An Approximation Scheme for Evolutive Hamilton-Jacobi Equations

Abstract We establish sufficient conditions under which the rate function for the Euler approximation scheme for a solution of a one-dimensional stochastic differential equation on the torus is close to that for an exact solution of this equation.

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A new variation for the relativistic Euler equations

Advances in Difference Equations ◽

10.1186/s13662-020-02990-6 ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Mahmoud A. E. Abdelrahman ◽

Hanan A. Alkhidhr

Keyword(s):

Euler Equations ◽

Nonlinear Waves ◽

Approximation Scheme ◽

Strength Function ◽

Convergent Sequence ◽

Approximate Solutions ◽

Large Initial Data ◽

Initial Value ◽

Relativistic Euler Equations ◽

Glimm Scheme

Abstract The Glimm scheme is one of the so famous techniques for getting solutions of the general initial value problem by building a convergent sequence of approximate solutions. The approximation scheme is based on the solution of the Riemann problem. In this paper, we use a new strength function in order to present a new kind of total variation of a solution. Based on this new variation, we use the Glimm scheme to prove the global existence of weak solutions for the nonlinear ultra-relativistic Euler equations for a class of large initial data that involve the interaction of nonlinear waves.

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Empowering the configuration-IP: new PTAS results for scheduling with setup times

Mathematical Programming ◽

10.1007/s10107-021-01694-3 ◽

2021 ◽

Author(s):

Klaus Jansen ◽

Kim-Manuel Klein ◽

Marten Maack ◽

Malin Rau

Keyword(s):

Approximation Scheme ◽

Constant Factor ◽

Setup Times ◽

Linear Programs ◽

Scheduling Problems ◽

Design Of Algorithms ◽

Packing Problems ◽

Integer Linear Programs ◽

One Machine ◽

Target Locations

AbstractInteger linear programs of configurations, or configuration IPs, are a classical tool in the design of algorithms for scheduling and packing problems where a set of items has to be placed in multiple target locations. Herein, a configuration describes a possible placement on one of the target locations, and the IP is used to choose suitable configurations covering the items. We give an augmented IP formulation, which we call the module configuration IP. It can be described within the framework of n-fold integer programming and, therefore, be solved efficiently. As an application, we consider scheduling problems with setup times in which a set of jobs has to be scheduled on a set of identical machines with the objective of minimizing the makespan. For instance, we investigate the case that jobs can be split and scheduled on multiple machines. However, before a part of a job can be processed, an uninterrupted setup depending on the job has to be paid. For both of the variants that jobs can be executed in parallel or not, we obtain an efficient polynomial time approximation scheme (EPTAS) of running time $$f(1/\varepsilon )\cdot \mathrm {poly}(|I|)$$ f ( 1 / ε ) · poly ( | I | ) . Previously, only constant factor approximations of 5/3 and $$4/3 + \varepsilon $$ 4 / 3 + ε , respectively, were known. Furthermore, we present an EPTAS for a problem where classes of (non-splittable) jobs are given, and a setup has to be paid for each class of jobs being executed on one machine.

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