scholarly journals A Survey of Trefftz Methods for the Helmholtz Equation

2016 ◽  
pp. 237-279 ◽  
Author(s):  
Ralf Hiptmair ◽  
Andrea Moiola ◽  
Ilaria Perugia
Keyword(s):  
Helmholtz Equation ◽  
Trefftz Methods

A preconditioned method for the solution of the Robbins problem for the Helmholtz equation

2011 ◽  
Vol 51 ◽  
pp. 87
Author(s):  
Jiang Le ◽  
Huang Jin ◽  
Xiao-Guang Lv ◽  
Qing-Song Cheng
Keyword(s):  
Helmholtz Equation

scholarly journals Gravitational Interaction in the Chimney Lattice Universe

Universe ◽  
2021 ◽  
Vol 7 (4) ◽  
pp. 101
Author(s):  
Maxim Eingorn ◽  
Andrew McLaughlin ◽  
Ezgi Canay ◽  
Maksym Brilenkov ◽  
Alexander Zhuk
Keyword(s):  
Helmholtz Equation ◽  
Gravitational Potential ◽  
Fourier Expansion ◽  
Gravitational Interaction ◽  
Alternative Solution ◽  
The Third ◽  
Present Universe ◽  
Delta Functions ◽  
Yukawa Potentials ◽  
The Universe

We investigate the influence of the chimney topology T×T×R of the Universe on the gravitational potential and force that are generated by point-like massive bodies. We obtain three distinct expressions for the solutions. One follows from Fourier expansion of delta functions into series using periodicity in two toroidal dimensions. The second one is the summation of solutions of the Helmholtz equation, for a source mass and its infinitely many images, which are in the form of Yukawa potentials. The third alternative solution for the potential is formulated via the Ewald sums method applied to Yukawa-type potentials. We show that, for the present Universe, the formulas involving plain summation of Yukawa potentials are preferable for computational purposes, as they require a smaller number of terms in the series to reach adequate precision.

scholarly journals A Helmholtz equation solver using unsupervised learning: Application to transcranial ultrasound

2021 ◽  
pp. 110430
Author(s):  
Antonio Stanziola ◽  
Simon R. Arridge ◽  
Ben T. Cox ◽  
Bradley E. Treeby
Keyword(s):  
Helmholtz Equation ◽  
Unsupervised Learning ◽  
Transcranial Ultrasound
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