − A 0 k ∞ ≤ 1 2 C N,A in (8.7), we deduce that h L − 1 A p, p i L 2 h p, p i L 2 (1 + C N ,A k A − A 0 k ∞ ) ≥ h L − 1 A p, p i L 2 , N. Garc ´ ıa T rillos, and D. Sanz-Alonso 21 and so com bining with (8.8) and the fact that k L − 1 A 0 p 0 − L − 1 A p 0 k L 2 ≤ C A k A − A 0 k ∞ w e conclude that 1 λ N + C N ,A k A − A 0 k ∞ ≥ h L − 1 A 0 p 0 , p 0 i L 2 . T aking the maximum ov er unit norm p 0 1 λ N + C N ,A k A − A 0 k ∞ ≥ 1 λ 0 N . W e may now switch the roles of A and A 0 and obtain a similar inequality which then implies that     1 λ N − 1 λ 0 N     ≤ C N ,A k A − A 0 k ∞ . (8.9) Let us now establish the statement about the eigenfunctions. Let ψ 0 N b e a unit norm eigenfunction of L − 1 A 0 with eigen v alue 1 λ 0 N . Let us denote by 1 ˜ λ k N , 1 ˜ λ k N +1 , . . . the differ ent eigenv alues of L − 1 A greater than or equal to 1 λ N . Also, for i ≥ k N w e let E i b e the eigenspace of L − 1 A asso ciated to the eigen v alue 1 ˜ λ i , and denote by P i the pro jection on to E i . Finally , we let P − b e the pro jection on to E k N \ span { ψ 1 , . . . , ψ N − 1 } . Then, L − 1 A ψ 0 N := N − 1 X i =1 1 λ i h ψ 0 N , ψ i i L 2 ψ i + 1 λ N P − ψ 0 N + ∞ X i = k N +1 1 ˜ λ i P i ψ 0 N , and in particular L − 1 A ψ 0 N − 1 λ N ψ 0 N = N − 1 X i =1  1 λ i − 1 λ N  h ψ 0 N , ψ i i L 2 ψ i + ∞ X i = k N +1  1 ˜ λ i − 1 λ N  P i ψ 0 N . Using the induction hypothesis and the fact that ψ 0 N is orthogonal to all ψ 0 1 , . . . , ψ 0 N − 1 , w e deduce that    L − 1 A ψ 0 N − 1 λ N ψ 0 N    2 L 2 ≥ − C N ,A k A − A 0 k 2 ∞ +  1 ˜ λ k N +1 − 1 λ N  2 ∞ X i = k N +1 k P i ψ 0 N k 2 L 2 . (8.10) Com bining with (8.9), and arguing as in the base case N = 1, it follows that    ψ 0 N − P − ψ 0 N − N − 1 X i =1 h ψ 0 N , ψ i i L 2 ψ i    L 2 =  ∞ X i = k N +1 k P i ψ 0 N k 2 L 2  1 / 2 ≤ C A,N k A − A 0 k ∞ . Using one more time the induction hypothesis and the fact that ψ 0 N is orthogonal to all ψ 0 1 , . . . , ψ 0 N − 1 , w e deduce that k ψ 0 N − P − ψ 0 N k L 2 ≤ C A,N k A − A 0 k ∞ . 22 The Bay esian F ormulation and W ell-P osedness of F ractional Elliptic Inv erse Problems As in the base case, we see that provided that k A − A 0 k ∞ is small enough, then k ψ 0 N − ψ N k L 2 ≤ C A,N k A − A 0 k ∞ , where ψ N := 1 k P − ψ 0 N k L 2 P − ψ 0 N . REFERENCES [1] G. L. Acosta and J. P . Borthagaray . A fractional Laplace equation: regularity of solutions and Finite Element approximations. arXiv pr eprint arXiv:1507.08970 , 2015. [2] D. Applebaum. L´ evy pr o c esses and sto chastic calculus . Cambridge universit y press, 2009. [3] A. Atangana and N. Bildik. The use of fractional order deriv ative to predict the groundwater flow. Mathematical Pr oblems in Engineering , 2013, 2013. [4] R. L. Bagley and J. T orvik. F ractional calculus-a different approach to the analysis of visco elas- tically damp ed structures. AIAA journal , 21(5):741–748, 1983. [5] J. Bertoin. L´ evy pr o c esses , volume 121. Cambridge univ ersity press, 1998. [6] S. Bo chner. Diffusion equation and sto chastic pro cesses. Pr o ce e dings of the National Ac ademy of Scienc es , 35(7):368–370, 1949. [7] L. Caffarelli and L. Silvestre. An extension problem related to the fractional Laplacian. Commu- nic ations in p artial differ ential e quations , 32(8):1245–1260, 2007. [8] L. A. Caffarelli and P . R. Stinga. F ractional elliptic equations, Cacciopp oli estimates and regu- larity . In Annales de l’Institut Henri P oinc are (C) Non Linear Analysis , volume 33, pages 767–807. Elsevier, 2016. [9] D. Calvetti, E. Somersalo, and R. Spies. V ariable order smo othness priors for ill-p osed inv erse problems. Mathematics of Computation , 84(294):1753–1773, 2015. [10] M. Dashti and A. M. Stuart. The ba yesian approach to inv erse problems. Handbo ok of Uncertain ty Quantification. [11] M. Dashti and A. M. Stuart. Uncertaint y quantification and weak approximation of an elliptic inv erse problem. SIAM Journal on Numeric al Analysis , 49(6):2524–2542, 2011. [12] A. de Pablo, F. Quir´ os, A. Ro dr ´ ıguez, and J. L. V´ azquez. A fractional p orous medium equation. A dvances in Mathematics , 226(2):1378–1409, 2011. [13] E. Di Nezza, G. Palatucci, and E. V aldino ci. Hitchhik ers guide to the fractional Sob olev spaces. Bul letin des Sciences Math´ ematiques , 136(5):521–573, 2012. [14] H. W. Engl, M. Hank e, and A. Neubauer. Re gularization of inverse pr oblems , volume 375. Springer Science & Business Media, 1996. [15] B. Jin and W. Rundell. A tutorial on in verse problems for anomalous diffusion pro cesses. Inverse Pr oblems , 31(3):035003, 2015. [16] M. S. Jolly , V. R. Martinez, and E. S. Titi. A data assimilation algorithm for the sub critical surface quasi-geostrophic equation. arXiv pr eprint arXiv:1607.08574 , 2016. [17] J. Kaipio and E. Somersalo. Statistical and computational inverse pr oblems , volume 160. Springer Science & Business Media, 2006. [18] J. Klafter and I. G. Sokolo v. Anomalous diffusion spreads its wings. Physics world , 18(8):29, 2005. [19] R. H. No chetto, E. Ot´ arola, and A. J. Salgado. A PDE approach to fractional diffusion in general domains: a priori error analysis. F oundations of Computational Mathematics , 15(3):733–791, 2015. [20] M. Sasso, G. P almieri, and D. Amodio. Application of fractional deriv ative mo dels in linear viscoelastic problems. Mechanics of Time-Dep endent Materials , 15(4):367–387, 2011. [21] P . R. Stinga. F ractional powers of second order partial differential op erators: extension problem and regularity theory . 2010. [22] P . R. Stinga and J. L. T orrea. Extension problem and Harnack’s inequality for some fractional operators. Communic ations in Partial Differ ential Equations , 35(11):2092–2122, 2010. [23] A. M. Stuart. Inv erse problems: a Bay esian p erspective. A cta Numeric a , 19:451–559, 2010. [24] S. J. V ollmer. Posterior consistency for Ba y esian in v erse problems through stabilit y and regression results. Inverse Problems , 29(12):125011, 2013. [25] D. Xi u. Numerical metho ds for stochastic computations: a sp e ctr al metho d appr o ach . Princeton Universit y Press, 2010. [26] D. Xiu and G. E. Karniadakis. The Wiener–Askey p olynomial chaos for sto chastic differential equations. SIAM journal on scientific c omputing , 24(2):619–644, 2002. N. Garc ´ ıa T rillos, and D. Sanz-Alonso 23 [27] M. Zay ernouri and G. E. Karniadakis. F ractional sp ectral collo cation metho ds for linear and nonlinear v ariable order FPDEs. Journal of Computational Physics , 293:312–338, 2015. [28] Z. Zhang. An undetermined co efficient problem for a fractional diffusion equation. Inverse Pr oblems , 32(1):015011, 2015.