Related papers: Consistent Computation of First- and Second-Order …
A high-order quadrature algorithm is presented for computing integrals over curved surfaces and volumes whose geometry is implicitly defined by the level sets of (one or more) multivariate polynomials. The algorithm recasts the implicitly…
In this paper we investigate continuity properties of first and second order shape derivatives of functionals depending on second order elliptic PDE's around nonsmooth domains, essentially either Lipschitz or convex, or satisfying a uniform…
Finite volume methods for problems involving second order operators with full diffusion matrix can be used thanks to the definition of a discrete gradient for piecewise constant functions on unstructured meshes satisfying an orthogonality…
The usual approach to developing and analyzing first-order methods for smooth convex optimization assumes that the gradient of the objective function is uniformly smooth with some Lipschitz constant $L$. However, in many settings the…
Computational analysis with the finite element method requires geometrically accurate meshes. It is well known that high-order meshes can accurately capture curved surfaces with fewer degrees of freedom in comparison to low-order meshes.…
The problem of finding of analytical gradients (derivatives over atoms coordinates) of solvation energies can be decomposed on two subtasks: at the first stage we search for parameters of the superficial devices (three coordinates, three…
We propose Differentiable Surface Splatting (DSS), a high-fidelity differentiable renderer for point clouds. Gradients for point locations and normals are carefully designed to handle discontinuities of the rendering function.…
Large scale optimization problems are ubiquitous in machine learning and data analysis and there is a plethora of algorithms for solving such problems. Many of these algorithms employ sub-sampling, as a way to either speed up the…
Fractional derivatives are a well-studied generalization of integer order derivatives. Naturally, for optimization, it is of interest to understand the convergence properties of gradient descent using fractional derivatives. Convergence…
In the context of the optimization of Deep Neural Networks, we propose to rescale the learning rate using a new technique of automatic differentiation. This technique relies on the computation of the {\em curvature}, a second order…
In this paper we shall introduce a simple, effective numerical method for finding differential operators for scalar and vector-valued functions on surfaces. The key idea of our algorithm is to develop an intrinsic and unified way to compute…
This paper presents a method for computing two-dimensional constant mean curvature surfaces. The method in question uses the variational aspect of the problem to implement an efficient algorithm. In principle it is a flow like method in…
Standard first-order stochastic optimization algorithms base their updates solely on the average mini-batch gradient, and it has been shown that tracking additional quantities such as the curvature can help de-sensitize common…
We study the dynamics of a droplet moving on an inclined rough surface in the absence of inertial and viscous stress effects. In this case, the dynamics of the droplet is a purely geometric motion in terms of the wetting domain and the…
This paper presents a high-order accurate numerical quadrature algorithm for evaluating integrals over curved surfaces and regions defined implicitly via a level set of a given function restricted to a hyperrectangle. The domain is divided…
Discontinuities in spatial derivatives appear in a wide range of physical systems, from creased thin sheets to materials with sharp stiffness transitions. Accurately modeling these features is essential for simulation but remains…
Curvature principal directions on geometric surfaces are a ubiquitous concept of Geometry Processing techniques. However they only account for order 2 differential quantities, oblivious of higher order differential behaviors. In this paper,…
The differentiable programming paradigm is a cornerstone of modern scientific computing. It refers to numerical methods for computing the gradient of a numerical model's output. Many scientific models are based on differential equations,…
Several machine learning applications involve the optimization of higher-order derivatives (e.g., gradients of gradients) during training, which can be expensive in respect to memory and computation even with automatic differentiation. As a…
Computing intrinsic distances on discrete surfaces is at the heart of many minimization problems in geometry processing and beyond. Solving these problems is extremely challenging as it demands the computation of on-surface distances along…