Fast convolution with free-space Green's functions

Vico, Felipe; Greengard, Leslie; Ferrando, M.

doi:10.1016/j.jcp.2016.07.028

Cited by 84 publications

(113 citation statements)

References 18 publications

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“…To the best of our knowledge, this observation has to be attributed to Vainikko [31] but has then been revitalized by Vico et al [34]. It is essential to a proper discretization of the Lippmann-Schwinger equation (1).…”

Section: Green's Function Discretization For the Volume: Gmentioning

confidence: 97%

Three-Dimensional Optical Diffraction Tomography With Lippmann-Schwinger Model

Pham

Soubies

Ayoub

et al. 2020

IEEE Trans. Comput. Imaging

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A broad class of imaging modalities involve the resolution of an inverse-scattering problem. Among them, three-dimensional optical diffraction tomography (ODT) comes with its own challenges. These include a limited range of views, a large size of the sample with respect to the illumination wavelength, and optical aberrations that are inherent to the system itself. In this work, we present an accurate and efficient implementation of the forward model. It relies on the exact (nonlinear) Lippmann-Schwinger equation. We address several crucial issues such as the discretization of the Green function, the computation of the far field, and the estimation of the incident field. We then deploy this model in a regularized variational-reconstruction framework and show on both simulated and real data that it leads to substantially better reconstructions than the approximate models that are traditionally used in ODT. IntroductionOptical diffraction tomography (ODT) is a noninvasive quantitative imaging modality [1,2]. This label-free technique allows one to determine a three-dimensional map of the refractive index (RI) of samples, which is of particular interest for applications that range from biology [3] to nanotechnologies [4]. The acquisition setup sequentially illuminates the sample from different angles. For each illumination, the outgoing complex wave field (i.e., the scattered field) is recorded by a digital-holography microscope [5,6]. Then, from this set of measurements, the RI of the sample can be reconstructed by solving an inverse-scattering problem. However, its resolution is very challenging due to the nonlinear nature of the interaction between the light and the sample. Related WorksTo simplify the reconstruction problem, pioneering works focused on linearized models. These include Born [1] and Rytov [7] approximations, which are valid for weakly scattering samples [8]. Although originally used to deploy direct inversion methods, these linearized models have been later combined with iterative regularization techniques to improve their robustness to noise and to alleviate the missing-cone problem [9,10].Nonlinear models that adhere more closely to the physic of the acquisition are needed to recover samples with higher variations of their refractive index. For instance, beam-propagation methods (BPM) [11][12][13][14] rely on a slice-by-slice propagation model that accounts for multiple scatterings within the direction of propagation (no reflection). Other nonlinear models include the contrast source-inversion method [15] or the recursive Born approximation [16]. Although more accurate, all these models come at the price of a large computational cost.The theory of scalar diffraction recognizes the Lippmann-Schwinger (LS) model to be the most faithful. It accounts for multiple scatterings, both in transmission and reflection. Iterative forward models that solve the LS equation have

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Section: Green's Function Discretization For the Volume: Gmentioning

confidence: 97%

Three-Dimensional Optical Diffraction Tomography With Lippmann-Schwinger Model

Pham

Soubies

Ayoub

et al. 2020

IEEE Trans. Comput. Imaging

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“…Consequently, the only remaining difficulty is the computation of the convolution in (18), which however is a ubiquitous problem in computational physics and scientific computing in general. In fact, the solution of (18) A variety of efficient numerical methods has been developed to solve the Poisson equation in unbounded domains [50][51][52][53][54][55][56][57]. While most of them are based on formulation (18) it is also possible to employ (19) directly.…”

Section: B Evaluation Of the Dipolar Interaction Potentialmentioning

confidence: 99%

“…However, in our application these limitations are not an issue which is why we employ the second algorithm. The second algorithm is based on the observation that the solution of (18) is indistinguishable from the solution of is an entire function [56] (and thus C ∞ ). Application of the convolution theorem to (20) yields…”

Section: B Evaluation Of the Dipolar Interaction Potentialmentioning

confidence: 99%

“…During normal operation the algorithm requires the application of two FFTs (forward and inverse) of size 2J x 2J y 2J z and hence the algorithm is of the same complexity as the simple strategy presented at the very beginning of this section. All considerations in [56] are based on the assumption that Ω is a unit box with an aspect ratio ζ = max(L x , L y , L z )/ min(L x , L y , L z ) equal to one. In the application considered below the aspect ratio of the computational domain is given by ζ = 2.…”

Section: B Evaluation Of the Dipolar Interaction Potentialmentioning

confidence: 99%

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Optimal control of the self-bound dipolar droplet formation process

Mennemann

Langen

Exl

et al. 2019

Computer Physics Communications

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Dipolar Bose-Einstein condensates have recently attracted much attention in the world of quantum many body experiments. While the theoretical principles behind these experiments are typically supported by numerical simulations, the application of optimal control algorithms could potentially open up entirely new possibilities. As a proof of concept, we demonstrate that the formation process of a single dipolar droplet state could be dramatically accelerated using advanced concepts of optimal control. More specifically, our optimization is based on a multilevel B-spline method reducing the number of required cost function evaluations and hence significantly reducing the numerical effort. Moreover, our strategy allows to consider box constraints on the control inputs in a concise and efficient way. To further improve the overall efficiency, we show how to evaluate the dipolar interaction potential in the generalized Gross-Pitaevskii equation without sacrificing the spectral convergence rate of the underlying time-splitting spectral method.

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“…For our next example, we consider a lens problem based on one in [58]. We use this example to demonstrate the method on a problem with more complicated spatial variation of permittivity and also to show how the method behaves when increasing the frequency of the problem relative to the size of the domain.…”

Section: Eaton Lensmentioning

confidence: 99%

An iterative domain decomposition, spectral finite element method on non-conforming meshes suitable for high frequency Helmholtz problems

Galagusz

McFee

2019

Journal of Computational Physics

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The purpose of this research is to describe an efficient iterative method suitable for obtaining high accuracy solutions to high frequency time-harmonic scattering problems. The method allows for both refinement of local polynomial degree and non-conforming mesh refinement, including multiple hanging nodes per edge. Rather than globally assemble the finite element system, we describe an iterative domain decomposition method which can use element-wise fast solvers for elements of large degree. Since continuity between elements is enforced through moment equations, the resulting constraint equations are hierarchical. We show that, for high frequency problems, a subset of these constraints should be directly enforced, providing the coarse space in the dual-primal domain decomposition method. The subset of constraints is chosen based on a dispersion criterion involving mesh size and wavenumber. By increasing the size of the coarse space according to this criterion, the number of iterations in the domain decomposition method depends only weakly on the wavenumber. We demonstrate this convergence behaviour on standard domain decomposition test problems and conclude the paper with application of the method to electromagnetic problems in two dimensions. These examples include beam steering by lenses and photonic crystal waveguides, as well as radar cross section computation for dielectric, perfect electric conductor, and electromagnetic cloak scatterers.

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Fast convolution with free-space Green's functions

Cited by 84 publications

References 18 publications

Three-Dimensional Optical Diffraction Tomography With Lippmann-Schwinger Model

Three-Dimensional Optical Diffraction Tomography With Lippmann-Schwinger Model

Optimal control of the self-bound dipolar droplet formation process

An iterative domain decomposition, spectral finite element method on non-conforming meshes suitable for high frequency Helmholtz problems

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