Exponential Krylov time integration for modeling multi-frequency optical response with monochromatic sources

Abstract

Light incident on a layer of scattering material such as a piece of sugar or white paper forms a characteristic speckle pattern in transmission and reflection. The information hidden in the correlations of the speckle pattern with varying frequency, polarization and angle of the incident light can be exploited for applications such as biomedical imaging and high-resolution microscopy. Conventional computational models for multifrequency optical response involve multiple solution runs of Maxwell's equations with monochromatic sources. Exponential Krylov subspace time solvers are promising candidates for improving efficiency of such models, as single monochromatic solution can be reused for the other frequencies without performing full time-domain computations at each frequency. However, we show that the straightforward implementation appears to have serious limitations. We further propose alternative ways for efficient solution through Krylov subspace methods. Our methods are based on two different splittings of the unknown solution into different parts, each of which can be computed efficiently. Experiments demonstrate a significant gain in computation time with respect to the standard solvers.

An accurate restarting for shift-and-invert Krylov subspaces computing matrix exponential actions of nonsymmetric matrices

“…The residual as a function of t in the SAI Krylov subspace method exhibits a much more irregular behavior than in the polynomial Krylov method ( 5)- (8), see Figure 2. If the RT restarting is applied for the SAI Krylov subspace method then it can happen that δ such that r k (s) is below a certain tolerance for s ∈ [0, δ] is too small to be used in practice for an efficient restarting.…”

“…Other methods for computing matrix exponential actions for large matrices include Chebyshev polynomials [4] and the scaling and squaring method combined with Taylor series [2]. Krylov subspace computations of the matrix exponential and other matrix functions has been an active research area, with many important developments such as rational Krylov subspace methods [31,38,11,17,18], restarting techniques [10,37,1,14,19,26] and interesting large-scale computational applications [22,11,24,5,8].…”

A nested Schur complement solver with mesh-independent convergence for the time domain photonics modeling

“…Numerical solution of the time-dependent Maxwell equations is an important computational problem arising in various scientific and engineering fields such as photonic crystal modeling, gas and oil industry, biomedical simulations, and astrophysics. Rather often the application environment suggests that the Maxwell equations have to be solved many times, for instance, for different source functions or different medium parameters) [6]. The size of the spatial computational domain as well as the necessity to solve the equations many times make this task very demanding in terms of computational costs.…”

“…Therefore, advanced computational techniques have to be applied, such as modern finite element discretizations [9,28] in space and efficient integration schemes in time. Along with multirate and implicit time integration schemes [31,10], exponential time integration schemes [18] have recently been shown promising for solving the Maxwell equations in time [19,5,6].…”

A residual concept for Krylov subspace evaluation of the $φ$ matrix function

“…Introduction. Exponential time integration is an actively developing research field [41], with numerous successful applications in challenging large scale computations, such as elastic wave equations [43], wind farm simulations [30], power delivery network analysis [69], large circuit analysis [70] vector finite element discretizations of Maxwell's equations [12] and photonic crystal modeling [9,11].…”