Related papers: Characteristic Evolution and Matching
We survey some of our recent results on inverse problems for evolution equations. The goal is to provide a unified approach to solve various types of evolution equations. The inverse problems we consider consist in determining unknown…
We study the generation of singularities from the initial datum for a solution of the Cauchy problem for a class of Hamilton-Jacobi equations of evolution. For such equations, we give conditions for the existence of singular generalized…
A chain regularization method is combined with special purpose computer hardware to study the evolution of massive black hole binaries at the centers of galaxies. Preliminary results with up to N=260,000 particles are presented. The decay…
We characterize the behavior of the solutions of linear evolution partial differential equations on the half line in the presence of discontinuous initial conditions or discontinuous boundary conditions, as well as the behavior of the…
The formulation of the initial value problem for the Einstein equations is at the heart of obtaining interesting new solutions using numerical relativity and still very much under theoretical and applied scrutiny. We develop a specialised…
We investigate the initial-value problem of the non-linear Liouville hierarchy. For the general form of the interaction potential we construct an explicit solution in terms of an expansion over particle clusters whose evolution is described…
Molecular traits, such as gene expression levels or protein binding affinities, are increasingly accessible to quantitative measurement by modern high-throughput techniques. Such traits measure molecular functions and, from an evolutionary…
We present a new method of extracting gravitational radiation from three-dimensional numerical relativity codes and providing outer boundary conditions. Our approach matches the solution of a Cauchy evolution of Einstein's equations to a…
We consider a class of partial differential equations with Carlitz derivatives over a local field of positive characteristic, for which an analog of the Cauchy problem is well-posed. Equations of such type correspond to quasi-holonomic…
We model a radiating, moving black hole in terms of a worldtube-nullcone boundary value problem. We evolve this data in the region interior to the worldtube but exterior to a trapped surface by means of a characteristic evolution based upon…
Although the traditional form of the Einstein field equations is intrinsically four-dimensional, the field of numerical general relativity focuses on the reformulation of these equations as a 3 + 1-dimensional Cauchy problem, in which…
Global properties of maximal future Cauchy developments of stationary, m-dimensional asymptotically flat initial data with an outer trapped boundary are analyzed. We prove that, whenever the matter model is well posed and satisfies the null…
Evolutionary algorithms are well suited for solving the knapsack problem. Some empirical studies claim that evolutionary algorithms can produce good solutions to the 0-1 knapsack problem. Nonetheless, few rigorous investigations address the…
The paper provides a direct proof the uniqueness of solutions to the Camassa-Holm equation, based on characteristics. Given a conservative solution $u=u(t,x)$, an equation is introduced which singles out a unique characteristic curve…
To observe the dynamic formation of black holes in general relativity, one essentially needs to prove that closed trapped surfaces form during evolution from initial data that do not already contain trapped surfaces. We discuss the recent…
The mathematical runtime analysis of evolutionary algorithms traditionally regards the time an algorithm needs to find a solution of a certain quality when initialized with a random population. In practical applications it may be possible…
Over the past 30 years many researchers in the field of evolutionary computation have put a lot of effort to introduce various approaches for solving hard problems. Most of these problems have been inspired by major industries so that…
Recently, it has been proven that evolutionary algorithms produce good results for a wide range of combinatorial optimization problems. Some of the considered problems are tackled by evolutionary algorithms that use a representation which…
We develop and calibrate a characteristic waveform extraction tool whose major improvements and corrections of prior versions allow satisfaction of the accuracy standards required for advanced LIGO data analysis. The extraction tool uses a…
Numerical relativity codes now being developed will evolve initial data representing colliding black holes at a relatively late stage in the collision. The choice of initial data used for code development has been made on the basis of…