Damped Dynamical Systems for Solving Equations and Optimization Problems

Gulliksson, Mårten; Ögren, Magnus; Oleynik, Anna; Zhang, Ye

doi:10.1007/978-3-319-70658-0_32-1

Cited by 7 publications

(25 citation statements)

References 47 publications

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“…In this section we give a brief description of the Discrete Functional Particle Method (DFPM) and describe how DFPM can be used to solve (1). For a more comprehensive discussion of DFPM see Gulliksson et al (2019).…”

Section: The Discrete Functional Particle Methodsmentioning

confidence: 99%

“…In order to ensure fast convergence of the iterative scheme (13) one must choose the time step and damping such that for convergence G < 1 and for efficiency G is as small as possible. The otpimal choice of parameters can be summarized in the following theorem, see Gulliksson (2017), Gulliksson et al (2019) and references therein.…”

Section: Convergence Analysis and Choice Of Parametersmentioning

confidence: 99%

“…Iterative methods for linear ill-posed problems are certainly not new and include classical methods like Landweber iteration with Nesterov acceleration and Conjugate gradient methods, see Neubauer (2000Neubauer ( , 2017. Very recently a new approach has been developed based on second order damped dynamical systems, see Gulliksson et al (2019) for an introduction and overview and specifically Zhang and Hofmann (2018) for the ill-posed linear case. In Zhang and Hofmann (2018) it is shown that the method is a regularization method, i.e., loosely speaking for an ill-posed linear problem there exists a unique solution when the number of iterations tend to infinity and the error tends to zero.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

An Iterative Approach to Ill-Conditioned Optimal Portfolio Selection

Gulliksson

Mazur

2019

Comput Econ

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Covariance matrix of the asset returns plays an important role in the portfolio selection. A number of papers is focused on the case when the covariance matrix is positive definite. In this paper, we consider portfolio selection with a singular covariance matrix. We describe an iterative method based on a second order damped dynamical systems that solves the linear rank-deficient problem approximately. Since the solution is not unique, we suggest one numerical solution that can be chosen from the iterates that balances the size of portfolio and the risk. The numerical study confirms that the method has good convergence properties and gives a solution as good as or better than the solutions that are based on constrained least norm Moore-Penrose, Lasso, and naive equal-weighted approaches. Finally, we complement our result with an empirical study where we analyze a portfolio with actual returns listed in S&P 500 index. Keywords Mean-variance portfolio • Singular covariance matrix • Linear ill-posed problems • Second order damped dynamical systems B Stepan Mazur

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Section: The Discrete Functional Particle Methodsmentioning

confidence: 99%

Section: Convergence Analysis and Choice Of Parametersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An Iterative Approach to Ill-Conditioned Optimal Portfolio Selection

Gulliksson

Mazur

2019

Comput Econ

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“…The first step in the Dynamical Functional Particle Method [17], DFPM, is to introduce an artificial time τ making all the functions and functionals in the preceeding section depend on τ . Thus, ψ(r) become, say, Ψ(r, τ ) and we drop all the tildes in the notation used above to indicate the new dependence on τ , e.g., E( ψ) will become E( Ψ) .…”

Section: General Formulation Of Dfpm With Constraintsmentioning

confidence: 99%

“…3 we introduce the damped dynamical system to be used to attain a stationary solution that solves our original constrained minimization problem. This approach to obtain the solution is an extension of the Dynamical Functional Particle Method (DFPM), see [17], able to handle nonlinear constraints. Our version of DFPM involves finding the Lagrange parameters and we derive the linear equations determining those.…”

Section: Introductionmentioning

confidence: 99%

Dynamical representations of constrained multicomponent nonlinear Schrödinger equations in arbitrary dimensions

Gulliksson,

Ogren

2021

Preprint

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We present new approaches for solving constrained multicomponent nonlinear Schrödinger equations in arbitrary dimensions. The idea is to introduce an artificial time and solve an extended damped second order dynamic system whose stationary solution is the solution to the timeindependent nonlinear Schrödinger equation. Constraints are often considered by projection onto the constraint set, here we include them explicitly into the dynamical system. We show the applicability and efficiency of the methods on examples of relevance in modern physics applications.

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