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Publications (10 of 26) Show all publications
Rousse, F., Fasi, M., Dmytryshyn, A., Gulliksson, M. & Ögren, M. (2024). Simulations of quantum dynamics with fermionic phase-space representations using numerical matrix factorizations as stochastic gauges. Journal of Physics A: Mathematical and Theoretical, 57(1), Article ID 015303.
Open this publication in new window or tab >>Simulations of quantum dynamics with fermionic phase-space representations using numerical matrix factorizations as stochastic gauges
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2024 (English)In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 57, no 1, article id 015303Article in journal (Refereed) Published
Abstract [en]

The Gaussian phase-space representation can be used to implement quantum dynamics for fermionic particles numerically. To improve numerical results, we explore the use of dynamical diffusion gauges in such implementations. This is achieved by benchmarking quantum dynamics of few-body systems against independent exact solutions. A diffusion gauge is implemented here as a so-called noise-matrix, which satisfies a matrix equation defined by the corresponding Fokker-Planck equation of the phase-space representation. For the physical systems with fermionic particles considered here, the numerical evaluation of the new diffusion gauges allows us to double the practical simulation time, compared with hitherto known analytic noise-matrices. This development may have far reaching consequences for future quantum dynamical simulations of many-body systems. 

Place, publisher, year, edition, pages
Institute of Physics (IOP), 2024
Keywords
phase-space representations, quantum dynamics, diffusion gauges
National Category
Computational Mathematics Condensed Matter Physics
Research subject
Mathematics; Physics
Identifiers
urn:nbn:se:oru:diva-110059 (URN)10.1088/1751-8121/ad0e2b (DOI)001113350500001 ()2-s2.0-85180071987 (Scopus ID)
Funder
Carl Tryggers foundation , CTS 19:431Wenner-Gren Foundations, UPD 2019-0067Swedish Research Council, 2021-05393
Available from: 2023-12-05 Created: 2023-12-05 Last updated: 2024-02-05Bibliographically approved
Gulliksson, M., Oleynik, A. & Mazur, S. (2023). Portfolio Selection with a Rank-Deficient Covariance Matrix. Computational Economics
Open this publication in new window or tab >>Portfolio Selection with a Rank-Deficient Covariance Matrix
2023 (English)In: Computational Economics, ISSN 0927-7099, E-ISSN 1572-9974Article in journal (Refereed) Epub ahead of print
Abstract [en]

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.

Place, publisher, year, edition, pages
Springer, 2023
Keywords
Mean-variance portfolio, Rank-deficient covariance matrix, Linear ill-posed problems, Second order damped dynamical systems
National Category
Economics Computational Mathematics Probability Theory and Statistics
Identifiers
urn:nbn:se:oru:diva-106824 (URN)10.1007/s10614-023-10404-4 (DOI)001011973000002 ()2-s2.0-85162625424 (Scopus ID)
Available from: 2023-07-28 Created: 2023-07-28 Last updated: 2023-12-08Bibliographically approved
Dmytryshyn, A., Fasi, M. & Gulliksson, M. (2022). The dynamical functional particle method for multi-term linear matrix equations. Applied Mathematics and Computation, 435, Article ID 127458.
Open this publication in new window or tab >>The dynamical functional particle method for multi-term linear matrix equations
2022 (English)In: Applied Mathematics and Computation, ISSN 0096-3003, E-ISSN 1873-5649, Vol. 435, article id 127458Article in journal (Refereed) Published
Abstract [en]

Recent years have seen a renewal of interest in multi-term linear matrix equations, as these have come to play a role in a number of important applications. Here, we consider the solution of such equations by means of the dynamical functional particle method, an iterative technique that relies on the numerical integration of a damped second order dy-namical system. We develop a new algorithm for the solution of a large class of these equations, a class that includes, among others, all linear matrix equations with Hermi-tian positive definite or negative definite coefficients. In numerical experiments, our MAT -LAB implementation outperforms existing methods for the solution of multi-term Sylvester equations. For the Sylvester equation AX + XB = C, in particular, it can be faster and more accurate than the built-in implementation of the Bartels-Stewart algorithm, when A and B are well conditioned and have very different size.

Place, publisher, year, edition, pages
Elsevier, 2022
Keywords
Linear matrix equation, Discrete functional particle method, Lyapunov equation, Sylvester equation, Generalized Sylvester equation
National Category
Mathematics
Identifiers
urn:nbn:se:oru:diva-101785 (URN)10.1016/j.amc.2022.127458 (DOI)000863293600002 ()2-s2.0-85135936262 (Scopus ID)
Funder
Swedish Research Council, 2021-05393Wenner-Gren Foundations, UPD2019-0067
Available from: 2022-10-17 Created: 2022-10-17 Last updated: 2022-10-17Bibliographically approved
Gulliksson, M. & Ögren, M. (2021). Dynamical representations of constrained multicomponent nonlinear Schrödinger equations in arbitrary dimensions. Journal of Physics A: Mathematical and Theoretical, 54(27), Article ID 275304.
Open this publication in new window or tab >>Dynamical representations of constrained multicomponent nonlinear Schrödinger equations in arbitrary dimensions
2021 (English)In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, Vol. 54, no 27, article id 275304Article in journal (Refereed) Published
Abstract [en]

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 time-independent 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.

Place, publisher, year, edition, pages
IOP Publishing, 2021
Keywords
dynamical systems, Lagrange parameters, vortices, constrained optimization, multicomponent nonlinear Schrödinger equation, stationary states
National Category
Computational Mathematics Atom and Molecular Physics and Optics
Research subject
Mathematics; Physics
Identifiers
urn:nbn:se:oru:diva-92367 (URN)10.1088/1751-8121/ac0506 (DOI)000659666500001 ()2-s2.0-85108968426 (Scopus ID)
Available from: 2021-06-14 Created: 2021-06-14 Last updated: 2021-07-27Bibliographically approved
Gulliksson, M., Oleynik, A. & Mazur, S. (2021). Portfolio Selection with a Rank-deficient Covariance Matrix. Örebro: Örebro University, School of Business
Open this publication in new window or tab >>Portfolio Selection with a Rank-deficient Covariance Matrix
2021 (English)Report (Other academic)
Abstract [en]

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.

Place, publisher, year, edition, pages
Örebro: Örebro University, School of Business, 2021. p. 25
Series
Working Papers, School of Business, ISSN 1403-0586 ; 12
Keywords
Mean–variance portfolio, Rank-deficient covariance matrix, Linear ill-posed problems, Second order damped dynamical systems
National Category
Economics Computational Mathematics Probability Theory and Statistics
Identifiers
urn:nbn:se:oru:diva-92154 (URN)
Available from: 2021-06-04 Created: 2021-06-04 Last updated: 2022-10-27Bibliographically approved
Ögren, M. & Gulliksson, M. (2020). A numerical damped oscillator approach to constrained Schrödinger equations. European journal of physics, 41(6), Article ID 065406.
Open this publication in new window or tab >>A numerical damped oscillator approach to constrained Schrödinger equations
2020 (English)In: European journal of physics, ISSN 0143-0807, E-ISSN 1361-6404, Vol. 41, no 6, article id 065406Article in journal (Refereed) Published
Abstract [en]

This article explains and illustrates the use of a set of coupled dynamical equations, second order in a fictitious time, which converges to solutions of stationary Schrödinger equations with additional constraints. In fact, the method is general and can solve constrained minimization problems in many fields. We present the method for introductory applications in quantum mechanics including three qualitative different numerical examples: the radial Schrödinger equation for the hydrogen atom; the two-dimensional harmonic oscillator with degenerate excited states; and a non-linear Schrödinger equation for rotating states. The presented method is intuitive, with analogies in classical mechanics for damped oscillators, and easy to implement, either in own coding, or with software for dynamical systems. Hence, we find it suitable to introduce it in a continuation course in quantum mechanics or generally in applied mathematics courses which contain computational parts. The undergraduate student can for example use our derived results and the code (supplemental material) to study the Schrödinger equation in 1D for any potential. The graduate student and the general physicist can work from our three examples to derive their own results for other models including other global constraints.

Place, publisher, year, edition, pages
Institute of Physics (IOP), 2020
Keywords
minimization, Schrodinger equation, constraints, excited states, degenerate states, non-linear Schrodinger equation
National Category
Other Physics Topics Computational Mathematics Didactics
Research subject
Physics; Mathematics
Identifiers
urn:nbn:se:oru:diva-83658 (URN)10.1088/1361-6404/aba70b (DOI)000578315900001 ()2-s2.0-85094588173 (Scopus ID)
Available from: 2020-06-26 Created: 2020-06-26 Last updated: 2023-12-08Bibliographically approved
Gulliksson, M. & Mazur, S. (2020). An Iterative Approach to Ill-Conditioned Optimal Portfolio Selection. Computational Economics, 56, 773-794
Open this publication in new window or tab >>An Iterative Approach to Ill-Conditioned Optimal Portfolio Selection
2020 (English)In: Computational Economics, ISSN 0927-7099, E-ISSN 1572-9974, Vol. 56, p. 21p. 773-794Article in journal (Refereed) Published
Abstract [en]

Covariance matrix of the asset returns plays an important role in the portfolioselection. A number of papers is focused on the case when the covariance matrixis positive definite. In this paper, we consider portfolio selection with a singu-lar covariance matrix. We describe an iterative method based on a second orderdamped dynamical systems that solves the linear rank-deficient problem approxi-mately. Since the solution is not unique, we suggest one numerical solution that canbe chosen from the iterates that balances the size of portfolio and the risk. The nu-merical study confirms that the method has good convergence properties and givesa solution as good as or better than the constrained least norm Moore-Penrose solu-tion. Finally, we complement our result with an empirical study where we analyzea portfolio with actual returns listed in S&P 500 index.

Place, publisher, year, edition, pages
Springer, 2020. p. 21
Keywords
Mean-variance portfolio, singular covariance matrix, linear ill-posed problems, second order damped dynamical systems
National Category
Probability Theory and Statistics Economics Other Mathematics
Research subject
Statistics; Economics; Mathematics
Identifiers
urn:nbn:se:oru:diva-74364 (URN)10.1007/s10614-019-09943-6 (DOI)000574523100001 ()2-s2.0-85075334337 (Scopus ID)
Funder
The Jan Wallander and Tom Hedelius Foundation, P18-0201
Note

Funding Agency:

Internal research Grants at Örebro University

Available from: 2019-05-22 Created: 2019-05-22 Last updated: 2023-12-08Bibliographically approved
Zhang, Y., Gong, R., Gulliksson, M. & Cheng, X. (2019). A coupled complex boundary expanding compacts method for inverse source problems. Journal of Inverse and Ill-Posed Problems, 27(1), 67-86
Open this publication in new window or tab >>A coupled complex boundary expanding compacts method for inverse source problems
2019 (English)In: Journal of Inverse and Ill-Posed Problems, ISSN 0928-0219, E-ISSN 1569-3945, Vol. 27, no 1, p. 67-86Article in journal (Refereed) Published
Abstract [en]

In this paper, we consider an inverse source problem for elliptic partial differential equations with both Dirichlet and Neumann boundary conditions. The unknown source term is to be determined by additional boundary data. This problem is ill-posed since the dimensionality of the boundary is lower than the dimensionality of the inner domain. To overcome the ill-posed nature, using the a priori information (sourcewise representation), and based on the coupled complex boundary method, we propose a coupled complex boundary expanding compacts method (CCBECM). A finite element method is used for the discretization of CCBECM. The regularization properties of CCBECM for both the continuous and discrete versions are proved. Moreover, an a posteriori error estimate of the obtained finite element approximate solution is given and calculated by a projected gradient algorithm. Finally, numerical results show that the proposed method is stable and effective.

Place, publisher, year, edition, pages
Walter de Gruyter, 2019
Keywords
Inverse source problem, expanding compacts method, finite element method, error estimation
National Category
Computational Mathematics
Identifiers
urn:nbn:se:oru:diva-68829 (URN)10.1515/jiip-2017-0002 (DOI)000457195600006 ()2-s2.0-85053166222 (Scopus ID)
Funder
Knowledge Foundation, 20170059
Note

Funding Agencies:

Alexander von Humboldt foundation  

Natural Science Foundation of China  11571311  11401304 

Fundamental Research Funds for the Central Universities  NS2018047 

Available from: 2018-09-11 Created: 2018-09-11 Last updated: 2019-02-13Bibliographically approved
Gulliksson, M., Ögren, M., Oleynik, A. & Zhang, Y. (2019). Damped Dynamical Systems for Solving Equations and Optimization Problems. In: Bharath Sriraman (Ed.), Handbook of the Mathematics of the Arts and Sciences: . Springer
Open this publication in new window or tab >>Damped Dynamical Systems for Solving Equations and Optimization Problems
2019 (English)In: Handbook of the Mathematics of the Arts and Sciences / [ed] Bharath Sriraman, Springer , 2019Chapter in book (Other academic)
Abstract [en]

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.

Place, publisher, year, edition, pages
Springer, 2019
Keywords
Optimization, damped dynamical systems, convex problems, eigenvalue problems, image analysis, inverse problems, quantum physics, Schrödinger equation
National Category
Computational Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:oru:diva-71881 (URN)10.1007/978-3-319-70658-0 (DOI)978-3-319-70658-0 (ISBN)
Available from: 2019-01-29 Created: 2019-01-29 Last updated: 2019-04-01Bibliographically approved
Baravdish, G., Svensson, O., Gulliksson, M. & Zhang, Y. (2019). Damped second order flow applied to image denoising. IMA Journal of Applied Mathematics, 84(6), 1082-1111
Open this publication in new window or tab >>Damped second order flow applied to image denoising
2019 (English)In: IMA Journal of Applied Mathematics, ISSN 0272-4960, E-ISSN 1464-3634, Vol. 84, no 6, p. 1082-1111Article in journal (Refereed) Published
Abstract [en]

In this paper, we introduce a new image denoising model: the damped flow (DF), which is a second order nonlinear evolution equation associated with a class of energy functionals of an image. The existence, uniqueness and regularization property of DF are proven. For the numerical implementation, based on the Störmer–Verlet method, a discrete DF, SV-DDF, is developed. The convergence of SV-DDF is studied as well. Several numerical experiments, as well as a comparison with other methods, are provided to demonstrate the efficiency of SV-DDF.

Place, publisher, year, edition, pages
Oxford University Press, 2019
Keywords
Nonlinear flow, image denoising, p-parabolic, p-Laplace, inverse problems, regularization, damped Hamiltonian system, symplectic method, Störmer–Verlet.
National Category
Computational Mathematics
Identifiers
urn:nbn:se:oru:diva-79218 (URN)10.1093/imamat/hxz027 (DOI)000509388900002 ()2-s2.0-85082102558 (Scopus ID)
Note

Funding Agency:

Alexander von Humboldt Foundation

 

Available from: 2020-01-16 Created: 2020-01-16 Last updated: 2020-04-03Bibliographically approved
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Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0003-0332-2315

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