1,721,024 research outputs found
Numerical hints for insulation problems
Full text linkIn this work we analyze a problem of thermal insulation from the numerical point of view via finite element method. Physically, we are considering a domain of given temperature, thermally insulated by surrounding it with a constant amount of thermal insulator. From the mathematical point of view, this problem is composed by an elliptic partial differential equation with Robin–Dirichlet boundary conditions. Our question is related to the best (or worst) shape for the external domain, in terms of heat dispersion (of course, under prescribed geometrical constraints)
Perspective Shape-from-Shading Problem: A Unified Convergence Result for Several Non-Lambertian Models
Full text linkShape-from-Shading represents the problem of computing the three-dimensional shape of a surface given a single gray-value image of it as input. In a recent paper, we showed that the introduction of an attenuation factor in the brightness equations related to various perspective Shape-from-Shading models allows us to ensure the well-posedness of the corresponding differential problems. Here, we propose a unified convergence result of a numerical scheme for several non-Lambertian reflectance models. This result is interesting since it can be easily extended to other non-Lambertian models in a unified and, therefore, powerful framework
A Convergent Semi-Lagrangian Scheme for the Game ∞-Laplacian
Full text linkWe propose a new semi-Lagrangian scheme for the game infinity-Laplacian. We demonstrate the convergence of the scheme to the viscosity solution of the given problem, showing its consistency, monotonicity, and stability. The proof of this result is established following the Barles-Souganidis analysis. This analysis assumes convergence at the boundary in a strong sense and is applied to our proposed scheme, augmented with an artificial viscosity term
A scheme for the game p-Laplacian and its application to image inpainting
Full text linkWe propose a new numerical scheme for the game p-Laplacian, based on a semi-Lagrangian
approximation. We focus on the 2D version of the game p-Laplacian, with the aim to apply
the new scheme in the context of image processing. Specifically, we want to solve the so-called
inpainting problem, which consists in reconstructing one or more missing parts of an image using
information taken from the known part. The numerical tests show the reliability of the proposed
method and the advantages of taking a p > 1 in terms of execution time and accurac
Multidimensional smoothness indicators for first-order Hamilton-Jacobi equations
Full text linkThe lack of smoothness is a common feature of weak solutions of nonlinear hyperbolic equations and is a crucial issue in their approximation. This has motivated several efforts to define appropriate indicators, based on the values of the approximate solutions, in order to detect the most troublesome regions of the domain. This information helps to adapt the approximation scheme in order to avoid spurious oscillations when using high-order schemes. In this paper we propose a genuinely multidimensional extension of the WENO procedure in order to overcome the limitations of indicators based on dimensional splitting. Our aim is to obtain new regularity indicators for problems in 2D and apply them to a class of “adaptive filtered” schemes for first order evolutive Hamilton-Jacobi equations. According to the usual procedure, filtered schemes are obtained by a simple coupling of a high-order scheme and a monotone scheme. The mixture is governed by a filter function F and by a switching parameter εn=εn(Δt,Δx)>0 which goes to 0 as (Δt,Δx) is going to 0. The adaptivity is related to the smoothness indicators and allows to tune automatically the switching parameter ε^n_j in time and space. Several numerical tests on critical situations in 1D and 2D are presented and confirm the effectiveness of the proposed indicators and the efficiency of our scheme
Convergence of adaptive filtered schemes for first order evolutionary Hamilton–Jacobi equations
Full text linkWe consider a class of “filtered” schemes for first order time dependent Hamilton–Jacobi equations and prove a general convergence result for this class of schemes. A typical filtered scheme is obtained mixing a high-order scheme and a monotone scheme according to a filter function F which decides where the scheme has to switch from one scheme to the other. A crucial role for this switch is played by a parameter ε= ε(Δ t, Δ x) > 0 which goes to 0 as the time and space steps (Δ t, Δ x) are going to 0 and does not depend on the time tn, for each iteration n. The tuning of this parameter in the code is rather delicate and has an influence on the global accuracy of the filtered scheme. Here we introduce an adaptive and automatic choice of ε= εn(Δ t, Δ x) at every iteration modifying the classical set up. The adaptivity is controlled by a smoothness indicator which selects the regions where we modify the regularity threshold εn. A convergence result and some error estimates for the new adaptive filtered scheme are proved, this analysis relies on the properties of the scheme and of the smoothness indicators. Finally, we present some numerical tests to compare the adaptive filtered scheme with other methods
A comprehensive introduction to photometric 3D-reconstruction
Full text linkPhotometric 3D-reconstruction techniques aim at inferring the geometry of a scene from one or several images, by inverting a physical model describing the image formation. This chapter presents an introductory overview of the main photometric 3D-reconstruction techniques which are shape-from-shading, photometric stereo and shape-from-polarisation
A Conservative a-Posteriori Time-Limiting Procedure in Quinpi Schemes
Full text linkThe superior stability properties of implicit time schemes allow to avoid small time steps required to satisfy restrictive stability conditions for stiff hyperbolic systems. In Puppo et al. (Commun Appl Math Comput 2022) an implicit third order finite volume scheme based on a third order DIRK combined with a third order CWENO reconstruction for the space-limiting was proposed. The originality of the proposed method, named Quinpi, lies in the computation of a first order implicit predictor which is used to fix the nonlinear weights of the space reconstruction, thus simplifying considerably the non-linearity of the scheme. However, the time-limiting in the above mentioned paper, which is necessary to control spurious oscillations in the implicit time integration, requires a conservative correction. In this work, we address this problem and we propose a conservative a-posteriori time-limiting procedure inspired by the MOOD method. The numerical experiments show the reliability of the proposed scheme and include both linear and nonlinear scalar conservation laws
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