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This paper introduces a novel neural network-based approach to solving the Monge-Amp\`ere equation with the transport boundary condition, specifically targeted towards optical design applications. We leverage multilayer perceptron networks to learn approximate solutions by minimizing a loss function that encompasses the equation's residual, boundary conditions, and convexity constraints. Our main results demonstrate the efficacy of this method, optimized using L-BFGS, through a series of test cases encompassing symmetric and asymmetric circle-to-circle, square-to-circle, and circle-to-flower reflector mapping problems. Comparative analysis with a conventional least-squares finite-difference solver reveals the competitive, and often superior, performance of our neural network approach on the test cases examined here. A comprehensive hyperparameter study further illuminates the impact of factors such as sampling density, network architecture, and optimization algorithm. While promising, further investigation is needed to verify the method's robustness for more complicated problems and to ensure consistent convergence. Nonetheless, the simplicity and adaptability of this neural network-based approach position it as a compelling alternative to specialized partial differential equation solvers.

This manuscript will unify inverse freeform reflector design and surface light scattering to design freeform reflectors with a scattering surface. We use microfacets, which are small, tilted mirrors superimposed on a smooth surface. We form a simple model of surface roughness and light scattering based on the orientations of the microfacets. Using a least-squares solver to compute the smooth reflector as a starting point, we can subsequently alter the surface using an optimization procedure to account for the scattering. After optimization, the resulting reflector surface produces the desired scattered light distribution. We verify the resulting reflector using raytracing.

Concatenated backward ray mapping is an alternative for ray tracing in 2D. It is based on the phase-space description of an optical system. Phase space is the set of position and direction coordinates of light rays intersecting a surface. The original algorithm is limited to optical systems consisting of only straight surfaces; we generalize it to accommodate curved surfaces. The algorithm is applied to a standard optical system, the compound parabolic concentrator. We compare the accuracy and speed of the generalized algorithm, the original algorithm and Monte Carlo ray tracing. The results show that the generalized algorithm outperforms both other methods., Comment: 23 pages, 16 figures, submitted to "Selected topics from 22nd ECMI Conference on Industrial and Applied Mathematics in The Journal of Mathematics in Industry"

We introduce a novel approach to calculating three-dimensional freeform reflectors with a scattering surface. Our method is based on optimal transport and utilizes a Fredholm integral equation to express scattering. By solving this integral equation through a process similar to deconvolution, which we call `unfolding,' we can recover a typical specular design problem. Consequently, we consider freeform reflector design with a scattering surface as a two-step process wherein the target distribution is first altered to account for scattering, and then the resulting specular problem is solved. We verify our approach using a custom raytracer that implements the surface scattering model we used to derive the Fredholm integral.

A least-squares method for solving the hyperbolic Monge-Amp\`ere equation with transport boundary condition is introduced. The method relies on an iterative procedure for the gradient of the solution, the so-called mapping. By formulating error functionals for the interior domain, the boundary, both separately and as linear combination, three minimization problems are solved iteratively to compute the mapping. After convergence, a fourth minimization problem, to compute the solution of the Monge-Amp\`ere equation, is solved. The approach is based on a least-squares method for the elliptic Monge-Amp\`ere equation by Prins et al., and is improved upon by the addition of analytical solutions for the minimization on the interior domain and by the introduction of two new boundary methods. Lastly, the iterative method is tested on a variety of examples. It is shown that, when the iterative method converges, second-order global convergence as function of the spatial discretization is obtained., Comment: 30 pages, 24 figures

We combine two-dimensional freeform reflector design with a scattering surface modelled using microfacets, i.e., small specular surfaces representing surface roughness. The model results in a convolution integral for the scattered light intensity distribution, which yields an inverse specular problem after deconvolution. Thus, the shape of a reflector with a scattering surface may be computed using deconvolution, followed by solving the typical inverse problem of specular reflector design.

We present a novel approach of modelling surface light scattering in the context of freeform optical design. The model relies on energy conservation and optimal transport theory. For isotropic scattering in cylindrically or rotationally symmetric systems with in-plane scattering, the scattered light distribution can be expressed as a convolution between a scattering function, which characterises the optical properties of the surface, and a specular light distribution. Deconvolving this expression allows for traditional specular reflector design procedures to be used, whilst accounting for scattering.

We present three alternative derivations of the method of characteristics (MOC) for a second order nonlinear hyperbolic partial differential equation. The MOC gives rise to two mutually coupled systems of ordinary differential equations. As a special case we consider the Monge-Amp\`ere equation, for which we solve the system of ODE's using explicit one-step methods (Euler, Runge-Kutta) and spline interpolation. Numerical examples demonstrate the performance of the methods., Comment: 43 pages, 29 figures

Conventional stochastic rounding (CSR) is widely employed in the training of neural networks (NNs), showing promising training results even in low-precision computations. We introduce an improved stochastic rounding method, that is simple and efficient. The proposed method succeeds in training NNs with 16-bit fixed-point numbers and provides faster convergence and higher classification accuracy than both CSR and deterministic rounding-to-the-nearest method.

Due to the limited number of bits in floating-point or fixed-point arithmetic, rounding is a necessary step in many computations. Although rounding methods can be tailored for different applications, round-off errors are generally unavoidable. When a sequence of computations is implemented, round-off errors may be magnified or accumulated. The magnification of round-off errors may cause serious failures. Stochastic rounding (SR) was introduced as an unbiased rounding method, which is widely employed in, for instance, the training of neural networks (NNs), showing a promising training result even in low-precision computations. Although the employment of SR in training NNs is consistently increasing, the error analysis of SR is still to be improved. Additionally, the unbiased rounding results of SR are always accompanied by large variances. In this study, some general properties of SR are stated and proven. Furthermore, an upper bound of rounding variance is introduced and validated. Two new probability distributions of SR are proposed to study the trade-off between variance and bias, by solving a multiple objective optimization problem. In the simulation study, the rounding variance, bias, and relative errors of SR are studied for different operations, such as summation, square root calculation through Newton iteration and inner product computation, with specific rounding precision.