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Classification Techniques
Published in Harry G. Perros, An Introduction to IoT Analytics, 2021
This is a term to indicate the various problems that arise when dealing with data with high-dimensionality, which typically do not occur with data with lower dimensions. These problems arise in many areas such as numerical analysis, sampling, combinatorics, Machine Learning, data mining, and databases. The common theme of these problems is that when the data dimensionality increases, the volume of the space within which the data lie increases as well so that the data becomes sparse. For example, let us assume that we have 1000 one-dimensional data points uniformly distributed within a unit interval. Now, let us change this data to two-dimensional data also uniformly distributed in the equivalent space which is now a unit square. Intuitively, we can see that the two-dimensional data is not as densely distributed in the unit square as the one-dimensional data in the unit interval. Likewise, if we assume that the data is now three-dimensional within a unit cube.
Finite Element Interpolation
Published in Dr Arzhang Angoshtari, Ali Gerami Matin, Finite Element Methods in Civil and Mechanical Engineering, 2020
Dr Arzhang Angoshtari, Ali Gerami Matin
Meshes of some simple geometries can be directly defined in FEniCS. For example, the following code generates structured meshes of the unit square and the unit cube: Mesh_S = UnitSquareMesh(4,4) Mesh_C = UnitCubeMesh(4,4,4)
Wearable Compact Fractal Antennas for 5G and Medical Systems
Published in Albert Sabban, Wearable Systems and Antennas Technologies for 5G, IOT and Medical Systems, 2020
A curve, with endpoints, is represented by a continuous function whose domain is the unit interval [0, 1]. The curve may lie in a plane or in a three-dimensional space. A fractal curve is a densely self-intersecting curve that passes through every point of the unit square. A fractal curve is a continuous mapping from the unit interval to the unit square.
Discretization error estimates for discontinuous Galerkin isogeometric analysis
Published in Applicable Analysis, 2023
The parameterization of the computational domain using just one tensor-product spline function is possible only in simple cases. A necessary condition for this to be possible is that the computational domain is topologically equivalent to the unit square or the unit cube. This might not be the case for more complicated computational domains. Such domains are typically decomposed into subdomains, in IgA called patches, where each of them is parameterized by its own geometry function. The standard approach is to set up a conforming discretization. For a standard Poisson problem, this means that the overall discretization needs to be continuous. For higher order problems, like the biharmonic problem, even more regularity is required, conforming discretizations in this case are rather hard to construct, cf. Ref. [2] and references therein.
Parameter estimation with model order reduction for elliptic differential equations
Published in Inverse Problems in Science and Engineering, 2018
Axel Ariaan Lukassen, Martin Kiehl
The second example is the thermal block. The thermal block is describing the heat transport through a block of solid material. This block consists of disjunct subblocks of different heat conductivities . We assume, that for all i and assume, that the first parameter is equal 1. Hence, we restrict the parameters to . We use the setting of [8,9]. Then the domain is the unit square . The bottom boundary with unit outward normal n(x) is denoted by , the left and right boundary are and the upper boundary is named . The boundary conditions are similar to [9]. We have an unit flux into the domain on . The no-flux condition holds on and the boundary values are set zero on . Furthermore, the heat conductivity of the complete block is described by
Exponential high-order compact finite difference method for convection-dominated diffusion problems on nonuniform grids
Published in Numerical Heat Transfer, Part B: Fundamentals, 2019
We conduct numerical experiments with nine test problems. The 1D problems are defined on the unit interval and the 2 D problems are defined on the unit square We use the following function to generate computational grids: where Nx, Ny are the number of the subintervals along x- and y-directions, λ is the stretching parameter controlling the density of grid points in the x-direction, and θ is the control angle. It is worth to point out that the gird is uniform if and only if λ = 0. When if a nonuniform grid along the coordinate axis direction with grid clustering near the right boundary of the computational domain; if a nonuniform grid along the coordinate axis direction with grid clustering near the left boundary. When if a nonuniform grid along the coordinate axis direction with clustering grids near the two boundary; if a nonuniform grid along the coordinate axis direction with grid clustering near the interval middle of the computational domain. And much larger the absolute value of λ is, more grid points are clustered into the area required refinement and vice versa.