The Experts below are selected from a list of 35496 Experts worldwide ranked by ideXlab platform
Guofeng Zhang - One of the best experts on this subject based on the ideXlab platform.
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convergence behavior of generalized parameterized uzawa method for singular Saddle Point problems
2017Co-Authors: Zhaozheng Liang, Guofeng ZhangAbstract:In this paper, we will seek the least squares solution for singular Saddle-Point problems. The parameterized Uzawa (PU) method is further studied and a generalized PU (GPU) proper splitting is proposed. The convergence behavior of the corresponding GPU iteration is studied. It is proved that the GPU iteration method can converge to the best least squares solutions of the singular Saddle-Point problems. In addition, we prove that the GPU preconditioned GMRES for singular Saddle-Point problems will also determine the least squares solution at breakdown. The eigenvalue distributions of the GPU preconditioned matrix are derived. Numerical experiments are presented, which show that the convergence behavior of the singular preconditioning is significantly better than that of the corresponding nonsingular case and demonstrate that the GPU iteration has better convergence behavior than the PU iteration, both as a solver and a preconditioner of GMRES.
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variants of the accelerated parameterized inexact uzawa method for Saddle Point problems
2016Co-Authors: Zhaozheng Liang, Guofeng ZhangAbstract:In this paper, based on the SOR and SSOR splittings of the (1,1) part of Saddle-Point coefficient matrix, some variants of the accelerated parameterized inexact Uzawa (VAPIU) method are proposed for solving nonsingular and singular Saddle-Point problems. By choosing different parameter matrices, we derive some existing and new iterative methods. The corresponding convergence and semi-convergence of the VAPIU methods for solving nonsingular and singular Saddle-Point problems are studied in depth, respectively. The preconditioning strategies based on the VAPIU splittings of the coefficient matrices are presented. Numerical experiments are provided, which confirms that these new methods need less CPU times per iteration step comparing with some other methods for solving both nonsingular and singular Saddle-Point problems.
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pu sts method for non hermitian Saddle Point problems
2015Co-Authors: Zhaozheng Liang, Guofeng ZhangAbstract:Abstract Based on the skew-Hermitian triangular splitting (STS) of the (1,1) part of Saddle-Point coefficient matrix, a modified Uzawa method is proposed for solving non-Hermitian Saddle-Point problems with non-Hermitian positive definite and skew-Hermitian dominant (1,1) part. Convergence properties of this method are analyzed and the corresponding convergence result is derived under suitable conditions. Numerical experiments are provided to confirm the theoretical results, which demonstrate that this method is effective and feasible for Saddle-Point problems with non-Hermitian positive definite and skew-Hermitian dominant (1,1) part.
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on parameterized generalized skew hermitian triangular splitting iteration method for singular and nonsingular Saddle Point problems
2015Co-Authors: Guofeng Zhang, Lidan Liao, Zhaozheng LiangAbstract:Recently, Krukier et al. (2014) and Dou et al. (2014) have studied the generalized skew-Hermitian triangular splitting (GSTS) iteration method for nonsingular and singular Saddle Point problems, respectively. In this paper, we further extend the GSTS method to a parameterized GSTS (PGSTS) method for solving non-Hermitian nonsingular and singular Saddle Point problems. By singular value decomposition technique, we derive conditions of the new iterative method for guaranteeing the convergence for non-Hermitian nonsingular Saddle Point problems and its semi-convergence for singular Saddle Point problems, respectively. In addition, the choice of the acceleration parameters in a practical manner is studied. Numerical experiments are provided, which further confirm our theoretical results and show the new method is feasible and effective for non-Hermitian nonsingular or singular Saddle Point problems.
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on hss based sequential two stage method for non hermitian Saddle Point problems
2014Co-Authors: Muzheng Zhu, Guofeng Zhang, Zhong Zheng, Zhaozheng LiangAbstract:Abstract For large sparse Saddle Point problems with symmetric positive definite (1, 1)-block, Li et al. studied an efficient iterative method (see Li et al. (2011)) [25]. By making use of the same preconditioning technique and a new matrix splitting based on the Hermitian and skew-Hermitian splitting (HSS) of the (1, 1)-block of the preconditioned non-Hermitian Saddle Point systems, an efficient sequential two-stage method is proposed for solving the non-Hermitian Saddle Point problems. Theoretical analysis shows the proposed iterative method is convergent, and that the spectral radius of iterative matrix monotonically decreases and tends to 0 as the iterative parameter α approaches infinity. Numerical experiments arising from Naiver–Stokes problem are provided to show that the new iterative method is feasible, effective and robust.
Lukas Novotny - One of the best experts on this subject based on the ideXlab platform.
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optical phonon resonances with Saddle Point excitons in twisted bilayer graphene
2014Co-Authors: A Jorio, Mark Kasperczyk, Nick Clark, Elke Neu, Patrick Maletinsky, Aravind Vijayaraghavan, Lukas NovotnyAbstract:Twisted-bilayer graphene (tBLG) exhibits van Hove singularities in the density of states that can be tuned by changing the twisting angle θ. A θ-defined tBLG has been produced and characterized with optical reflectivity and resonance Raman scattering. The θ-engineered optical response is shown to be consistent with persistent Saddle-Point excitons. Separate resonances with Stokes and anti-Stokes Raman scattering components can be achieved due to the sharpness of the two-dimensional Saddle-Point excitons, similar to what has been previously observed for one-dimensional carbon nanotubes. The excitation power dependence for the Stokes and anti-Stokes emissions indicate that the two processes are correlated and that they share the same phonon.
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optical phonon resonances with Saddle Point excitons in twisted bilayer graphene
2014Co-Authors: A Jorio, Mark Kasperczyk, Nick Clark, Elke Neu, Patrick Maletinsky, Aravind Vijayaraghavan, Lukas NovotnyAbstract:Twisted-bilayer graphene (tBLG) exhibits van Hove singularities in the density of states that can be tuned by changing the twisting angle $\theta$. A $\theta$-defined tBLG has been produced and characterized with optical reflectivity and resonance Raman scattering. The $\theta$-engineered optical response is shown to be consistent with persistent Saddle-Point excitons. Separate resonances with Stokes and anti-Stokes Raman scattering components can be achieved due to the sharpness of the two-dimensional Saddle-Point excitons, similar to what has been previously observed for one-dimensional carbon nanotubes. The excitation power dependence for the Stokes and anti-Stokes emissions indicate that the two processes are correlated and that they share the same phonon.
Jorge Cortes - One of the best experts on this subject based on the ideXlab platform.
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Distributed Coordination for Nonsmooth Convex Optimization via Saddle-Point Dynamics
2019Co-Authors: Jorge Cortes, Simon K. NiederländerAbstract:This paper considers continuous-time coordination algorithms for networks of agents that seek to collectively solve a general class of nonsmooth convex optimization problems with an inherent distributed structure. Our algorithm design builds on the characterization of the solutions of the nonsmooth convex program as Saddle Points of an augmented Lagrangian. We show that the associated Saddle-Point dynamics are asymptotically correct but, in general, not distributed because of the presence of a global penalty parameter. This motivates the design of a discontinuous Saddle-Point-like algorithm that enjoys the same convergence properties and is fully amenable to distributed implementation. Our convergence proofs rely on the identification of a novel global Lyapunov function for Saddle-Point dynamics. This novelty also allows us to identify mild convexity and regularity conditions on the objective function that guarantee the exponential convergence rate of the proposed algorithms for convex optimization problems subject to equality constraints. Various examples illustrate our discussion.
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distributed Saddle Point subgradient algorithms with laplacian averaging
2017Co-Authors: David Mateosnunez, Jorge CortesAbstract:We present distributed subgradient methods for min-max problems with agreement constraints on a subset of the arguments of both the convex and concave parts. Applications include constrained minimization problems where each constraint is a sum of convex functions in the local variables of the agents. In the latter case, the proposed algorithm reduces to primal-dual updates using local subgradients and Laplacian averaging on local copies of the multipliers associated to the global constraints. For the case of general convex-concave Saddle-Point problems, our analysis establishes the convergence of the running time-averages of the local estimates to a Saddle Point under periodic connectivity of the communication digraphs. Specifically, choosing the gradient step-sizes in a suitable way, we show that the evaluation error is proportional to $1/\sqrt{t}$ , where $t$ is the iteration step. We illustrate our results in simulation for an optimization scenario with nonlinear constraints coupling the decisions of agents that cannot communicate directly.
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Saddle Point dynamics conditions for asymptotic stability of Saddle Points
2017Co-Authors: Ashish Cherukuri, Bahman Gharesifard, Jorge CortesAbstract:This paper considers continuously differentiable functions of two vector variables that have (possibly a continuum of) min-max Saddle Points. We study the asymptotic convergence properties of the associated Saddle-Point dynamics (gradient descent in the first variable and gradient ascent in the second one). We identify a suite of complementary conditions under which the set of Saddle Points is asymptotically stable under the Saddle-Point dynamics. Our first set of results is based on the convexity-concavity of the function defining the Saddle-Point dynamics to establish the convergence guarantees. For functions that do not enjoy this feature, our second set of results relies on properties of the linearization of the dynamics, the function along the proximal normals to the Saddle set, and the linearity of the function in one variable. We also provide global versions of the asymptotic convergence results. Various examples illustrate our discussion.
Zhaozheng Liang - One of the best experts on this subject based on the ideXlab platform.
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convergence behavior of generalized parameterized uzawa method for singular Saddle Point problems
2017Co-Authors: Zhaozheng Liang, Guofeng ZhangAbstract:In this paper, we will seek the least squares solution for singular Saddle-Point problems. The parameterized Uzawa (PU) method is further studied and a generalized PU (GPU) proper splitting is proposed. The convergence behavior of the corresponding GPU iteration is studied. It is proved that the GPU iteration method can converge to the best least squares solutions of the singular Saddle-Point problems. In addition, we prove that the GPU preconditioned GMRES for singular Saddle-Point problems will also determine the least squares solution at breakdown. The eigenvalue distributions of the GPU preconditioned matrix are derived. Numerical experiments are presented, which show that the convergence behavior of the singular preconditioning is significantly better than that of the corresponding nonsingular case and demonstrate that the GPU iteration has better convergence behavior than the PU iteration, both as a solver and a preconditioner of GMRES.
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variants of the accelerated parameterized inexact uzawa method for Saddle Point problems
2016Co-Authors: Zhaozheng Liang, Guofeng ZhangAbstract:In this paper, based on the SOR and SSOR splittings of the (1,1) part of Saddle-Point coefficient matrix, some variants of the accelerated parameterized inexact Uzawa (VAPIU) method are proposed for solving nonsingular and singular Saddle-Point problems. By choosing different parameter matrices, we derive some existing and new iterative methods. The corresponding convergence and semi-convergence of the VAPIU methods for solving nonsingular and singular Saddle-Point problems are studied in depth, respectively. The preconditioning strategies based on the VAPIU splittings of the coefficient matrices are presented. Numerical experiments are provided, which confirms that these new methods need less CPU times per iteration step comparing with some other methods for solving both nonsingular and singular Saddle-Point problems.
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pu sts method for non hermitian Saddle Point problems
2015Co-Authors: Zhaozheng Liang, Guofeng ZhangAbstract:Abstract Based on the skew-Hermitian triangular splitting (STS) of the (1,1) part of Saddle-Point coefficient matrix, a modified Uzawa method is proposed for solving non-Hermitian Saddle-Point problems with non-Hermitian positive definite and skew-Hermitian dominant (1,1) part. Convergence properties of this method are analyzed and the corresponding convergence result is derived under suitable conditions. Numerical experiments are provided to confirm the theoretical results, which demonstrate that this method is effective and feasible for Saddle-Point problems with non-Hermitian positive definite and skew-Hermitian dominant (1,1) part.
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on parameterized generalized skew hermitian triangular splitting iteration method for singular and nonsingular Saddle Point problems
2015Co-Authors: Guofeng Zhang, Lidan Liao, Zhaozheng LiangAbstract:Recently, Krukier et al. (2014) and Dou et al. (2014) have studied the generalized skew-Hermitian triangular splitting (GSTS) iteration method for nonsingular and singular Saddle Point problems, respectively. In this paper, we further extend the GSTS method to a parameterized GSTS (PGSTS) method for solving non-Hermitian nonsingular and singular Saddle Point problems. By singular value decomposition technique, we derive conditions of the new iterative method for guaranteeing the convergence for non-Hermitian nonsingular Saddle Point problems and its semi-convergence for singular Saddle Point problems, respectively. In addition, the choice of the acceleration parameters in a practical manner is studied. Numerical experiments are provided, which further confirm our theoretical results and show the new method is feasible and effective for non-Hermitian nonsingular or singular Saddle Point problems.
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on hss based sequential two stage method for non hermitian Saddle Point problems
2014Co-Authors: Muzheng Zhu, Guofeng Zhang, Zhong Zheng, Zhaozheng LiangAbstract:Abstract For large sparse Saddle Point problems with symmetric positive definite (1, 1)-block, Li et al. studied an efficient iterative method (see Li et al. (2011)) [25]. By making use of the same preconditioning technique and a new matrix splitting based on the Hermitian and skew-Hermitian splitting (HSS) of the (1, 1)-block of the preconditioned non-Hermitian Saddle Point systems, an efficient sequential two-stage method is proposed for solving the non-Hermitian Saddle Point problems. Theoretical analysis shows the proposed iterative method is convergent, and that the spectral radius of iterative matrix monotonically decreases and tends to 0 as the iterative parameter α approaches infinity. Numerical experiments arising from Naiver–Stokes problem are provided to show that the new iterative method is feasible, effective and robust.
A Jorio - One of the best experts on this subject based on the ideXlab platform.
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optical phonon resonances with Saddle Point excitons in twisted bilayer graphene
2014Co-Authors: A Jorio, Mark Kasperczyk, Nick Clark, Elke Neu, Patrick Maletinsky, Aravind Vijayaraghavan, Lukas NovotnyAbstract:Twisted-bilayer graphene (tBLG) exhibits van Hove singularities in the density of states that can be tuned by changing the twisting angle θ. A θ-defined tBLG has been produced and characterized with optical reflectivity and resonance Raman scattering. The θ-engineered optical response is shown to be consistent with persistent Saddle-Point excitons. Separate resonances with Stokes and anti-Stokes Raman scattering components can be achieved due to the sharpness of the two-dimensional Saddle-Point excitons, similar to what has been previously observed for one-dimensional carbon nanotubes. The excitation power dependence for the Stokes and anti-Stokes emissions indicate that the two processes are correlated and that they share the same phonon.
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optical phonon resonances with Saddle Point excitons in twisted bilayer graphene
2014Co-Authors: A Jorio, Mark Kasperczyk, Nick Clark, Elke Neu, Patrick Maletinsky, Aravind Vijayaraghavan, Lukas NovotnyAbstract:Twisted-bilayer graphene (tBLG) exhibits van Hove singularities in the density of states that can be tuned by changing the twisting angle $\theta$. A $\theta$-defined tBLG has been produced and characterized with optical reflectivity and resonance Raman scattering. The $\theta$-engineered optical response is shown to be consistent with persistent Saddle-Point excitons. Separate resonances with Stokes and anti-Stokes Raman scattering components can be achieved due to the sharpness of the two-dimensional Saddle-Point excitons, similar to what has been previously observed for one-dimensional carbon nanotubes. The excitation power dependence for the Stokes and anti-Stokes emissions indicate that the two processes are correlated and that they share the same phonon.
