| 1. |
- Gulliksson, Mårten, 1963-, et al.
(författare)
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Damped Dynamical Systems for Solving Equations and Optimization Problems
- 2019
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Ingår i: Handbook of the Mathematics of the Arts and Sciences. - Cham : Springer. - 9783319706580
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Bokkapitel (övrigt vetenskapligt/konstnärligt)abstract
- We present an approach for solving optimization problems with or without constrains which we call Dynamical Functional Particle Method (DFMP). The method consists of formulating the optimization problem as a second order damped dynamical system and then applying symplectic method to solve it numerically. In the first part of the chapter, we give an overview of the method and provide necessary mathematical background. We show that DFPM is a stable, efficient, and given the optimal choice of parameters, competitive method. Optimal parameters are derived for linear systems of equations, linear least squares, and linear eigenvalue problems. A framework for solving nonlinear problems is developed and numerically tested. In the second part, we adopt the method to several important applications such as image analysis, inverse problems for partial differential equations, and quantum physics. At the end, we present open problems and share some ideas of future work on generalized (nonlinear) eigenvalue problems, handling constraints with reflection, global optimization, and nonlinear ill-posed problems.
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| 2. |
- Gulliksson, Mårten, 1963-, et al.
(författare)
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Portfolio Selection with a Rank-deficient Covariance Matrix
- 2021
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- In this paper, we consider optimal portfolio selection when the covariance matrix of the asset returns is rank-deficient. For this case, the original Markowitz’ problem does not have a unique solution. The possible solutions belong to either two subspaces namely the range- or nullspace of the covariance matrix. The former case has been treated elsewhere but not the latter. We derive an analytical unique solution, assuming the solution is in the null space, that is risk-free and has minimum norm. Furthermore, we analyse the iterative method which is called the discrete functional particle method in the rank-deficient case. It is shown that the method is convergent giving a risk-free solution and we derive the initial condition that gives the smallest possible weights in the norm. Finally, simulation results on artificial problems as well as real-world applications verify that the method is both efficient and stable.
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| 3. |
- Gulliksson, Mårten, 1963-, et al.
(författare)
-
Portfolio Selection with a Rank-Deficient Covariance Matrix
- 2024
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Ingår i: Computational Economics. - : Springer. - 0927-7099 .- 1572-9974. ; 63, s. 2247-2269
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we consider optimal portfolio selection when the covariance matrix of the asset returns is rank-deficient. For this case, the original Markowitz' problem does not have a unique solution. The possible solutions belong to either two subspaces namely the range- or nullspace of the covariance matrix. The former case has been treated elsewhere but not the latter. We derive an analytical unique solution, assuming the solution is in the null space, that is risk-free and has minimum norm. Furthermore, we analyse the iterative method which is called the discrete functional particle method in the rank-deficient case. It is shown that the method is convergent giving a risk-free solution and we derive the initial condition that gives the smallest possible weights in the norm. Finally, simulation results on artificial problems as well as real-world applications verify that the method is both efficient and stable.
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| 4. |
- Gulliksson, Mårten, 1963-, et al.
(författare)
-
Portfolio Selection with a Rank-Deficient Covariance Matrix
- 2023
-
Ingår i: Computational Economics. - : Springer. - 0927-7099 .- 1572-9974. ; 63, s. 2247-2269
-
Tidskriftsartikel (refereegranskat)abstract
- In this paper, we consider optimal portfolio selection when the covariance matrix of the asset returns is rank-deficient. For this case, the original Markowitz' problem does not have a unique solution. The possible solutions belong to either two subspaces namely the range- or nullspace of the covariance matrix. The former case has been treated elsewhere but not the latter. We derive an analytical unique solution, assuming the solution is in the null space, that is risk-free and has minimum norm. Furthermore, we analyse the iterative method which is called the discrete functional particle method in the rank-deficient case. It is shown that the method is convergent giving a risk-free solution and we derive the initial condition that gives the smallest possible weights in the norm. Finally, simulation results on artificial problems as well as real-world applications verify that the method is both efficient and stable.
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| 5. |
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| 6. |
- Kostykin, Vadim, et al.
(författare)
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On the Existence of Unstable Bumps in Neural Networks
- 2013
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Ingår i: Integral equations and operator theory. - : Springer Science and Business Media LLC. - 0378-620X .- 1420-8989. ; 75:4, s. 445-458
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Tidskriftsartikel (refereegranskat)abstract
- We study the neuronal field equation, a nonlinear integro-differential equation of Hammerstein type. By means of the Amann three fixed point theorem we prove the existence of bump solutions to this equation. Using the Krein-Rutman theorem we show their Lyapunov instability.
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| 7. |
- Oleynik, Anna, et al.
(författare)
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Iterative Schemes for Bump Solutions in a Neural Field Model
- 2015
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Ingår i: Differential Equations and Dynamical Systems. - : Springer Science and Business Media LLC. - 0971-3514 .- 0974-6870. ; 23:1, s. 79-98
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Tidskriftsartikel (refereegranskat)abstract
- We develop two iteration schemes for construction of localized stationary solutions (bumps) of a one-population Wilson-Cowan model with a smooth firing rate function. The first scheme is based on the fixed point formulation of the stationary Wilson-Cowan model. The second one is formulated in terms of the excitation width of a bump. Using the theory of monotone operators in ordered Banach spaces we justify convergence of both iteration schemes.
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| 8. |
- Oleynik, Anna, et al.
(författare)
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On the properties of nonlinear nonlocal operators arising in neural field models
- 2013
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Ingår i: Journal of Mathematical Analysis and Applications. - : Elsevier BV. - 0022-247X .- 1096-0813. ; 398:1, s. 335-351
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Tidskriftsartikel (refereegranskat)abstract
- We study the existence and continuous dependence of stationary solutions of the one-population Wilson-Cowan model on the steepness of the firing rate functions. We investigate the properties of the nonlinear nonlocal operators which arise when formulating the stationary one-population Wilson-Cowan model as a fixed point problem. The theory is used to study the existence and continuous dependence of localized stationary solutions of this model on the steepness of the firing rate functions. The present work generalizes and complements previously obtained results as we relax on the assumptions that the firing rate functions are given by smoothed Heaviside functions.
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