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Register a new userForward-Mode Differentiation of Maxwells Equations
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We present a previously unexplored forward-mode differentiation method for Maxwells equations, with applications in the field of sensitivity analysis. This approach yields exact gradients and is similar to the popular adjoint variable method, but provides a significant improvement in both memory and speed scaling for problems involving several output parameters, as we analyze in the context of finite-difference time-domain (FDTD) simulations. Furthermore, it provides an exact alternative to numerical derivative methods, based on finite-difference approximations. To demonstrate the usefulness of the method, we perform sensitivity analysis of two problems. First we compute how the spatial near-field intensity distribution of a scatterer changes with respect to its dielectric constant. Then, we compute how the spectral power and coupling efficiency of a surface grating coupler changes with respect to its fill factor.
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We derive the discretized Maxwells equations using the discrete variational derivative method (DVDM), calculate the evolution equation of the constraint, and confirm that the equation is satisfied at the discrete level. Numerical simulations showed that the results obtained by the DVDM are superior to those obtained by the Crank-Nicolson scheme. In addition, we study the two types of the discretized Maxwells equations by the DVDM and conclude that if the evolution equation of the constraint is not conserved at the discrete level, then the numerical results are also unstable.
We propose a controllability method for the numerical solution of time-harmonic Maxwells equations in their first-order formulation. By minimizing a quadratic cost functional, which measures the deviation from periodicity, the controllability method determines iteratively a periodic solution in the time domain. At each conjugate gradient iteration, the gradient of the cost functional is simply computed by running any time-dependent simulation code forward and backward for one period, thus leading to a non-intrusive implementation easily integrated into existing software. Moreover, the proposed algorithm automatically inherits the parallelism, scalability, and low memory footprint of the underlying time-domain solver. Since the time-periodic solution obtained by minimization is not necessarily unique, we apply a cheap post-processing filtering procedure which recovers the time-harmonic solution from any minimizer. Finally, we present a series of numerical examples which show that our algorithm greatly speeds up the convergence towards the desired time-harmonic solution when compared to simply running the time-marching code until the time-harmonic regime is eventually reached.
We introduce in this work the normalizing field flows (NFF) for learning random fields from scattered measurements. More precisely, we construct a bijective transformation (a normalizing flow characterizing by neural networks) between a Gaussian random field with the Karhunen-Lo`eve (KL) expansion structure and the target stochastic field, where the KL expansion coefficients and the invertible networks are trained by maximizing the sum of the log-likelihood on scattered measurements. This NFF model can be used to solve data-driven forward, inverse, and mixed forward/inverse stochastic partial differential equations in a unified framework. We demonstrate the capability of the proposed NFF model for learning Non Gaussian processes and different types of stochastic partial differential equations.
We demonstrate that soliton-plasmon bound states appear naturally as propagating eigenmodes of nonlinear Maxwells equations for a metal/dielectric/Kerr interface. By means of a variational method, we give an explicit and simplified expression for the full-vector nonlinear operator of the system. Soliplasmon states (propagating surface soliton-plasmon modes) can be then analytically calculated as eigenmodes of this non-selfadjoint operator. The theoretical treatment of the system predicts the key features of the stationary solutions and gives physical insight to understand the inherent stability and dynamics observed by means of finite element numerical modeling of the time independent nonlinear Maxwell equations. Our results contribute with a new theory for the development of power-tunable photonic nanocircuits based on nonlinear plasmonic waveguides.
A matrix basis formulation is introduced to represent the 3 x 3 dyadic Greens functions in the frequency domain for the Maxwells equations and the elastic wave equation in layered media. The formulation can be used to decompose the Maxwells Greens functions into independent TE and TM components, each satisfying a Helmholtz equation, and decompose the elastic wave Greens function into the S-wave and the P-wave components. In addition, a derived vector basis formulation is applied to the case for acoustic wave sources from a non-viscous fluid layer.
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