Stochastic and Deterministic Solution of the Model

1981 ◽  
pp. 126-160
Author(s):  
Dipak R. Basu
Keyword(s):  
Deterministic Solution

scholarly journals Maximum Entropy Analysis of Flow Networks: Theoretical Foundation and Applications

Entropy ◽  
2019 ◽  
Vol 21 (8) ◽  
pp. 776 ◽  
Author(s):  
Robert K. Niven ◽  
Markus Abel ◽  
Michael Schlegel ◽  
Steven H. Waldrip
Keyword(s):  
Maximum Entropy ◽  
Joint Probability ◽  
Financial Economic ◽  
Flow Networks ◽  
Flow Network ◽  
Physical Constraints ◽  
Generalized Maximum Entropy ◽  
Deterministic Solution ◽  
Network Properties ◽  
The Cost

The concept of a “flow network”—a set of nodes and links which carries one or more flows—unites many different disciplines, including pipe flow, fluid flow, electrical, chemical reaction, ecological, epidemiological, neurological, communications, transportation, financial, economic and human social networks. This Feature Paper presents a generalized maximum entropy framework to infer the state of a flow network, including its flow rates and other properties, in probabilistic form. In this method, the network uncertainty is represented by a joint probability function over its unknowns, subject to all that is known. This gives a relative entropy function which is maximized, subject to the constraints, to determine the most probable or most representative state of the network. The constraints can include “observable” constraints on various parameters, “physical” constraints such as conservation laws and frictional properties, and “graphical” constraints arising from uncertainty in the network structure itself. Since the method is probabilistic, it enables the prediction of network properties when there is insufficient information to obtain a deterministic solution. The derived framework can incorporate nonlinear constraints or nonlinear interdependencies between variables, at the cost of requiring numerical solution. The theoretical foundations of the method are first presented, followed by its application to a variety of flow networks.

scholarly journals Feasibility of a Multigroup Deterministic Solution Method for Three-Dimensional Radiotherapy Dose Calculations

2008 ◽  
Vol 72 (1) ◽  
pp. 220-227 ◽  
Author(s):  
Oleg N. Vassiliev ◽  
Todd A. Wareing ◽  
Ian M. Davis ◽  
John McGhee ◽  
Douglas Barnett ◽  
...  
Keyword(s):  
Solution Method ◽  
Three Dimensional ◽  
Dose Calculations ◽  
Deterministic Solution ◽  
Radiotherapy Dose

scholarly journals Probability Study on the Thermal Stress Distribution in Thick HK40 Stainless Steel Pipe Using Finite Element Method

Designs ◽  
2019 ◽  
Vol 3 (1) ◽  
pp. 9
Author(s):  
Sujith Bobba ◽  
Shaik Abrar ◽  
Shaik Mujeebur Rehman
Keyword(s):  
Finite Element ◽  
Stainless Steel ◽  
High Temperature ◽  
Material Properties ◽  
Thermal Stresses ◽  
Steel Pipe ◽  
Stress Distributions ◽  
Stainless Steel Pipe ◽  
Deterministic Solution ◽  
Analytical Expressions

The present work deals with the development of a finite element methodology for obtaining the stress distributions in thick cylindrical HK40 stainless steel pipe that carries high-temperature fluids. The material properties and loading were assumed to be random variables. Thermal stresses that are generated along radial, axial, and tangential directions are generally computed using very complex analytical expressions. To circumvent such an issue, probability theory and mathematical statistics have been applied to many engineering problems, which allows determination of the safety both quantitatively and objectively based on the concepts of reliability. Monte Carlo simulation methodology is used to study the probabilistic characteristics of thermal stresses, and was implemented to estimate the probabilistic distributions of stresses against the variations arising due to material properties and load. A 2-D probabilistic finite element code was developed in MATLAB, and the deterministic solution was compared with ABAQUS solutions. The values of stresses obtained from the variation of elastic modulus were found to be low compared to the case where the load alone was varying. The probability of failure of the pipe structure was predicted against the variations in internal pressure and thermal gradient. These finite element framework developments are useful for the life estimation of piping structures in high-temperature applications and for the subsequent quantification of the uncertainties in loading and material properties.

scholarly journals Robust Maximum Coverage Facility Location Problem with Drones Considering Uncertainties in Battery Availability and Consumption

2020 ◽  
pp. 036119812096809
Author(s):  
Darshan R. Chauhan ◽  
Avinash Unnikrishnan ◽  
Miguel Figliozzi ◽  
Stephen D. Boyles
Keyword(s):  
Robust Optimization ◽  
Facility Location Problem ◽  
Computational Time ◽  
Facility Locations ◽  
Maximum Coverage ◽  
Robust Model ◽  
Average Maximum ◽  
Deterministic Solution ◽  
Battery Consumption ◽  
Battery Energy

Given a set of a spatially distributed demand for a specific commodity, potential facility locations, and drones, an agency is tasked with locating a pre-specified number of facilities and assigning drones to them to serve the demand while respecting drone range constraints. The agency seeks to maximize the demand served while considering uncertainties in initial battery availability and battery consumption. The facilities have a limited supply of the commodity being distributed and also act as a launching site for drones. Drones undertake one-to-one trips (from located facility to demand location and back) until their available battery energy is exhausted. This paper extends the work done by Chauhan et al. and presents an integer linear programming formulation to maximize coverage using a robust optimization framework. The uncertainty in initial battery availability and battery consumption is modeled using a penalty-based approach and gamma robustness, respectively. A novel robust three-stage heuristic (R3SH) is developed which provides objective values which are within 7% of the average solution reported by MIP solver with a median reduction in computational time of 97% on average. Monte Carlo simulation based testing is performed to assess the value of adding robustness to the deterministic problem. The robust model provides higher and more reliable estimates of actual coverage under uncertainty. The average maximum coverage difference between the robust optimization solution and the deterministic solution is 8.1% across all scenarios.

scholarly journals Numerical treatment of stochastic heroin epidemic model

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
M. Rafiq ◽  
Ali Raza ◽  
M. Usman Iqbal ◽  
Zubair Butt ◽  
Hafiza Anum Naseem ◽  
...  
Keyword(s):  
Numerical Analysis ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Explicit Methods ◽  
Numerical Treatment ◽  
Deterministic Solution ◽  
The Mean ◽  
Heroin Model

Abstract We have presented the numerical analysis of a stochastic heroin epidemic model in this paper. The mean of stochastic heroin model is itself a deterministic solution. The effect of reproduction number has also been observed in the stochastic heroin epidemic model. We have developed some stochastic explicit and implicitly driven explicit methods for this model. But stochastic explicit methods have flopped for certain values of parameters. In support, some theorems and graphical illustrations are presented.

The principle of the diffusion of arbitrary constants

1972 ◽  
Vol 9 (3) ◽  
pp. 519-541 ◽  
Author(s):  
Andrew D. Barbour
Keyword(s):  
Central Limit ◽  
Continuous Time ◽  
Large Population ◽  
Biological Models ◽  
Deterministic Solution ◽  
Lattice Processes

Equations are derived describing a central limit type large population approximation for continuous time Markov lattice processes in one or more dimensions, such as are commonly encountered in biological models. A method of solving the equations using only the deterministic solution of the process is explained, and it is extended by the use of a martingale argument to provide more detailed information about the process.

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