Uncertainty quantification and weak approximation of an elliptic inverse problem

University of Warwick institutional repository: http://go.warwick.ac.uk/wrap

This paper is made available online in accordance with

publisher policies. Please scroll down to view the document

itself. Please refer to the repository record for this item and our

policy information available from the repository home page for

further information.

To see the final version of this paper please visit the publisher’s website.

Access to the published version may require a subscription.

Author(s): M. DASHTI AND A. M. STUART

Article Title: UNCERTAINTY QUANTIFICATION AND WEAK

APPROXIMATION OF AN ELLIPTIC INVERSE PROBLEM

Year of publication: 2011

Link to published article:

http://dx.doi.org/10.1137/100814664

UNCERTAINTY QUANTIFICATION AND WEAK

APPROXIMATION OF AN ELLIPTIC INVERSE PROBLEM∗

M. DASHTI† AND A. M. STUART†

Abstract. We consider the inverse problem of determining the permeability from the pressure in a Darcy model of flow in a porous medium. Mathematically the problem is to find the diffusion coefficient for a linear uniformly elliptic partial differential equation in divergence form, in a bounded domain in dimensiond≤3, from measurements of the solution in the interior. We adopt a Bayesian approach to the problem. We place a prior random field measure on the log permeability, specified through the Karhunen–Lo`eve expansion of its draws. We consider Gaussian measures constructed this way, and study the regularity of functions drawn from them. We also study the Lipschitz properties of the observation operator mapping the log permeability to the observations. Combining these regularity and continuity estimates, we show that the posterior measure is well defined on a suitable Banach space. Furthermore the posterior measure is shown to be Lipschitz with respect to the data in the Hellinger metric, giving rise to a form of well posedness of the inverse problem. Determining the posterior measure, given the data, solves the problem of uncertainty quantification for this inverse problem. In practice the posterior measure must be approximated in a finite dimensional space. We quantify the errors incurred by employing a truncated Karhunen–Lo`eve expansion to represent this meausure. In particular we study weak convergence of a general class of locally Lipschitz functions of the log permeability, and apply this general theory to estimate errors in the posterior mean of the pressure and the pressure covariance, under refinement of the finite-dimensional Karhunen–Lo`eve truncation.

Key words. inverse problems, Bayesian approach, uncertainty quantification, elliptic inverse problem

AMS subject classifications.35J99, 62G99

DOI.10.1137/100814664

1. Introduction. There is a growing interest in uncertainty quantification for differential equations in which the input data is uncertain. In the context of elliptic partial differential equations much of this work has concentrated on the problem of groundwater flow in which uncertainty enters the diffusion coefficient in a divergence form elliptic partial differential equation. Here there has been substantial work in the numerical analysis community devoted to quantifying the error in the solution of the problem in the case where the diffusion coefficient is a random field speci-fied through a Karhunen–Lo`eve or polynomial chaos expansion which is truncated [2, 3, 4, 7, 6, 16, 20, 21, 23, 24, 25, 29]. However, in practice the unknown diffusion coefficient is often conditioned by observational data, leading to an inverse problem [22]. This gives rise to a far more complicated measure on the diffusion coefficient. The purpose of this paper is to study this inverse problem and, in particular, the effect of approximating the underlying probability measure via a finite, but large, set of real valued random variables. Much of the existing numerical analysis concerning groundwater flow with random permeability requires uniform upper and lower bounds over the probability space, and hence excludes the log normal permeability distribu-tions widely used in applicadistribu-tions. An exception is the recent paper [6] in which the effect of log normal diffusion coefficient on the pressure is studied, and weak

approx-∗_{Received by the editors November 15, 2010; accepted for publication (in revised form) August}

22, 2011; published electronically December 15, 2011. http://www.siam.org/journals/sinum/49-6/81466.html

†_{Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (M.Dashti@warwick.}

ac.uk, [email protected]). The second author’s work was supported by the EPSRC (UK) and ERC.

imation quantified. For the inverse problem we study here we also use log normal priors; these are attractive from an inverse modeling perspective precisely because no prior bounds on the permeability may be known. A key tool when working with log normal distributions, and hence Gaussian measures, is the Fernique theorem which facilitates functional integration of a wide class of functions, including the exponen-tial of quadratics, against Gaussian measures [12]. The paper [6] exemplifies the key role of the Fernique theorem and this theorem will also be used extensively in our developments of the inverse problem.

We consider the elliptic equation

−∇ ·(eu∇p) =f+∇ ·g, x∈D, (1.1a)

p=φ, x∈∂D, (1.1b)

with D an open, bounded, and connected subset ofRd_{, d≤3,p,u,f, and φscalar}

functions and g a vector function on D. Given any u∈L∞(D) we define λ(u) and Λ(u) by

λ(u) = ess inf

x∈De

u(x)_{, Λ(u) = ess sup} x∈De

u(x)_.

(1.2)

Where it causes no confusion we will simply write λ or Λ. Equation (1.1) arises as a model for flow in a porous medium with pthe pressure (or the head) and eu _the

permeability (or the transmissivity); the velocityvis given by the formulav∝ −eu_∇p.

Consider making noisy observations of a set of linear functionalsl_jof the pressure fieldp, so thatl_j :p→l_j(p)∈R. We write the observations as

(1.3) y_j =l_j(p) +η_j, j= 1, . . . , K.

We assume, for simplicity, that η = {η_j}K

j=1 is a mean zero Gaussian observational

noise with covariance Γ. In this paper we considerl_j to be either

(a) pointwise evaluation of pat a point x_j ∈D (assuming enough regularity for f,g andφso that this makes sense almost everywhere inD); or

(b) l_j :H1(D)→ R, a functional onH1(D) (again assuming enough regularity forf,g, andφso thatp∈H1(D)).

Our objective is to determineufromy={y_j}K_j=1∈RK. We adopt a probabilistic approach which we now outline. In the sequel we derive conditions under which we may viewl_j(p) as a function ofu. Then, concatenating the data, we have

y=G(u) +η,

with

(1.4) G(u) =l₁(p), . . . , l_K(p)T.

Here the observation operatorG mapsX into RK _{where X is a Banach space which}

we specify below in various scenarios and is determined by the forward model. From the properties ofη we see that the likelihood of the datay givenuis

P(y|u)∝exp

−1 2

Γ−12y− G(u)2

,

where | · | is the standard Euclidean norm. Let P(u) denote a prior distribution on the functionu. Ifuwere finite dimensional, the posterior distribution, by Bayes’ rule, would be given by

For infinite-dimensional spaces however, there is no density with respect to the Lebesgue measure. In this context Bayes rule should be interpreted as providing the Radon– Nikodym derivative between the posterior measure μy_{(du) = P(du|y) and the prior}

measureμ₀(du) =P(du):

(1.5) dμ

y

dμ₀(u)∝exp

−1 2

Γ−12_y− G_(u)2

.

The problem of making sense of Bayes rule for probability measures on function spaces with a Gaussian prior is addressed in [8, 9, 27]. In section 2 we recall these results, and then, in subsection 2.3, we state and prove a new result concerning stability properties of the posterior measure with respect to finite-dimensional approximation of the prior.

In section 3 we show that the observation operator G of the elliptic problem described above satisfies boundedness and Lipschitz continuity conditions for appro-priate choices of the Banach space X. In section 4 we combine the results of the preceding two sections to show that formula (1.5) holds for the posterior measure, and to study its approximation with respect to finite dimensional specification of the prior and posterior. Section 5 contains some concluding remarks.

2. Bayesian approach to inverse problems for functions. In this section we recall various theoretical results related to the development of Bayesian statistics on function space. We also state and prove a new result on the weak approximation of the posterior using finite-dimensional truncation of the Karhunen–Lo`eve expansion. We assume that we are given two Banach spacesX andY, a function Φ :X×Y →R and a probability measureμ₀supported onX. Consider the putative Radon–Nikodym derivative (derived by the conditioning of the joint measure on (u, y)∈X×Y ony, see [13])

dμy dμ₀(u) =

1 Z(y)exp

−Φ(u;y), (2.1a)

Z(y) =

X

exp(−Φ(u;y))μ₀(du). (2.1b)

Our aim is to find conditions on Φ andμ₀under whichμyis a well-defined probability measure on X, which is continuous in the datay, and to describe an approximation result for μy _{with respect to approximation of Φ. Remarkably these results may all}

be proved simply by establishing properties of the operator Φ and its approximation onX, and then choosing the prior Gaussian measure so thatμ₀(X) = 1. This clearly separates the analytic and probabilistic aspects of the Bayesian formulation of inverse problems for functions. The results of this section are independent of the specific inverse problem described in section 1 and have wide applicability. Note, however, that (1.5) is a particular case of the general set-up of (2.1), withY =RK_{. But the level}

of generality we adopt allows us to work with infinite dimensional data (functions) and/or non-Gaussian observational errorη. In particular if the data y =G(u) +η, whereG :X →Y is the observation operator,Y is a Hilbert space, and η is a mean zero Gaussian random field on Y with covariance operator Γ and Cameron–Martin spaceH(Y)⊂Y given byH(Y), Γ−12·,Γ−12·_Y,Γ−12 · _Y, then we define Φ as

Φ(u;y) = 1 2Γ

−1

2(y− G(u))2_Y −1 2Γ

−1 2y2_Y

= 1 2Γ

−1

On the other hand ifY =RK _{andη has Lebesgue densityρ, then we define Φ by the}

identity exp(−Φ(u;y)) =ρ(y− G(u)).Note that these two definitions agree, up to an additive constant depending only ony, whenηis Gaussian andY is finite dimensional; such a constant simply amounts to adjusting the normalizationZ(y).The subtraction of the term 1₂Γ−12y2_Y in the infinite-dimensional data setting is required to make sure that Φ(·;y) is almost surely finite with respect toη [27]. For simplicity we work in the case where X comprises periodic functions on the d-dimensional torus Td_;

generalizations are possible.

2.1. Well-defined and well-posed Bayesian inverse problems. In [8, 9, 27], it is shown that some appropriate properties of the log likelihood Φ together with an appropriate choice of a Gaussian prior measure imply the existence of a well-posed Bayesian inverse problem. Here, we recall these results. To this end we assume the following conditions on Φ.

Assumption 2.1. _{Let X andY be Banach spaces. The functionΦ :X}×_Y →R

satisfies

(i) for every >0 andr >0 there is M =M(, r)∈Rsuch that for all u∈X, and for ally∈Y such thaty_Y < r,

Φ(u, y)≥M−u2_X;

(ii) for every r > 0 there exists K =K(r)>0 such that for all u∈ X, y ∈Y withmax{u_X,y_Y}< r

Φ(u, y)≤K;

(iii) for every r >0 there exists L=L(r)>0 such that for all u₁, u₂ ∈X and y∈Y withmax{u_1X,u_2X,y_Y}< r

|Φ(u₁, y)−Φ(u₂, y)| ≤Lu₁−u_2X;

(iv) for every >0andr >0, there isC=C(, r)∈Rsuch that for ally₁, y₂∈Y withmax{y_1Y,y_2Y}< r and for everyu∈X

|Φ(u, y₁)−Φ(u, y₂)| ≤exp(u2_X+C)y₁−y_2Y.

We now recall two results from [8] concerning well definedness and well posedness of the posterior measure.

Theorem 2.2 (_{see [8]}). _{Let Φ satisfy Assumptions2.1(i)–(iii). Assume that μ0}

is a Gaussian measure with μ₀(X) = 1. Then μy _{given by (2.1) is a well-defined}

probability measure.

One can also show continuity of the posterior in the Hellinger metric dHell with

respect to the data y. For any two measures μ and μ both absolutely continuous with respect to the same reference measureν, the Hellinger metricd_Hellis defined by

d_Hell(μ, μ) =

1

2

dμ dν −

dμ dν

2

dν.

The Hellinger metric is independent of the choice of reference measureν, the measure with respect to which bothμandμ are absolutely continuous. For functionG:X → S withS a Banach space, we have

Eμ_G(u)−Eμ_G(u) S ≤C

Eμ_G(u)2 S+Eμ

G(u)2_S12_d

Hell(μ, μ).

In our elliptic inverse problem, for example, we will chooseSto be the spaceH₀1where the pressure lives, orH₀1⊗H₀1, the natural space for pressure covariance. Theorem 2.3 which follows is hence quite useful: for example, it implies Lipschitz continuity of the posterior mean or posterior covariance with respect to data. (See [8, section 2]).

Theorem 2.3 (see [8]). Let Φsatisfy Assumptions 2.1(i)–(iv). Assume that μ₀

is a Gaussian measure with μ₀(X) = 1. Then

dHell(μy, μy

)≤Cy−y_Y,

whereC=C(r) withmax{y_Y,y_Y} ≤r.

We note that under the assumptions of Theorem 2.3, one can also show that

u∗= arg min

u∈E

Φ(u;y) +1 2u

2 E

,

with (E, · _E) the Cameron–Martin space of a centered Gaussian measureμ₀, is well defined (see Theorem 5.2 of [27]). WhenX, and henceE, are finite dimensional it is readily seen from (2.1a) that the maximum a posterior estimator ofμy_equalsu∗_{. This}

can be shown to be true for the infinite-dimensional case as well [14]. The functionu∗ is the solution of the classical Tikhonov regularization of the inverse problem prob-lem of finding u, given Φ(u;y), when the regularization term is chosen to be 1₂u2_E. Thus there is a direct link between Bayesian regularization and classical regulariza-tion. However, a fully Bayesian approach, which would include probing the posterior distribution by, for example, Markov chain-Monte Carlo methods (MCMC), allows for the possibility of quantifying uncertainty. The conditions on Φ allow for the construc-tion of MCMC methods in the infinite-dimensional setting, leading to methods which are robust under mesh refinement; thus we are following the approach of “sample then discretize,” which is the Bayesian analogue of the optimization approach “optimize then discretize”; the advantages of adopting this approach are illustrated in [10] for an inverse problem arising in fluid mechanics and discussed in greater generality in [11]. In addition, a discussion of the relationship between the classical and Bayesian approach to inverse problems and the numerical algorithms used in each context, can be found in [27, section 5].

2.2. Approximation of the posterior. In this section we recall a result con-cerning approximation of μy on the Banach space X when the function Φ is ap-proximated. This will be used in the next subsection for approximation ofμy on a finite-dimensional space. Consider ΦN to be an approximation of Φ. Here we state a result which quantifies the effect of this approximation in the posterior measure in terms of the approximation error in Φ.

Defineμy,N _by

dμy,N

dμ₀ (u) = 1 ZN_(y)exp

−ΦN(u), (2.3a)

ZN(y) =

X

exp−ΦN(u)dμ₀(u). (2.3b)

We suppress the dependence of Φ and ΦN _{onyin this section as it is considered fixed.}

Theorem 2.4 (see [9]). Assume that the measuresμ andμN are both absolutely

N, and that for any >0there exists C=C()∈Rsuch that

|Φ(u)−ΦN(u)| ≤exp(u2_X+C)ψ(N),

where ψ(N)→ 0 as N → ∞. Then there exists a constant independent of N such that

d_Hell(μ, μN)≤Cψ(N).

2.3. Approximating the posterior measure in a finite-dimensional space.

In this section, we again consider approximation of the posterior measure for the in-verse problem onX ⊆L2(Td_{) whereTdenotes the unit cube [−π, π)}d _{with opposite}

faces identified. But here we additionally assume that the approximation is made in a finite-dimensional subspace and hence corresponds to something that can be im-plemented computationally. Our approximation space will be defined by truncating the Karhunen–Lo`eve basis {ϕ_l}∞_l=1 comprising the eigenfunctions of the covariance operator of the Gaussian measureμ₀.For simplicity we assume from now on thatμ₀ is centered (zero mean).

Define the subspaceWN spanned by the{ϕ_l}N_l=1and letW⊥ denote the comple-ment ofWN in L2(Td). For anyu∈X let

uN =

N

l=1

u_lϕ_l, withu_l= (u, ϕ_l), (2.4)

where (·,·) is the L2-inner product. The approximate posterior measure will induce a measure on the coefficients {u_l}N

l=1 appearing in (2.4), and hence a measure νN

onuN _∈WN_{.Our interest is in quantifying the error incurred when approximating}

expectations underμy _{by expectations underν}N_.

For simplicity we consider the case where the datayis finite dimensional and the posterior measure is defined via (2.1) with

(2.5) Φ(u) :=1

2

Γ−12y− G(u)2.

Here | · |denotes the Euclidean norm and Γ is assumed positive and symmetric. We drop explicitydependence in Φ (and approximation ΦN) throughout this section.

Note that μ₀ factors as the product of two independent measuresμN₀ ⊗μ⊥₀ on WN ⊕W⊥. Let PN be the orthogonal projection ofL2(Td) onto WN, and P⊥ = I−PN. Any u ∈ X can be written as u = uN +u⊥, where uN = PNu is given by (2.4) and u⊥ =P⊥u. We defineGN(·) =G(PN·) and consider the approximate measure (2.3) with ΦN given by

ΦN(u) =1 2Γ

−1/2_{(y− G}N_(u))2_.

(2.6)

BecauseGN(u) depends only onuN, and becauseμ₀=μN₀ ⊗μ⊥₀ the resulting measure μy,N onX can be factored asμy,N =νN ⊗μ⊥, where

dνN dμN 0

(u)∝exp

−1 2|Γ

−1/2_{(y− G}N_(u))|2

(2.7)

and μ⊥ = μ⊥₀. The measure νN _{given by (2.7) is finite dimensional and amenable}

we assume that expectations with respect to νN _{on W}N _{can be computed exactly.}

Discussion of MCMC methods which are robust to increasingN, can be found at the end of subsection 2.1.

We are interested in approximating expectations underμy_{of functionsG:X→S,} S a Banach space. For example,G(u) may denote the pressure field, or covariance of the pressure field for the elliptic inverse in section 1. Abusing notation, we will sometimes writeG(u) =G(uN, u⊥). In practice we are able to compute expectations ofG(uN,0) underνN. Thus we are interested in estimating the weak error

e=EμyG(uN, u⊥)−EνNG(uN,0)_S. (2.8)

We now state and prove a theorem concerning this error, under the following assump-tions onG andG.

Assumption 2.5. _{Assume that X, X, and X are Banach spaces, and X is}

continuously embedded into X and X is continuously embedded into X. Suppose also that the centered Gaussian probability measureμ₀ satisfiesμ₀(X) = 1. Then, for all >0, there isK=K()∈(0,∞), such that, for allu₁, u₂∈X,

|G(u₁)− G(u₂)| ≤Kexpmax{u₁2_X,u₂2_X}

u₁−u_2X,

G(u₁)−G(u₂)_S≤Kexpmax{u₁2_X,u₂2_X}

u₁−u_2X.

Furthermore, P⊥_L(X,X) =ψ(N) → 0 as N → ∞, and P⊥_L(X,X) is bounded

independently of N.

Theorem 2.6. Let Assumption2.5hold and assume thatΦandΦN are given by

(2.5)and(2.6), respectively. Then the probability measuresμy _andμy,N_{are absolutely}

continuous with respect to μ₀ and given by (2.1) and (2.3). Furthermore, the weak error (2.8)satisfiese≤Cψ(N)asN → ∞.

Proof. Since X → X → X we have, for some constants C₁, C₂ > 0, · X ≤ C₁ · _X ≤ C₂ · _X. From the assumptions on G and P⊥ we deduce that

Assumptions 2.1(i)–(iii) hold for Φ and ΦN _{given by (2.5) and (2.6), respectively.}

Thusμy _andμy,N _{are well-defined probability measures, both absolutely continuous}

with respect to μ₀, and satisfyingμy_{(X) = μ}y,N_{(X) = 1. By the triangle inequality}

we havee≤e₁+e₂ where

e₁=EμyG(uN, u⊥)−Eμy,NG(uN, u⊥)_S, (2.9)

e₂=Eμy,NG(uN, u⊥)−EνNG(uN,0)_S.

We first estimatee₁. Note that, by use of the assumptions onGandGN_{, we have}

that for any >0 there isK=K()∈(0,∞) such that

|Φ(u)−ΦN(u)| ≤ 1 2Γ

−1

22y− G(u)− GN(u)Γ−12G(u)− GN(u)

≤Kexpu2_Xψ(N).

By Theorem 2.4 we deduce that the Hellinger distance between μy _{and μ}y,N _tends

to zero like ψ(N) and hence, by (2.2), that e₁ ≤Cψ(N). This last bound follows after noting that the required integrability ofG(u)2_S andG(uN₎2

S follows from the

Lipschitz bound onG, the operator norm bound onP⊥and the Fernique theorem [12]. We now estimatee₂. BecauseEμy,N_G(uN_{,0) =E}νN_G(uN_{,0) we obtain}

From the Lipschtiz properties ofGwe deduce that

e₂≤K₁()Eμy,N

expmax{u2_X,PNu2_X}P⊥u_X

≤K₂()Eμy,N

expCu2_Xu_X

ψ(N)

≤K₃()Eμy,N

exp2Cu2_Xψ(N),

whereC is independent of. The result follows by the Fernique theorem.

The results of this section are quite general, concerning a wide class of Bayesian inverse problems for functions, using Gaussian priors. These results can also be gen-eralized to the case of Besov priors introduced in [18] and this approach is pursued in [13]. In the next section we establish conditions which enable application of Theo-rems 2.2, 2.3, 2.4, and 2.6 to the specific elliptic inverse problem described in section 1. We then apply these results in section 4.

3. Estimates on the observation operator. In order to apply Theorems 2.2, 2.3, 2.4, and 2.6 to the elliptic inverse problem described in section 1 we need to prove certain properties of the observation operatorG given by (1.4), viewed as a mapping from a Banach spaceXintoRm_{, and the functionG:X→S. Then the prior measure} μ₀must be chosen so thatμ₀(X) = 1. It is hence desirable to find spacesX which are as large as possible, so as not to unduly restrict the prior, but for which the desired properties of the observation operator hold. As discussed in section 1 we consider the observation operator obtained from either the pointwise measurements of p or bounded linear functionals ofp∈H1(D). We obtain the bounds on the observation operator for each of these cases in the following.

3.1. Measurements from bounded linear functionals of p∈ H1(D). In this case, using standard energy estimates, we have the following result.

Proposition 3.1. Consider (1.1) with D ⊂Rd, d = 2,3, a bounded domain,

the boundary of D, ∂D, C1-regular, f ∈ H−1(D), g ∈ L2(D), and assume that φ may be extended to a function φ ∈ H1(D). Then there exists a constant C = C(f_H−1,g_L2,φ_H1_(D))such that

(i) if u∈L∞(D), then

p_H1_(D) ≤ C exp(2u_L∞_(D));

(ii) if u₁, u₂∈L∞(D), andp₁, p₂ are the corresponding solutions of(1.1), then

p₁−p_2H1_(D)≤Cexp(4 max{u_1L∞,u_2L∞})u₁−u_2L∞.

Corollary 3.2. _{Let the assumptions of Proposition 3.1hold. Consider} G_{(u) =}

(l₁(p), . . . , l_K(p))with l_j, j = 1, . . . , K, bounded linear functionals on H1(D). Then for any u, u₁, u₂∈L∞(D)we have

|G(u)| ≤C exp(2u_L∞_(D))

and

|G(u₁)− G(u₂)| ≤ C exp(4 max{u_1L∞,u_2L∞})u₁−u_2L∞_(D),

Proof of Proposition 3.1. (i) Substitutingq=p−φin (1.1) and taking the inner product withq, we obtain

e−uL∞∇_q

L2 ≤ euL∞∇φL2+fH−1+gL2

and therefore

∇p_L2≤(1 + e2uL∞₎∇_φ

L2+ euL∞(fH−1+gL2),

(3.1)

which implies the result of part (i). (ii) The differencep₁−p₂satisfies

∇ ·(eu1∇_(p

1−p2)) =∇ ·((eu2−eu1)∇p2).

Taking the inner product of the equation withp₁−p₂gives

e−u1L∞∇_(p1−_p2)2

L2 ≤ eu2−eu1L∞∇p2L2∇(p1−p2)L2.

For anyx∈D we can write

eu2(x)_−eu1(x)_=u

2(x)−u1(x) 1

0

esu2+(1−s)u1_ds

≤ u₂−u_1L∞emax{u1L∞,u2L∞}_.

Hence using the estimate for∇p_2L2 from part (i), the result follows.

3.2. Pointwise measurements ofp. Here we obtain bounds on theL∞-norm of the pressurep.

Proposition 3.3. _{Consider (1.1) with D} ⊂Rd_{, d = 2,3, a bounded domain,}

the boundary of D, ∂D, C1-regular, f ∈ Lr_{(D), g ∈ L}2r_{(D) with r > d/2, and} φ∈L∞(∂D). There exists C=C(K, d, r, D,sup_∂D|φ|,f_Lr,g_L2r)such that

p_L∞_(D) ≤ C exp(u_L∞_(D)).

Proposition 3.4. Let the assumptions of Proposition 3.3 hold. Suppose also

that φmay be extended to a functionφ∈W1,2r_{(D). Then}

(i) the mappingu→pfromC( ¯D) intoL∞(D)is locally Lipschitz continuous; (ii) assume that u₁ ∈ L∞(D), u₂ ∈ Ct_{(D) for some t > 0, and p}

1, p2 are the

corresponding solutions of(1.1). Then for any >0there existsCdepending onK, d, t, ,f_Lr,g_L2r, andφ_W1,2r such that

p₁−p_2L∞_(D)≤C expc₀max{u_1L∞_(D),u_2Ct_(D)}

u₁−u_2L∞_(D),

wherec₀= 4 + (4 + 2d)/t+.

Corollary 3.5. _{Let the assumptions of Proposition 3.3hold. Consider} G_{(u) =}

(l₁(p), . . . , l_K(p)) with l_j, j = 1, . . . , K, pointwise evaluations of p atx_j ∈ D. Then for any u∈L∞(D)there existsC=C(K, d, r, D,sup_∂D|φ|,f_Lr,g_L2r)such that

|G(u)| ≤C exp(u_L∞_(D)).

If u₁ ∈ L∞(D), u₂ ∈ Ct_{(D) for some t > 0 and φ can be extended to a function} φ∈W1,2r_{(D), then, for any >0,}

|G(u₁)− G(u₂)| ≤C expc₀max{u_1L∞_(D),u_2Ct_(D)}

u₁−u_2L∞_(D)

with C = C(K, d, t, ,f_Lr,g_L2r,φ_W1,2r) and c₀ = 4 + (4 + 2d)/t+ for any

Proof of Proposition 3.3. By Theorems 8.15 and 8.17 of [17] we have, recalling the definition ofλin (1.2),

sup

D

|p| ≤sup

∂D

|φ|+C(d, r,|D|)

p_0L2+ 1

λfLr+ 1 λgL2r

,

where p₀ is the solution of (1.1) with φ= 0. Taking the inner product of (1.1) with p₀we obtain, for c_i=c_i(d, r,|D|),

∇p_0L2≤ c1

λ (fH−1+gL2) ≤ c2

λ (fLr+gL2).

Hence

sup

D

|p| ≤ 1

λC(d, r, D)

sup

∂D

|φ|+f_Lr+g_L2r

(3.2)

and the result follows sinceλ≥e−uL∞(D)_.

Proof of Proposition 3.4. The differencep₁−p₂ satisfies

∇ ·(eu1∇_(p

1−p2)) =∇ ·F, x∈D,

p₁(x) =p₂(x), x∈∂D, (3.3)

where

F =−(eu1(x)_−eu2(x)_)∇p 2.

Let

λ_m= min{λ(u₁), λ(u₂)} and Λ_m= max{Λ(u₁),Λ(u₂)}.

By (3.2) we have

sup

D

|p₁−p₂| ≤ C(d, r, D) λ_m FL2r (3.4)

forr > d/2. Now, F_L2r may be estimated as follows, using Theorem A.3,

F_L2r≤ (eu1−eu2)∇p2_L2r

≤Λ_mu₁−u_2L∞∇p_2L2r

≤Λ_mC(d, r, D, u₂)(f_Lr+g_L2r+φ_W1,2r)u₁−u_2L∞_(D),

where in the last line we have used the fact that 2rd/(2r+d) < r. This gives the Lipschitz continuity.

To show the second part, we use (3.4) and Corollary A.4 to write

p₁−p_2L∞_(D)≤

1

λ_mC(d, r, D)FL2r

≤Λm

λ_mC(d, r, D)∇p2L2ru1−u2L∞(D) ≤C(d, D, r, t)Λm(1 + Λm)

λ2_m

1 + (Λ_m/λ_m)1+tdu₂ 1+d

t

Ct_(D)

×1 + (Λ_m/λ_m)1t_u2 1 t

Ct_(D)

Since 1/λ_m,Λ_m≤exp{u_1Ct,u_2Ct} and for any >0 there existsC() such that

Λ_m(1 + Λ_m) λ2_m

1 + (Λ_m/λ_m)1+tdu₂ 1+d

t

Ct_(D)

1 + (Λ_m/λ_m)1tu₂ 1 t

Ct_(D)

≤ C() exp (c₀(t) max{u_1L∞,u_2Ct})

withc₀= 4 + (4 + 2d)/t+. The result follows.

We can now summarize our assumptions on the forcing function and boundary conditions of (1.1) for our two choices of the observation operator G as follows. We will use these assumptions in subsequent sections. In both cases covered by these assumptions it is a consequence that Φ andG satisfy Assumptions 2.1 and 2.5.

Assumption 3.6. We consider the observation operatorG(u) = (l₁(p), . . . , l_K(p))

defined as in(1.4) withD⊂Rd _{a bounded open set withC}1 _{regular boundary. Then}

either

(i) the mapping l_j is a bounded linear functional on H1(D) for j = 1, . . . , K and we assume that f ∈ H−1(D), g ∈ L2(D), and φ may be extended to φ∈H1(D); or

(ii) the mapping l_j is the pointwise evaluation of p at a point x_j ∈ D for j = 1, . . . , Kand we assume thatf ∈Lr(D),g∈L2r(D), and φmay be extended toφ∈W1,2r(D), withr > d/2.

4. Properties of the posterior measure for the elliptic inverse problem.

We use the properties of the observation operatorGfor the elliptic problem to establish well posedness of the inverse problem, and to study approximation of the posterior measure in finite-dimensional spaces defined via Fourier truncation.

4.1. Well posedness of the posterior measure. Here we show the well de-finedness of the posterior measure and its continuity with respect to the data for the elliptic problem. We have the following theorem.

Theorem 4.1. Consider the inverse problem for finding u from noisy

obser-vations of p in the form of (1.3) and with p solving (1.1) in D = (−π, π)d_{. Let}

Assumption 3.6 hold and consider μ₀ to be distributed asN(0,(−Δ)−s) with Δ the Laplacian operator acting on H2(Td_{)Sobolev functions with zero average onT}d_{, and} s > d/2. Then the measureμ(du|y) =P(du|y)is absolutely continuous with respect to μ₀ with Radon–Nikodym derivative given by (2.1),(2.5). Furthermore, the posterior measure is continuous in the Hellinger metric with respect to the data:

dHell(μy, μy

)≤C|y−y|.

Proof. LetX =Ct_{(D). We first define measuresπ}

0andπonX×RK as follows.

Define π₀(du,dy) =μ₀(du)⊗ρ(dy), whereρis a centered Gaussian with covariance matrix Γ, assumed positive. SinceG :X→RK _{is continuous and since Lemma 6.25 of}

[27] shows thatμ₀(X) = 1 for anyt∈(0, s−d/2), we deduce thatG isμ₀measurable fort∈(0, s−d/2). We defineρ(dy|u) =N(G(u),Γ) and then byμ₀-measurability ofG, π(du,dy) =ρ(dy|u)μ₀(du). From the properties of Gaussian measure onRK _{we have}

dπ

dπ₀(u, y)∝exp(−Φ(u;y))

with Φ(u;y) =1₂|Γ−1/2_{(y− G(u))|}2_{. We now show that Assumptions 2.1(i)–(iv) hold}

for this Φ. Assumption 2.1(i) is automatic because

Φ(u;y) := 1 2

By Corollaries 3.2 and 3.5,Gis bounded on bounded sets inX, and Lipschitz onX, for anyt >0, proving Assumptions 2.1(ii), (iii). To prove Assumption 2.1(iv) note that

|Φ(u;y₁)−Φ(u;y₂)| ≤ 1 2Γ

−1

2y₁+y₂− G(u)Γ−12y₁−y₂

≤c₁expc₂u_X|y₁−y₂|

and

exp(cu_X)≤c₁() exp(u2_X).

By Assumption 2.1(ii) we deduce that

X

exp(−Φ(u;y))μ₀(dy)≥exp(−K(r))μ₀(u_X< r)>0.

By conditioning, as in Theorem 6.31 of [27], we deduce that the regular conditional probabilityμy_{(du) =P}π_{(du|y) is absolutely continuous with respect toμ}

0and is given

by (1.5). Continuity in the Hellinger metric follows from Theorem 2.3.

In the case that the spatial domainD⊂Rn _{is open and bounded with Lipschitz}

boundary, the prior measureμ₀∼ N(0,A−s) withs > d−1/2 andAis the Laplacian operator acting on D(A) = {u ∈ H2(D) : ∇u·n|_∂D = 0 and _Du = 0} satisfies μ₀(Ct(D)) = 1,t < s−d+ 1/2. Hence the techniques of the preceding theorem can be used to show well posedness of the posterior in this case provideds > d−1₂.

Generalizations to the case that instead of the scalar diffusion constanteuin (1.1), we have a diffusion matrixA= (a_ij)_i,j=1,...,d with

λ(A)|ξ|2≤ d

i,j=1

a_ijξ_iξ_j≤Λ(A)|ξ|2, ξ∈Rd,

for λ(A) >0 and Λ(A)<∞, are possible. In fact, one can prove estimates similar to those of Corollaries 3.2 and 3.5 for G : (Ct_(D))d×d _{→ R}K_{, and then choose an}

appropriate prior withμ₀(Ct_(D)d×d_{) = 1 in a similar way to the scalar case.}

4.2. Weak approximation in a Fourier basis. Let x = (x₁, . . . , x_d) ∈ Td_.

Withϕ_k1,...,kd = ei(k1x1+···+kdxd)_{, the set}{_ϕ

k1,...,kd}k1,...,kd∈Zforms a basis forL2(Td).

Define

WN = span{ϕ_k1,...,kd,|k₁| ≤N, . . . ,|k_d| ≤N}

and recall the notation established in subsection 2.3.1 We let p denote the H1 -valued random variable found from the solution of the elliptic problem (1.1) with u distributed according to the posteriorμy _andpN _{the analogous random variable with} u=uN

F distributed according toνN with the Fourier truncation described above. We

have the following approximation theorem.

Theorem 4.2. _{Consider the inverse problem of findingufrom noisy observations}

of pin the form of (1.3) and with p solving (1.1) and D = (−π, π)d. Let Assump-tion3.6 hold and considerμ₀ to be distributed asN(0,(−Δ)−s)withΔthe Laplacian operator acting on H2(Td) Sobolev functions with zero average on Td, and s > d/2. Then, for anyt < s−d₂,

Eμy_p−EνN_pN

H1 ≤C N−t,

and, withp¯=Eμy_p,p¯N _=EνN_pN_{, andS=H}1

0 ⊗H01, we have

Eμy

(p−p¯)⊗(p−p¯)−EνN(pN−p¯N)⊗(pN −p¯N)_S ≤C N−t.

Note that the mean and covariance of the pressure field, when conditioned by the data, may be viewed as giving a quantification of uncertainty in the pressure when conditioned on data. The previous theorem thus estimates errors arising in this quantification of uncertainty when Karhunen–Lo`eve truncation is used to represent the posterior measure.

Proof of Theorem 4.2. We apply Theorem 2.6 withG(u) =pandS=H₀1(D) or G(u) = p⊗pand S =H₀1⊗H₀1.We choose X =X =L∞ and X =Ct _{for any} t < s−d

2.ThenX is continuously embedded intoX andXinto X and, under the

assumptions of the theorem,μ₀(X) = μ₀(X) =μ₀(X) = 1. From Proposition 3.1 it is straightforward to show the required Lipschitz condition on G (in either the pressure or pressure covariance cases) whilst the required Lipschitz condition on G follows from Assumptions 3.6. It remains to prove the operator norm bounds onP⊥. We consider dimensiond= 2 first. We write

uN_F(x, y) =

−N≤k≤N

−N≤j≤N

ˆ

u(j, k)ei(xj+ky)

=

−N≤k≤N

−N≤j≤N

1 4π2

_π

−π _π

−π

u(ζ, ξ)e−i(ζj+ξk)dζdξei(xj+ky)

= 1 π2 π −π π −π

u(ζ, ξ)D_N(x−ζ)D_N(y−ξ) dζdξ

= 1 π2 π −π π −π

u(x−ζ, y−ξ)D_N(ζ)D_N(ξ) dζdξ,

where, as shown in [28],

D_N(x) = 1 2

−N≤n≤N

eixn=1 2

sin(N+ 1/2)x sin(x/2) . (4.1)

Noting that_−ππ D_N(x) dx=π, we have

uN_F(x, y)−u(x, y) = 1 π2 _π −π _π −π

u(x−ζ, y−ξ)−u(x, y)

D_N(ζ)D_N(ξ) dζdξ

= 1 π2 _π −π _π −π

u(x−ζ, y−ξ)−u(x−ζ, y)

+u(x−ζ, y)−u(x, y)

D_N(ζ)D_N(ξ) dζdξ

= 1 π2 π −π π −π

u(x−ζ, y−ξ)−u(x−ζ, y)

D_N(ξ) dξ

D_N(ζ) dζ

+ 1

π2

π

−π

u(x−ζ, y)−u(x, y)

D_N(ζ) dζ

π

−π

D_N(ξ) dξ.

x, ζ, y∈T, and noting thatf(ξ) is periodic we write

2

_π

−π

f(ξ)D_N(ξ) dξ=

_π

−π f(ξ)

sin(ξ/2) sin ((N+ 1/2)ξ) dξ

=−

_π− π N+1/2

−π− π N+1/2

f(ξ+_N+1π_/2)

sin(ξ₂+_2Nπ₊₁)sin ((N+ 1/2)ξ) dξ

=−

_π

−π

f(ξ+_N+1π_/2)

sin(ξ₂+_2Nπ₊₁)sin ((N+ 1/2)ξ) dξ

= 1 2

_π

−π

f(ξ) sin(ξ/2) −

f(ξ+_N+1π_/2)

sin(ξ₂+ π 2N+1)

sin ((N+ 1/2)ξ) dξ.

In the third line we have used _−π−α−π g(ξ) dξ = _π−απ g(ξ) dξ for any 2π periodic function gand anyα∈R. Let h= _2N2π₊₁. We can write

4

_π

−π

f(ξ)D_N(ξ) dξ≤

_π

−π

f(ξ) sin(ξ/2) −

f(ξ+h)

sin(ξ+₂h)

dξ

=I₁+I₂+I₃

withI₁=₀π,I₂=_−π−h, and I₃ =_−h0 of the integrand in the right-hand side of the above inequality. Noting thatf(0) = 0, we have

I₁=

_π

0

f(ξ)−f(ξ+h)

sin(ξ+₂h) +f(ξ)

1 sin(ξ/2)−

1

sin(ξ+₂h)

dξ

≤c htf_Ct

_π

0

1 ξ+hdξ+

_π

0

f_Ctξt(sin(ξ/2 +h/2)−sin(ξ/2))

sin(ξ/2) sin(ξ/2 +h/2) dξ

≤c htf_Ctlog

1 h+c h

_π

0

f_Ctξt

ξ(ξ+h)dξ

≤c htf_Ctlog

1

h+c hfCt

h

0

1

ξ1−th+c hfCt

π

h

1 h1−t(ξ+h)

≤c htf_Ct log

1 h+c h

t_f Ct.

Similarly

I₂=

_−h

−π

f(ξ)−f(ξ+h)

sin(ξ₂) −f(ξ+h)

1

sin(ξ+₂h)− 1 sin(ξ/2)

dξ

≤c htf_Ct

_−h

−π

1 −ξdξ+

_−h

−π

f_Ct|ξ+h|t|sin(ξ/2)−sin((ξ+h)/2)|

|sin(ξ/2)| |sin((ξ+h)/2)| dξ

≤c htf_Ctlog

1 h+c h

_−h

−π

f_Ct|ξ+h|t

|ξ| |ξ+h| dξ

≤c htf_Ctlog

1

h+c hfCt

_−2h

−π

1

|ξ|h1−tdξ+c hfCt

_−h

−2h

1

h|ξ+h|1−tdξ

≤c htf_Ct log

1 h+c h

Finally

I₃≤c

₀

−h f(ξξ)

+f(ξ+h) ξ+h

dξ

≤cf_Ct

₀

−h

1

|ξ|1−tdξ+cfCt

₀

−h

1 |ξ+h|1−tdξ ≤c htf_Ct.

Hence we have

_π

−π

u(x−ζ, y−ξ)−u(x−ζ, y)

D_N(ξ) dξ≤C N−tu_Ct logN,

and

_π

−π

u(x−ζ, y)−u(x, y)

D_N(ζ) dζ≤C N−tu_Ct logN.

Now since for fixed >0 sufficiently small,

_π

0

|sin (N+ 1/2)x| |sin(x/2)| dx≤

_/N

0

(2N+ 1) dx+

/N

1 x/2dx+

_π

|sin (N+ 1/2)x| |sin(x/2)| dx

≤

2 + 1 N

+ 2 logN+C()

≤clogN asN → ∞,

we haveD_N(ξ)_L1_(−π,π)=O(logN) and therefore

|uN_F(x, y)−u(x, y)| ≤Cu_Ct_(D)N−t(logN)2 for any x, y∈T2.

Similarly in the three-dimensional case one can show that

uN_F −u_L∞_(D)≤Cu_Ct_(D)N−t(logN)3.

Since t can be chosen arbitrarily close to s−d₂ we obtain P⊥_L(X,X) =O(N−t)

for any t < s− d₂. This, since X = X, implies that P⊥_L(X,X) is bounded

independently ofN. The result follows by Theorem 2.6.

Appendix A. LetD ⊂Rd _{be open and bounded and letp ∈W}1,q

0 (D) satisfy

the following integral identity:

D

∇v·(a∇p+e) +f vdx= 0 (A.1)

for anyv ∈C₀1(D) and witha satisfying 0< λ ≤a≤Λ<∞. We find an estimate for the W1,q _{norm of pwith special attention on how the upper bound depends on}

the diffusion coefficient a. The results of this appendix are obtained using slight modification of the proof of Shaposhnikov [26] for our purpose here.

In the following, Lemma A.1 gives an estimate for the W1,q _{norm of pover B,}

a ball of sufficiently small radius in Rd _{and of center 0 ∈ R}d_{. Lemma A.2 gives a}

similar result for the case that the domain is B∩ {(x1, . . . , xd_)∈Rd _:xd _{≥0}. This}

lemma allows us to consider the effect of the boundary when generalizing to a bounded domainD. Lemma A.1 and A.2 are then used to prove Theorem A.3 which gives an estimate forp_W1,q_(D)in a general bounded domainD. Finally in Corollary A.4 we

considera∈Ct(D) and obtain an estimate forp_W1,q with a polynomial dependence

ona_Ct_(D).

Notation. In this appendix we use the operatorP_r, which forga scalar function is defined as

P_r(g)(y) =

B(0,r)

K(x−y)g(x) dx,

where

K(y) =

₋₁

d(d−1)α(d)|y|d−2, d >2, 1

2πln|y|, d= 2.

Also for anyr >0 we define

B(0, r) ={x∈Rd:|x|< r}.

Lemma A.1. Let a ∈ C(B(0,1)) and a(0) = a₀. Suppose that for a positive

r < 1, p ∈ W₀1,q(B(0, r)) satisfies (A.1) with D ≡ B(0, r), P_rf ∈ W1,q_{(B(0, r)),} e∈Lq_{(B(0, r)), andsuppp,suppf,suppe⊂B(0, r). Then, lettingB=B(0, r), we}

have

(A.2) p_W1,q_(B)≤

1 λ

C₀ C₁(1 +r)

e_Lq_(B)+P_r(f)_W1,q_(B)

,

withC₀ a constant depending only on dandq and

C₁= 1−(1 +r)C0

λ sup_B |a−a0|>0.

Lemma A.2. Leta∈C(B(0,1)),a(0) =a₀, andG(0, r) =B(0, r)∩{(x1, . . . , xd) :

xd _{≥ 0} for positive r < 1. Suppose that p ∈ W}1,q

0 (G(0, r)) satisfies (A.1) with

D ≡G(0, r), P_rf ∈ W1,q_{(G(0, r)), and e∈ L}q_{(G(0, r)). Assume that the functions} p, f,andevanish in a neighborhood of the spherical part of the hemisphereG(0, r)but not necessarily onG(0, r)∩{xd= 0}. Then the result of LemmaA.1withB=G(0, r) holds.

Theorem A.3. Assume that D ⊂Rd, d= 2,3, is a bounded C1 domain, and

a∈C(D). Suppose also that e∈Lq_{(D) andf ∈L}2_{(D). Then for2< q <6}

p_W1,q ≤

C(D, d, q) λ

1 + 1

δ1+d

(1 +δ)(1 + Λ)(e_Lq_(D)+f_L2_(D)),

whereδ is a positive constant that satisfies the following:

max

y∈B(x,δ)|

a(y)−a(x)| ≤ λ

4C₀(d, q) for any x∈D,

whereC₀(d, q)is the constant in Lemma A.1.

Proof. Chooser_D so that for any x∈D there exists a ball of radius r_D inside D that contains x. Letr= min{r_D, δ}. Corresponding to r, consider {x_j}J

j=1 ⊂D

so that the set of the neighborhoods of these points, {U(x_j)}J

j=1 defined as follows,

forms a cover ofD:

• forx_j ∈D,U(x_j) =B(x_j, r),

• forx_j∈∂D,U(x_j) =B(x_j, r)∩D and there existsC1mappingψ_j such that ψ_j(U(x_j)) =G(x_j, r), whereG(x_j, r) =B(x_j, r)∩ {(x1, . . . , xd_{) :x}d_≥xd

j},

• and there exists a partition of unity{ξ_j}J

j=1 subordinate to{U(xj)}.

Letw_j =ξ_jp. Hencep=J_j=1w_j. In the following we will apply the result of Lemmas A.1 and A.2 to estimate ξ_jp_W1,q, which then results in an estimate for

p_W1,q ≤_jJ₌₁w_jW1,q. Define

˜

e=ξ_je−a p∇ξ_j and f˜=ξ_jf +a∇ξ_j· ∇p.

OnU(x_j) withx_j∈D,w_j satisfies

B(xj,r)

∇v·(a∇w_j+ ˜e) +vf˜dx= 0.

By Lemma A.1, withB_j =B(x_j, r), we have

w_jW1,q_(B j)≤

C(d, q) λ (1 +r)

e˜_Lq_(B

j)+Pr( ˜f)W1,q(Bj)

.

To estimatee˜_Lq_(Bj) we write

e˜_Lq_(B

j)≤ ξjeLq(Bj)+a∇ξjpLq(Bj)

≤(1 + Λ)C(ξ_j,∇ξ_j) (e_Lq+p_Lq₎

≤c(1 + Λ)

r (eLq+pLq).

ForP_r( ˜f)_W1,q_(B

j), by Sobolev embedding theorem (assuming that 2≤q≤6) and

since by Theorem 9.9 of [17]P_rg_H2 ≤C(d)g_L2, we have

P_r( ˜f)_W1,q_(D)≤ C(D)P_r( ˜f)_H2

≤ C(D, d)f˜_L2

≤ C(D, d)(ξ_jf_L2+a∇ξ_j· ∇p_L2)

≤ C(D, d, ξ_j,∇ξ_j)(1 + Λ)(f_L2+∇p_L2)

≤ C(D, d)(1 + Λ)

r (fL2+∇pL2). Since 2< q <6 andD is bounded,