A numerical solution of Burgers' equation based on least squares approximation

WOS: 000237866800027Burgers' equation which is one-dimensional non-linear partial differential equation was converted to p non-linear ordinary differential equations by using the method of discretization in time. Each of them was solved by the least squares method. For various values of viscosity at different time steps, the numerical solutions obtained were compared with the exact solutions. It was seen that both of them were in excellent agreement. While the exact solution was not available for viscosity smaller than 0.01, it was shown that mathematical structure of the equation for the obtained numerical solutions did not decay. (c) 2005 Elsevier Inc. All rights reserved

A finite element approach for solution of Burgers' equation

WOS: 000180511400017In this study, for the Burgers' equation which is used a model problem in turbulence and shock wave theory was given a finite element solution. Considering the numerical results obtained for different values of viscosity compared with Cole's analytical solution, it was seen that they were in good agreement. (C) 2002 Published by Elsevier Science Inc

LiP-Flow: Learning Inference-Time Priors for Codec Avatars via Normalizing Flows in Latent Space

Neural face avatars that are trained from multi-view data captured in camera domes can produce photo-realistic 3D reconstructions. However, at inference time, they must be driven by limited inputs such as partial views recorded by headset-mounted cameras or a front-facing camera, and sparse facial landmarks. To mitigate this asymmetry, we introduce a prior model that is conditioned on the runtime inputs and tie this prior space to the 3D face model via a normalizing flow in the latent space. Our proposed model, LiP-Flow, consists of two encoders that learn representations from the rich training-time and impoverished inference-time observations. A normalizing flow bridges the two representation spaces and transforms latent samples from one domain to another, allowing us to define a latent likelihood objective. We trained our model end-to-end to maximize the similarity of both representation spaces and the reconstruction quality, making the 3D face model aware of the limited driving signals. We conduct extensive evaluations where the latent codes are optimized to reconstruct 3D avatars from partial or sparse observations. We show that our approach leads to an expressive and effective prior, capturing facial dynamics and subtle expressions better. Check out our project page for an overview.ISSN:0302-9743ISSN:1611-334

Selection shapes the robustness of ligand-binding amino acids

The phenotypes of biological systems are to some extent robust to genotypic changes. Such robustness exists on multiple levels of biological organization. We analyzed this robustness for two categories of amino acids in proteins. Specifically, we studied the codons of amino acids that bind or do not bind small molecular ligands. We asked to what extent codon changes caused by mutation or mistranslation may affect physicochemical amino acid properties or protein folding. We found that the codons of ligand-binding amino acids are on average more robust than those of non-binding amino acids. Because mistranslation is usually more frequent than mutation, we speculate that selection for error mitigation at the translational level stands behind this phenomenon. Our observations suggest that natural selection can affect the robustness of very small units of biological organization

High Order C0C^0 C 0 -Continuous Galerkin Schemes for High Order PDEs, Conservation of Quadratic Invariants and Application to the Korteweg-de Vries Model

International audienceWe address the Korteweg-de Vries equation as an interesting model of high order partial differential equation, and show that it is possible to develop reliable and effective schemes, in terms of accuracy, computational efficiency, simplicity of implementation and, if required, conservation of the lower invariants, on the basis of a (only) $H^1$-conformal Galerkin approximation, namely the Spectral Element Method. The proposed approach is {\it a priori} easily extensible to other partial differential equations and to multidimensional problems