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In this article, we propose the Fast Algorithms for Maxwell’s Equations (FAME) package for solving Maxwell’s equations for modeling three-dimensional photonic crystals. FAME combines the null-space free method with fast Fourier transform (FFT)-based matrix-vector multiplications to solve the generalized eigenvalue problems (GEPs) arising from Yee’s discretization. The GEPs are transformed into a null-space free standard eigenvalue problem with a Hermitian positive-definite coefficient matrix. The computation times for FFT-based matrix-vector multiplications with matrices of dimension 7 million are only 0.33 and 3.6 × 10 − 3 seconds using MATLAB with an Intel Xeon CPU and CUDA C++ programming with a single NVIDIA Tesla P100 GPU, respectively. Such multiplications significantly reduce the computational costs of the conjugate gradient method for solving linear systems. We successfully use FAME on a single P100 GPU to solve a set of GEPs with matrices of dimension more than 19 million, in 127 to 191 seconds per problem. These results demonstrate the potential of our proposed package to enable large-scale numerical simulations for novel physical discoveries and engineering applications of photonic crystals.

Utilizing the optimal mass transportation (OMT) technique to convert an irregular 3D brain image into a cube, a required input format for the U-net algorithm, is a brand new idea for medical imaging research. We develop a cubic volume-measure-preserving OMT (V-OMT) model for the implementation of this conversion. The contrast-enhanced histogram equalization grayscale of fluid attenuated inversion recovery (FLAIR) in a brain image creates the corresponding density function. We then propose an effective two-phase U-net algorithm combined with the V-OMT algorithm for training and validation. First, we use the U-net and V-OMT algorithms to precisely predict the whole tumor (WT) region. Second, we expand this predicted WT region with dilation and create a smooth function by convoluting the step-like function associated with the WT region in the brain image with a 5×5×5 blur tensor. Then, a new V-OMT algorithm with mesh refinement is constructed to allow the U-net algorithm to effectively train Net1--Net3 models. Finally, we propose ensemble voting postprocessing to validate the final labels of brain images. We randomly choose 1000 and 251 brain samples from theBraTS 2021 training dataset, which contains 1251 samples, for training and validation, respectively. The Dice scores of the WT, tumor core (TC) and enhanced tumor (ET) regions for validation computed by Net1--Net3 were 0.93705, 0.90617 and 0.87470, respectively. A significant improvement in brain tumor detection and segmentation with higher accuracy is achieved.

Optimal mass transportation has been widely applied in various fields, such as data compression, generative adversarial networks, and image processing. In this paper, we adopt the projected gradient method, combined with the homotopy technique, to find a minimal volume-measure-preserving solution for a 3-manifold optimal mass transportation problem. The proposed projected gradient method is shown to be sublinearly convergent at a rate of O(1/k). Several numerical experiments indicate that our algorithms can significantly reduce transportation costs. Some applications of the optimal mass transportation maps—to deformations and canonical normalizations between brains and solid balls—are demonstrated to show the robustness of our proposed algorithms.

The standard Yee's scheme for the Maxwell eigenvalue problem places the discrete electric field variable at the midpoints of the edges of the grid cells. It performs well when the permittivity is a scalar field. However, when the permittivity is a Hermitian full tensor field it would generate un-physical complex eigenvalues or frequencies. In this paper, we propose a finite element method which can be interpreted as a modified Yee's scheme to overcome this difficulty. This interpretation enables us to create a fast FFT eigensolver that can compute very effectively the band structure of the anisotropic photonic crystal with SC and FCC lattices. Furthermore, we overcome the usual large null space associated with the Maxwell eigenvalue problem by deriving a null-space free discrete eigenvalue problem which involves a crucial Hermitian positive definite linear system to be solved in each of the iteration steps. It is demonstrated that the CG method without preconditioning converges in 37 iterations even when the dimension of a matrix is as large as 5 , 184 , 000 .