Numerical tensor calculus

Acta Numerica ◽  
2014 ◽  
Vol 23 ◽  
pp. 651-742 ◽  
Author(s):  
Wolfgang Hackbusch
Keyword(s):  
Large Scale ◽  
Spatial Dimension ◽  
Grid Function ◽  
Numerical Representation ◽  
Regular Grid ◽  
Tensor Calculus ◽  
Spatial Dimensions ◽  
Tensor Spaces ◽  
Numerical Performance ◽  
Grid Points

The usual large-scale discretizations are applied to two or three spatial dimensions. The standard methods fail for higher dimensions because the data size increases exponentially with the dimension. In the case of a regular grid withngrid points per direction, a spatial dimensiondyieldsndgrid points. A grid function defined on such a grid is an example of a tensor of orderd. Here, suitable tensor formats help, since they try to approximate these huge objects by a much smaller number of parameters, which increases only linearly ind. In this way, data of sizend= 10001000can also be treated.This paper introduces the algebraic and analytical aspects of tensor spaces. The main part concerns the numerical representation of tensors and the numerical performance of tensor operations.

scholarly journals Renormalization group theory of molecular dynamics

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Daiji Ichishima ◽  
Yuya Matsumura
Keyword(s):  
Molecular Dynamics ◽  
Renormalization Group ◽  
Critical Exponent ◽  
Computational Efficiency ◽  
Large Scale ◽  
Dissipative Particle Dynamics ◽  
Computational Cost ◽  
Coarse Graining ◽  
Spatial Dimension ◽  
Deborah Number

AbstractLarge scale computation by molecular dynamics (MD) method is often challenging or even impractical due to its computational cost, in spite of its wide applications in a variety of fields. Although the recent advancement in parallel computing and introduction of coarse-graining methods have enabled large scale calculations, macroscopic analyses are still not realizable. Here, we present renormalized molecular dynamics (RMD), a renormalization group of MD in thermal equilibrium derived by using the Migdal–Kadanoff approximation. The RMD method improves the computational efficiency drastically while retaining the advantage of MD. The computational efficiency is improved by a factor of $$2^{n(D+1)}$$ 2 n ( D + 1 ) over conventional MD where D is the spatial dimension and n is the number of applied renormalization transforms. We verify RMD by conducting two simulations; melting of an aluminum slab and collision of aluminum spheres. Both problems show that the expectation values of physical quantities are in good agreement after the renormalization, whereas the consumption time is reduced as expected. To observe behavior of RMD near the critical point, the critical exponent of the Lennard-Jones potential is extracted by calculating specific heat on the mesoscale. The critical exponent is obtained as $$\nu =0.63\pm 0.01$$ ν = 0.63 ± 0.01 . In addition, the renormalization group of dissipative particle dynamics (DPD) is derived. Renormalized DPD is equivalent to RMD in isothermal systems under the condition such that Deborah number $$De\ll 1$$ D e ≪ 1 .

A new three-term spectral conjugate gradient algorithm with higher numerical performance for solving large scale optimization problems based on Quasi-Newton equation

2021 ◽  
pp. 2150053
Author(s):  
Jie Guo ◽  
Zhong Wan
Keyword(s):  
Conjugate Gradient ◽  
Large Scale ◽  
Line Search ◽  
Optimization Problems ◽  
Gradient Algorithm ◽  
Large Scale Optimization ◽  
Newton Equation ◽  
Numerical Performance ◽  
Scale Optimization ◽  
Quasi Newton

A new spectral three-term conjugate gradient algorithm in virtue of the Quasi-Newton equation is developed for solving large-scale unconstrained optimization problems. It is proved that the search directions in this algorithm always satisfy a sufficiently descent condition independent of any line search. Global convergence is established for general objective functions if the strong Wolfe line search is used. Numerical experiments are employed to show its high numerical performance in solving large-scale optimization problems. Particularly, the developed algorithm is implemented to solve the 100 benchmark test problems from CUTE with different sizes from 1000 to 10,000, in comparison with some similar ones in the literature. The numerical results demonstrate that our algorithm outperforms the state-of-the-art ones in terms of less CPU time, less number of iteration or less number of function evaluation.

Forming Canonical 8D Data Clouds to Explore the Transient Dynamics of Physical Stiffened Plates: A Machine Learning Approach for Complex Mechanical Structures

2020 ◽  
Author(s):  
Ioannis T. Georgiou
Keyword(s):  
Machine Learning ◽  
Large Scale ◽  
Complex Structure ◽  
Spatial Dimension ◽  
Physical Structure ◽  
Transient Dynamics ◽  
Alloy Plate ◽  
Considerable Uncertainty ◽  
Acceleration Signal ◽  
The Impact

Abstract This work presents a data-driven explorative study of the physics of the dynamics of a physical structure of complicated geometry. The geometric complexity of the physical system renders the typical single sensor acceleration signal quite complicated for a physics interpretation. We need the spatial dimension to resolve the single sensory signal over its entire time horizon. Thus we are introducing the spatial dimension by the canonical eight-dimensional data cloud (Canonical 8D-Data Cloud) concept to build methods to explore the impact-induced free dynamics of physical complex mechanical structures. The complex structure in this study is a large scale aluminum alloy plate stiffened by a frame made of T-section beams. The Canonical 8D-Data Cloud is identified with the simultaneous acceleration measurements by eight piezoelectric sensors equally spaced and attached on the periphery of a circular material curve drawn on the uniform surface of the stiffened plate. The Data Cloud approach leads to a systematic exploration-discovery-quantification of uncertainty in this physical complex structure. It is found that considerable uncertainty is stemming from the sensitivity of transient dynamics on the parameters of space-time localized force pulses, the latter being used as a means to diagnose the presence of structural anomalies. The Data Cloud approach leads to aspects of machine learning such as reduced dynamics analytics of big sensory data by means of heavenly machine-assisted computations to carry out the unparalleled data reduction analysis enabled by the Advanced Proper Orthogonal Decomposition Transform. Emphasized is the connection between the characteristic geometric features of high-dimensional datasets as a whole, the Data Cloud, and the modal physics of the dynamics.

A New Business Dimension

2001 ◽  
pp. 75-91
Author(s):  
Andre L. Brandao
Keyword(s):  
Antenna Arrays ◽  
Large Scale ◽  
Digital Signal ◽  
Spatial Dimension ◽  
Link Quality ◽  
Radio Link ◽  
Wireless Internet ◽  
Space Division Multiple Access ◽  
New Business ◽  
Market Opportunities

Space division multiple access (SDMA) is a promising technique useful for increasing capacity, reducing interference and improving overall wireless communication link quality. With a large-scale penetration expected for wireless Internet, the radio link will require significant reduction in cost and increase in capacity, benefits that the proper exploitation of the spatial dimension can offer. Market opportunities with SDMA are significant, as a number of companies have been recently formed to bring products based on this new concept to the wireless marketplace. The approach to SDMA is broad, ranging from "switched-beam techniques" to "adaptive antennas." Basically the technique employs antenna arrays and digital signal processing to achieve the necessary increases incapacity and quality needed in the wireless world.

scholarly journals Partition Function in One, Two, and Three Spatial Dimensions from Effective Lagrangian Field Theory

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Christoph P. Hofmann
Keyword(s):  
Partition Function ◽  
Degrees Of Freedom ◽  
Spatial Dimension ◽  
Lagrangian Method ◽  
Point Of View ◽  
Low Energy ◽  
Effective Theory ◽  
Goldstone Bosons ◽  
Effective Lagrangian ◽  
Spatial Dimensions

The systematic effective Lagrangian method was first formulated in the context of the strong interaction; chiral perturbation theory (CHPT) is the effective theory of quantum chromodynamics (QCD). It was then pointed out that the method can be transferred to the nonrelativistic domain—in particular, to describe the low-energy properties of ferromagnets. Interestingly, whereas for Lorentz-invariant systems the effective Lagrangian method fails in one spatial dimension (ds=1), it perfectly works for nonrelativistic systems in ds=1. In the present brief review, we give an outline of the method and then focus on the partition function for ferromagnetic spin chains, ferromagnetic films, and ferromagnetic crystals up to three loops in the perturbative expansion—an accuracy never achieved by conventional condensed matter methods. We then compare ferromagnets in ds=1, 2, 3 with the behavior of QCD at low temperatures by considering the pressure and the order parameter. The two apparently very different systems (ferromagnets and QCD) are related from a universal point of view based on the spontaneously broken symmetry. In either case, the low-energy dynamics is described by an effective theory containing Goldstone bosons as basic degrees of freedom.

scholarly journals Determination of Forecast Errors Arising from Different Components of Model Physics and Dynamics

2004 ◽  
Vol 132 (11) ◽  
pp. 2570-2594 ◽  
Author(s):  
T. N. Krishnamurti ◽  
J. Sanjay ◽  
A. K. Mitra ◽  
T. S. V. Vijaya Kumar
Keyword(s):  
Large Scale ◽  
Forecast Model ◽  
Regression Coefficients ◽  
Second Step ◽  
Cumulus Convection ◽  
Forecast Errors ◽  
Radiative Processes ◽  
Grid Points ◽  
Model Equations

Abstract This paper addresses a procedure to extract error estimates for the physical and dynamical components of a forecast model. This is a two-step process in which contributions to the forecast tendencies from individual terms of the model equations are first determined using an elaborate bookkeeping of the forecast. The second step regresses these estimates of tendencies from individual terms of the model equations against the observed total tendencies. This process is executed separately for the entire horizontal and vertical transform grid points of a global model. The summary of results based on the corrections to the physics and dynamics provided by the regression coefficients highlights the component errors of the model arising from its formulation. This study provides information on geographical and vertical distribution of forecast errors contributed by features such as nonlinear advective dynamics, the rest of the dynamics, deep cumulus convection, large-scale condensation physics, radiative processes, and the rest of physics. Several future possibilities from this work are also discussed in this paper.

scholarly journals Integer Programming Formulations for Approximate Packing Circles in a Rectangular Container

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Igor Litvinchev ◽  
Edith Lucero Ozuna Espinosa
Keyword(s):  
Large Scale ◽  
Optimization Problem ◽  
Continuous Optimization ◽  
Packing Problem ◽  
Regular Grid ◽  
Food Industries ◽  
Rectangular Container ◽  
Binary Problem ◽  
Weighted Number ◽  
Grid Nesting

A problem of packing a limited number of unequal circles in a fixed size rectangular container is considered. The aim is to maximize the (weighted) number of circles placed into the container or minimize the waste. This problem has numerous applications in logistics, including production and packing for the textile, apparel, naval, automobile, aerospace, and food industries. Frequently the problem is formulated as a nonconvex continuous optimization problem which is solved by heuristic techniques combined with local search procedures. New formulations are proposed for approximate solution of packing problem. The container is approximated by a regular grid and the nodes of the grid are considered as potential positions for assigning centers of the circles. The packing problem is then stated as a large scale linear 0-1 optimization problem. The binary variables represent the assignment of centers to the nodes of the grid. Nesting circles inside one another is also considered. The resulting binary problem is then solved by commercial software. Numerical results are presented to demonstrate the efficiency of the proposed approach and compared with known results.

General variational model reduction applied to incompressible viscous flows

2008 ◽  
Vol 617 ◽  
pp. 31-50 ◽  
Author(s):  
THORSTEN BOGNER
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Reynolds Numbers ◽  
Spatial Dimension ◽  
Reduced Model ◽  
Navier Stokes ◽  
Variational Model ◽  
Navier Stokes Equation ◽  
Nonlinear Dynamical ◽  
Density Matrix Renormalization ◽  
Spatial Dimensions

In this paper, a method is introduced that allows calculation of an approximate proper orthogonal decomposition (POD) without the need to perform a simulation of the full dynamical system. Our approach is based on an application of the density matrix renormalization group (DMRG) to nonlinear dynamical systems, but has no explicit restriction on the spatial dimension of the model system. The method is not restricted to fluid dynamics. The applicability is exemplified on the incompressible Navier–Stokes equation in two spatial dimensions. Merging of two equal-signed vortices with periodic boundary conditions is considered for low Reynolds numbers Re≤800 using a spectral method. We compare the accuracy of a reduced model, obtained by our method, with that of a reduced model obtained by standard POD. To this end, error functionals for the reductions are evaluated. It is observed that the proposed method is able to find a reduced system that yields comparable or even superior accuracy with respect to standard POD method results.

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