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Reduced Wiener Chaos representation of random fields via basis adaptation and projection

Panagiotis Tsilifis, Roger G. Ghanem
2017 Journal of Computational Physics  
We take advantage of the adaptation capabilities of these expansions where the core idea is to rotate the basis of the underlying Gaussian Hilbert space, in order to achieve reduced functional representations  ...  A new characterization of random fields appearing in physical models is presented that is based on their well-known Homogeneous Chaos expansions.  ...  Basis adaptation in Homogeneous Chaos expansions of random fields The Homogeneous (Wiener) Chaos We consider a probability space (Ω, F , P) and G a d-dimensional Gaussian Hilbert space, that is a closed  ... 
doi:10.1016/j.jcp.2017.04.009 fatcat:vqxewfnw2bgdlbt75qwik3kzdy

Efficient polynomial chaos approximations: Active, local and basis-adapted

Ziad Georges Ghauch
2021 ANZIAM Journal  
Basis adaptation in homogeneous chaos spaces. J. Comput. Phys. 259:304–317, 2014. doi: 10.1016/j.jcp.2013.12.009. P. Tsilifis and R. G. Ghanem.  ...  Homogeneous chaos basis adaptation for design optimization under uncertainty: Application to the oil well placement problem. AI EDAM 31(3):265–276, 2017. doi: 10.1017/S0890060417000166. R.  ...  Basis-adapted polynomial chaos Consider a probability space (Ω, F, P) consisting of a sample space Ω, a σ-algebra F on Ω, and a corresponding probability measure P on (Ω, F).  ... 
doi:10.21914/anziamj.v62.15833 fatcat:cv2tjjydgbholm7h2bma2fuh6y

Strong turbulent convection: distributed chaos and large-scale circulation [article]

A. Bershadskii
2016 arXiv   pre-print
Two types of spontaneous breaking of the space translational symmetry in distributed chaos have been considered for turbulent thermal convection at large values of Rayleigh number.  ...  The analysis suggests that in this case the large-scale circulation can be considered as a natural (harmonic) part of the distributed chaos.  ...  Space homogeneity is hardly expected in such cell. The distributed chaos is a basis for turbulence (both homogeneous and inhomogeneous) [2] .  ... 
arXiv:1604.07762v2 fatcat:l3r3cjlhrnagfjz6ztqwmmcbxq

DIFFERENT FACETS OF CHAOS IN QUANTUM MECHANICS

V. R. MANFREDI, L. SALASNICH
1999 International Journal of Modern Physics B  
Then we analyze the problem of dynamical chaos and the time scales associated with chaos suppression in quantum mechanics. Summary: 1. Introduction 2. Quantum Chaology and Spectral Statistics 3.  ...  From Poisson to GOE Transition: Comparison with Experimental Data 3.1 Atomic Nuclei 3.2 The Hydrogen Atom in the Strong Magnetic Field 4. Quantum Chaos and Field Theory 5.  ...  Figure 8 : 8 Exact eigenstates |i > expressed in the |µ, ν > basis states: a) Quantum Integrable States; b) Quantum KAM States; c) Quantum Chaotic States (adapted from Ref. 35).  ... 
doi:10.1142/s0217979299002447 fatcat:ncs6fi6dmjhdzmavwh2dajx6vy

Compressive sensing adaptation for polynomial chaos expansions

Panagiotis Tsilifis, Xun Huan, Cosmin Safta, Khachik Sargsyan, Guilhem Lacaze, Joseph C. Oefelein, Habib N. Najm, Roger G. Ghanem
2019 Journal of Computational Physics  
Basis adaptation in Homogeneous Chaos spaces rely on a suitable rotation of the underlying Gaussian germ.  ...  In this paper we present a new adaptation mechanism that builds on compressive sensing algorithms, resulting in a reduced polynomial chaos approximation with optimal sparsity.  ...  Section 2.1 describes the use of PC expansion as a response surface, including Hermite (Homogeneous) Chaos for both standard Gaussian variables and rotated Gaussian variables produced from the basis adaptation  ... 
doi:10.1016/j.jcp.2018.12.010 fatcat:sfugdffucnfozh7kup2ctnp25y

Page 680 of Astronomy and Astrophysics Vol. 320, Issue 2 [page]

1998 Astronomy and Astrophysics  
Thus, if the phase-space were roughly homogeneous, the extent of each trajectory in Fig. 14 would increase by one order.  ...  The region around the low-eccentricity chaos is sensible to the value of Jupiter’s eccentricity, the chaos grown for e; = 0.061 but still left a great portion of the phase space regular.  ... 

From Self-Organization to Evolution of RNA Molecules: The Origin of Biological Information

Peter Schuster
2004 Solid State Phenomena  
Evolution of RNA molecules is considered as pattern formation in sequence space, which manifests itself as another pattern in the space of minimum-free-energy structures.  ...  Evolutionary optimization of RNA molecules occurs in steps: Short adaptive periods are interrupted by long epochs of quasi-stationarity during which the mean replication rate of the populations is essentially  ...  reaction in the flow reactor Deterministic chaos in space and time Pattern formation in autocatalytic third order reactions G.Nicolis, I.Prigogine.  ... 
doi:10.4028/www.scientific.net/ssp.97-98.27 fatcat:6jioadlfl5ferou77latm3gdoq

Interactions destroy dynamical localization with strong and weak chaos

G. Gligorić, J. D. Bodyfelt, S. Flach
2011 Europhysics letters  
We observe strong and weak chaos regimes of wave packet spreading in momentum space. In the intermediate strong chaos regime the condensate energy grows as t^1/2.  ...  Noninteracting rotors show dynamical localization in momentum space.  ...  Image: Ornamental multiplication of space-time figures of temperature transformation rules (adapted from T. S. Bíró and P.  ... 
doi:10.1209/0295-5075/96/30004 fatcat:jsxqdmddwnhuzcotn6xjc5c5ti

Page 4107 of Mathematical Reviews Vol. , Issue 2004e [page]

2004 Mathematical Reviews  
It is known that homogeneous solutions in the theory of elasticity do not form a basis in the classical sense, that is, on a segment.  ...  From the text (translated from the Russian): “We study the basis properties of a characteristic system of homogeneous solutions in the theory of elasticity.  ... 

Bayesian adaptation of chaos representations using variational inference and sampling on geodesics

P. Tsilifis, R. G. Ghanem
2018 Proceedings of the Royal Society A  
A novel approach is presented for constructing polynomial chaos representations of scalar quantities of interest (QoI) that extends previously developed methods for adaptation in Homogeneous Chaos spaces  ...  In this work, we develop a Bayesian formulation of the problem that characterizes the posterior distributions of the series coefficients and the adaptation rotation matrix acting on the Gaussian input  ...  Homogeneous chaos with adapted input basis (a) Homogeneous chaos Throughout this paper, we assume that our observable quantities of interest (QoI's) lie in the space of square integrable random variables  ... 
doi:10.1098/rspa.2018.0285 pmid:30333707 pmcid:PMC6189593 fatcat:cv76eailkna6ngmelzikkmurue

Adaptive stochastic Galerkin FEM with hierarchical tensor representations

Martin Eigel, Max Pfeffer, Reinhold Schneider
2016 Numerische Mathematik  
The adaptive adjustment includes the refinement of the FE mesh based on a residual estimator, the problem-adapted stochastic discretization in anisotropic Legendre Wiener chaos and the successive increase  ...  train format of the problem and a refinement algorithm on the basis of a posteriori error estimates to adaptively adjust the different employed discretizations.  ...  Rank adaptivity. As explained above, the ALS algorithm finds a stationary point in a subset M ≤r of the fully discretized tensor space R N ×d1×...×d M .  ... 
doi:10.1007/s00211-016-0850-x fatcat:ptnvxsbclzgdvgfndhp6nzaoni

Beyond Wiener–Askey Expansions: Handling Arbitrary PDFs

Xiaoliang Wan, George Em Karniadakis
2005 Journal of Scientific Computing  
In this paper we present a Multi-Element generalized Polynomial Chaos (ME-gPC) method to deal with stochastic inputs with arbitrary probability measures.  ...  Based on the decomposition of the random space of the stochastic inputs, we construct numerically a set of orthogonal polynomials with respect to a conditional probability density function (PDF) in each  ...  An effective approach based on the the homogeneous chaos theory of Wiener [1] is polynomial chaos, which has been successfully used in mechanics [2, 3] .  ... 
doi:10.1007/s10915-005-9038-8 fatcat:3c5yikacfbdw7grmzx7esjojgu

Chaos Theory and Self-Organization Systems in Recovery Medicine: A Scientific Review

Alexander A. Khadartsev, Valeriy M. Eskov
2015 Integrative Medicine International  
Furthermore, the importance of modulation in adaptation programs in the control of the functional systems of a human body, confirmed in the data submitted for 14 patents covering inventions that have led  ...  In this review, the theoretical principles, which provide the possibility of correction of acute stress pathology, are formulated in terms of the theory of chaos and self-organization.  ...  Adaptation systems became flexible and provide adaptation to the environment as a consequence of chaos.  ... 
doi:10.1159/000377679 fatcat:zcwuwun6qfgt3ckkfqi6cufpii

Polynomial Chaos Expansion method as a tool to evaluate and quantify field homogeneities of a novel waveguide RF Wien filter

J. Slim, F. Rathmann, A. Nass, H. Soltner, R. Gebel, J. Pretz, D. Heberling
2017 Nuclear Instruments and Methods in Physics Research Section A : Accelerators, Spectrometers, Detectors and Associated Equipment  
Since Monte-Carlo simulations are computationally very expensive, we discuss here an efficient surrogate modeling scheme based on the Polynomial Chaos Expansion method to compute the field quality in the  ...  In reality, mechanical tolerances and misalignments decrease the simulated field quality, and it is therefore important to consider them in the simulations.  ...  Acknowledgment This work has been performed in the framework of the JEDI collaboration (Jülich Electric Dipole moment Investigations), and is supported by an ERC Advanced-Grant of the European Union (proposal  ... 
doi:10.1016/j.nima.2017.03.040 fatcat:ffu33sbwzbeghasdk6s6iabl5q

ENERGY LEVEL QUASI-CROSSINGS: ACCIDENTAL DEGENERACIES OR SIGNATURE OF QUANTUM CHAOS?

V. R. MANFREDI, L. SALASNICH
2000 International Journal of Modern Physics E  
In the field of quantum chaos, the study of energy levels plays an important role.  ...  In particular, we analyze in detail degeneracies and quasi-crossings in the eigenvalues of quantum Hamiltonians which are classically non-integrable. Summary: 1. Introduction; 2.  ...  Some simplifications can be made by working in the (2+1)-dimensional Minkowski space (β = 0, 1, 2) and choosing spatially homogenous Yang-Mills and the Higgs fields.  ... 
doi:10.1142/s0218301300000283 fatcat:wdych5rtbjdrpduotvypsyutqq
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