An (n +1)-D coefficient inverse problem for the stationary radiative transport equation is considered for the first time. A globally convergent so-called convexification numerical method is developed and its convergen...
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An (n +1)-D coefficient inverse problem for the stationary radiative transport equation is considered for the first time. A globally convergent so-called convexification numerical method is developed and its convergence analysis is provided. The analysis is based on a Carleman estimate. Extensive numerical studies in the two-dimensional case are presented.
We propose a globally convergent numerical method, called the convexification, to numerically compute the viscosity solution to first-order Hamilton-Jacobi equations through the vanishing viscosity process where the v...
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We propose a globally convergent numerical method, called the convexification, to numerically compute the viscosity solution to first-order Hamilton-Jacobi equations through the vanishing viscosity process where the viscosity parameter is a fixed small number. By convexification, we mean that we employ a suitable Carleman weight function to convexify the cost functional defined directly from the form of the Hamilton-Jacobi equation under consideration. The strict convexity of this functional is rigorously proved using a new Carleman estimate. We also prove that the unique minimizer of this strictly convex functional can be reached by the gradient descent method. Moreover, we show that the minimizer well approximates the viscosity solution of the Hamilton-Jacobi equation as the noise contained in the boundary data tends to zero. Some interesting numerical illustrations are presented. (C) 2021 Elsevier Inc. All rights reserved.
In this paper,firstly,we give a counterexample to point out there exist deficiencies in our previous works(Wu et *** J Glob Optim 31:45-60,2005).In addition,we improve the corresponding ***,an example is presented to ...
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In this paper,firstly,we give a counterexample to point out there exist deficiencies in our previous works(Wu et *** J Glob Optim 31:45-60,2005).In addition,we improve the corresponding ***,an example is presented to illustrate how a monotone non-convex optimization problem can be transformed into an equivalent convex minimization problem.
A new 2-D implementation of the globally convergent convexification numerical method for coefficient inverse problems is described. Based on a simplified mathematical model, numerical results from imaging of targets w...
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A new 2-D implementation of the globally convergent convexification numerical method for coefficient inverse problems is described. Based on a simplified mathematical model, numerical results from imaging of targets which mimic antipersonnel land mines demonstrate that the algorithm can detect the location(s) of the inclusion(s) from the background medium, as well as identify the material property of the inclusion(s) and the background. With the 'tails' incorporated in the new development, the computational efficiency of the algorithm is dramatically improved relatively to the previous 'tail-free' implementation.
Systems reliability plays an important role in systems design, operation and management. Systems reliability can be improved by adding redundant components or increasing the reliability levels of subsystems. Determina...
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Systems reliability plays an important role in systems design, operation and management. Systems reliability can be improved by adding redundant components or increasing the reliability levels of subsystems. Determination of the optimal amount of redundancy and reliability levels among various subsystems under limited resource constraints leads to a mixed-integer nonlinear programming problem. The continuous relaxation of this problem in a complex system is a nonconvex nonseparable optimization problem with certain monotone properties. In this paper, we propose a convexification method to solve this class of continuous relaxation problems. Combined with a branch-and-bound method, our solution scheme provides an efficient way to find an exact optimal solution to integer reliability optimization in complex systems.
A convexification method is proposed for solving a class of global optimization problems with certain monotone properties. It is shown that this class of problems can be transformed into equivalent concave minimizatio...
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A convexification method is proposed for solving a class of global optimization problems with certain monotone properties. It is shown that this class of problems can be transformed into equivalent concave minimization problems using the proposed convexification schemes. An outer approximation method can then be used to find the global solution of the transformed problem. Applications to mixed-integer nonlinear programming problems arising in reliability optimization of complex systems are discussed and satisfactory numerical results are presented.
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