Approximate Primal-Dual Fixed-Point based Langevin Algorithms for Non-smooth Convex Potentials

The Langevin algorithms are frequently used to sample the posterior distributions in Bayesian inference. In many practical problems, however, the posterior distributions often consist of non-differentiable components, posing challenges for the standard Langevin algorithms, as they require to evaluate the gradient of the energy function in each iteration. To this end, a popular remedy is to utilize the proximity operator, and as a result one needs to solve a proximity subproblem in each iteration. The conventional practice is to solve the subproblems accurately, which can be exceedingly expensive, as the subproblem needs to be solved in each iteration. We propose an approximate primal-dual fixed-point algorithm for solving the subproblem, which only seeks an approximate solution of the subproblem and therefore reduces the computational cost considerably. We provide theoretical analysis of the proposed method and also demonstrate its performance with numerical examples

The absorption of oral morroniside in rats: In vivo, in situ and in vitro studies

Morroniside is one of the most important iridoid glycosides from Cornus officinalis Sieb. et Zucc. In the present study, the pharmacokinetics and bioavailability studies of morroniside were conducted on Sprague-Dawley (SD) rats. A rat in situ intestinal perfusion model was used to characterize the absorption of morroniside. Caco-2 cells were used to examine the transport mechanisms of morroniside. The pharmacokinetic study of morroniside exhibited linear dose-proportional pharmacokinetic characteristics and low bioavailability (4.3 %) in SD rats. Its average Peff value for transport across the small intestinal segments changed from (3.09 ± 2.03) × 10–6 to (4.53 ± 0.94) × 10–6 cm s–1. In Caco-2 cells, the Papp values ranged from (1.61 ± 0.53) × 10–9 to (1.19 ± 0.22) × 10–7 cm s–1 for the apical to basolateral side and the Pratio values at three concentrations were all lower than 1.16. Morroniside showed poor absorption and it might not be a specific substrate of P-glycoprotein (P-gp)

Generating Functions of Switched Linear Systems: Analysis, Computation, and Stability Applications

In this paper, a unified framework is proposed to study the exponential stability of discrete-time switched linear systems, and more generally, the exponential growth rates of their trajectories, under three types of switching rules: arbitrary switching, optimal switching, and random switching. It is shown that the maximum exponential growth rates of system trajectories over all initial states under these three switching rules are completely characterized by the radii of convergence of three suitably defined families of functions called the strong, the weak, and the mean generating functions, respectively. In particular, necessary and sufficient conditions for the exponential stability of the switched linear systems are derived based on these radii of convergence. Various properties of the generating functions are established and their relations are discussed. Algorithms for computing the generating functions and their radii of convergence are also developed and illustrated through examples

NF-ULA: Langevin Monte Carlo with Normalizing Flow Prior for Imaging Inverse Problems

Bayesian methods for solving inverse problems are a powerful alternative to classical methods since the Bayesian approach offers the ability to quantify the uncertainty in the solution. In recent years, data-driven techniques for solving inverse problems have also been remarkably successful, due to their superior representation ability. In this work, we incorporate data-based models into a class of Langevin-based sampling algorithms for Bayesian inference in imaging inverse problems. In particular, we introduce NF-ULA (Normalizing Flow-based Unadjusted Langevin algorithm), which involves learning a normalizing flow (NF) as the image prior. We use NF to learn the prior because a tractable closed-form expression for the log prior enables the differentiation of it using autograd libraries. Our algorithm only requires a normalizing flow-based generative network, which can be pre-trained independently of the considered inverse problem and the forward operator. We perform theoretical analysis by investigating the well-posedness and non-asymptotic convergence of the resulting NF-ULA algorithm. The efficacy of the proposed NF-ULA algorithm is demonstrated in various image restoration problems such as image deblurring, image inpainting, and limited-angle X-ray computed tomography (CT) reconstruction. NF-ULA is found to perform better than competing methods for severely ill-posed inverse problems

NF-ULA: Normalizing Flow-Based Unadjusted Langevin Algorithm for Imaging Inverse Problems

Bayesian methods for solving inverse problems are a powerful alternative to classical methods since the Bayesian approach offers the ability to quantify the uncertainty in the solution. In recent years, data-driven techniques for solving inverse problems have also been remarkably successful, due to their superior representation ability. In this work, we incorporate data-based models into a class of Langevin-based sampling algorithms for Bayesian inference in imaging inverse problems. In particular, we introduce NF-ULA (Normalizing Flow-based Unadjusted Langevin algorithm), which involves learning a normalizing flow (NF) as the image prior. We use NF to learn the prior because a tractable closed-form expression for the log prior enables the differentiation of it using autograd libraries. Our algorithm only requires a normalizing flow-based generative network, which can be pre-trained independently of the considered inverse problem and the forward operator. We perform theoretical analysis by investigating the well-posedness and non-asymptotic convergence of the resulting NF-ULA algorithm. The efficacy of the proposed NF-ULA algorithm is demonstrated in various image restoration problems such as image deblurring, image inpainting, and limited-angle X-ray computed tomography (CT) reconstruction. NF-ULA is found to perform better than competing methods for severely ill-posed inverse problems

NF-ULA: Normalizing Flow-Based Unadjusted Langevin Algorithm for Imaging Inverse Problems

Bayesian methods for solving inverse problems are a powerful alternative to classical methods since the Bayesian approach offers the ability to quantify the uncertainty in the solution. In recent years, data-driven techniques for solving inverse problems have also been remarkably successful, due to their superior representation ability. In this work, we incorporate data-based models into a class of Langevin-based sampling algorithms for Bayesian inference in imaging inverse problems. In particular, we introduce NF-ULA (Normalizing Flow-based Unadjusted Langevin algorithm), which involves learning a normalizing flow (NF) as the image prior. We use NF to learn the prior because a tractable closed-form expression for the log prior enables the differentiation of it using autograd libraries. Our algorithm only requires a normalizing flow-based generative network, which can be pre-trained independently of the considered inverse problem and the forward operator. We perform theoretical analysis by investigating the well-posedness and non-asymptotic convergence of the resulting NF-ULA algorithm. The efficacy of the proposed NF-ULA algorithm is demonstrated in various image restoration problems such as image deblurring, image inpainting, and limited-angle X-ray computed tomography (CT) reconstruction. NF-ULA is found to perform better than competing methods for severely ill-posed inverse problems

Bayesian inference and uncertainty quantification for image reconstruction with Poisson data

We provide a complete framework for performing infinite-dimensional Bayesian inference and uncertainty quantification for image reconstruction with Poisson data. In particular, we address the following issues to make the Bayesian framework applicable in practice. We first introduce a positivity-preserving reparametrization, and we prove that under the reparametrization and a hybrid prior, the posterior distribution is well-posed in the infinite dimensional setting. Second we provide a dimension-independent MCMC algorithm, based on the preconditioned Crank-Nicolson Langevin method, in which we use a primal-dual scheme to compute the offset direction. Third we give a method combining the model discrepancy method and maximum likelihood estimation to determine the regularization parameter in the hybrid prior. Finally we propose to use the obtained posterior distribution to detect artifacts in a recovered image. We provide an example to demonstrate the effectiveness of the proposed method

NLTG Priors in Medical Image: Nonlocal TV-Gaussian (NLTG) prior for Bayesian inverse problems with applications to Limited CT Reconstruction

Bayesian inference methods have been widely applied in inverse problems, {largely due to their ability to characterize the uncertainty associated with the estimation results.} {In the Bayesian framework} the prior distribution of the unknown plays an essential role in the Bayesian inference, {and a good prior distribution can significantly improve the inference results.} In this paper, we extend the total~variation-Gaussian (TG) prior in \cite{Z.Yao2016}, and propose a hybrid prior distribution which combines the nonlocal total variation regularization and the Gaussian (NLTG) distribution. The advantage of the new prior is two-fold. The proposed prior models both texture and geometric structures present in images through the NLTV. The Gaussian reference measure also provides a flexibility of incorporating structure information from a reference image. Some theoretical properties are established for the NLTG prior. The proposed prior is applied to limited-angle tomography reconstruction problem with difficulties of severe data missing. We compute both MAP and CM estimates through two efficient methods and the numerical experiments validate the advantages and feasibility of the proposed NLTG prior