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Mathematical Models with Nonlocal Initial Conditions: An Exemplification from Quantum Mechanics. MATHEMATICAL AND COMPUTATIONAL APPLICATIONS 2021. [DOI: 10.3390/mca26040073] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/19/2022]
Abstract
Nonlocal models are ubiquitous in all branches of science and engineering, with a rapidly expanding range of mathematical and computational applications due to the ability of such models to capture effects and phenomena that traditional models cannot. While spatial nonlocalities have received considerable attention in the research community, the same cannot be said about nonlocality in time, in particular when nonlocal initial conditions are present. This paper aims at filling this gap, providing an overview of the current status of nonlocal models and focusing on the mathematical treatment of such models when nonlocal initial conditions are at the heart of the problem. Specifically, our representative example is given for a nonlocal-in-time problem for the abstract Schrödinger equation. By exploiting the linear nature of nonlocal conditions, we derive an exact representation of the solution operator under assumptions that the spectrum of Hamiltonian is contained in the horizontal strip of the complex plane. The derived representation permits us to establish the necessary and sufficient conditions for the problem’s well-posedness and the existence of its solution under different regularities. Furthermore, we present new sufficient conditions for the existence of the solution that extend the existing results in this field to the case when some nonlocal parameters are unbounded. Two further examples demonstrate the developed methodology and highlight the importance of its computer algebra component in the reduction procedures and parameter estimations for nonlocal models. Finally, a connection of the considered models and developed analysis is discussed in the context of other reduction techniques, concentrating on the most promising from the viewpoint of data-driven modelling environments, and providing directions for further generalizations.
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Petromichelakis I, Kougioumtzoglou IA. Addressing the curse of dimensionality in stochastic dynamics: a Wiener path integral variational formulation with free boundaries. Proc Math Phys Eng Sci 2020; 476:20200385. [PMID: 33362412 DOI: 10.1098/rspa.2020.0385] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/15/2020] [Accepted: 10/21/2020] [Indexed: 11/12/2022] Open
Abstract
A Wiener path integral variational formulation with free boundaries is developed for determining the stochastic response of high-dimensional nonlinear dynamical systems in a computationally efficient manner. Specifically, a Wiener path integral representation of a marginal or lower-dimensional joint response probability density function is derived. Due to this a priori marginalization, the associated computational cost of the technique becomes independent of the degrees of freedom (d.f.) or stochastic dimensions of the system, and thus, the 'curse of dimensionality' in stochastic dynamics is circumvented. Two indicative numerical examples are considered for highlighting the capabilities of the technique. The first relates to marine engineering and pertains to a structure exposed to nonlinear flow-induced forces and subjected to non-white stochastic excitation. The second relates to nano-engineering and pertains to a 100-d.f. stochastically excited nonlinear dynamical system modelling the behaviour of large arrays of coupled nano-mechanical oscillators. Comparisons with pertinent Monte Carlo simulation data demonstrate the computational efficiency and accuracy of the developed technique.
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Affiliation(s)
- Ioannis Petromichelakis
- Department of Civil Engineering and Engineering Mechanics, Columbia University, 500 W 120th Street, New York, NY 10027, USA
| | - Ioannis A Kougioumtzoglou
- Department of Civil Engineering and Engineering Mechanics, Columbia University, 500 W 120th Street, New York, NY 10027, USA
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Falk J, Bronstein L, Hanst M, Drossel B, Koeppl H. Context in synthetic biology: Memory effects of environments with mono-molecular reactions. J Chem Phys 2019; 150:024106. [DOI: 10.1063/1.5053816] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Johannes Falk
- Institut für Festkörperphysik, Technische Universität Darmstadt, Hochschulstr. 6, 64289 Darmstadt, Germany
| | - Leo Bronstein
- Bioinspired Communication Systems, Technische Universität Darmstadt, Rundeturmstr. 12, 64283 Darmstadt, Germany
- Institute of Biostatistics and Clinical Research, University of Münster, Schmeddingstr. 56, 48149 Münster, Germany
| | - Maleen Hanst
- Bioinspired Communication Systems, Technische Universität Darmstadt, Rundeturmstr. 12, 64283 Darmstadt, Germany
| | - Barbara Drossel
- Institut für Festkörperphysik, Technische Universität Darmstadt, Hochschulstr. 6, 64289 Darmstadt, Germany
| | - Heinz Koeppl
- Bioinspired Communication Systems, Technische Universität Darmstadt, Rundeturmstr. 12, 64283 Darmstadt, Germany
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Gouasmi A, Parish EJ, Duraisamy K. A priori estimation of memory effects in reduced-order models of nonlinear systems using the Mori-Zwanzig formalism. Proc Math Phys Eng Sci 2017; 473:20170385. [PMID: 28989314 DOI: 10.1098/rspa.2017.0385] [Citation(s) in RCA: 24] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/02/2017] [Accepted: 09/01/2017] [Indexed: 11/12/2022] Open
Abstract
Reduced models of nonlinear dynamical systems require closure, or the modelling of the unresolved modes. The Mori-Zwanzig procedure can be used to derive formally closed evolution equations for the resolved physics. In these equations, the unclosed terms are recast as a memory integral involving the time history of the resolved variables. While this procedure does not reduce the complexity of the original system, these equations can serve as a mathematically consistent basis to develop closures based on memory approximations. In this scenario, knowledge of the memory kernel is paramount in assessing the validity of a memory approximation. Unravelling the memory kernel requires solving the orthogonal dynamics, which is a high-dimensional partial differential equation that is intractable, in general. A method to estimate the memory kernel a priori, using full-order solution snapshots, is proposed. The key idea is to solve a pseudo orthogonal dynamics equation, which has a convenient Liouville form, instead. This ersatz arises from the assumption that the semi-group of the orthogonal dynamics is a composition operator for one observable. The method is exact for linear systems. Numerical results on the Burgers and Kuramoto-Sivashinsky equations demonstrate that the proposed technique can provide valuable information about the memory kernel.
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Affiliation(s)
- Ayoub Gouasmi
- Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI, USA
| | - Eric J Parish
- Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI, USA
| | - Karthik Duraisamy
- Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI, USA
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Kelly A, Montoya-Castillo A, Wang L, Markland TE. Generalized quantum master equations in and out of equilibrium: When can one win? J Chem Phys 2016; 144:184105. [DOI: 10.1063/1.4948612] [Citation(s) in RCA: 42] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/03/2023] Open
Affiliation(s)
- Aaron Kelly
- Department of Chemistry, Stanford University, Stanford, California 94305, USA
| | | | - Lu Wang
- Department of Chemistry and Chemical Biology, Rutgers University, Piscataway, New Jersey 08854, USA
| | - Thomas E. Markland
- Department of Chemistry, Stanford University, Stanford, California 94305, USA
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Athanassoulis GA, Tsantili IC, Kapelonis ZG. Beyond the Markovian assumption: response–excitation probabilistic solution to random nonlinear differential equations in the long time. Proc Math Phys Eng Sci 2015. [DOI: 10.1098/rspa.2015.0501] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Uncertainty quantification for dynamical systems under non-white excitation is a difficult problem encountered across many scientific and engineering disciplines. Difficulties originate from the lack of Markovian character of system responses. The response–excitation (RE) theory, recently introduced by Sapsis & Athanassoulis (Sapsis & Athanassoulis 2008
Probabilistic Eng. Mech.
23, 289–306 (
doi:10.1016/j.probengmech.2007.12.028
)) and further studied by Venturi
et al.
(Venturi
et al.
2012
Proc. R. Soc. A
468, 759–783 (
doi:10.1098/rspa.2011.0186
)), is a new approach, based on a simple differential constraint which is exact but non-closed. The evolution equation obtained for the RE probability density function (pdf) has the form of a generalized Liouville equation, with the excitation time frozen in the time-derivative term. In this work, the missing information of the RE differential constraint is identified and a closure scheme is developed for the long-time, stationary, limit-state of scalar nonlinear random differential equations (RDEs) under coloured excitation. The closure scheme does not alter the RE evolution equation, but collects the missing information through the solution of local statistically linearized versions of the nonlinear RDE, and interposes it into the solution scheme. Numerical results are presented for two examples, and compared with Monte Carlo simulations.
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Affiliation(s)
- G. A. Athanassoulis
- School of Naval Architecture and Marine Engineering, National Technical University of Athens, Athens, Greece
- Research Center for High Performance Computing, ITMO University, St. Petersburg, Russian Federation
| | - I. C. Tsantili
- School of Naval Architecture and Marine Engineering, National Technical University of Athens, Athens, Greece
- Geostatistics Laboratory, School of Mineral Resources Engineering, Technical University of Crete, Chania, Greece
| | - Z. G. Kapelonis
- School of Naval Architecture and Marine Engineering, National Technical University of Athens, Athens, Greece
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Harlim J, Li X. Parametric reduced models for the nonlinear Schrödinger equation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:053306. [PMID: 26066278 DOI: 10.1103/physreve.91.053306] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/15/2015] [Indexed: 06/04/2023]
Abstract
Reduced models for the (defocusing) nonlinear Schrödinger equation are developed. In particular, we develop reduced models that only involve the low-frequency modes given noisy observations of these modes. The ansatz of the reduced parametric models are obtained by employing a rational approximation and a colored-noise approximation, respectively, on the memory terms and the random noise of a generalized Langevin equation that is derived from the standard Mori-Zwanzig formalism. The parameters in the resulting reduced models are inferred from noisy observations with a recently developed ensemble Kalman filter-based parametrization method. The forecasting skill across different temperature regimes are verified by comparing the moments up to order four, a two-time correlation function statistics, and marginal densities of the coarse-grained variables.
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Affiliation(s)
- John Harlim
- Department of Mathematics, the Pennsylvania State University, University Park, Pennsylvania 16802-6400, USA
- Department of Meteorology, the Pennsylvania State University, University Park, Pennsylvania 16802-5013, USA
| | - Xiantao Li
- Department of Mathematics, the Pennsylvania State University, University Park, Pennsylvania 16802-6400, USA
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