Search

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 .

This article focuses on numerically studying the eigenstructure behavior of generalized eigenvalue problems (GEPs) arising in three dimensional (3D) source-free Maxwell's equations with magnetoelectric coupling effects which model 3D reciprocal chiral media. It is challenging to solve such a large-scale GEP efficiently. We combine the null-space free method with the inexact shift-invert residual Arnoldi method and MINRES linear solver to solve the GEP with a matrix dimension as large as 5,308,416. The eigenstructure is heavily determined by the chirality parameter γ . We show that all the eigenvalues are real and finite for a small chirality γ . For a critical value γ = γ ⁎ , the GEP has 2 × 2 Jordan blocks at infinity eigenvalues. Numerical results demonstrate that when γ increases from γ ⁎ , the 2 × 2 Jordan block will first split into a complex conjugate eigenpair, then rapidly collide with the real axis and bifurcate into positive (resonance) and negative eigenvalues with modulus smaller than the other existing positive eigenvalues. The resonance band also exhibits an anticrossing interaction. Moreover, the electric and magnetic fields of the resonance modes are localized inside the structure, with only a slight amount of field leaking into the background (dielectric) material.

This paper focuses on studying the bifurcation analysis of the eigenstructure of the $\gamma$-parameterized generalized eigenvalue problem ($\gamma$-GEP) arising in three-dimensional (3D) source-free Maxwell's equations with Pasteur media, where $\gamma$ is the magnetoelectric chirality parameter. For the weakly coupled case, namely, $\gamma < \gamma_{*} \equiv$ critical value, the $\gamma$-GEP is positive definite, which has been well-studied by Chern et.\ al, 2015. For the strongly coupled case, namely, $\gamma > \gamma_{*}$, the $\gamma$-GEP is no longer positive definite, introducing a totally different and complicated structure. For the critical strongly coupled case, numerical computations for electromagnetic fields have been presented by Huang et.\ al, 2018. In this paper, we build several theoretical results on the eigenstructure behavior of the $\gamma$-GEPs. We prove that the $\gamma$-GEP is regular for any $\gamma > 0$, and the $\gamma$-GEP has $2 \times 2$ Jordan blocks of infinite eigenvalues at the critical value $\gamma_{*}$. Then, we show that the $2 \times 2$ Jordan block will split into a complex conjugate eigenvalue pair that rapidly goes down and up and then collides at some real point near the origin. Next, it will bifurcate into two real eigenvalues, with one moving toward the left and the other to the right along the real axis as $\gamma$ increases. A newly formed state whose energy is smaller than the ground state can be created as $\gamma$ is larger than the critical value. This stunning feature of the physical phenomenon would be very helpful in practical applications. Therefore, the purpose of this paper is to clarify the corresponding theoretical eigenstructure of 3D Maxwell's equations with Pasteur media., Comment: 26 pages, 5 figures

This work is devoted to the numerical computation of complex band structure $$\mathbf {k}=\mathbf {k}(\omega )\in {\mathbb {C}}^3$$, with $$\omega $$ being positive frequencies, of three dimensional isotropic dispersive or non-dispersive photonic crystals from the perspective of structured quadratic eigenvalue problems (QEPs). Our basic strategy is to fix two degrees of freedom in $$\mathbf {k}$$ and to view the remaining one as the eigenvalue of a complex gyroscopic QEP which stems from Maxwell’s equations discretized by Yee’s scheme. We reformulate this gyroscopic QEP into a $$\top $$-palindromic QEP, which is further transformed into a structured generalized eigenvalue problem for which we have established a structure-preserving shift-and-invert Arnoldi algorithm. Moreover, to accelerate the inner iterations of the shift-and-invert Arnoldi algorithm, we propose an efficient preconditioner which makes most of the fast Fourier transforms. The advantage of our method is discussed in detail and corroborated by several numerical results.

Full spectrum of a large sparse ⊤-palindromic quadratic eigenvalue problem (⊤-PQEP) is considered arguably for the first time in this article. Such a problem is posed by calculation of surface Green's functions (SGFs) of mesoscopic transistors with a tremendous non-periodic cross-section. For this problem, general purpose eigensolvers are not efficient, nor is advisable to resort to the decimation method etc. to obtain the Wiener–Hopf factorization. After reviewing some rigorous understanding of SGF calculation from the perspective of ⊤-PQEP and nonlinear matrix equation, we present our new approach to this problem. In a nutshell, the unit disk where the spectrum of interest lies is broken down adaptively into pieces small enough that they each can be locally tackled by the generalized ⊤-skew-Hamiltonian implicitly restarted shift-and-invert Arnoldi (G⊤SHIRA) algorithm with suitable shifts and other parameters, and the eigenvalues missed by this divide-and-conquer strategy can be recovered thanks to the accurate estimation provided by our newly developed scheme. Notably the novel non-equivalence deflation is proposed to avoid as much as possible duplication of nearby known eigenvalues when a new shift of G⊤SHIRA is determined. We demonstrate our new approach by calculating the SGF of a realistic nanowire whose unit cell is described by a matrix of size 4000 × 4000 at the density functional tight binding level, corresponding to a 8 × 8 nm 2 cross-section. We believe that quantum transport simulation of realistic nano-devices in the mesoscopic regime will greatly benefit from this work.

In this paper, we consider the two-dimensional Maxwell's equations with the TM mode in pseudo-chiral media. The system can be reduced to the acoustic equation with a negative index of refraction. We first study the transmission eigenvalue problem (TEP) for this equation. By the continuous finite element method, we discretize the reduced equation and transform the study of TEP to a quadratic eigenvalue problem by deflating all nonphysical zeros. We then estimate half of the eigenvalues are negative with order of \begin{document}$O(1)$\end{document} and the other half of eigenvalues are positive with order of \begin{document}$O(10^2)$\end{document} . In the second part of the paper, we present a practical numerical method to reconstruct the support of the inhomogeneity by the near-field measurements, i.e., Cauchy data. Based on the linear sampling method, we propose the truncated singular value decomposition to solve the ill-posed near-field integral equation, at one wave number which is not a transmission eigenvalue. By carefully chosen an indicator function, this method produce different jumps for the sampling points inside and outside the support. Numerical results show that our method is able to reconstruct the support reliably.

Numerical simulations play a significant role in studying the properties of dispersive metallic photonic crystals. The dispersive photonic crystals are modeled by the Maxwell equations, and the equations are then discretized by the widely-used Yee’s scheme. After applying certain similarity transformations to the discretized system, the original simulation problem becomes a non-Hermitian eigenvalue problem with clustered eigenvalues. An efficient contour integral (CI) based eigensolver is developed to overcome the difficulties of applying existing methods to solve eigenvalues in designated regions. This efficient method combines the contour integral, the fast matrix–vector multiplication, and efficient linear system solver. The numerical results illustrate the efficiency of our algorithm.

This paper focuses on studying the eigenstructure of generalized eigenvalue problems (GEPs) arising in the three-dimensional source-free Maxwell equations for bi-anisotropic complex media with a 3-by-3 permittivity tensor e > 0 , a permeability tensor μ > 0 , and scalar magnetoelectric coupling constants ξ = ζ = i γ . The bi-Lebedev scheme is appealing because it preserves the symmetry inherent to the Maxwell eigenvalue problem exactly and because full degrees of freedom of electromagnetic fields at each grid point are taken into account in the discretization. The resulting GEP has eigenvalues appearing in quadruples { ± ω , ± ω } . We consider two main scenarios, where γ γ ∗ and γ > γ ∗ with γ ∗ as a critical value. In the former case, all the eigenvalues are real. In the latter case, the GEP has complex eigenvalues, and we particularly focus on the bifurcation of the eigenstructure of the GEPs. Numerical results demonstrate that the newborn ground state has occurred after γ = γ > γ ∗ , and the associated eigenvector has an exotic phenomenon of localization. Moreover, the Poynting vectors of the newborn eigenvector not only are concentrated in the material but also display exciting patterns.

Summary Numerical solution of nonlinear eigenvalue problems (NEPs) is frequently encountered in computational science and engineering. The applicability of most existing methods is limited by the matrix structures, properties of the eigen-solutions, sizes of the problems, etc. This paper aims to remove those limitations and develop robust and universal NEP solvers for large-scale engineering applications. The novelty lies in two aspects. First, a rational interpolation approach (RIA) is proposed based on the Keldysh theorem for holomorphic matrix functions. Comparing with the existing contour integral approach, the RIA provides the possibility to select sampling points in more general regions and has advantages in improving the accuracy and reducing the computational cost. Second, a resolvent sampling scheme using the RIA is proposed to construct reliable search spaces for the Rayleigh-Ritz procedure, based on which a robust eigen-solver, called RSRR, is developed for solving general NEPs. The RSRR can be easily implemented and parallelized. The advantages of the RIA and the performance of the RSRR are demonstrated by a variety of benchmark and application examples. This article is protected by copyright. All rights reserved.

We study how to efficiently solve the eigenvalue problems in computing band structure of three-dimensional dispersive metallic photonic crystals with face-centered cubic lattices based on the lossless Drude model. The discretized Maxwell equations result in large-scale standard eigenvalue problems whose spectrum contains many zero and cluster eigenvalues, both prevent existed eigenvalue solver from being efficient. To tackle this computational difficulties, we propose a hybrid Jacobi–Davidson method (hHybrid) that integrates harmonic Rayleigh–Ritz extraction, a new and hybrid way to compute the correction vectors, and a FFT-based preconditioner. Intensive numerical experiments show that the hHybrid outperforms existed eigenvalue solvers in terms of timing and convergence behaviors.

The numerical simulation of the band structure of three-dimensional dispersive metallic photonic crystals with face-centered cubic lattices leads to large-scale nonlinear eigenvalue problems, which are very challenging due to a high-dimensional subspace associated with the eigenvalue zero and the fact that the desired eigenvalues (with smallest real part) cluster and are close to the zero eigenvalues. For the solution of the nonlinear eigenvalue problem, a Newton-type iterative method is proposed and the nullspace-free method is applied to exclude the zero eigenvalues from the associated generalized eigenvalue problem. To find the successive eigenvalue/eigenvector pairs, we propose a new nonequivalence deflation method to transform converged eigenvalues to infinity, while all other eigenvalues remain unchanged. The deflated problem is then solved by the same Newton-type method, which is used as a hybrid method that combines with the Jacobi--Davidson and the nonlinear Arnoldi methods to compute the cluster...

A null-space free method with the FFT-based matrix–vector multiplications was proposed to solve the Maxwell equations that model the three-dimensional photonic crystals. The most time-consuming parts of this method were the FFT-based matrix–vector multiplications. In this article, we propose new mathematical formulas to compute the FFT-based matrix–vector multiplications and derive highly efficient algorithms on top of the NVIDIA GPU architecture. The resulting algorithms are approximately two- to threefold faster than the previous algorithms. We have successfully used a single NVIDIA Tesla P100 GPU to solve a set of generalized eigenvalue problems of 5 , 184 , 000 dimensions in 17 to 22 seconds for each problem. Furthermore, we ported the codes to a GPU cluster and achieved near linear scalability. To our knowledge, these GPU implementations of the proposed algorithms are the fastest implementations. The schemes can be applied to simulate a three-dimensional photonic crystal with all 14 Bravais lattices. These highly efficient schemes and codes raise possibilities for large-scale and near real-time numerical simulations for novel physical discoveries and engineering applications of photonic crystals.

Summary Among numerous iterative methods for solving the minimal nonnegative solution of an M-matrix algebraic Riccati equation, the structure-preserving doubling algorithm (SDA) stands out owing to its overall efficiency as well as accuracy. SDA is globally convergent and its convergence is quadratic, except for the critical case for which it converges linearly with the linear rate 1/2. In this paper, we first undertake a delineatory convergence analysis that reveals that the approximations by SDA can be decomposed into two components: the stable component that converges quadratically and the rank-one component that converges linearly with the linear rate 1/2. Our analysis also shows that as soon as the stable component is fully converged, the rank-one component can be accurately recovered. We then propose an efficient hybrid method, called the two-phase SDA, for which the SDA iteration is stopped as soon as it is determined that the stable component is fully converged. Therefore, this two-phase SDA saves those SDA iterative steps that previously have to have for the rank-one component to be computed accurately, and thus essentially, it can be regarded as a quadratically convergent method. Numerical results confirm our analysis and demonstrate the efficiency of the new two-phase SDA. Copyright © 2015 John Wiley & Sons, Ltd.

We study a robust and efficient eigensolver for computing a few smallest positive eigenvalues of the three-dimensional Maxwell's transmission eigenvalue problem. The discretized governing equations by the Nedelec edge element result in a large-scale quadratic eigenvalue problem (QEP) for which the spectrum contains many zero eigenvalues and the coefficient matrices consist of patterns in the matrix form $XY^{-1}Z$, both of which prevent existing eigenvalue solvers from being efficient. To remedy these difficulties, we rewrite the QEP as a particular nonlinear eigenvalue problem and develop a secant-type iteration, together with an indefinite locally optimal block preconditioned conjugate gradient (LOBPCG) method, to sequentially compute the desired positive eigenvalues. Furthermore, we propose a novel method to solve the linear systems in each iteration of LOBPCG. Intensive numerical experiments show that our proposed method is robust, although the desired real eigenvalues are surrounded by complex eigenv...

We present an efficient null space free Jacobi--Davidson method to compute the positive eigenvalues of time harmonic Maxwell's equations. We focus on a class of spatial discretizations that guarantee the existence of discrete vector potentials, such as Yee's scheme and the edge elements. During the Jacobi--Davidson iteration, the correction process is applied to the vector potential instead. The correction equation is solved approximately as in the standard Jacobi--Davidson approach. The computational cost of the transformation from the vector potential to the corrector is negligible. As a consequence, the expanding subspace automatically stays out of the null space and no extra projection step is needed. Numerical evidence confirms that the proposed scheme indeed outperforms the standard and projection-based Jacobi--Davidson methods by a significant margin.

To numerically determine the band structure of three-dimensional photonic crystals with face-centered cubic lattices, we study how the associated large-scale generalized eigenvalue problem (GEP) can be solved efficiently. The main computational challenge is due to the complexity of the coefficient matrix and the fact that the desired eigenvalues are interior. For solving the GEP by the shift-and-invert Lanczos method, we propose a preconditioning for the associated linear system therein. Recently, a way to reformat the GEP to the null space free eigenvalue problem (NFEP) is proposed. For solving the NFEP, we analyze potential advantages and disadvantages of the null space free inverse Lanczos method, the shift-invert residual Arnoldi method, and the Jacobi-Davidson method from theoretical viewpoints. These four approaches are compared numerically to find out their properties. The numerical results suggest that the shift-invert residual Arnoldi method with an initialization scheme is the fastest and the most robust eigensolver for the target eigenvalue problems. Our findings promise to play an essential role in simulating photonic crystals.