scholarly journals Numerical Approximation of Poisson Problems in Long Domains

2021 ◽  
Author(s):  
Michel Chipot ◽  
Wolfgang Hackbusch ◽  
Stefan Sauter ◽  
Alexander Veit
Keyword(s):  
Numerical Experiments ◽  
Exponential Sums ◽  
Differential Operators ◽  
Cartesian Product ◽  
Computational Effort ◽  
Tensor Rank ◽  
Tensor Structure ◽  
Order Tensor ◽  
Tensor Approximation ◽  
The Right

AbstractIn this paper, we consider the Poisson equation on a “long” domain which is the Cartesian product of a one-dimensional long interval with a (d − 1)-dimensional domain. The right-hand side is assumed to have a rank-1 tensor structure. We will present and compare methods to construct approximations of the solution which have tensor structure and the computational effort is governed by only solving elliptic problems on lower-dimensional domains. A zero-th order tensor approximation is derived by using tools from asymptotic analysis (method 1). The resulting approximation is an elementary tensor and, hence has a fixed error which turns out to be very close to the best possible approximation of zero-th order. This approximation can be used as a starting guess for the derivation of higher-order tensor approximations by a greedy-type method (method 2). Numerical experiments show that this method is converging towards the exact solution. Method 3 is based on the derivation of a tensor approximation via exponential sums applied to discretized differential operators and their inverses. It can be proved that this method converges exponentially with respect to the tensor rank. We present numerical experiments which compare the performance and sensitivity of these three methods.

scholarly journals A 3-D RBF-FD solver for modeling the atmospheric global electric circuit with topography (GEC-RBFFD v1.0)

2015 ◽  
Vol 8 (10) ◽  
pp. 3007-3020 ◽  
Author(s):  
V. Bayona ◽  
N. Flyer ◽  
G. M. Lucas ◽  
A. J. G. Baumgaertner
Keyword(s):  
Numerical Model ◽  
Differential Operators ◽  
Lower Boundary ◽  
Elliptic Partial Differential Equation ◽  
Electric Circuit ◽  
Global Electric Circuit ◽  
Mesh Free ◽  
Conductivity Profile ◽  
The Right ◽  
Upper Boundary

Abstract. A numerical model based on radial basis function-generated finite differences (RBF-FD) is developed for simulating the global electric circuit (GEC) within the Earth's atmosphere, represented by a 3-D variable coefficient linear elliptic partial differential equation (PDE) in a spherically shaped volume with the lower boundary being the Earth's topography and the upper boundary a sphere at 60 km. To our knowledge, this is (1) the first numerical model of the GEC to combine the Earth's topography with directly approximating the differential operators in 3-D space and, related to this, (2) the first RBF-FD method to use irregular 3-D stencils for discretization to handle the topography. It benefits from the mesh-free nature of RBF-FD, which is especially suitable for modeling high-dimensional problems with irregular boundaries. The RBF-FD elliptic solver proposed here makes no limiting assumptions on the spatial variability of the coefficients in the PDE (i.e., the conductivity profile), the right hand side forcing term of the PDE (i.e., distribution of current sources) or the geometry of the lower boundary.

Transient Natural Convection Between Two Zones in an Insulated Enclosure

1988 ◽  
Vol 110 (1) ◽  
pp. 116-125 ◽  
Author(s):  
P. A. Litsek ◽  
A. Bejan
Keyword(s):  
Heat Transfer ◽  
Natural Convection ◽  
Numerical Experiments ◽  
Thermal Stratification ◽  
Convection Flow ◽  
Natural Convection Flow ◽  
Flow And Heat Transfer ◽  
Vertical Opening ◽  
The Right ◽  
Rayleigh Numbers

The natural convection flow and heat transfer between two enclosures that communicate through a vertical opening is studied by considering the evolution of an enclosed fluid in which the left half is originally at a different temperature than the right half. Numerical experiments show that at sufficiently high Rayleigh numbers the ensuing flow is oscillatory. This and other features are anticipated on the basis of scale analysis. The time scales of the oscillation, the establishment of thermal stratification, and eventual thermal equilibrium are determined and tested numerically. At sufficiently high Rayleigh numbers the heat transfer between the communicating zones is by convection, in accordance with the constant-Stanton-number trend pointed out by Jones and Otis (1986). The range covered by the numerical experiments is 102 < Ra < 107, 0.71 < Pr < 100, and 0.25 < H/L < 1.

scholarly journals Locating Critical Circular and Unconstrained Failure Surface in Slope Stability Analysis with Tailored Genetic Algorithm

2017 ◽  
Vol 39 (4) ◽  
pp. 87-98
Author(s):  
Tomasz Pasik ◽  
Raymond van der Meij
Keyword(s):  
Genetic Algorithm ◽  
Slope Stability ◽  
Failure Surface ◽  
Computational Effort ◽  
Engineering Practice ◽  
Initial Population ◽  
Detailed Result ◽  
Population Type ◽  
Slip Planes ◽  
The Right

Abstract This article presents an efficient search method for representative circular and unconstrained slip surfaces with the use of the tailored genetic algorithm. Searches for unconstrained slip planes with rigid equilibrium methods are yet uncommon in engineering practice, and little publications regarding truly free slip planes exist. The proposed method presents an effective procedure being the result of the right combination of initial population type, selection, crossover and mutation method. The procedure needs little computational effort to find the optimum, unconstrained slip plane. The methodology described in this paper is implemented using Mathematica. The implementation, along with further explanations, is fully presented so the results can be reproduced. Sample slope stability calculations are performed for four cases, along with a detailed result interpretation. Two cases are compared with analyses described in earlier publications. The remaining two are practical cases of slope stability analyses of dikes in Netherlands. These four cases show the benefits of analyzing slope stability with a rigid equilibrium method combined with a genetic algorithm. The paper concludes by describing possibilities and limitations of using the genetic algorithm in the context of the slope stability problem.

scholarly journals X-ray jets and magnetic flux emergence in the Sun

2008 ◽  
Vol 4 (S259) ◽  
pp. 201-210
Author(s):  
Fernando Moreno-Insertis
Keyword(s):  
Magnetic Flux ◽  
Solar Atmosphere ◽  
Numerical Experiments ◽  
High Speed ◽  
Computational Effort ◽  
Plasma Jets ◽  
Solar Interior ◽  
Magnetized Plasma ◽  
Flux Emergence ◽  
X Ray

AbstractMagnetized plasma is emerging continually from the solar interior into the atmosphere. Magnetic flux emergence events and their consequences in the solar atmosphere are being observed with high space, time and spectral resolution by a large number of space missions in operation at present (e.g. SOHO, Hinode, Stereo, Rhessi). The collision of an emerging and a preexisting magnetic flux system in the solar atmosphere leads to the formation of current sheets and to field line reconnection. Reconnection under solar coronal conditions is an energetic event; for the field strengths, densities and speeds involved in the collision of emerging flux systems, the reconnection outflows lead to launching of high-speed (hundreds of km/s), high-temperature (107 K) plasma jets. Such jets are being observed with the X-Ray and EUV detectors of ongoing satellite missions. On the other hand, the spectacular increase in computational power in recent years permits to carry out three-dimensional numerical experiments of the time evolution of flux emerging systems and the launching of jets with a remarkable degree of detail.In this review, observation and modeling of the solar X-Ray jets are discussed. A two-decade long computational effort to model the magnetic flux emergence events by different teams has led to numerical experiments which explain, even quantitatively, many of the observed features of the X-ray jets. The review points out that, although alternative mechanisms must be considered, flux emergence is a prime candidate to explain the launching of the solar jets.

Perturbation of nonlinear partial differential variational inequalities, II

1978 ◽  
Vol 82 (1-2) ◽  
pp. 153-170
Author(s):  
Elena Stroescu
Keyword(s):  
Variational Inequality ◽  
Differential Operators ◽  
Cartesian Product ◽  
Divergence Form ◽  
Partial Differential ◽  
Convergence Of Solutions ◽  
Direct Sum ◽  
Partial Differential Operators ◽  
Compactness Theorems ◽  
Differential Variational Inequalities

SynopsisThis paper is devoted to the study of the weak respectively strong convergence of solutions of a variational inequality, with nonlinear partial differential operators of the generalized divergence form and of semimonotone type, under a perturbation of the domain of definition. In this study we use abstract convergence theorems given by Stroescu and Vivaldi, convergence concepts defined according to Stummel and compactness theorems of the natural imbedding of the Cartesian product of Sobolev spaces into the direct sum of Lp spaces, also by Stummel.

scholarly journals Analytic tadpole coefficients of one-loop integrals

2021 ◽  
Vol 2021 (9) ◽  
Author(s):  
Bo Feng ◽  
Tingfei Li ◽  
Xiaodi Li
Keyword(s):  
Differential Equations ◽  
Recurrence Relations ◽  
Differential Operators ◽  
Tensor Structure ◽  
Loop Integral ◽  
Cut Method ◽  
Analytic Expressions

Abstract One remaining problem of unitarity cut method for one-loop integral reduction is that tadpole coefficients can not be straightforward obtained through this way. In this paper, we reconsider the problem by applying differential operators over an auxiliary vector R. Using differential operators, we establish the corresponding differential equations for tadpole coefficients at the first step. Then using the tensor structure of tadpole coefficients, we transform the differential equations to the recurrence relations for undetermined tensor coefficients. These recurrence relations can be solved easily by iteration and we can obtain analytic expressions of tadpole coefficients for arbitrary one-loop integrals.

Numerical Experiments in Turbulent Natural Convection Using Two-Equation Eddy-Viscosity Models

2008 ◽  
Vol 130 (7) ◽  
Author(s):  
X. Albets-Chico ◽  
A. Oliva ◽  
C. D. Pérez-Segarra
Keyword(s):  
Natural Convection ◽  
Aspect Ratio ◽  
Numerical Experiments ◽  
Eddy Viscosity ◽  
Computational Effort ◽  
Simulation Data ◽  
Turbulent Natural Convection ◽  
Viscosity Models ◽  
Buoyancy Driven Flows ◽  
Rayleigh Numbers

This work is focused on the simulation and prediction of turbulent natural convection flows by means of two-equation eddy-viscosity models. In order to show the generality, precision, and numerical issues related to these models under natural convection, three different buoyancy-driven cavities have been simulated: a tall cavity with a 30:1 aspect ratio, a cavity with a 5:1 aspect ratio, and, finally, a 4:1 aspect ratio cavity. All cases are solved under moderate and∕or transitional Rayleigh numbers (2.43×1010, 5×1010, and 1×1010, respectively) and all simulations are compared to experimental and∕or direct numerical simulation data available in literature. These different situations allow to check the applicability of two-equation eddy-viscosity models in buoyancy-driven flows, giving criteria on computational effort∕precision and their physical behavior.

scholarly journals The Vortex-like Behavior of the Riemann Zeta Function to the Right of the Critical Strip

2021 ◽  
Vol 77 (1) ◽  
Author(s):  
J. M. Sepulcre ◽  
T. Vidal
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Exponential Sums ◽  
Critical Strip ◽  
Real Numbers ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
The Right ◽  
Class Of Functions ◽  
Dense Set

AbstractBased on an equivalence relation that was established recently on exponential sums, in this paper we study the class of functions that are equivalent to the Riemann zeta function in the half-plane $$\{s\in {\mathbb {C}}:\mathrm{Re}\, s>1\}$$ { s ∈ C : Re s > 1 } . In connection with this class of functions, we first determine the value of the maximum abscissa from which the images of any function in it cannot take a prefixed argument. The main result shows that each of these functions experiments a vortex-like behavior in the sense that the main argument of its images varies indefinitely near the vertical line $$\mathrm{Re}\, s=1$$ Re s = 1 . In particular, regarding the Riemann zeta function $$\zeta (s)$$ ζ ( s ) , for every $$\sigma _0>1$$ σ 0 > 1 we can assure the existence of a relatively dense set of real numbers $$\{t_m\}_{m\ge 1}$$ { t m } m ≥ 1 such that the parametrized curve traced by the points $$(\mathrm{Re} (\zeta (\sigma +it_m)),\mathrm{Im}(\zeta (\sigma +it_m)))$$ ( Re ( ζ ( σ + i t m ) ) , Im ( ζ ( σ + i t m ) ) ) , with $$\sigma \in (1,\sigma _0)$$ σ ∈ ( 1 , σ 0 ) , makes a prefixed finite number of turns around the origin.

scholarly journals An orthogonal equivalence theorem for third order tensors

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Liqun Qi ◽  
Chen Ling ◽  
Jinjie Liu ◽  
Chen Ouyang
Keyword(s):  
Singular Values ◽  
Equivalence Theorem ◽  
Low Rank ◽  
Third Order ◽  
Order Tensor ◽  
Rank Tensor ◽  
Tensor Recovery ◽  
Rank One ◽  
Tensor Approximation ◽  
Value Decomposition

<p style='text-indent:20px;'>In 2011, Kilmer and Martin proposed tensor singular value decomposition (T-SVD) for third order tensors. Since then, T-SVD has applications in low rank tensor approximation, tensor recovery, multi-view clustering, multi-view feature extraction, tensor sketching, etc. By going through the Discrete Fourier Transform (DFT), matrix SVD and inverse DFT, a third order tensor is mapped to an f-diagonal third order tensor. We call this a Kilmer-Martin mapping. We show that the Kilmer-Martin mapping of a third order tensor is invariant if that third order tensor is taking T-product with some orthogonal tensors. We define singular values and T-rank of that third order tensor based upon its Kilmer-Martin mapping. Thus, tensor tubal rank, T-rank, singular values and T-singular values of a third order tensor are invariant when it is taking T-product with some orthogonal tensors. Some properties of singular values, T-rank and best T-rank one approximation are discussed.</p>

scholarly journals Moments and oscillations of exponential sums related to cusp forms

2016 ◽  
Vol 162 (3) ◽  
pp. 479-506 ◽  
Author(s):  
ESA V. VESALAINEN
Keyword(s):  
Exponential Sums ◽  
Upper Bounds ◽  
Cusp Forms ◽  
Average Order ◽  
Small Denominators ◽  
Moment Estimates ◽  
Holomorphic Cusp Forms ◽  
Order Of Magnitude ◽  
The Mean ◽  
The Right

AbstractWe consider large values of long linear exponential sums involving Fourier coefficients of holomorphic cusp forms. The sums we consider involve rational linear twistse(nh/k) with sufficiently small denominators. We prove both pointwise upper bounds and bounds for the frequency of large values. In particular, thek-aspect is treated. As an application we obtain upper bounds for all the moments of the sums in question. We also give the asymptotics with the right main term for fourth moments.We also consider the mean square of very short sums, proving that on average short linear sums with rational additive twists exhibit square root cancellation. This result is also proved in a slightly sharper form.Finally, the consideration of moment estimates for both long and short exponential sums culminates in a result concerning the oscillation of the long linear sums. Essentially, this result says that for a positive proportion of time, such a sum stays in fairly long intervals, where its order of magnitude does not drop below the average order of magnitude and where its argument is in a given interval of length 3π/2+ϵ and so cannot wind around the origin.

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