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Showing 1 to 13 of 13 for “"Homotopy Methods"”.
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Parallel Sparse Linear Algebra for Homotopy Methods
Globally convergent homotopy methods are used to solve difficult nonlinear systems of equations by tracking the zero curve of a homotopy map. Homotopy curve tracking involves solving a sequence of linear systems, which often vary greatly in difficulty. In this research, a popular iterative solution …
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Comparison of modified Riks/Wempner and homotopy methods
… used for tracking the equilibrium paths by the homotopy method. Two subroutines were written as required by HOMPACK. Four different structures were analyzed by these two programs. Comparison of the two methods has been carried out based on the number of Jacobian matrix evaluations and the CPU …
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Homotopy methods for solving the optimal projection equations for the reduced order model problem
… The proposed algorithms utilize probability-one homotopy theory as the main tool. It is shown that there is a family of systems (the homotopy) that make a continuous transformation from some initial system to the final system. With a carefully chosen initial problem a theorem guarantees that all …
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POLSYS_PLP: A Partitioned Linear Product Homotopy Code for Solving Polynomial Systems of Equations
Globally convergent, probability-one homotopy methods have proven to be very effective for finding all the isolated solutions to polynomial systems of equations. After many years of development, homotopy path trackers based on probability-one homotopy methods are reliable and fast. Now, theoretical …
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Convergence and Boundedness of Probability-One Homotopies for Model Order Reduction
… examines the requirements of probability-one homotopy methods to guarantee global convergence. Homotopy algorithms for nonlinear systems of equations construct a continuous family of systems, and solve the given system by tracking the continuous curve of solutions to the family. The main …
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Solution of Constrained Clustering Problems through Homotopy Tracking
Modern machine learning methods are dependent on active optimization research to improve the set of methods available for the efficient and effective extraction of information from large datasets. This, in turn, requires an intense and rigorous study of optimization methods and their possible …
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Improving shock-capturing robustness for higher-order finite element solvers
… Grid adaptation, along with shock-capturing methods such as artificial viscosity, can be used to resolve the shock by targeting the key flow features for grid refinement. This is a powerful tool, but cannot proceed without first converging on an initially coarse, unrefined mesh. These coarse …
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Optimal control with structure constraints and its application to the design of passive mechanical systems
… synthesis, linear matrix inequalities (LMI), homotopy methods, gradient- and subgradientbased optimization are used. Some new algorithms and extensions are proposed, such as a subgradient-based method to maximize the minimal damping with structured feedback, a multiplier method for structured …
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Homotopy algorithms for the H² and the combined H²/H<sup>∞</sup> model order reduction problems
… problem. Without the global convergence of homotopy methods, both the H² optimal and the combined H²/H<sup>∞</sup> model reduction problems are very difficult. For both problems homotopy algorithms based on several formulations input normal form; Ly, Bryson, and Cannon's 2 X 2 block …
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Parallel homotopy curve tracking on a hypercube
Probability-one homotopy methods are a class of methods for solving non-linear systems of equations that are globally convergent with probability one from an arbitrary starting point. The essence of these algorithms is the construction of an appropriate homotopy map and subsequent tracking of some …
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Continuous Low-Thrust Trajectory Optimization: Techniques and Applications
… scheme based on numerical and analytical methods. Whereas other conventional optimization packages rely on numerical solution approaches, we employ analytical and semi-analytical techniques such as symmetry and homotopy methods to assist in the solution-finding process. The first objective …
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Homotopy algorithms for H²/H<sup>∞</sup> control analysis and synthesis
… problem. Without the global convergence of homotopy methods, both the H² optimal and the combined H²/H<sup>∞</sup> model reduction problems are very difficult. For both problems homotopy algorithms based on several formulations—input normal form; Ly, Bryson, and Cannon's 2 X 2 block …
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Global solution to parametric complementarity constrained programs and applications in optimal parameter selection
… mapping, element-by-element inverse, and homotopy methods. These methods, however, are time-consuming if an exact solution is required, and are limited to solving specific types of numerical examples. In this chapter, we construct a hierarchical mathematical program to formulate the …