In computer graphics, the efficient rendering of a clear sky may greatly enhance the realism of synthesised interactive virtual environments. However, light atmospheric scattering, lying behind a reliable sky synthesis, is a computationally demanding process. Obtaining the highest possible accuracy and quality of the synthesised image in an efficient manner is not trivial as even minor integration errors may result in huge image artefacts. In this paper we propose two new approaches, namely the first based on the splines and the second on the Taylor expansion, in order to obtain the efficient computing of the transmittance (optical depth) component of the physically based single scattering model. We used Bruneton's framework [1] to compare the proposed solution with state-of-the-art methods in a single, scattering-based clear sky synthesis, in terms of image quality and performance (time). We managed to improve the quality of the single scattering effect and time performance and we removed the necessity of pre-calculating values to look-up tables for the transmittance part of the scattering integral, reducing exploited memory usage.

We consider the following BVP ẍ (t) = f (t, ẋ (t) , x (t)) − h (t), x (0) = x (1) = 0, where f is continuous and satisfies some other conditions, h ∈ H 1 0 (0, 1) together with its discretizationUsing monotonicity methods we obtain the convergence of a solutions to a family of discrete problems to the solution of a continuous one, i.e. the existence of non-spurious solutions to the above problems is considered. Continuous dependence on parameters for the continuous problem is also investigated.

We use the finite dimensional monotonicity methods in order to investigate problems connected with the discrete sx,·-Laplacian on simple, connected, undirected, weighted, and finite graphs with nonlinearities given in a non-potential form. Positive solutions are also considered.