# Basis functions

## Top PDF Basis functions:

### DGTD methods using modal basis functions and symplectic local time-stepping: application to wave propagation problems

3.1 Criteria for the choice of local basis functions When a real implementation is considered, all choices for basis fields are not equivalent, be- cause the numerical computation of integrals present in matrix terms, and then the numerical solution of linear systems (or the inversion/factorization of local mass matrix) cannot be ex- act in general. Indeed, the condition number of the mass matrices as well as the quadrature formulae used for volume or interface integrals play an important role in the actual numerical accuracy and stability of the software developed [9]. The use of simplicial elements makes the construction of an optimal set of basis functions more difficult.
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### Method of Moments Simulation of Modulated Metasurface Antennas With a Set of Orthogonal Entire-Domain Basis Functions

specific sub-domains. The main advantage of the Fourier- Bessel basis with respect to Gaussian Ring basis Functions (GRBFs) resides in its orthogonality and its completeness, allowing an analysis of a general class of disk current distribu- tion in an effective manner i.e., more compactly. In comparison with the well known family of orthogonal Zernike functions, FBBFs possess a very limited bandwidth, leading to an easy and fast calculation of the reaction integrals in the spectral domain. Moreover we have observed that the Fourier-Bessel basis is better suited than the Zernike basis in describing typical current distributions on circular metasurfaces. The proposed basis can be extended to analyze the more general class of elliptical apertures, while maintaining the orthogo- nality properties. The basis chosen here is not limited to a particular kind of excitation even if the examples proposed in this paper used a dipole excitation. When the MTS is excited from the center, GRBFs allow a better representation of the currents in the immediate vicinity of the feed. Finally, it is expected that, owing to the orthogonality of the basis, the FBBF decomposition will provide new insight into the design procedure of circular metasurfaces without directly referring to their implementation using patches.
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### A generalized finite element formulation for arbitrary basis functions : from isogeometric analysis to XFEM

2. The compact support guarantees the sparseness of the global stiffness and mass matrices used in implicit solution algorithms. Elements, in this traditional setting, define regions used for both numerical quadrature and generating the basis functions. The simplicity of the Lagrange polynomials encourages a strategy, where every choice of basis function is implemented separately as its own element type to optimize its efficiency. The boundaries of the elements also define the geometry used to evaluate surface tractions, contact, and other boundary conditions. Again, the simplicity of the basis functions encourages the implementation of each type of boundary condition independently to optimize efficiency. To obtain the necessary generality in our formulation, some efficiency is necessarily sacrificed; however, the savings in development time and the use of code that has been validated using a broad class of basis functions more than compensates for this in our opinion.
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### Trefftz Discontinuous Galerkin basis functions for a class of Friedrichs systems coming from linear transport

1 Introduction Trefftz methods use exact solutions of a given partial differential equation (PDEs) as basis functions for the numerical approximation. Nevertheless, despite their great potential for the improvement of Finite Element Methods (FEM) solvers [ 34 ] or for the enrichment of Discontinuous Galerkin (DG) approximations [ 33 ], Trefftz methods are scarcely used nowadays [ 24 ]. One reason is the limited number of different Trefftz basis functions available in the literature. Indeed, for Trefftz methods, one has to construct new basis functions for every different class of PDEs: only after this task is accomplished, then it is possible to measure the gain in accuracy with respect to more conventional methods, or to incorporate them in general FEM or DG solvers [ 9 , 11 , 17 , 23 , 29 ]. One of the difficulty when applying the TDG method is therefore to find solutions to the system considered. Our objective in this work is precisely to contribute to this global program by constructing new exact solutions to general Friedrichs systems with relaxation, in view of their use as basis functions for an original Trefftz Discontinuous Galerkin (TDG) method. The Friedrichs systems with have in mind can be seen as extensions of wave like models already treated in [ 18 , 29 ]. With respect to the literature on Trefftz methods, we introduce a new generic non-zero relaxation matrix which is representative of dissipation mechanisms often encountered in complex physics.
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### RWG Basis Functions for Accurate Modeling of Substrate Integrated Waveguide Slot-Based Antennas

When comparing the accuracy of the solutions, one should keep in mind that the proper evaluation of the post diffraction in both HFSS and CST depend on the accuracy of the 3D mesh, the former being hexahedral, the latter cubic. While the chosen mesh is important also with RWG basis functions to describe currents on slots - in this case, 18 of them are sufficient to provide these results -, the post contribution is exact, as the full cylindrical shape is included analytically in the MM boundaries.

### Calculation of Generalized Polynomial-Chaos Basis Functions and Gauss Quadrature Rules in Hierarchical Uncertainty Quantification

Therefore, a question is how to determine the generalized polynomial-chaos basis functions and Gauss quadrature rules from a general surrogate model. This paper aims to partly answer this key question in hierarchical uncertainty quan- tification. Our method is based on the ideas of changing variables and monotone interpolation [34]–[37]. Specifically, we represent the random input as a linear function of a new parameter, and treat such parameter as a new random input. Using two monotone interpolation schemes, physically consis- tent closed-form cumulative density functions and probability density functions can be constructed for the new random input. Due to the special forms of the obtained density functions, we can easily determine a proper Gauss quadrature rule and the basis functions for a generalized polynomial chaos expansion. We focus on the general framework and verify our method by using both synthetic and performance-level circuit surrogate models. Our method can be employed to handle a wide variety of surrogate models, including device-level models for SPICE- level simulators [1]–[3], circuit-level performance models for behavior-level simulation [8], as well as gate-level statistical models for the timing analysis of digital VLSI [16]–[18]. In this paper we will focus only on the derivation of the basis functions and Gauss quadrature rules and refer the reader interested to the extensive literature on how to use them in a stochastic spectral simulator (see [1]–[5], [20]–[27] and the references therein).
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### Policy search with cross-entropy optimization of basis functions

preferred to their artificial intelligence counterparts. For instance, the term ‘controller’ will be used instead of ‘agent’, and ‘process’ instead of ‘environment’. or state-action pair. The value function is used to compute a policy. Unfortunately, most state-of-the-art value-function techniques do not scale well to high-dimensional problems: they are typically applied to problems with only up to six continuous state variables [3]–[6]. Moreover, designing value function approximators is often a difficult, counterintuitive task [3]. Motivated by these shortcomings, direct policy search algorithms have been proposed [7]–[11]. These al- gorithms parameterize the policy and search for an optimal parameter vector which maximizes the return, without using a value function. In the literature, typically ad-hoc policy parameterizations with a few parameters are designed for specific problems, using intuition and prior knowledge about the optimal policy [8], [9]. On the other hand, in the area of value function approximation, significant efforts have been invested to develop techniques that automatically find good approximators, without relying on prior knowledge [3], [5], [6], [12]. Most of these techniques represent value functions using a linear combination of basis functions (BFs), and automatically find BFs that lead to an accurate value function representation.
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### Environmental effects with Frozen Density Embedding in Real-Time Time-Dependent Density Functional Theory using localized basis functions

I (t), ρ II ] ). Again we need its mapping onto the real-space grid as shown in the rst green box (box a). Then, we utilize the methods reported in the rectangular orange box (box b) to calculate the non-additive part of the embedding potential at each grid point. Finally we add the non-additive (kinetic and exchange-correlation) poten- tial to the electrostatic potential of the environment calculated again at each grid point. It should be noted that because the density of the environment is frozen, thus the correspond- ing electrostatic potential remains constant during the time propagation. In the next phase its matrix representation in the localized Gaussian basis functions is obtained as in Eq.20, (box c, in Figure). The active system is evolved (box d) using an eective time-dependent Kohn-Sham matrix, which contains the usual implicit and explicit time-dependent terms, respectively (J[ρ I (t)] +V XC [ρ I (t)] ) and v ext (t) , plus the time-dependent embedding potential
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### Cross-entropy optimization of control policies with adaptive basis functions

Cross-Entropy Optimization of Control Policies With Adaptive Basis Functions Lucian Bu¸soniu, Damien Ernst, Member, IEEE, Bart De Schutter, Member, IEEE, and Robert Babuška Abstract—This paper introduces an algorithm for direct search of control policies in continuous-state discrete-action Markov deci- sion processes. The algorithm looks for the best closed-loop policy that can be represented using a given number of basis functions (BFs), where a discrete action is assigned to each BF. The type of the BFs and their number are specified in advance and determine the complexity of the representation. Considerable flexibility is achieved by optimizing the locations and shapes of the BFs, to- gether with the action assignments. The optimization is carried out with the cross-entropy method and evaluates the policies by their empirical return from a representative set of initial states. The return for each representative state is estimated using Monte Carlo simulations. The resulting algorithm for cross-entropy pol- icy search with adaptive BFs is extensively evaluated in problems with two to six state variables, for which it reliably obtains good policies with only a small number of BFs. In these experiments, cross-entropy policy search requires vastly fewer BFs than value- function techniques with equidistant BFs, and outperforms policy search with a competing optimization algorithm called DIRECT.
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### Parameter optimization of orthonormal basis functions for efficient rational approximations

Keywords: Müntz-Laguerre functions; orthogonal projection; pole placement; model order reduction. 1. Introduction Rational orthogonal basis functions (OBF) are useful tools in the identification and modeling of linear dynamical systems and found numerous applications in control and signal processing [1]. In approximation problems using OBF, one of the major difficulties is the choice of the poles defining the functions. Due to their simplicity, Laguerre basis functions are often used. They have a real multiple-order pole whose choice is of great importance for computing low-order and good quality models. Much work has been done on the subject and optimal methods [2-5] or sub-optimal methods [6-8] have been proposed in literature. However Laguerre functions are poorly suited to compact modeling of systems possessing several time constants or resonant characteristics. Two-parameter Kautz functions are more adequate for modeling such systems but their efficiency is also limited. Techniques for an optimal or a suboptimal choice of the two- parameter Kautz poles are respectively presented in [9] and [10].
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### Multifidelity surrogate modeling based on Radial Basis Functions

Another possible choice in this context is the Radial Ba- sis Functions (RBF) based interpolation method (Pow- ell 1987; Dyn et al 1986). Among metamodeling ap- proaches, it is one of the most effective multidimen- sional approximation methods (Jin et al 2001), as the dimension of the input space does not alter its perfor- mance (Powell 2001). It is also suitable for optimization since an adaptive sampling strategy has been developed by Gutmann (2001) to sequentially enrich the training database of the RBF metamodel, in order to find an optimal design of the model simulated.

### Radial Basis Functions and Kriging Metamodels for Aerodynamic Optimization

Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex France Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vi[r]

### Three-Dimensional Swirl Flow Velocity-Field Reconstruction Using a Neural Network With Radial Basis Functions

statistical distribution (zero mean and variance one) to prevent numerical approximation problem during the learning step due to the use of data having different order of magnitude. 3 – 2 Learning the neural network Algorithms of “standard ridge regression” and “forward selection” are implemented using Matlab software on a personal computer Pentium II 333 MHz. To initialize the neural network method, only radii of activation functions φ have to be previously defined. They are arbitrarily initialized to give an a priori good representation of the desired evolution. Though this step is not essential because these values will be next optimized during the neural network learning step, computation-time can be decreased in this way. Initial values of radii are given in Table 2.
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### The Cellular Basis for Animal Regeneration

2008 ). Second, cells deriving from the different tissues may respond to injury and regeneration cues differently. A host of molecular signaling factors including WNTs, BMP/TGF- bs, IGFs, and FGFs, have been identified as involved in appendage regeneration based on the inhibition of regeneration upon their inactivation, but detailed analysis of these phenotypes has been limited (for reviews see Antos and Tanaka, 2010; Poss, 2010; Stoick-Cooper et al., 2007; Yokoyama, 2008 ). Precisely how these pathways affect cells from each different tissue is crucially lacking. Third, the similarity in lineage restriction between limb blastema cells and progenitors found in the devel- oping limb bud suggests that morphogenesis and patterning events occurring during limb development and regeneration may be more similar than previously appreciated ( Nacu and Tanaka, 2011 ). Finally, it is interesting that in vertebrate limb regeneration, blastema cells show restriction not only in their tissue fates, but also in the positional identity they can adopt along the proximal distal axis ( Butler, 1955; Kragl et al., 2009 ). Understanding the molecular basis of positional identity and its restriction during regeneration is also an important future goal ( Tamura et al., 2010 ).
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### On the reduction of a random basis

of the lattice. The lattice basis reduction problem deals with finding a basis of a given lattice, whose vectors are “short” and ”almost orthogonal”. The problem is old and there are numerous notions of reduction (for a general survey, see for example [15, 26, 14]). Solving even approximately the lattice basis reduction problem has numerous theoretical and practical applications in integer optimization [19], computational number theory [18] and cryptog- raphy [23]. In 1982, Lenstra, Lenstra and Lov´ asz [18] introduced for the first time an efficient (polynomial with respect to the length of the input) approx- imation reduction algorithm. It depends on a real approximation parameter s ∈]0, √ 3/2[ and is called LLL(s). The output basis of the LLL algorithm is called an LLL(s) reduced or s-reduced basis. The next definition (character- izing a s-reduced basis) and the LLL-algorithm itself make a broad use of the
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### On the reduction of a random basis

In this paper we study the asymptotics (with respect to the dimension n of the ambient space) of the random variables M g n and I n g under spherical models and for general codimensions of the random basis. The variable M g n is the supremum of the set of those s for which the basis is s 2 - reduced. As mentioned earlier an LLL(s) reduced basis satisfies a set of local conditions. The second variable I n g is the place where the satisfied local condition is the weakest. This indicates

### Border Basis relaxation for polynomial optimization

The optimization problem can be reformulated as solving polynomial equations related to the (minimal) critical value of the polynomial f on a semi-algebraic set. Polynomial solvers based, for instance, on Gröbner basis or border basis computation can then be used to recover the real critical points from the complex solutions of (zero-dimensional) polynomial systems (see e.g. (Parrilo and Sturmfels, 2003; Safey El Din, 2008; Greuet and Safey El Din, 2011)). This type of methods relies entirely on polynomial algebra and univariate root finding. So far, there is no clear comparison of these elimination methods and the relaxation approaches.
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### Structure as a basis of peat classification

Structural model of Moonbeam (axonic) peat. 3 ) A macro- and microscopic examination of these gross struc- tural types should result in the development of a mathem[r]

### Entire functions sharing small functions with their difference operators

Faculty of Exact Sciences, University of Bechar-(Algeria) elfarissi.abdallah@yahoo.fr Abstract. We investigate uniqueness problems for an entire function that shares two small functions of finite order with their difference operators. In particular, we give a generalization of a result in [2].

### Minimum Implicational Basis for -Semidistributive Lattices

Instance: The set of meet-irreducible elements M(C L ) of the closure system C L . Question: Find a minimum basis Σ for C L . This problem remains open for general lattices. Duquenne [2] has given a latticial version of this prob- lem and shown that it is polynomial for upper locally distributive lattices or antimatroid. Recently, Wild [14] has proposed a polynomial time algorithm to compute