Mathematical modelling for high precision ray tracing in optical design

Wu, Changmao; Xia, Yuanyuan; Xu, Zhengwei; Liu, Li; Tang, Xiongxin; Chen, Qiao; Xu, Fanjiang

doi:10.1016/j.apm.2024.01.012

Cited by 1 publication

(2 citation statements)

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“….}. In order to solve Equation (7), we approximate w = Va using the approximation scheme of Equation ( 6), and as the vector a is constant, we obtain w x 4 =V x 4 a, w y 4 =V y 4 a, and w x 2 y 2 =V x 2 y 2 a, with w x k y l denoting the partial derivative of w of order k over x and l over y, ∂ k+l w ∂x k ∂y l , for all given x i , y j with i, j ∈ (1, 2, . .…”

Section: Solution Of Partial Differential Equationsmentioning

confidence: 99%

“…The effect of round-off errors when utilizing a standard accuracy for reduction algorithms is highlighted in [6], and a high-precision "RingAllreduce" algorithm was proposed. A high-precision ray-tracing algorithm is presented in [7], reducing round-off errors in the numerical examples. A high arithmetic precision is also significant in the design of Field Programmable Gate Arrays (FPGAs), and a new representation to tackle programming challenges is proposed in [8].…”

Section: Introductionmentioning

confidence: 99%

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Taylor Polynomials in a High Arithmetic Precision as Universal Approximators

Bakas

2024

Computation

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Function approximation is a fundamental process in a variety of problems in computational mechanics, structural engineering, as well as other domains that require the precise approximation of a phenomenon with an analytic function. This work demonstrates a unified approach to these techniques, utilizing partial sums of the Taylor series in a high arithmetic precision. In particular, the proposed approach is capable of interpolation, extrapolation, numerical differentiation, numerical integration, solution of ordinary and partial differential equations, and system identification. The method employs Taylor polynomials and hundreds of digits in the computations to obtain precise results. Interestingly, some well-known problems are found to arise in the calculation accuracy and not methodological inefficiencies, as would be expected. In particular, the approximation errors are precisely predictable, the Runge phenomenon is eliminated, and the extrapolation extent may a priory be anticipated. The attained polynomials offer a precise representation of the unknown system as well as its radius of convergence, which provides a rigorous estimation of the prediction ability. The approximation errors are comprehensively analyzed for a variety of calculation digits and test problems and can be reproduced by the provided computer code.

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Section: Solution Of Partial Differential Equationsmentioning

confidence: 99%