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In this section, we present the Cauchy problem for the Helmholtz equation as an example to which the theoretical analysis of this chapter can be applied. For sim- plicity, we only consider the small wave number case (0< k <1). For more details regarding the more general case, we refer to [88].

Consider the following boundary value problem for the Helmholtz equation:            ∆v(x1, x2) +k2v(x1, x2) = 0, (x1, x2)∈(0, π)×(0,1), vx2(x1,0) = 0, x1∈[0, π], v(x1,1) =u(x1), x1∈[0, π], v(0, x2) =v(π, x2) = 0, x2∈[0,1]. (3.5.1)

Problem (3.5.1) is well-posed since it corresponds to inversion of a negative-definite elliptic operator with mixed Dirichlet/Neumann data. In fact, by the method of separation of variables, the solution v(x1, x2) in the domain (0, π)×(0,1) can be expressed as v(x1, x2) = ∞ X j=1 cosh(x2 p j2_−k2₎ cosh(pj2_−k2₎ ujφj(x1), (3.5.2) whereφj(x1) = q 2 π sin(jx1) and uj =hu, φji.

Define the forward mappingK:D(K)⊂L2(0, π)→L2(0, π) by Ku(x1) =v(x1,0) = ∞ X j=1 1 cosh(pj2_−k2₎ujφj(x1),

which maps the boundary data of (3.5.1) onx2= 1 into the solution onx2= 0.Then K is a self-adjoint, positive-definite, linear operator, with eigenvalues behaving as

lj =

1

cosh(pj2_−k2₎ ∼exp(−j). (3.5.3)

The inverse problem is to find the function u, given noisy observations of

v(·,0).More precisely the data y is given by

y=v(·,0) +√1

nη,

=Ku+√1

nη.

If we place a Gaussian measureN(0, τ2C₀) as prior on u and assume thatη is also GaussianN(0,C1), then we may apply the theory developed in this chapter. Under

Assumption 3.2.1, Theorem 3.4.3 can be applied to this problem withb= 1 ands= 1 to obtain the contraction rate of the conditional Gaussian posterior distribution.

We now present a numerical simulation for obtaining the rate of the MISE of the posterior mean as the noise disappears (n→ ∞), when α= 2, γ= 1 and we have a fixed τ = 1. In this case, our theory predicts that

MISE ln(√n)−2(α∧γ)= ln(√n)−2.

To simulate MISE we average the error over a thousand realizations of the noise

η, for n = 10k, k = 1, ...,100. We denote the simulated MISE by MISE. The\ true solution u† ∈ Hγ _{is a fixed draw from a Gaussian measure N(0,Σ), where Σ} has eigenvaluesσj =j−2γ−1−ε, forε= 10−10. We use the first 105 Fourier modes. In Figure 3.1 we plot −1

2ln MISE\

against ln ln(√n) in the case β = 0. The solid line is the relation predicted by Theorem 3.4.1, that is, a line with slope 1. A least squares fit to the simulated points gives a slope of 1.0341 with coefficient of determination 0.9884. In Figure 3.2 we haveβ= 2 and all the other parameters the same. The least squares fit gives a slope 0.9723 with coefficient of determination 0.9916, confirming that the regularity of the noise as determined byβdoes not affect the rate of convergence.

0 1 2 3 4 5 6 0 1 2 3 4 5 6 l n! l n(√_n )" − 1 ln 2 ! d M I S E " Figure 3.1: −1 2ln MISE\

plotted against ln ln(√n) forn = 10k, k = 1, ...,100 in the caseb=s= 1, α= 2, β= 0, γ= 1, for fixed τ= 1.

0 1 2 3 4 5 6 0 1 2 3 4 5 6 l n! l n(√_n )" − 1 ln 2 ! d M I S E " Figure 3.2: −1 2ln MISE\ plotted against ln ln(√n) forn = 10k_{, k = 1, ...,100 in the}

3.6

Conclusions

We have considered a class of Bayesian severely ill-posed linear inverse problems with Gaussian additive noise and Gaussian priors in a diagonal setting, that is a setting in which the three operators defining the problem are simultaneously diagonalizable. In particular, we assumed that the forward operator K has singular values which decay like exp(−sjb) for s, b > 0. In addition to the problem of determining the initial condition of the heat equation considered in [45] (b= 2), our theory covers a range of other severely ill-posed inverse problems such as the Cauchy problem for the Helmholtz equation (b= 1).

We showed that in our severely ill-posed setting the posterior is absolutely continuous with respect to the prior almost surely with respect to the joint distri- bution of the unknown and the data (Theorem 3.3.2). This is in contrast to the mildly ill-posed case where it is possible to have that the posterior and the prior are mutually singular independently of the data; this happens if the prior is not sufficiently regularizing (see Proposition 3.3.3).

We also showed rates of posterior contraction in the small noise limit (The- orem 3.4.3) and in particular generalized the sharp rates obtained in [45] for the caseb= 2 to our generalized setup. Our analysis is inspired by the techniques used in [45], however our more general setting leads to technical improvements in the proofs (for example Lemma 3.4.5). As in [45], we have that the posterior contracts at the minimax rate if either the prior is oversmoothing the truth (in our notation

α≥γ +1₂) and the scaling of the prior is fixed, or for a prior of any regularity by rescaling it appropriately as the noise disappears.

Finally, we presented a numerical simulation supporting the obtained con- vergence rate of the mean integrated squared error of the posterior mean, in the case of the Cauchy problem for the Helmholtz equation.