Triangle meshes remain the most popular data representation for surface geometry. This ubiquitous is essentially a hybrid one that decouples continuous vertex locations from discrete topological triangulation. Unfortunately, combinatorial nature of triangulation prevents taking derivatives over space possible meshings any given surface. As result, to date, mesh processing and optimization techn...

Based on minimizing a piecewise differentiable lp function subject to a single inequality constraint, this paper discusses algorithms for a discretized regularization problem for ill-posed inverse problems. We examine computational challenges of solving this regularization problem. Possible minimization algorithms such as the steepest descent method, iteratively weighted least squares (IRLS) me...

This position paper focuses on the problem of stabilization of underactuated and nonholonomic autonomous vehicles at a given posture. The stabilization problem for those systems represents a challenge for nonlinear control theory because the linearization of such systems is not controllable. In fact, as shown by Brockett [6], for this class of systems there does not exist a continuously differe...

By introducing quadratic penalty terms, a convex nonseparable quadratic network program can be reduced to an unconstrained optimization problem whose objective function is a piecewise quadratic and continuously differentiable function. A conjugate gradient method is applied to the reduced problem and its convergence is proved. The computation exploits the special network data structures origina...

We present a method for smoothing discontinuous dynamics involving contact and friction, thereby facilitating the use of local optimization techniques for control. The method replaces the standard Linear Complementarity Problem with a Stochastic Linear Complementarity Problem. The resulting dynamics are continuously differentiable, and the resulting controllers are robust to disturbances. We de...

In the classic paper [1] it is proven that the mesh-functions obtained by solving a discrete analogue of the Dirichlet problem converge, as the mesh-width approaches zero, to a harmonic function which solves the Dirichlet problem in a somewhat generalized sense. More precisely, let the function / be continuously differentiable in a bounded plane domain G and continuous in the closure G of G, an...

Solving equations of the form H(x)=0 is one most faced problem in mathematics and other science fields such as chemistry or physics. This kind cannot be solved without use iterative methods. The Steffensen-type methods, defined using divided differences are derivative free, usually considered to solve these problems when H a non-differentiable operator due its accuracy efficiency. However, gene...

We construct two counter-examples related to Fréchet differentiability in infinite dimensions. The first one gives a convex Lipschitzian function on a Banach space such that its convolution with a given measure is Fréchet differentiable only on a measure zero set. The second one gives a Borel function on a space with a Gaussian measure such that it is Lipschitzian along the Cameron–Martin subsp...