The Experts below are selected from a list of 129 Experts worldwide ranked by ideXlab platform
Philip Korman - One of the best experts on this subject based on the ideXlab platform.
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a global Solution Curve for a class of periodic problems including the relativistic pendulum
arXiv: Analysis of PDEs, 2016Co-Authors: Philip KormanAbstract:Using continuation methods, we study the global Solution structure of periodic Solutions for a class of periodically forced equations, generalizing the case of relativistic pendulum. We obtain results on the existence and multiplicity of periodic Solutions. Our approach is suitable for numerical computations, and in fact we present some numerically computed bifurcation diagrams illustrating our results.
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a global Solution Curve for a class of free boundary value problems arising in plasma physics
arXiv: Analysis of PDEs, 2016Co-Authors: Philip KormanAbstract:We study the existence and multiplicity of Solutions and the global Solution Curve of the following free boundary value problem, arising in plasma physics, see R. Temam [18], and H. Berestycki and H. Brezis [3]: find a function $u(x)$ and a constant $b$, satisfying \[ \Delta u+g(x,u)=p(x) \;\; \mbox{in $D$} \] \[ u \,| \, _{\partial D}=b, \;\;\;\; \int_{\partial D} \frac{\partial u}{\partial n} \, ds=0 \,. \] Here $D \subset R^n$, is a bounded domain, with a smooth boundary. This problem can be seen as a PDE generalization of the periodic problem for one-dimensional pendulum-like equations. We use continuation techniques. Our approach is suitable for numerical computations.
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a global Solution Curve for a class of free boundary value problems arising in plasma physics
Applied Mathematics and Optimization, 2015Co-Authors: Philip KormanAbstract:We study the existence and multiplicity of Solutions and the global Solution Curve of the following free boundary value problem, arising in plasma physics, see Temam (Arch Ration Mech Anal 60(1):51---73, 1975---1976), and Berestycki and Brezis (Nonlinear Anal. 4(3):415---436, 1980): find a function $$u(x)$$ u ( x ) and a constant $$b$$ b , satisfying $$\begin{aligned}&\Delta u+g(x,u)=p(x) \; \;\text{ in }\, D \\u \;\; \;\int \limits _{\partial D} \frac{\partial u}{\partial n} \, ds=0. \end{aligned}$$ Δ u + g ( x , u ) = p ( x ) in D u | ? D = b , ? ? D ? u ? n d s = 0 . Here $$D \subset R^n$$ D ? R n , is a bounded domain, with a smooth boundary. This problem can be seen as a PDE generalization of the periodic problem for one-dimensional pendulum-like equations. We use continuation techniques. Our approach is suitable for numerical computations.
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a global Solution Curve for a class of periodic problems including the pendulum equation
Zeitschrift für Angewandte Mathematik und Physik, 2007Co-Authors: Philip KormanAbstract:Using continuation methods and bifurcation theory, we study the exact multiplicity of periodic Solutions, and the global Solution structure, for a class of periodically forced pendulum-like equations. Our results apply also to the first order equations. We also show that by choosing a forcing term, one can produce periodic Solutions with any number of Fourier coefficients arbitrarily prescribed.
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on the oscillations of the Solution Curve for a class of semilinear equations
Journal of Mathematical Analysis and Applications, 2006Co-Authors: Anahit Galstian, Philip KormanAbstract:Abstract We consider positive Solutions of the Dirichlet problem Δ u + λ ( u + sin u ) = 0 , x ∈ B , u = 0 for x ∈ ∂ B , where B is unit ball in R n , λ is a positive parameter. Let λ 1 denote the principal eigenvalue of the Laplacian on B with zero boundary conditions. We show that for 1 ⩽ n ⩽ 5 the problem has infinitely many positive Solutions at λ = λ 1 , while for n ⩾ 6 the problem has at most finitely many Solutions at any λ .
Isao Ono - One of the best experts on this subject based on the ideXlab platform.
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uniform sampling of local pareto optimal Solution Curves by pareto path following and its applications in multi objective ga
Genetic and Evolutionary Computation Conference, 2007Co-Authors: Ken Harada, Jun Sakuma, Shigenobu Kobayashi, Isao OnoAbstract:Although multi-objective GA (MOGA) is an efficient multi-objective optimization (MOO) method, it has some limitations that need to be tackled, which include unguaranteed uniformity of Solutions and uncertain finding of periphery of Pareto-optimal Solutions. It has been shown that, on bi-objective problems, which are the subject of this paper, local Pareto-optimal Solutions form Curves. In this case, some of the limitations of MOGA can be resolved by sampling the Curves uniformly in the variable space and in the objective space. This paper proposes Pareto Path Following (PPF) which does the sampling by extending the framework of Numerical Path Following, verifies that PPF exhibits the desired behaviors, and addresses the extension of PPF for problems with more than two objective functions.Application of PPF is not limited to refinement of Solutions obtained with MOGA. PPF makes it natural to have a local Pareto-optimal Solution Curve as the unit of search, which leads to Curve-based MOGA. PPF also enables examination of which Pareto-optimal Solution Curves are found by MOO methods, and performance metrics based on it can be defined. This paper proposes these applications of PPF in MOGA and compares standard MOGA and Curve-based MOGA using the metrics to reveal their characteristics.
Jan F. Van Impe - One of the best experts on this subject based on the ideXlab platform.
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a novel algorithm for fast representation of a pareto front with adaptive reSolution application to multi objective optimization of a chemical reactor
Computers & Chemical Engineering, 2017Co-Authors: Ihab Hashem, Dries Telen, Philippe Nimmegeers, Filip Logist, Jan F. Van ImpeAbstract:Abstract Solving a multi-objective optimization problem yields an infinite set of points in which no objective can be improved without worsening at least another objective. This set is called the Pareto front. A Pareto front with adaptive reSolution is a representation where the number of points at any segment of the Pareto front is directly proportional to the curvature of this segment. Such representations are attractive since steep segments, i.e., knees, are more significant to the decision maker as they have high trade-off level compared to the more flat segments of the Solution Curve. A simple way to obtain such representation is the a posteriori analysis of a dense Pareto front by a smart filter to keep only the points with significant trade-offs among them. However, this method suffers from the production of a large overhead of insignificant points as well as the absence of a clear criterion for determining the required density of the initial dense representation of the Pareto front. This paper's contribution is a novel algorithm for obtaining a Pareto front with adaptive reSolution. The algorithm overcomes the pitfalls of the smart filter strategy by obtaining the Pareto points recursively while calculating the trade-off level between the obtained points before moving to a deeper recursive call. By using this approach, once a segment of trade-offs insignificant to the decision maker's needs is identified, the algorithm stops exploring it further. The improved speed of the proposed algorithm along with its intuitively simple Solution process make it a more attractive route to solve multi-objective optimization problems in a way that better suits the decision maker's needs.
Ken Harada - One of the best experts on this subject based on the ideXlab platform.
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uniform sampling of local pareto optimal Solution Curves by pareto path following and its applications in multi objective ga
Genetic and Evolutionary Computation Conference, 2007Co-Authors: Ken Harada, Jun Sakuma, Shigenobu Kobayashi, Isao OnoAbstract:Although multi-objective GA (MOGA) is an efficient multi-objective optimization (MOO) method, it has some limitations that need to be tackled, which include unguaranteed uniformity of Solutions and uncertain finding of periphery of Pareto-optimal Solutions. It has been shown that, on bi-objective problems, which are the subject of this paper, local Pareto-optimal Solutions form Curves. In this case, some of the limitations of MOGA can be resolved by sampling the Curves uniformly in the variable space and in the objective space. This paper proposes Pareto Path Following (PPF) which does the sampling by extending the framework of Numerical Path Following, verifies that PPF exhibits the desired behaviors, and addresses the extension of PPF for problems with more than two objective functions.Application of PPF is not limited to refinement of Solutions obtained with MOGA. PPF makes it natural to have a local Pareto-optimal Solution Curve as the unit of search, which leads to Curve-based MOGA. PPF also enables examination of which Pareto-optimal Solution Curves are found by MOO methods, and performance metrics based on it can be defined. This paper proposes these applications of PPF in MOGA and compares standard MOGA and Curve-based MOGA using the metrics to reveal their characteristics.
Ihab Hashem - One of the best experts on this subject based on the ideXlab platform.
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a novel algorithm for fast representation of a pareto front with adaptive reSolution application to multi objective optimization of a chemical reactor
Computers & Chemical Engineering, 2017Co-Authors: Ihab Hashem, Dries Telen, Philippe Nimmegeers, Filip Logist, Jan F. Van ImpeAbstract:Abstract Solving a multi-objective optimization problem yields an infinite set of points in which no objective can be improved without worsening at least another objective. This set is called the Pareto front. A Pareto front with adaptive reSolution is a representation where the number of points at any segment of the Pareto front is directly proportional to the curvature of this segment. Such representations are attractive since steep segments, i.e., knees, are more significant to the decision maker as they have high trade-off level compared to the more flat segments of the Solution Curve. A simple way to obtain such representation is the a posteriori analysis of a dense Pareto front by a smart filter to keep only the points with significant trade-offs among them. However, this method suffers from the production of a large overhead of insignificant points as well as the absence of a clear criterion for determining the required density of the initial dense representation of the Pareto front. This paper's contribution is a novel algorithm for obtaining a Pareto front with adaptive reSolution. The algorithm overcomes the pitfalls of the smart filter strategy by obtaining the Pareto points recursively while calculating the trade-off level between the obtained points before moving to a deeper recursive call. By using this approach, once a segment of trade-offs insignificant to the decision maker's needs is identified, the algorithm stops exploring it further. The improved speed of the proposed algorithm along with its intuitively simple Solution process make it a more attractive route to solve multi-objective optimization problems in a way that better suits the decision maker's needs.