In this paper, we study a broad class of structured monotone inclusion problems in real Hilbert spaces. We propose a novel primal-dual splitting algorithm for solving such inclusions, which accommodates multiple monotone operators and cocoercive terms, as well as a composite monotone operator involving the linear map. The algorithm combines forward evaluations for the cocoercive components with backward resolvent steps for the monotone operators and employs a dual update for the linear composition term. It generalizes and unifies several existing methods, while requiring only a single resolvent or operator evaluation per iteration. We prove weak convergence of the iterates under standard assumptions on monotonicity and cocoercivity. Furthermore, we establish strong convergence under a mild regularity condition, such as uniform monotonicity. Numerical experiments on image deblurring and denoising problems demonstrate the efficiency and flexibility of the proposed algorithm.
. In this paper, we present a novel and general framework for color image denoising by integrating a traditional variational approach and learningbased denoisers. Specifically, we incorporate the deep residual convolutional neural network (CNN) prior into the traditional saturation-value total variation (SVTV) based variational model to integrate the advantages of both approaches. Saturation regularization is applied to process the saturation channel of a color image, while CNN based denoiser handles the value channel of a color image. Then, the denoising process is performed in parallel on saturation channel and value channel by using saturation regularization and deep CNN prior respectively. Theoretically, we investigate the properties of the proposed model, and give a detailed discussion based on the mathematical foundation about the solution. Numerically, we formulate an efficient and effective algorithm to solve the proposed model based on the framework of alternating direction method of multipliers (ADMM). Numerical examples are presented to demonstrate that the performance of the proposed framework is competitive in terms of visual quality and some criteria such as peak signal-to-noise ratio (PSNR), structure similarity (SSIM), quaternion structural similarity (QSSIM), and feature similarity (FSIM).
We consider a two-component semilinear reaction-diffusion system in a bounded spatial domain Omega over a time interval (0, T), which governs the water density u(x, t) and the vegetation biomass density v(x, t) for x is an element of Omega and 0 < t < T. In this system, called the Klausmeier-Gray-Scott model, we assume that an unknown source depends only on the spatial variable and appears in the reaction-diffusion equation for u. The main subject is the inverse source problem of determining a source term from limited data on (u, v). We establish two kinds of stability estimates by means of Carleman estimates. First, a Carleman estimate with a singular weight yields a Lipschitz stability estimate for the inverse source problem from data consisting of a snapshot u(& centerdot;, t(0)) in Omega and (u, v) in a subdomain omega over a time interval. Second, without assuming boundary data, we prove a Ho & uml;lder stability estimate in any interior sub domain Omega(0) satisfying Omega(-)(0) subset of Omega. We further study how much the observation data can be reduced while preserving uniqueness and stability in the inverse problem under suitable additional conditions.
