Complex, Lorentzian, and Euclidean simplicial quantum gravity: numerical methods and physical prospects

Evaluating gravitational path integrals in the Lorentzian has been a long-standing challenge due to the numerical sign problem. We show that this challenge can be overcome in simplicial quantum gravity. By deforming the integration contour into the complex, the sign fluctuations can be suppressed, for instance using the holomorphic gradient flow algorithm. Working through simple models, we show that this algorithm enables efficient Monte Carlo simulations for Lorentzian simplicial quantum gravity. In order to allow complex deformations of the integration contour, we provide a manifestly holomorphic formula for Lorentzian simplicial gravity. This leads to a complex version of simplicial gravity that generalizes the Euclidean and Lorentzian cases. Outside the context of numerical computation, complex simplicial gravity is also relevant to studies of singularity resolving processes with complex semi-classical solutions. Along the way, we prove a complex version of the Gauss-Bonnet theorem, which may be of independent interest.

Application of Meshless local Petrov-Galerkin approach for steady state groundwater flow modeling

The complicated behavior of groundwater system in an arid aquifer is generally studied by solving the governing equations using either analytical or numerical methods. In this regard, analytical methods are just for some aquifers with regular boundaries. Numerical methods used for this aim are finite difference (FDM) and finite element methods (FEM) which are engaged for some simple aquifers. Using them in the complex cases with irregular boundaries has some shortcomings, depended on meshes. In this study, meshless local Petrov-Galerkin (MLPG) method based on the moving kriging (MK) approximation function is used to simulate groundwater flow in steady state over three aquifers, two standard and a real field aquifer. Moving kriging function known as new function which reduces the uncertain parameter. For the first aquifer, a simple rectangular aquifer, MLPG-MK indicates good agreement with analytical solutions. In the second one, aquifer conditions get more complicated. However, MLPG-MK reveals results more accurate than FDM. RMSE for MLPG-MK and FDM is 0.066 and 0.322 m respectively. In the third aquifer, Birjand unconfined aquifer located in Iran is investigated. In this aquifer, there are 190 extraction wells. The geometry of the aquifer is irregular as well. With this challenging issues, MLPG-MK again shows satisfactory accuracy. As the RMSE for MLPG-MK and FDM are 0.483 m and 0.566 m. therefore, planning for this aquifer based on the MLPG-MK is closer to reality.

AUTOMATION OF CONSTRUCTION OF A SCHEME FOR SOLVING COMPUTE- INTENSIVE PROBLEMS OF MATHEMATICAL PHYSICS ON SUPERCOMPUTERS

В статье представлен подход к разработке информационно-аналитической системы, помогающей исследователю решать вычислительно сложные задачи математической физики на суперкомпьютерах. Система автоматически строит схему решения задачи по спецификации пользователя, введенной им в режиме диалога. Схема включает наиболее подходящие математические модели для решения задачи, численные методы, алгоритмы и параллельные архитектуры, ссылки на доступные фрагменты параллельного кода, которые пользователь может использовать при разработке собственного кода. Построение схемы осуществляется на основе онтологии проблемной области «Решение вычислительно сложных задач математической физики», онтологии заданной предметной области и экспертных правил, построенных с использованием технологии Semantic Web. The paper presents an approach to the development of an information-analytical system that helps a researcher to solve compute-intensive problems of mathematical physics on supercomputers. The system automatically builds a scheme for solving the problem according to the user's specification entered by him in the dialogue mode. The scheme includes the most suitable mathematical models for solving the problem, numerical methods, algorithms and parallel architectures, links to available fragments of parallel code that the user can use when developing their own code. The construction of the scheme is carried out on the basis of the ontology of the problem area "Solving compute-intensive problems of mathematical physics", the ontology of a given subject area and expert rules built using the Semantic Web technology.

Construction Solutions of Ordinary and Partial Differential Equations using the Analytical and Numerical Methods

In this paper we studied the solution of partial differential equations using numerical methods. The paper includes study of the solving partial differential equations of the type of parabolic, elliptic and hyperbolic, and the method of the net was used for the numerical nods, which represents a case of finite differences. We have two types of solution which are the internal solution and boundary solution. The internal solution is based on the internal nodes of the net. The boundary solution depends on the boundary nodes of the net, in addition to finding the analytical solution of the equations to compare the results. We also discussed solving the problem of Laplace, Poisson, for the importance of these equations in the applied side; Mat lab was used to find the values of tables for the values of border differences. We have derived a new formula for the solution of partial differential equations containing three independent variables.