99 research outputs found
Localization and Pattern Formation in Quantum Physics. II. Waveletons in Quantum Ensembles
In this second part we present a set of methods, analytical and numerical,
which can describe behaviour in (non) equilibrium ensembles, both classical and
quantum, especially in the complex systems, where the standard approaches
cannot be applied. The key points demonstrating advantages of this approach
are: (i) effects of localization of possible quantum states; (ii) effects of
non-perturbative multiscales which cannot be calculated by means of
perturbation approaches; (iii) effects of formation of complex/collective
quantum patterns from localized modes and classification and possible control
of the full zoo of quantum states, including (meta) stable localized patterns
(waveletons). We demonstrate the appearance of nontrivial localized (meta)
stable states/patterns in a number of collective models covered by the
(quantum)/(master) hierarchy of Wigner-von Neumann-Moyal-Lindblad equations,
which are the result of ``wignerization'' procedure (Weyl-Wigner-Moyal
quantization) of classical BBGKY kinetic hierarchy, and present the explicit
constructions for exact analytical/numerical computations (fast convergent
variational-wavelet representation). Numerical modeling shows the creation of
different internal structures from localized modes, which are related to the
localized (meta) stable patterns (waveletons), entangled ensembles (with
subsequent decoherence) and/or chaotic-like type of behaviour.Comment: LaTeX2e, spie.cls, 13 pages, 6 figures, submitted to Proc. of SPIE
Meeting, The Nature of Light: What is a Photon? Optics & Photonics, SP200,
San Diego, CA, July-August, 200
Localization and Pattern Formation in Quantum Physics. I. Phenomena of Localization
In these two related parts we present a set of methods, analytical and
numerical, which can illuminate the behaviour of quantum system, especially in
the complex systems. The key points demonstrating advantages of this approach
are: (i) effects of localization of possible quantum states, more proper than
"gaussian-like states"; (ii) effects of non-perturbative multiscales which
cannot be calculated by means of perturbation approaches; (iii) effects of
formation of complex quantum patterns from localized modes or classification
and possible control of the full zoo of quantum states, including (meta) stable
localized patterns (waveletons). We'll consider calculations of Wigner
functions as the solution of Wigner-Moyal-von Neumann equation(s) corresponding
to polynomial Hamiltonians. Modeling demonstrates the appearance of (meta)
stable patterns generated by high-localized (coherent) structures or
entangled/chaotic behaviour. We can control the type of behaviour on the level
of reduced algebraical variational system. At the end we presented the
qualitative definition of the Quantum Objects in comparison with their
Classical Counterparts, which natural domain of definition is the category of
multiscale/multiresolution decompositions according to the action of
internal/hidden symmetry of the proper realization of scales of functional
spaces. It gives rational natural explanation of such pure quantum effects as
``self-interaction''(self-interference) and instantaneous quantum interaction.Comment: LaTeX2e, spie.cls, 13 pages, 15 figures, submitted to Proc. of SPIE
Meeting, The Nature of Light: What is a Photon? Optics & Photonics, SP200,
San Diego, CA, July-August, 200
Tensor approximation in visualization and graphics
In this course, we will introduce the basic concepts of tensor approximation (TA) – a higher-order generalization of the SVD and PCA methods – as well as its applications to visual data representation, analysis and visualization, and bring the TA framework closer to visualization and computer graphics researchers and practitioners. The course will cover the theoretical background of TA methods, their properties and how to compute them, as well as practical applications of TA methods in visualization and computer graphics contexts. In a first theoretical part, the attendees will be instructed on the necessary mathematical background of TA methods to learn the basics skills of using and applying these new tools in the context of the representation of large multidimensional visual data. Specific and very noteworthy features of the TA framework are highlighted which can effectively be exploited for spatio-temporal multidimensional data representation and visualization purposes. In two application oriented sessions, compact TA data representation in scientific visualization and computer graphics as well as decomposition and reconstruction algorithms will be demonstrated. At the end of the course, the participants will have a good basic knowledge of TA methods along with a practical understanding of its potential application in visualization and graphics related projects
Multiresolution Tensor Learning for Efficient and Interpretable Spatial Analysis
Efficient and interpretable spatial analysis is crucial in many fields such as geology, sports, and climate science. Large-scale spatial data often contains complex higher-order correlations across features and locations. While tensor latent factor models can describe higher-order correlations, they are inherently computationally expensive to train. Furthermore, for spatial analysis, these models should not only be predictive but also be spatially coherent. However, latent factor models are sensitive to initialization and can yield inexplicable results. We develop a novel Multi-resolution Tensor Learning (MRTL) algorithm for efficiently learning interpretable spatial patterns. MRTL initializes the latent factors from an approximate full-rank tensor model for improved interpretability and progressively learns from a coarse resolution to the fine resolution for an enormous computation speedup. We also prove the theoretical convergence and computational complexity of MRTL. When applied to two real-world datasets, MRTL demonstrates 4 ~ 5 times speedup compared to a fixed resolution while yielding accurate and interpretable models
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