Let H be a bounded and Lipschitz continuous function. We consider discontinuous viscosity solutions of the Hamilton–Jacobi equation $$U_{t}+H(U_x)=0$$ signed Radon measure valued entropy conservation law $$u_{t}+[H(u)]_x=0$$ . After having proved precise statement formal relation $$U_x=u$$ , we establish estimates for (strictly positive!) times at which singularities disappear. Here are jump di...

We study the fine properties of a class weak solutions $u$ eikonal equation arising as asymptotic domain family energy functionals introduced in (Rivi\`ere T, Serfaty S. Limiting wall for problem related to micromagnetics. Comm Pure Appl Math 2001; 54(3):294-338). In particular we prove that entropy defect measure associated is concentrated on 1-rectifiable set, which detects jump-type disconti...

We present a numerical method for the variable coefficient Poisson equation in three dimensional irregular domains and with interfacial discontinuities. The discretization embeds the domain and interface into a uniform Cartesian grid augmented with virtual degrees of freedom to provide accurate treatment of jump and boundary conditions. The matrix associated with the discretization is symmetric...

Abstract In phase change simulations, material properties such as density, viscosity, or thermal conductivity may exhibit jump discontinuities, possibly of several orders magnitude. These discontinuities represent interfaces between the phases, and they emerge naturally during simulation; thus, their exact location is generally unknown a priori. Our goal to simulate processes with meshfree gene...

When modelling a chemical process, a modeller is usually required to handle a wide variations in time and/or length scales of its underlying differential equations by eliminating either the faster or slower dynamics. When compelled to deal with both and simultaneously simplify model structure, he/she is sometimes forced to make decisions that render the resulting model discontinuous. Discontinu...

We consider the estimation of nonparametric regression function with long memory data and investigate the asymptotic rates of convergence of estimators based on block thresholding. We show that the estimators achieve optimal minimax convergence rates over a large class of functions that involve many irregularities of a wide variety of types, including chip and Doppler functions, and jump discon...

Abstract We consider a prototypical nonlinear parabolic equation whose flux has three distinguished features: it is with respect to both the unknown and its gradient, homogeneous, depends only on direction of gradient. For such equation, we obtain existence uniqueness entropy solutions Dirichlet problem, homogeneous Neumann Cauchy problem. Qualitative properties solutions, as finite speed propa...

Tikhonov regularization is a popular and effective method for the approximate solution of illposed problems, including Fredholm equations of the first kind. The Tikhonov method works well when the solution of the equation is well-behaved, but fails for solutions with irregularities, such as jump discontinuities. In this paper we develop a method that overcomes the limitations of the standard Ti...

Segmentation is ajimdamental process affecting the overall quality and utility of a machine vision system. Range Profile Tracking (RPT) is a systematic approach for stable, accurate and high speed segmentation of range images that is based on Kalman filtering. Tests of RPT have produced stable decompositions of second order surfaces bounded by jump and crease discontinuities, having a volumetri...

The Immersed Interface Method is employed to solve the time-varying electric field equations around a three-dimensional vesicle. To achieve second-order accuracy the implicit jump conditions for the electric potential, up to the second normal derivative, are derived. The trans-membrane potential is determined implicitly as part of the algorithm. The method is compared to an analytic solution ba...