Gradient Algorithm with a Smooth Objective Function

2020 ◽  
pp. 127-150
Author(s):  
Alexander J. Zaslavski
Keyword(s):  
Objective Function ◽  
Gradient Algorithm

scholarly journals Optimal Control of Thermal Damage to Targetted Regions in a Biological Material

Volume 4 ◽  
2004 ◽  
Author(s):  
F. Scott Gayzik ◽  
Elaine P. Scott ◽  
Tahar Loulou
Keyword(s):  
Objective Function ◽  
Thermal Damage ◽  
Numerical Technique ◽  
Damage Model ◽  
Gradient Algorithm ◽  
Bioheat Equation ◽  
Hyperthermia Treatment ◽  
Potential Applications ◽  
Damage Field ◽  
The Relationship

A numerical technique with potential applications in hyperthermia treatment planning is presented. The treatment is simulated using a 2D transient computational model of the Pennes bioheat equation within an optimization algorithm. The algorithm recovers the heating protocol which will lead to a desired damage field. The relationship between temperature, time and thermal damage is expressed as a first order rate process using the Arrhenius equation. The objective function of the control problem is based on this thermal damage model. The adjoint method in conjunction with the conjugate gradient algorithm is used to minimize the objective function. The results from a numerical simulation show good agreement between the optimal damage field and the damage field recovered by the algorithm. A comparison between the recovered damage field and the commonly used thermal dose is also made.

scholarly journals Riemannian Gradient Algorithm for the Numerical Solution of Linear Matrix Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Xiaomin Duan ◽  
Huafei Sun ◽  
Xinyu Zhao
Keyword(s):  
Numerical Solution ◽  
Objective Function ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Geodesic Distance ◽  
Convergence Speed ◽  
Gradient Algorithm ◽  
Linear Matrix ◽  
Geodesic Curve

A Riemannian gradient algorithm based on geometric structures of a manifold consisting of all positive definite matrices is proposed to calculate the numerical solution of the linear matrix equationQ=X+∑i=1mAiTXAi. In this algorithm, the geodesic distance on the curved Riemannian manifold is taken as an objective function and the geodesic curve is treated as the convergence path. Also the optimal variable step sizes corresponding to the minimum value of the objective function are provided in order to improve the convergence speed. Furthermore, the convergence speed of the Riemannian gradient algorithm is compared with that of the traditional conjugate gradient method in two simulation examples. It is found that the convergence speed of the provided algorithm is faster than that of the conjugate gradient method.

Nonlinear inversion of geoelectric data acquired across 3D objects using a finite-element approach

Geophysics ◽  
2008 ◽  
Vol 73 (3) ◽  
pp. F121-F133 ◽  
Author(s):  
Laurent Marescot ◽  
Sérgio Palma Lopes ◽  
Stéphane Rigobert ◽  
Alan G. Green
Keyword(s):  
Finite Element ◽  
Objective Function ◽  
Reference Model ◽  
Element Model ◽  
Gradient Algorithm ◽  
Complex Object ◽  
Nonlinear Inversion ◽  
Finite Element Approach ◽  
Element Approach ◽  
Geoelectric Data

We introduce a new finite-element-based scheme for the fast nonlinear inversion of large 3D geoelectric data sets acquired around isolated objects or across the earth’s surface. The principal novelty of this scheme is the combination of a versatile finite-element approach with (1) a method involving minimization of an objective function using a conjugate-gradient algorithm that includes an adjoint-field technique for efficiently establishing the objective-function gradient and (2) parabolic interpolation for estimating suitable inversion step lengths. This scheme is capable of handling large volumes of data acquired using diverse electrode configurations located around or across 3D structures. Only three solutions to the forward problem are required for each iteration. Computation of the Jacobian matrix, which might require computers with a large amount of memory, is not necessary. To minimize artificial irregularities in the inverted models, particularly near the electrodes, we smooth the model parametersafter each iteration. By including the influence of a reference model in the objective function, a priori information can be incorporated in the inversion process. Our new scheme is tested successfully on synthetic data generated for current and potential electrodes distributed around the surface of a complex object of finite extent. We also demonstrate the utility of the new scheme on geoelectric data acquired around a laboratory-scale object. Tomographic inversion of the 52,272 simulated voltage values in terms of an 8775-element model requires less than 45 minutes on a relatively slow Sun workstation. For the inversion of the 1016 observed voltage values in terms of an 81,480-element model, approximately 60 minutes of computer time is required. The rapid and flexible inversion scheme opens up new possibilities for resistivity imaging in geology, hydrology, engineering, nondestructive testing, and even biology and medicine, fields of study in which finite-element models are already used to represent complicated targets.

scholarly journals Inexact Version of Bregman Proximal Gradient Algorithm

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
S. Kabbadj
Keyword(s):  
Objective Function ◽  
Convex Functions ◽  
Gradient Algorithm ◽  
Simple Condition ◽  
Proximal Gradient Algorithm

The Bregman Proximal Gradient (BPG) algorithm is an algorithm for minimizing the sum of two convex functions, with one being nonsmooth. The supercoercivity of the objective function is necessary for the convergence of this algorithm precluding its use in many applications. In this paper, we give an inexact version of the BPG algorithm while circumventing the condition of supercoercivity by replacing it with a simple condition on the parameters of the problem. Our study covers the existing results, while giving other.

Pretension Optimization Method for Complex Cablenet System

2014 ◽  
Vol 574 ◽  
pp. 143-146
Author(s):  
Guo Qiang You ◽  
Ying Bai Xie
Keyword(s):  
Objective Function ◽  
Axial Symmetry ◽  
Mathematic Model ◽  
Optimization Method ◽  
Gradient Algorithm ◽  
Matrix Analysis ◽  
Analysis Method ◽  
Design Variables ◽  
Reduced Gradient ◽  
Generalized Reduced Gradient

Based on balance matrix analysis method and considered the evenness of pretension distribution as objective function, a pretension optimization method is proposed for complex cablenet system of large span structure. In this method, the whole cablenet system is firstly divided into several groups according to its axial symmetry to simplify its balance matrix, and then balance matrix analysis method is used to analyze balance matrix of grouped cablenet system. Next, the corresponding optimum mathematic model for grouped cablenet system can be established with pretension solutions coefficients as design variables and evenness of pretension distribution as objective function. Finally, generalized reduced gradient algorithm is used to solve the optimum mathematic model of an example, and the result is satisfactory.

Power Generation for India with Higher Efficiency

2018 ◽  
Vol 6 (7) ◽  
pp. 267
Author(s):  
Umeshkannan P ◽  
Muthurajan KG
Keyword(s):  
Power Generation ◽  
Objective Function ◽  
Fossil Fuels ◽  
Programming Model ◽  
Developed Countries ◽  
Power Requirement ◽  
Average Efficiency ◽  
Potential Demand ◽  
The Developed Countries ◽  
Developed Nations

The developed countries are consuming more amount of energy in all forms including electricity continuously with advanced technologies.  Developing  nation’s  energy usage trend rises quickly but very less in comparison with their population and  their  method of generating power is not  seems  to  be  as  advanced  as  developed  nations. The   objective   function   of   this   linear   programming model is to maximize the average efficiency of power generation inIndia for 2020 by giving preference to energy efficient technologies. This model is subjected to various constraints like potential, demand, running cost and Hydrogen / Carbon ratio, isolated load, emission and already installed capacities. Tora package is used to solve this linear program. Coal,  Gas,  Hydro  and  Nuclear  sources can are  supply around 87 %  of  power  requirement .  It’s concluded that we can produce power  at  overall  efficiency  of  37%  while  meeting  a  huge demand  of  13,00,000  GWh  of  electricity.  The objective function shows the scenario of highaverage efficiency with presence of 9% renewables. Maximum value   is   restricted   by   low   renewable   source’s efficiencies, emission constraints on fossil fuels and cost restriction on some of efficient technologies. This    model    shows    that    maximum    18%    of    total requirement   can   be   met   by   renewable itself which reduces average efficiency to 35.8%.   Improving technologies  of  renewable  sources  and  necessary  capacity addition  to  them in  regular  interval  will  enhance  their  role and existence against fossil fuels in future. The work involves conceptualizing, modeling, gathering information for data’s to be used in model for problem solving and presenting different scenarios for same objective.

Developing Schedule With Linear Programming (Case Study: STTF II Project Komplek Sukamukti Banjaran)

2020 ◽  
Vol 4 (02) ◽  
pp. 34-45
Author(s):  
Naufal Dzikri Afifi ◽  
Ika Arum Puspita ◽  
Mohammad Deni Akbar
Keyword(s):  
Linear Programming ◽  
Objective Function ◽  
Project Scheduling ◽  
Time Limit ◽  
Minimum Time ◽  
Service Cost ◽  
Trade Off ◽  
Overtime Work ◽  
The Cost

Shift to The Front II Komplek Sukamukti Banjaran Project is one of the projects implemented by one of the companies engaged in telecommunications. In its implementation, each project including Shift to The Front II Komplek Sukamukti Banjaran has a time limit specified in the contract. Project scheduling is an important role in predicting both the cost and time in a project. Every project should be able to complete the project before or just in the time specified in the contract. Delay in a project can be anticipated by accelerating the duration of completion by using the crashing method with the application of linear programming. Linear programming will help iteration in the calculation of crashing because if linear programming not used, iteration will be repeated. The objective function in this scheduling is to minimize the cost. This study aims to find a trade-off between the costs and the minimum time expected to complete this project. The acceleration of the duration of this study was carried out using the addition of 4 hours of overtime work, 3 hours of overtime work, 2 hours of overtime work, and 1 hour of overtime work. The normal time for this project is 35 days with a service fee of Rp. 52,335,690. From the results of the crashing analysis, the alternative chosen is to add 1 hour of overtime to 34 days with a total service cost of Rp. 52,375,492. This acceleration will affect the entire project because there are 33 different locations worked on Shift to The Front II and if all these locations can be accelerated then the duration of completion of the entire project will be effective

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