Computing stationary states is an important topic for phase field crystal (PFC) models. Great efforts have been made for energy dissipation of the numerical schemes when using gradient flows. However, it is always time-consuming due to the requirement of small effective time steps. In this paper, we propose an adaptive accelerated proximal gradient method for finding the stationary states of PFC models. The energy dissipation is guaranteed and the convergence property is established for the discretized energy functional. Moreover, the connections between generalized proximal operator with classical (semi-)implicit and explicit schemes for gradient flow are given. Extensive numerical experiments, including two three dimensional periodic crystals in Landau-Brazovskii (LB) model and a two dimensional quasicrystal in Lifshitz-Petrich (LP) model, demonstrate that our approach has adaptive time steps which lead to significant acceleration over semi-implicit methods for computing complex structures. Furthermore, our result reveals a deep physical mechanism of the simple LB model via which the sigma phase is first discovered.

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 [1] 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 [1]. We first established the sub-sequence convergence property of the iteration scheme proposed in [1], 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.

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.

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.

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.