Efficient generalized Golub–Kahan based methods for dynamic inverse problems

Abstract

We consider efficient methods for computing solutions to and estimating uncertainties in dynamic inverse problems, where the parameters of interest may change during the measurement procedure. Compared to static inverse problems, incorporating prior information in both space and time in a Bayesian framework can become computationally intensive, in part, due to the large number of unknown parameters. In these problems, explicit computation of the square root and/or inverse of the prior covariance matrix is not possible. In this work, we develop efficient, iterative, matrix-free methods based on the generalized Golub-Kahan bidiagonalization that allow automatic regularization parameter and variance estimation. We demonstrate that these methods can be more flexible than standard methods and develop efficient implementations that can exploit structure in the prior, as well as possible structure in the forward model. Numerical examples from photoacoustic tomography, deblurring, and passive seismic tomography demonstrate the range of applicability and effectiveness of the described approaches. Specifically, in passive seismic tomography, we demonstrate our approach on both synthetic and real data. To demonstrate the scalability of our algorithm, we solve a dynamic inverse problem with approximately 43, 000 measurements and 7.8 million unknowns in under 40 seconds on a standard desktop.

Temperature estimation of inflamed bowel by the photoacoustic inverse approach

“…Furthermore, in order to challenge the identification of IBD with the photoacoustic technique, a quite large standard deviation of 1.5°C was assigned to the prior distribution. It is noted that standard deviations smaller than 10 −2 °C were obtained for the posterior distributions obtained for local temperatures in Alaeian et al By following previous works available in the literature, the parameters of the Matérn prior covariance matrix, that is, the characteristic length scale and the smoothness parameter, were set to l = 0.0465 mm (the pixel size) and α = 1.5, respectively …”

“…It is noted that standard deviations smaller than 10 −2 C were obtained for the posterior distributions obtained for local temperatures in Alaeian et al 11 By following previous works available in the literature, the parameters of the Matérn prior covariance matrix, that is, the characteristic length scale and the smoothness parameter, were set to l = 0.0465 mm (the pixel size) and α = 1.5, respectively. 33,35,36 Initially, the situation with uniform inflammation in the mucosa is considered, with a laser pulse frequency of 100 Hz. Figure 5A,B presents the temperatures in the region, which were estimated with the independent prior (Equation 18a) and with the correlated prior given by the Matérn covariance matrix (Equation 18b), respectively.…”

“…However, in practice, there is no such independence, and the spatial correlation between the temperatures at different finite volumes must be taken into account in the covariance matrix of the prior information. Recent works related to imaging, based on inverse photoacoustic problems, have demonstrated that the Matérn class of covariance functions is appropriate for taking into account the correlation between the finite volumes. The Matérn covariance matrix for the prior Gaussian distribution used in this work can be written as where r i is the position vector of finite volume i , | r i − r j | is the distance between finite volumes i and j , α > 0 is a parameter that controls the smoothness of the random field, l is the characteristic length scale that controls the spatial range of correlation, G is the gamma function, and K α is the modified Bessel function of the second kind of order α .…”

“…However, in practice, there is no such independence, and the spatial correlation between the temperatures at different finite volumes must be taken into account in the covariance matrix of the prior information. Recent works related to imaging, based on inverse photoacoustic problems, 33,35,36 have demonstrated that the Matérn class 37 of covariance functions is appropriate for taking into account the correlation between the finite volumes. The Matérn covariance matrix for the prior Gaussian distribution used in this work can be written as 37…”

Modified truncated randomized singular value decomposition (MTRSVD) algorithms for large scale discrete ill-posed problems with general-form regularization

“…is A-weighted pseudoinverse of L and L † A = L † when p ≥ n; see [10] for details. This is computationally viable and attractive if not much effort is needed by applying L † A , e.g., when L is banded with small bandwidth and has a known null space; we refer the reader to, e.g., [4,5,8] for some available algorithms and codes. In many practical applications, however, such transformation is computationally unfeasible.…”