Chenglong BaoDepartment of Mathematics, National University of Singapore, Singapore 117543, SingaporeHui JiDepartment of Mathematics, National University of Singapore, Singapore 117543, SingaporeZuowei ShenDepartment of Mathematics, National University of Singapore, Singapore 117543, Singapore
Numerical Analysis and Scientific Computingmathscidoc:2206.25005
Applied and Computational Harmonic Analysis, 38, (3), 510-523, 2015.5
Sparse modeling/approximation of images plays an important role in image restoration. Instead of using a fixed system to sparsely model any input image, a more promising approach is using a system that is adaptive to the input image. A non-convex variational model is proposed in  for constructing a tight frame that is optimized for the input image, and an alternating scheme is used to solve the resulting non-convex optimization problem. Although it showed good empirical performance in image denoising, the proposed alternating iteration lacks convergence analysis. This paper aims at providing the convergence analysis of the method proposed in . We first established the sub-sequence convergence property of the iteration scheme proposed in , i.e., there exists at least one convergent sub-sequence and any convergent sub-sequence converges to a stationary point of the minimization problem. Moreover, we showed that the original method can be modified to have sequence convergence, i.e., the modified algorithm generates a sequence that converges to a stationary point of the minimization problem.
Jae Kyu ChoiInstitute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China Chenglong BaoYau Mathematical Sciences Center, Tsinghua University, Beijing 100084, ChinaXiaoqun ZhangInstitute of Natural Sciences, School of Mathematical Sciences, and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China
Numerical Analysis and Scientific Computingmathscidoc:2206.25004
SIAM Journal on Imaging Sciences, 11, (2), 1179-1204, 2018.5
Recent technical advances lead to the coupling of PET and MRI scanners, enabling one to acquire functional and anatomical data simultaneously. In this paper, we propose a tight frame based PET-MRI joint reconstruction model via the joint sparsity of tight frame coefficients. In addition, a nonconvex balanced approach is adopted to take the different regularities of PET and MRI images into account. To solve the nonconvex and nonsmooth model, a proximal alternating minimization algorithm is proposed, and the global convergence is present based on the Kurdyka--Łojasiewicz property. Finally, the numerical experiments show that our proposed models achieve better performance over the existing PET-MRI joint reconstruction models.
Chenglong BaoYau Mathematical Sciences Center, Tsinghua University, Beijing, 100084 China Jae Kyu ChoiSchool of Mathematical Sciences, Tongji University, Shanghai, 200092 ChinaBin DongBeijing International Center for Mathematical Research and Laboratory for Biomedical Image Analysis, Beijing Institute of Big Data Research, Peking University, Beijing, 100871 China
Numerical Analysis and Scientific Computingmathscidoc:2206.25003
SIAM Journal on Imaging Sciences, 12, (1), 492-520, 2019.2
Quantitative susceptibility mapping (QSM) uses the phase data in magnetic resonance signals to visualize a three-dimensional susceptibility distribution by solving the magnetic field to susceptibility inverse problem. Due to the presence of zeros of the integration kernel in the frequency domain, QSM is an ill-posed inverse problem. Although numerous regularization-based models have been proposed to overcome this problem, incompatibility in the field data, which leads to deterioration of the recovery, has not received enough attention. In this paper, we show that the data acquisition process of QSM inherently generates a harmonic incompatibility in the measured local field. Based on this discovery, we propose a novel regularization-based susceptibility reconstruction model with an additional sparsity-based regularization term on the harmonic incompatibility. Numerical experiments show that the proposed method achieves better performance than existing approaches.
Kai JiangSchool of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, Hunan, China, 411105Wei SiSchool of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, Hunan, China, 411105Chang ChenSchool of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, Hunan, China, 411105Chenglong BaoYau Mathematical Sciences Center, Tsinghua University, Beijing, China, 100084
Numerical Analysis and Scientific Computingmathscidoc:2206.25002
Finding the stationary states of a free energy functional is an important problem in phase field crystal (PFC) models. Many efforts have been devoted to designing numerical schemes with energy dissipation and mass conservation properties. However, most existing approaches are time-consuming due to the requirement of small effective step sizes. In this paper, we discretize the energy functional and propose efficient numerical algorithms for solving the constrained nonconvex minimization problem. A class of gradient-based approaches, which are the so-called adaptive accelerated Bregman proximal gradient (AA-BPG) methods, is proposed, and the convergence property is established without the global Lipschitz constant requirements. A practical Newton method is also designed to further accelerate the local convergence with convergence guarantee. One key feature of our algorithms is that the energy dissipation and mass conservation properties hold during the iteration process. Moreover, we develop a hybrid acceleration framework to accelerate the AA-BPG methods and most of the existing approaches through coupling with the practical Newton method. Extensive numerical experiments, including two three-dimensional periodic crystals in the Landau--Brazovskii (LB) model and a two-dimensional quasicrystal in the Lifshitz--Petrich (LP) model, demonstrate that our approaches have adaptive step sizes which lead to a significant acceleration over many existing methods when computing complex structures.
Chenglong BaoYau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China; Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, ChinaChang ChenYau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, ChinaKai JiangSchool of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, Hunan, 411105, China
Numerical Analysis and Scientific Computingmathscidoc:2206.25001
In this paper, we compute the stationary states of the multicomponent phase-field crystal model by formulating it as a block constrained minimization problem. The original infinite-dimensional non-convex minimization problem is approximated by a finite-dimensional constrained non-convex minimization problem after an appropriate spatial discretization. To efficiently solve the above optimization problem, we propose a so-called adaptive block Bregman proximal gradient (AB-BPG) algorithm that fully exploits the problem's block structure. The proposed method updates each order parameter alternatively, and the update order of blocks can be chosen in a deterministic or random manner. Besides, we choose the step size by developing a practical linear search approach such that the generated sequence either keeps energy dissipation or has a controllable subsequence with energy dissipation. The convergence property of the proposed method is established without the requirement of global Lipschitz continuity of the derivative of the bulk energy part by using the Bregman divergence. The numerical results on computing stationary ordered structures in binary, ternary, and quinary component coupled-mode Swift-Hohenberg models have shown a significant acceleration over many existing methods.