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1 SIAM J. SCI. COMPUT. Vol. 4, No., pp. A4 A7 c 8 Society for Indstrial and Applied Mathematics Downloaded 7/6/8 to Redistribtion sbject to SIAM license or copyright; see A RANDOMIZED MAXIMUM A POSTERIORI METHOD FOR POSTERIOR SAMPLING OF HIGH DIMENSIONAL NONLINEAR BAYESIAN INVERSE PROBLEMS KAINAN WANG, TAN BUI-THANH, AND OMAR GHATTAS Abstract. We present a randomized maximm a posteriori rmap method for generating approximate samples of posteriors in high dimensional Bayesian inverse problems governed by largescale forward problems. We derive the rmap approach by casting the problem of compting the MAP point as a stochastic optimization problem; interchanging optimization and expectation; and 3 approximating the expectation with a Monte Carlo method. For a specific randomized data and prior mean, rmap redces to the randomized maximm likelihood RML approach. It can also be viewed as an iterative stochastic Newton method. An analysis of the convergence of the rmap samples is carried ot for both linear and nonlinear inverse problems. Each rmap sample reqires soltion of a PDE-constrained optimization problem; to solve these problems, we employ a state-of-the-art trst region inexact Newton conjgate gradient method with sensitivity-based warm starts. An approximate Metropolization approach is presented to redce the bias in rmap samples. Varios nmerical methods will be presented to demonstrate the potential of the rmap approach in posterior sampling of nonlinear Bayesian inverse problems in high dimensions. Key words. randomized maximm a posteriori, inverse problems, ncertainty qantification, Markov chain Monte Carlo, trst region inexact Newton conjgate gradient AMS sbject classifications. 35Q6, 6F5, 35R3, 35Q93, 65C6 DOI..37/6M665. Introdction. We consider a class of inverse problems that seek to determine a distribted parameter in a partial differential eqation PDE model, from indirect observations of otpts of the model. We adopt the framework of Bayesian inference, which acconts for ncertainties in observations, the map from parameters to observables via soltion of the forward model, and prior information on the parameters. In particlar, we seek a statistical description of all possible sets of parameters that conform to the available prior knowledge and at the same time are consistent with the observations via the parameter-to-observable map. The soltion of a Bayesian inverse problem is the posterior measre, which encodes the degree of confidence on each set of parameters as the soltion to the inverse problem nder consideration. Mathematically, the posterior is a srface in high dimensional parameter space. Even when the prior and noise probability distribtions are Gassian, the posterior need not be de to the nonlinearity of the parameter-to-observable map. For large-scale inverse problems, exploring non-gassian posteriors in high dimensions to compte statistics sch as the mean, covariance, and/or higher moments is ex- Sbmitted to the jornal s Methods and Algorithms for Scientific Compting section Febrary 8, 6; accepted for pblication in revised form September 9, 7; pblished electronically Janary 6, 8. Institte for Comptational Engineering and Sciences, The University of Texas at Astin, Astin, TX 787. Crrent address: Sanchez Oil & Gas Corp., Hoston, TX 77 kennan.wong@ gmail.com. Department of Aerospace Engineering and Engineering Mechanics, and Institte for Comptational Engineering and Sciences, The University of Texas at Astin, Astin, TX 787 tanbi@ices. texas.ed. Institte for Comptational Engineering and Sciences, Jackson School of Geosciences, and Department of Mechanical Engineering, The University of Texas at Astin, Astin, TX 787 omar@ices.texas.ed. A4

2 rmap SAMPLING METHOD FOR BAYESIAN INVERSE PROBLEMS A43 Downloaded 7/6/8 to Redistribtion sbject to SIAM license or copyright; see tremely challenging. The sal method of choice for compting statistics is Markov chain Monte Carlo MCMC [, 3, 37, 43, 53, 54, 55], which jdiciosly samples the posterior distribtion, so that sample statistics can be sed to approximate the exact distribtions. The problem, however, is that standard MCMC methods often reqire millions of samples for convergence; since each sample reqires an evalation of the parameter-to-observable map, this entails millions of expensive forward PDE simlations a prohibitive proposition. On one hand, with the rapid development of parallel compting, parallel MCMC methods [6, 6, 58, 6, 6] are stdied to accelerate the comptation. While parallelization allows MCMC algorithms to prodce more samples in less time with mltiple processors, sch accelerations typically do not improve the mixing and convergence of MCMC algorithms. More sophisticated MCMC methods that exploit the gradient and higher derivatives of the log posterior and hence the parameter-to-observable map [, 3, 5,,, 5, 7, 4, 44, 5] can, on the other hand, improve the mixing, acceptance rate, and convergence of MCMC. Several of these methods exploit local crvatre in parameter space as captred by the Hessian operator of the negative logarithm of the posterior. This reqires maniplating the Hessian of the data misfit fnctional i.e., the negative log likelihood. The stochastic Newton method SN [3, 4, 5] makes these Hessian maniplations tractable by invoking a low rank approximation, motivated by the theoretically established or experimentally observed compactness of this operator for many large-scale ill-posed inverse problems. However, despite its sccessfl application to million-parameter problems governed by expensive-to-solve PDEs [9, 36], two barriers exist that prevent frther scaling of SN to challenging problems. First, even compting low rank Hessian information for every sample in parameter space can be prohibitive. Second, when the crvatre of the negative log posterior changes rapidly, SN s local Gassian approximation may not provide a good enogh model for the posterior, and hence the MCMC proposal may not be effective. This may reslt in low acceptance rates and excessive nmbers of forward PDE solves. In this paper, we consider an optimization boosted sampling framework, the randomized maximm a posteriori rmap method that is inspired by the randomized maximm likelihood RML [39, 48] and the randomize-then-optimize RTO approaches []. Throgh compting each sample by PDE-constrained optimization [3, 4, 4, 33], it can explore the parameter space more efficiently. It can also be viewed as a nonlinear SN method that exectes mltiple Newton iterations in every MCMC step to generate a better proposal and to allow an improved acceptance rate. On the other hand, solving optimization problems is expensive, and hence we discss several improvements and extensions to make the rmap method more applicable towards solving real problems. We present or discssions in the following order. Section introdces a statistical inversion setting based on the Bayesian framework in infinite dimensions. The core of the paper is section 3. In this section, we first convert the maximm a posteriori MAP problem into a stochastic programming problem, which is then solved sing sample average approximation. This rmap method rediscovers the RML method as a special case. Reslts for convergence of the rmap ensemble sing stochastic programming theory are presented, and the extension of the rmap to infinite dimensional problems is discssed at length. We also show that rmap is a generalization of SN for linear inverse problems, they become identical. It is worth noting that for nonlinear inverse problems, rmap samples are not the actal bt rather the approximate samples of the nderlying posterior distribtion. Therefore, in section 4 we also

3 A44 KAINAN WANG, TAN BUI-THANH, AND OMAR GHATTAS Downloaded 7/6/8 to Redistribtion sbject to SIAM license or copyright; see discss an approximate Metropolization techniqe to redce the bias between sample approximation and the tre posterior distribtion. We discss in section 5 a finite element discretization of the infinite dimensional Bayes inverse problem. We also describe how to solve the optimization problem efficiently at each sampling step. In particlar, we present a sensitivity approach to obtain good initial gesses for frther accelerating the optimization procedre. In section 6, varios nmerical reslts showing the efficiency of proposed strategies compared to state-of-the-art alternatives are presented for D analytical problems as well as D inverse problems governed by the Helmholtz eqation. Finally, we conclde the paper in section 7.. Infinite dimensional Bayesian inverse problem setting. We consider the following generic forward model: B, w = in Ω, which, for example, can be PDEs modeling the physical problem nder consideration. The forward problem involves solving for the forward state w given a modeling of the distribted parameter. In the inverse problem, the task is to reconstrct given some available observations of w on parts of the domain Ω. One widely accepted model for the relationship between model parameters and observations is the additive noise model, d = G + η, with d = [d,..., d K ] T denoting all observed data, G := [w x,..., w x K ] T denoting the parameter-to-observable or forward map, i.e., the map from the distribted parameter to the observables w x i at locations {x j }, j =,,..., K, and noise being represented by η, a random vector normally distribted by N, L with bonded covariance matrix L. For simplicity, we take L = σ I, where I is the identity matrix of appropriate dimension. For notational convenience, throghot the paper we se boldface italic letters for vectors and matrices and Roman letters for infinite dimensional conterparts. For example, denotes a fnction in L Ω, while represents its discrete conterpart. The inverse problem can be formlated as choosing model parameters that minimize the discrepancy between model prediction and observations: sbject to the forward problem min Φ, d := d G L 3 B, w =, where L := L denotes the weighted Eclidean norm indced by the inner prodct in R K. This optimization problem, however, is ill-posed. An intitive reason is that the dimension of vector of observations d is often mch smaller than that of the parameter typically infinite before discretization, and hence d provides limited information abot the distribted parameter. As a reslt, the nll space of the Jacobian of the parameter-to-observable map F is nonempty. In particlar, for a class of inverse problems, we have shown that the Gass Newton approximation of the Hessian which is the prodct of the Jacobian transpose and the Jacobian, and is also eqal to the fll Hessian of the misfit Φ evalated at the optimal parameter in the

4 rmap SAMPLING METHOD FOR BAYESIAN INVERSE PROBLEMS A45 Downloaded 7/6/8 to Redistribtion sbject to SIAM license or copyright; see zero residal case, i.e., when the data is noise free is a compact operator [,, ], and hence its range space is effectively finite dimensional. In this paper, we choose to tackle the ill-posedness sing a Bayesian framework [7, 6, 37, 4, 4, 5, 59]. We seek a statistical description of all possible parameter fields that conform to some prior knowledge and at the same time are consistent with the observations. The Bayesian approach accomplishes this throgh a statistical inference framework that incorporates ncertainties in the observations, the forward map G, and the prior information. To begin, we postlate the prior as a Gassian measre µ := N, C with mean fnction and covariance operator C on in L Ω, where C := α I s =: α A s, α >, with the domain of definition of A defined as { D A := H Ω : } = on Ω. n Here, H Ω is the sal Sobolev space. Assme that the mean fnction resides in the Cameron Martin space of µ; then one can show see, e.g., [59] that the prior measre µ is well-defined when s > d/ d is the spatial dimension, and in this case, any realization from the prior distribtion µ almost srely resides in the Hölder space X := C,β Ω with < β < s/. That is, µ X =, and the Bayesian posterior measre ν satisfies the Radon Nikodym derivative 4 ν d exp Φ, d µ if G is a continos map from X to R K. The MAP point see, e.g., [3, 59] for the definition of the MAP point in infinite dimensional settings is given by 5 MAP := arg min J ;, d := d G L + C, where C := C denotes the weighted L Ω norm indced by the L Ω inner prodct,. We shall also se, to denote the dality pairing on L Ω. It shold be pointed ot that the last term in 5 can be considered as a priorinspired reglarization; the MAP point is ths a soltion to the corresponding deterministic inverse problem. However, the Bayesian approach goes well beyond the deterministic soltion to provide a complete statistical description of the inverse soltion: the posterior encodes the degree of confidence probability in the estimate of all possible parameter fields. In addition to the MAP point, we also wish to interrogate the posterior distribtion for statistics sch as conditional mean and interval estimates. This reqires sampling of the distribtion, where empirical statistics from prodced samples can effectively approximate those of the posterior. Poplar sampling methods sally sffer from problems sch as the crse of dimensionality. On the other hand, sccessfl comptational methods for MAP estimation have been stdied extensively. These facts motivate s to explore sampling methods that are facilitated by MAP estimates, which we discss in detail below.

5 A46 KAINAN WANG, TAN BUI-THANH, AND OMAR GHATTAS Downloaded 7/6/8 to Redistribtion sbject to SIAM license or copyright; see 3. A randomized maximm a posteriori approach. In this section we present an approach, which we call the randomized maximm a posteriori rmap method, to compte approximate samples for the posterior distribtion. The idea is to first randomize the cost fnction to cast the MAP statement 5 into a stochastic programming problem, which is then solved sing the Monte Carlo method also known as the sample average approximation [57]. The reslting rmap method resembles the RML developed in [39, 48] as a special case. We therefore rediscover the RML method from a completely new, i.e., stochastic programing, viewpoint. It is this view that allows s to provide new theoretical reslts on the RML approach for nonlinear inverse problems that were previosly not available. Indeed, the fact that RML samples are exact samples of the posterior for linear inverse problems crrently seems to be the only available reslt on the RML method [, 39, 48]. We shall also show that the rmap method which from now on will be sed interchangeably with the RML method can be considered as a means to incorporate ncertainty into the soltion of deterministic inverse approaches. To begin, let s consider finite dimensional parameter space for simplicity of the exposition, i.e.,, R N. The posterior measre ν in this case has the density π post with respect to the Lebesge measre: the prior by π prior exp π post π like π prior, where the likelihood is given by π like exp Φ, d = exp C d G L and. The MAP problem 5 becomes 6 MAP := arg min J ;, d := d G L + C, where C R N N is the covariance matrix in this case. Throghot this paper, we denote by E the expectation. We now randomize the cost fnction and hence the MAP problem 6. Lemma 3.. Let θ R K and ε R N be two independent random vectors distribted by π θ and π ε with zero mean, i.e., E θ [θ] = and E ε [ε] =. The following reslt holds: ] J ;, d = E θ ε [J r ;, d, θ, ε] E θ [θ T [ θ E ε ε T ε ], where J r ;, d, θ, ε = d + θ G L + ε C, with E θ ε denoting the expectation with respect to the prodct measre π θ π ε indced by θ, ε. Conseqently, 7 MAP := arg min J ;, d = arg min E θ ε [J r ;, d, θ, ε]. Proof. Since θ and ε are independent, we have E θ ε [J r ;, d, θ, ε] = [ ] E θ d + θ G L + [ ] E ε ε C = J ;, d+e θ [θ T L [ d G ] E ε ε T C ] [ ] +E θ θ T [ θ +E ε ε T ε ], Finite dimensionality cold reslt from a discretization of distribted parameters see, e.g., [8] for a constrctive finite element discretization.

6 rmap SAMPLING METHOD FOR BAYESIAN INVERSE PROBLEMS A47 Downloaded 7/6/8 to Redistribtion sbject to SIAM license or copyright; see which proves the first assertion ] since E θ [θ] = and E ε [ε] =. The second assertion is obvios since E θ [θ T [ θ and E ε ε T ε ] are constants independent of. Lemma 3., particlarly identity 7, shows that the MAP point can be considered as the soltion of the following stochastic programming problem: [ ] 8 min E θ ε [J r ;, d, θ, ε] = E θ ε min J r ;, d, θ, ε, where we have interchanged the order of minimization and expectation. At this moment, 8 holds for finite dimensional cases, and whether it is also tre for infinite dimensional settings is nknown. Or next step is to approximate the expectation on the right-hand side of 8 sing the Monte Carlo approach also known as the sample average approximation [57]. In particlar, with n independent and identically distribted i.i.d. samples θ j, ε j from the prodct measre π θ π ε, we have 9 min E θ ε [J r ;, d, θ, ε] n min J r ;, d, θ j, ε j. n Let s define j := arg min J r ;, d, θ j, ε j = d + θ j G L + ε j C, and we are in a position to define the rmap method in Algorithm. As can be seen, the observation vector d and the prior mean are randomized in the first two steps, which are then followed by solving a randomized MAP problem in the third step. Finally, we take each pertrbed MAP point j as an approximate sample of the posterior π post. Algorithm The rmap algorithm. Inpt: Choose the sample size n : for j =,..., n do : Draw ε j π ε 3: Draw θ j π θ 4: Compte rmap sample j via 5: end for Throghot the paper, we choose the prodct measre to be π θ π ε = N, L N, C, and in this case the rmap approach becomes the RML method [, 38, 49]. That is, the RML method is a special case of or framework. In other words, by first casting the MAP comptation into a stochastic programming problem and then solving it sing the sample average approximation, we have arrived at a constrctive derivation of the RML method. One can show that the RML samples are exactly those of the posterior when the forward map G is linear [, 38, 49]. This seems to be the only theoretical reslt crrently available for RML. Or stochastic programming viewpoint shows that the RML method is nothing more than a sample average approximation to the stochastic optimization problem 8, whose soltion is the MAP point. However, the sample average does not converge to the MAP point, as we now show. Let s define S, d, θ, ε := arg min J r ;, d, θ, ε ; j= The conditions nder which the interchange is valid can be conslted in [56, Theorem 4.6].

7 A48 KAINAN WANG, TAN BUI-THANH, AND OMAR GHATTAS that is, S, d, θ, ε is the optimizer operator. Clearly, this operator maps a pair θ j, ε j to an RML sample Downloaded 7/6/8 to Redistribtion sbject to SIAM license or copyright; see j := arg min J r ;, d, θ j, ε j = S, d, θ j, ε j. Proposition 3.. Assme S, d, θ, ε is measrable with respect to the prodct measre π θ π ε ; then n n j= j a.s. E θ ε [S, d, θ, ε]. Proof. The reslt is a simple conseqence of the law of large nmbers. Note that setting θ = and ε = in reveals that S, d,, is the soltion of a deterministic inverse problem with prior-inspired reglarization. If we view θ and ε as the ncertainty in data d and the baseline the prior mean parameter, respectively, the rmap method can be considered as a Monte Carlo approach to propagate the ncertainty from d and to that of the inverse soltion. Corollary 3.3. When the forward map G is linear, the following holds: n n j= j a.s. MAP, and each rmap sample j is in fact the actal sample of the posterior. We now extend the rmap method to posterior distribtion in fnction spaces. In this case, C is a covariance operator from L Ω to L Ω, R K θ N, L, and L Ω ε N, C. For notational convenience, let s define ˆd := d + θ and û := + ε. The randomized MAP problem is now defined as û MAP := arg min := arg min J r ; û, ˆd ˆd G + L C +, û C. Note that the last two terms in are not the same as the last term in 5. The reason is that the Cameron Martin space of µ has zero measre [3, 5], and hence û almost srely does not belong to this space. As a reslt, the term û C is almost srely infinite, which shold be removed as done in. On the other hand, a soltion to 5 or is necessary in the Cameron Martin space since, otherwise, the term C is infinite. The existence of sch a soltion has been shown in [59], and hence is meaningfl. Frthermore, the last term, û C shold be nderstood in the limit sense see [5, 59] since û L Ω and the Cameron Martin space is dense in L Ω. Now we are in a position to analyze the rmap samples in fnction spaces. Lemma 3.4. If the forward map G is linear in, then û MAP is distribted by the posterior measre 4.

8 Downloaded 7/6/8 to Redistribtion sbject to SIAM license or copyright; see rmap SAMPLING METHOD FOR BAYESIAN INVERSE PROBLEMS A49 Proof. To begin, assme G = B. Taking the first variation of J ; r û, ˆd with respect to in the direction ũ in the Cameron Martin space gives J ; û, ˆd, ũ = L B L ˆd C û, ũ, where B : R K L Ω is the adjoint of B, and we have defined L := B L B + C. Again, the term C û, ũ shold be nderstood in the limit sense see [5, 59]. By definition, û MAP is a soltion of J ; û, ˆd, ũ = ũ. Conseqently, we have 3 û MAP = L B L ˆd + C û. Since both û and ˆd are Gassian, û MAP is also a Gassian random fnction. Assme that ˆd and û are independent; after some simple algebra and maniplation the mean of û MAP can be written as 4 E [ û MAP] = L B L d + C, which is exactly the MAP point in 5. Frthermore, the covariance operator of û MAP reads 5 E [ û MAP MAP û MAP MAP] = L. On the other hand, sing conditional Gassian measres [59], one can show that the posterior measre ν is a Gassian with mean fnction 6 ū = + CB L + BCB d B and covariance operator 7 C post = C CB L + BCB BC. The fact that 4 and 5 are identical to 6 and 7, respectively, follows directly from the matrix inversion lemma [8]. 3.. rmap as the SN method for linear inverse problems. We begin by extending the finite dimensional SN method in [4] to infinite dimensions. To that end, we define the SN proposal in fnction space as 8 v SN = [ J ;, d ] J ;, d + N, [ J ;, d ], where, from the definition of J in 5, we define 9a 9b J ;, d = G L [G d] + C, J ;, d = [ G ] L [G d] + G L G + C. Clearly, the infinite dimensional SN proposal redces to that proposed in [4] for finite dimensional problems. Here comes the relation between rmap and SN methods.

9 A5 KAINAN WANG, TAN BUI-THANH, AND OMAR GHATTAS Downloaded 7/6/8 to Redistribtion sbject to SIAM license or copyright; see Lemma 3.5. The rmap approach is identical to the SN method for linear inverse problems. Proof. Since the forward map is linear, i.e., G = B, the posterior is a Gassian measre as discssed above. A simple maniplation gives J ;, d = L B L d C and J ;, d = B L B + C. Conseqently, v SN = MAP + N, L, where MAP = L B L d + C as in the proof of Lemma 3.4. De to the linearity of G, we need only se one Newton iteration to obtain û MAP, and it is exactly given by 3. In order to show the eqivalence between rmap and SN, we need to prove that v SN and û MAP come from the same distribtion. Bt this is obvios by inspection: the mean fnction and the covariance fnction of v SN are exactly given by 4 and 5, i.e., the mean and the covariance of û MAP. 3.. rmap as an iterative SN method for nonlinear inverse problems. For a nonlinear forward map, rmap is no longer the same as the SN method. Instead, as we now show, it can be considered as an iterative SN method isn when the fll Hessian is approximated by the Gass Newton Hessian. To begin, we note that the rmap sample û MAP is a soltion of the eqation J ; û, ˆd =, which can be solved sing the Newton method. Each Newton iteration reads [ k+ = k J k ; û, ˆd ] J k ; û, ˆd, k =,.... Now, the Gass Newton part of the fll Hessian 9b is given by J g = G L G + C, which is independent of and d. The SN proposal in this case can be written as v SN = [ J g ] J ;, d + N, [ J g ], with denoting the crrent state of the SN Markov chain nder consideration. On the other hand, the rmap method with Gass Newton Hessian can be written as k+ = k [ J g k ] J k ; û, ˆd, k =,.... In particlar, as we define the initial gess to be =, we have = [ J g ] J ; û, ˆd. Now, by definition of û and ˆd, there exist ũ and d sch that û = + ũ and ˆd = d + d,

10 rmap SAMPLING METHOD FOR BAYESIAN INVERSE PROBLEMS A5 Downloaded 7/6/8 to Redistribtion sbject to SIAM license or copyright; see where ũ N, C and d N, L. Conseqently, by linearity of J ;, with respect to the last two argments see 9a we have J and becomes ; û, ˆd = J ;, d G L d C ũ, = [ J g ] J ;, d [ J g ] G L d + C ũ. } {{ } Next, the proof of Lemma 3.4 shows that is distribted by N, [ J g ]. Therefore, and v SN are identically distribted. The difference between the rmap and SN methods is now clear: the SN method ses as the MCMC proposal, while the rmap first contines to iterate ntil is approximately satisfied and then takes the last k as the proposal. In this sense, rmap can be viewed as an iterative SN method Relation between rmap and the randomize-then-optimize approach. This section draws a connection between the rmap method and the RTO approach []. We shall show that they are identical for a linear forward map linear inverse problems bt different if the forward map is nonlinear. We also propose a modification for the RTO method. The difference between RML and RTO is best demonstrated for finite dimensional parameter space. In this case, the jth rmap can be compted as 3 rmap j := arg min L d + θj G while the jth RTO sample [] can be written as 4 RT O j := arg min [ L G d θ j QT C ε j where Q is the first factor in the thin QR factorization of 5 G := G MAP := + C ε j, ], [ L G MAP, C ] T = QR evalated at the MAP point. De to the presence of C, G has fll colmn rank, and hence R is invertible. Clearly, rmap samples rmap j are not the same as RTO samples RT j O since they are extrema of different cost fnctions in general. Now, let s assme that the forward map is linear, i.e., G = B. Setting the derivative, with respect to, of the cost fnction in 3 to zero yields the following eqation for the jth rmap sample rmap j : G T [ L B d θ j C ε j ] =.

11 A5 KAINAN WANG, TAN BUI-THANH, AND OMAR GHATTAS Downloaded 7/6/8 to Redistribtion sbject to SIAM license or copyright; see Using 5 and the fact that Q is orthonormal, we arrive at [ ] G T QQ T L B d θ j C =, ε j which is exactly the eqation for the jth RTO sample RT j O if one sets the derivative, with respect to, of the cost fnction in 4 to zero. In other words, we have shown that RTO is identical to rmap for linear inverse problems. Up to this point we have observed that the RTO method reqires a QR factorization of G which cold be comptationally intractable for large-scale inverse problems in high dimensional parameter spaces. We propose to se G in place of Q. For a general forward map, the modified RTO problem reads compared to 4 6 RT O j := arg min [ L B d θ j GT C ε j and hence RTO samples now satisfy the eqation 7 G T G G T [ L B d θ j C ε j ] =. ], The modified approach has a cople of advantages: QR-factorization of possibly large-scale G is no longer needed, and there is no need to constrct G since all we need is its action, which can be compted efficiently sing an adjoint techniqe. The determinant of G RT O T G is necessary if the RTO density is needed, bt this is readily available from the MAP calclation and the comptation of RT O. 4. Metropolis-adjsted rmap method. Recall from Lemma 3.4 that for linear inverse problems, the rmap sample is exactly distribted by the posterior measre ν. When the forward map is nonlinear, Proposition 3. shows that this is no longer tre. In this case, rmap samples have bias which shold be removed via, for example, the standard Metropolization [53]. The work in [48] shows that for some nonlinear test problems, the acceptance rate is above 9%, and the athors proposed to accept all rmap samples. This simple strategy has been shown to work well in many cases see, e.g., [34, 49], thogh the reslting Markov chain can over/nderestimate the actal posterior. We shall show that this is the case for or inverse problem, and a debiasing procedre is necessary. An exact Metropolization has been proposed in [48], bt it is intractable except for problems with very small parameter dimension. We therefore propose an approximate Metropolized step, and this is done sing a finite dimensional framework. To that end, we replace û by finite dimensional vector, e.g., a vector of finite element nodal vales. Following [46], we begin by defining 8 δ = G û MAP ˆd. Note that û MAP also satisfies, which for the finite dimensional setting becomes 9 G û MAP [ L G û MAP ˆd ] + C û MAP û =. We can view 8 and 9 together as the definition of a map T : û, ˆd û MAP, δ, and we assme that this map is locally invertible. This allows s to

12 rmap SAMPLING METHOD FOR BAYESIAN INVERSE PROBLEMS A53 Downloaded 7/6/8 to Redistribtion sbject to SIAM license or copyright; see explicitly write T as [û ] ûmap + C G û MAP L δ 3 = T ˆd û MAP, δ = G û MAP. δ That is, we know T implicitly throgh its inverse. Given the fact that the distribtion of û, ˆd is available, the distribtion of û MAP, δ can be compted via the Jacobian of the transformation: J := û, ˆd û MAP, δ = I + C G û MAP L δ C G û MAP L G û MAP. I Simple algebra yields the determinant as 3 J = det I + C G û MAP L G û MAP + C G û MAP L δ, which is nothing bt the determinant of J scaled by a determinant of the prior covariance. Let s denote by h û MAP, δ the density of proposing the pair û MAP, δ via the map T described above. Clearly, it is the psh-forward of the probability of the pair û, ˆd. By the measre preservation property and the change of variables formla, we have h û MAP, δ = f T û MAP, δ J, where f is defined as 3 f û, ˆd [ exp û T C û ˆd ˆd T ] d L d. While the marginal distribtion of û MAP is desirable, the marginalization process on h û MAP, δ is not trivial as shall be shown. This sggests that we can condct the sampling in the agmented space defined by û MAP, δ. What remains is to constrct a joint posterior density of û MAP, δ sch that marginalizing ot δ yields exactly the posterior π post. Similar to [47, 6], we may choose the joint posterior as [ π û MAP, δ exp û T MAP C û MAP η Gû T MAP δ d L Gû MAP δ d η ] δt L δ. It can be shown that if η = η /η, the marginal distribtion of û MAP is exactly the posterior distribtion. Therefore, the conventional Metropolis Hastings algorithm can be applied to the joint distribtion π û MAP, δ, with h û MAP, δ as the proposal distribtion. The acceptance rate in this case reads π û MAP, δ h û MAP α û MAP k, δ k, δ = min,, π û MAP k, δ k h û MAP, δ

13 Downloaded 7/6/8 to Redistribtion sbject to SIAM license or copyright; see A54 KAINAN WANG, TAN BUI-THANH, AND OMAR GHATTAS where û MAP k, δ k is the previos sample and û MAP, δ is the proposed sample for the next state. While the above agmented space method garantees that û MAP k is correctly distribted by the posterior distribtion π post in the limit, it can be prohibitively expensive, especially for large-scale inverse problems. The reason is that adding the data δ increases dimensionality, which can be significant if the data dimension is large. Thogh MCMC methods are independent of the dimension, the nmber of samples cold be excessively large in order to obtain a reasonable reslt the crse of dimensionality. Moreover, the evalation of the fll Hessian as reqired in the comptation of the Jacobian can also be very expensive. To address these challenges, we now present an approximate marginalization of δ that allows s to carry ot the MCMC method in the original space of û MAP. A direct conseqence of this approximation is that û MAP k are no longer the trth samples of the posterior π post even in the limit. Nevertheless, or experiences, inclding the nmerical reslts presented in this paper, show that the reslts from the approximate Markov chain are very close to those from the genine Markov chain. To begin, we observe that f T û MAP, δ = p û MAP ζ δ η û MAP, where p û MAP = exp û MAP C G û MAP d is proportional to the posterior distribtion, ζ δ = exp δ HKT H δ HK, and where H and K are given by and η û MAP = exp KT HK, H = L + L G û MAP C G û MAP L K = L G û MAP d + G û MAP û MAP. Since the terms inclding δ constitte a Gassian kernel, sch a decomposition allows s to marginalize δ and obtain the probability of proposing û MAP : q û MAP = h û MAP, δ dδ = p û MAP η û MAP ω û MAP J, where ω û MAP reslts from integrating with respect to δ, and it possesses the explicit form ω û MAP H = L J. L

14 Downloaded 7/6/8 to Redistribtion sbject to SIAM license or copyright; see rmap SAMPLING METHOD FOR BAYESIAN INVERSE PROBLEMS A55 Sbstitting these formlas into the decomposition of q û MAP, we obtain the following ratio of posterior distribtion over proposal distribtion: θ û MAP = p q û MAP exp û MAP KT HK L J. With this ratio, we are able to compte the acceptance ratio between a newly proposed state û MAP and a crrent state û MAP k. One comptational consideration in practice wold be that directly compting the gradient of the forward map, G û MAP, can be expensive when the nmber of measrements is high. A frther practical simplification wold be approximating α with only the L J. Ths, the acceptance ratio we adopt has the form θ û MAP 33 α û MAP, û MAP Jû MAP k k = θ û MAP. k Jû MAP In addition, we drop the higher order term in the Jacobian. In other words, we replace the fll Hessian in 3 with a Gass Newton Hessian. These simplifications appear to be reasonable, as we will show in the nmerical reslts. It shold be pointed ot that we have recently shown that the misfit Gass Newton Hessian is a compact operator [, ]. Moreover, C is also a compact operator by the definition of the Gassian measre. It follows that C Φ g û MAP, ˆd C is compact and admits low rank approximation. This is in fact one of the key points that is exploited to constrct a scalable and mesh-independent method in or previos work on extreme scale Bayesian inversion [9, 4]. Ths, compting J can be done in a scalable manner independent of the mesh size sing the randomized SVD techniqe [3], for example. 5. Finite element discretization and optimization. For the practical problems we consider we assme the spatial dimension to be at least two; therefore we choose s > so that the infinite dimensional framework is well-defined as discssed in section. As a reslt, evalating the prior and/or generating a prior sample reqires s to discretize and/or solve a fractional PDE. Similarly to [8] and references therein we combine the finite element method FEM [8] and the matrix transfer techniqe see, e.g., [35] to discretize the trncated Karhnen Loève KL expansion of the prior. For the discretization of the forward eqation and hence the likelihood, we also se the same FEM. Using finite element approximation, the MAP problem 5 becomes a possibly high dimensional and nonlinear optimization problem. It is ths necessary to se the state-of-the-art scalable optimization solver to minimize the cost. Here we choose the trst region inexact Newton conjgate gradient TRINCG method, for which some of the main ideas can be fond in, e.g., [5, 7, 9, 45]. The method combines the rapid locally qadratic convergence rate properties of the Newton method, the effectiveness of trst region globalization for treating ill-conditioned problems, and the Eisenstat Walker idea of preventing oversolving. In the nmerical reslts section, we demonstrate the efficiency of this trst region method over poplar Levenberg Marqardt LM techniqes. As we shall see, in some difficlt examples, choosing TRINCG becomes critical in controlling comptation time for rmap sampling.

15 A56 KAINAN WANG, TAN BUI-THANH, AND OMAR GHATTAS Downloaded 7/6/8 to Redistribtion sbject to SIAM license or copyright; see Good initial gess for the rmap algorithm. One of the most important aspects of nmerical optimization, particlarly with the Newton method, is how to choose a good initial gess. The closer the initial gess is to the basin of attraction of a local minimm, the faster the convergence. This is clearly important since we desire to minimize the cost of compting rmap proposals. One way to achieve this is throgh sing sensitivity analysis, which we now describe. To begin, we distingish, the derivative with respect to, from derivatives with other variables: for example, ûi and denote derivatives with respect to û ˆdi i and ˆd i, respectively. Consider two consective rmap samples û MAP i and û MAP i+ that satisfy F Now, let s define û MAP i ; û i, ˆd i := J F û MAP i+ ; û i+, ˆd i+ := J û MAP i ; û i, ˆd i =, û MAP i+ ; û i+, ˆd i+ =. ũ = û i+ û i and d = ˆd i+ ˆd i. Assming that û MAP i is already compted from 34, we now constrct an initial gess for solving 35 sing the Newton method: 36 init = û MAP i + û MAP ˆdi i, d + ûi û MAP i, ũ, }{{} T which is simply the first order Taylor approximation of û MAP i+ arond û i, ˆd i. What remains is to compte T in 36. To this end, we expand the gradient in 35 sing the first order Taylor expansion to obtain the following eqation for T : 37 J û MAP i ; û i, ˆd i T G û MAP i L d + C ũ. Solving 37 reqires an adjoint solve to evalate the right-hand side and the inverse of J û MAP i ; û i, ˆd i the Hessian evalated at the ith rmap sample. If ûmap i û MAP i+ is small, init is a very good approximation of û MAP i+. Ths, solving 35 with init as the initial gess helps sbstantially redce the nmber of optimization iterations and hence the nmber of forward PDE solves. In practice, we linearize arond the MAP point 5, and this approach frther redces the nmber of PDE solves since J MAP ;, d is fixed and can be well approximated sing low rank approximation [9, 4]. 6. Nmerical reslts. In this section, we present sampling reslts sing several test cases. In section 6., we first demonstrate the effectiveness of sing the approximated Metropolization for efficient rmap sampling. We then se two analytical fnctions to compare the sampling efficiency of the approximated rmap to the RTO method and to the SN method described above. In section 6., we again se the approximated rmap method to sample a Bayesian inverse problem on a D Helmholtz forward model. Therein, we compare the comptational efficiency of the poplar LM method see, e.g., [49] to that of the TRINCG method for each rmap sample and the effectiveness of sing a good initial gess as discssed in section 5. In order to examine statistical convergence of rmap methods, we also compare rmap samples with those from the delayed rejection adaptive Metropolis DRAM sampler [9].

16 rmap SAMPLING METHOD FOR BAYESIAN INVERSE PROBLEMS A57 Downloaded 7/6/8 to Redistribtion sbject to SIAM license or copyright; see Analytical fnction example. Let s start by nmerically demonstrating how rmap and RTO cost fnctions in 3 and 4, respectively, change the original cost fnction in 5. To this end, we consider two analytical cost fnctions negative log posterior 38a 38b J :=.8 +., J := Comparison of agmented space Metropolization and its approximation. We first compare the accracy and statistical efficiency of the agmented space Metropolization and its approximation presented in section 4. To this end we choose to work with the cost fnction 38a bt with three different data variances σ = {.,.5,.}, i.e., J :=.8 + σ., to constrct three different posterior distribtions. We sample these distribtions with both agmented space Metropolized rmap and the approximated Metropolized rmap with 5 samples. We plot the histograms at,, and 5 samples in Figre. The reslts seem to indicate that the approximated method magenta histograms converges more rapidly than the agmented space conterpart cyan histograms. This is also reflected in a comparison of acceptance ratio in Figre, in which we observe that the acceptance rate for the approximate method is higher. One of the reasons for the low acceptance rate of the agmented space method is again de to the increase of dimensionality. For very skewed distribtion e.g., the fifth and the sixth row of Figre, however, the approximate method is less accrate than the agmented space conterpart. For the other cases, both methods are comparable. Recall that the approximate method is also mch less expensive. For these reasons, we will se the approximate method throghot the rest of the paper Comparing rmap and RTO methods. Figre 3 shows the original cost fnctionals J, J and their randomization with rmap and RTO methods. Note that both the original RTO and or modified version give identical reslts for all analytical reslts, and hence we do not distingish them. Here, we se the same θ and ε for both rmap and RTO. As can be seen, both randomized costs preserve the characteristics, e.g., mltimodality and skewness, of the original cost fnction. However, they differ from the original cost fnction as well as from one other, which agrees with or findings in section 3.3. We next examine the sensitivity of both rmap and RTO with mltimodality and optimization solvers. To that end, we first se the MATLAB rotine fminnc, the nconstrained optimization solver, and se the MAP point as the initial gess to compte rmap and RTO samples for the J cost fnctional. As can be seen in Figres 4a and 4d, both methods are stck in a mode. Instead, if we se û j := + ε j as an initial gess for compting the jth sample, we obtain the reslts in Figres 4b and 4e, respectively. Clearly, both methods explore both modes well. Ths, for rmap and RTO to work with the local optimization solver, it is important that initial gesses are well distribted in the parameter space. In fact, good initial gesses also help significantly redce the nmber of forward solves, as we will show in the following sbsection.

17 A58 KAINAN WANG, TAN BUI-THANH, AND OMAR GHATTAS Downloaded 7/6/8 to Redistribtion sbject to SIAM license or copyright; see Fig.. Comparison of sampling efficiency between the agmented space Metropolization and its approximation. From left to right, the nmber of samples in each colmn is,, and 5. From top to bottom, cyan plots are histograms prodced by agmented space methods and magenta plots are prodced by the approximated Metropolization. See online version for color. As a comparison, we employ the MATLAB constrained optimization solver fminbnd, with prescribed bond more than sfficient to cover the

18 rmap SAMPLING METHOD FOR BAYESIAN INVERSE PROBLEMS A59 Downloaded 7/6/8 to Redistribtion sbject to SIAM license or copyright; see Fig.. Comparison of acceptance ratios between the agmented space Metropolization and its approximation. In all cases, agmented space methods have a lower acceptance ratio de to the higher dimensionality. cost 5 5 original rmap RTO.5.5 a J, and its rmap and RTO. cost 5 5 original rmap RTO.5.5 b J, and its rmap and RTO. Fig. 3. Randomization of the cost fnctionals in 38 with rmap and RTO methods. modes. This optimization solver comptes initial gesses sing the golden section rle. The reslts for rmap and RTO are shown in Figres 4c and 4f: rmap still works well in this case, while RTO is stck in the left mode. Ths, rmap seems to be more robst with optimization solvers. From nmerical experiments we observe that rmap tends to displace the original fnction more than RTO does, and this may partially explain the robstness of the former. However, in this experiment, rmap seems to have introdced an artificial mode for the original fnction, as we now show in Figre 5, for cost fnction J. Note that the original cost fnction J has only one mode, bt it can become mltimodal for a range of ε and θ. As can be observed in Figres 5a and 5c, rmap pts a lot of samples in an artificial mode that was not in the original fnction, while RTO does not seem to generate the artifact. With the sqare root Jacobian correction in section 4, we can, in Figre 5b, both remove that artificial mode and improve the histogram for the actal mode. We can also improve the RTO samples by first taking the RTO density as the importance sampling density and then sing the importance weights to correct for RTO samples. The reslt in Figre 5d shows that this strategy effectively improves RTO s estimation of the posterior distribtion.

19 A6 KAINAN WANG, TAN BUI-THANH, AND OMAR GHATTAS Downloaded 7/6/8 to Redistribtion sbject to SIAM license or copyright; see 3 a rmap: MAP initial gess. 4 3 d RTO: MAP initial gess. 3 b rmap: random initial gess. 4 3 e RTO: random initial gess. 3 c rmap: Golden section initial gess. 4 3 f RTO: Golden section initial gess. Fig. 4. Sensitivity of rmap and RTO with local optimization solvers and initial gesses. Figres 4a and 4d show the reslts of fminnc and MAP initial gess. Figres 4b and 4e show the reslts of fminnc and random prior means as initial gesses. Figres 4c and 4f show the reslts of fminbnd and the defalt golden section rle initial gess. The cost fnctional J is sed to condct these experiments Comparing rmap and SN methods. In this section, we will nmerically confirm or discssion from section 3. on the improvement of rmap over the SN method. For concreteness, we choose J in 38a, a mltimodal fnction, for the comparison. We have shown in section 3. that rmap can be viewed as an iterative SN method. It is this deterministic iteration that can help rmap explore the sample space more rapidly. In particlar, rmap can be interpreted as a globalization strategy. It is in fact a move away from the inefficiencies of random-walk/diffsion processes toward powerfl optimization methods that se derivative information to traverse the posterior. For nmerical comparison, we compte samples from the Metropolis-adjsted rmap sampler, and in this case the total nmber of Newton iterations is approximately,. Since the parameter dimension is one, the total nmber of forward and adjoint PDE solves is 4,. For the SN Newton method, we take, samples. Three independent chains with three different initial states, namely the origin and the left and right modes of the posterior distribtion, are compted for both samplers. Figre 6 shows the histogram of each chain together with the exact density. We observe that rmap chains are capable of sampling both modes and the sampling reslts are independent of starting points. On the contrary, SN chains show dependency on the starting points, and they are stck in local minima Statistical convergence of rmap. We also nmerically examine Proposition 3. sing cost fnction J. First, we compte the expectation E θ ε [S, d, θ, ε]

20 rmap SAMPLING METHOD FOR BAYESIAN INVERSE PROBLEMS A6 5 5 Downloaded 7/6/8 to Redistribtion sbject to SIAM license or copyright; see a rmap..5 c RTO b Corrected rmap..5 d Corrected RTO. Fig. 5. Illstration of artificial mode created by rmap and correction strategies for both rmap and RTO. The correction for rmap was achieved by the sqare root of the Jacobian in section 4, and the correction for RTO was obtained via important sampling weights. The nmerical experiments were done for the cost fnctional J. sing a tensor prodct Gass Hermite qadratre. Ten independent rmap chains are compted, each of which has one million samples. We compte the averages { n n j= j} N n=, N = 6, over each chain, and the reslts are compared to the qadratre-based expectation. In Figre 7, it is shown that the approximate mean of rmap samples aligns well with the limit E θ ε [S, d, θ, ε] and hence confirms or theoretical reslt in Proposition Helmholtz problems. Althogh or proposed framework is valid for Bayesian inverse problems governed by any system of forward PDEs, here we illstrate the se of the framework on a freqency domain acostic wave eqation in the form of the Helmholtz eqation. Namely, the forward model B, w is defined, in an open and bonded domain Ω, as w e w = in Ω, w = g on Ω, n where w is the acostic field, is the logarithm of the distribted wave nmber field on Ω, n is the nit otward normal on Ω, and g is the prescribed Nemann sorce on the bondary. In sbsection 6.., we first discss the comptation of the gradient and Hessian of the objective fnction sing the adjoint method. The adjoint method enables tractable

21 A6 KAINAN WANG, TAN BUI-THANH, AND OMAR GHATTAS Downloaded 7/6/8 to Redistribtion sbject to SIAM license or copyright; see a rmap. c SN initial state at the origin. b SN initial state at the left mode. d SN initial state at the right mode. Fig. 6. Comparison of Metropolis-adjsted rmap and SN MCMC methods for sampling mltimodal problems. Three starting points are chosen for these two samplers, namely the left mode, zero, and the right mode. The histograms are the same irrespective of the starting points for the rmap method, and hence only one plot is shown here. While SN chains are trapped in local minima, rmap conterparts traverse the posterior very well. Fig. 7. Convergence test of rmap samples against a qadratre evalated expectation vale. Ble dashed lines show the errors of the rmap chains, each containing one million samples. They align well with the the solid red line, which represents the theoretical, n convergence rate from the central limit theorem. See online version for color. comptation of the MAP estimator, which is crcial to the rmap algorithm. In sbsection 6.., we analyze the sampling reslts sing the rmap algorithm. Throgh a comparison of the different optimization settings described above, we demonstrate