Genetic Exponentially Fitted Method for Solving Multi-dimensional Drift-diffusion Equations

Über die kreiseleigenschaften des atomkerns

In this paper, we develop a fully nonconforming (both $H(\operatorname{grad}\operatorname{curl})$-nonconforming and $H(\operatorname{curl})$-nonconforming) finite element on cubical meshes that can fit into the Stokes complex. The newly proposed element has 24 degrees of freedom. Unlike the fully $H(\operatorname{grad}\operatorname{curl})$-nonconforming tetrahedral finite elements in Huang (2023, Nonconforming finite element stokes complexes in three dimensions. Sci. China Math., 66, 1879–1902), the element in this paper yields a robust finite element method to solve the singularly perturbed quad-curl problem. To confirm this, we prove the optimal convergence of order $\mathscr O(h)$ for a fixed parameter $\varepsilon $ and the uniform convergence of order $\mathscr O(h^{1/2})$ for any value of $\varepsilon $ in the sense of an $H(\operatorname{grad}\operatorname{curl})$-norm. When $\varepsilon $ tends to 0, we employ the Nitsche’s method to improve the convergence order from $\mathscr O(h^{1/2})$ to $\mathscr O(h)$. Some numerical examples are used to verify the validity of the theoretical analysis.

This paper presents a versatile hyper-ellipsoidal basis function for function approximation in a given high dimensional space. This hyper-ellipsoidal basis function can be translated and rotated to cover the data based upon the distribution of data in a given high dimensional space. Based on this function, we propose a one-pass hyper-ellipsoidal learning algorithm for which any new incoming data can be fed for learning without involving the previously learned one. This learning algorithm is used to adjust the parameters of the versatile hyper-ellipsoidal basis function. In addition, we propose the hyper-ellipsoidal basis function (HEBF) neural network that uses the one-pass hyper-ellipsoidal neural learning algorithm. The structure of this neural network is similar to the radial basis function (RBF) neural networks. The hidden neurons in the HEBF neural network can be increased or decreased during learning process. The number of the hidden neurons in the network can be grown based on geometric growth criterion and can be reduced by merging the two hidden neurons into a new hidden neuron based on merging criterion during learning process. The merging process can be done independently without considering the learned data set.KeywordsRadial Basis FunctionHide NeuronRadial Basis Function Neural NetworkHigh Dimensional SpaceInitial WidthThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A family of nonlinear conservative finite difference schemes for the multidimensional Boussinesq Paradigm Equation is considered. A second order of convergence and a preservation of the discrete energy for this approach are proved. Existence and boundedness of the discrete solution on an appropriate time interval are established. The schemes have been numerically tested on the models of the propagation of a soliton and the interaction of two solitons. The numerical experiments demonstrate that the proposed family of schemes is about two times more accurate than the family of schemes studied in [Kolkovska N., Two families of finite difference schemes for multidimensional Boussinesq paradigm equation, In: Application of Mathematics in Technical and Natural Sciences, Sozopol, June 21–26, 2010, AIP Conf. Proc., 1301, American Institute of Physics, Melville, 2010, 395–403].

Locking-free finite element method for a bending moment formulation of Timoshenko beams

The derivation of minimal support basis functions for the discrete divergence operator

This paper considered the solution of special second order delay differential equations. The method was generated by interpolation and collocation techniques using a combination of power series and exponential function. The approximate basis function is interpolated at the first two grid points and collocated at both grid and off-grid points. The developed schemes and its derivatives were combined to form block methods to simultaneously solve special second order delay differential equations. The basic properties of the methods such as order, error constants, consistency and convergence were also examined. Numerical results are presented to show that the proposed method is suitable for solving second order delay differential equations.

Form invariant stress-strain relationships are presented for anisotropic media. The case of transversely isotropic media is then considered and the five elastic constants needed to describe such a medium are obtained in terms of the elements of the stiffness tensor. The dispersion equation is solved and approximate basis functions are constructed to expand the displacement fields inside such an anisotropic inclusion. The scattering of elastic waves by spherical or nonspherical inclusions is studied as a function of frequency and geometry and compared with the isotropic case.

In this paper, a robust L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> filtering method is proposed for a class of switched linear parameter varying (LPV) systems in which the state-space matrices and the time delay rely on real-time measured parameter. By using multiple delay-dependent Lyapunov-Krasovskii function, the delay-dependent L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> performance criterion for the switched LPV systems is first established. As there exists coupling between Lyapunov function and system parameter matrices, we utilized a slack matrix variable to decouple it. Because of the dependence on parameter, approximate basis function and gridding technique are utilized, and the solution for the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> filters can be transformed into finite dimensional convex optimization problem. Based on the switching logic with the minimum average dwell time (ADT) and parameter linear matrix inequalities (PLMIs) technique, the resulting filters guarantee the filtering error systems to be exponentially stable with a prespecified L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> disturbance attenuation level. A numerical example is presented to verify the effectiveness of the proposed method.

Typical real world machine control tasks have some characteristics which makes them difficult to solve: Their state spaces are high-dimensional and continuous, and it may be impossible to reach a satisfying target state by exploration or human control. To overcome these problems, in this paper, we propose (1) to use radial basis functions for value function approximation in continuous space reinforcement learning and (2) the use of learned inverse projection functions for state space exploration. We apply our approach to path planning in dynamic environments and to an aircraft autolanding simulation, and evaluate its performance.

Isogeometric analysis (IGA) is a computational technique that integrates computer-aided design (CAD) with finite element analysis (FEA) by employing the same basis functions for both geometry representation and solution approximation. While IGA offers numerous advantages, such as improved accuracy and efficiency, it also presents several challenges related to geometric modeling. Some of these challenges include accurately representing complex geometries with NURBS (Non-Uniform Rational B-Splines) or other basis functions used in IGA and generating high-quality meshes that conform to the complex geometry represented by NURBS curves/surfaces. This paper introduces an analytical framework to provide a more efficient and theoretically grounded method for generating curvilinear configurations and its analytical solution in IGA, bridging the gap between generated data and its physical representations. This innovative approach is distinguished by integrating the NURBS parameterization in curve generation and providing a corresponding framework to achieve a broader and more accurate explanation of meshes and properties, especially constructing new coordinates and calculating the physical displacements under these conditions. Our model enables the analytical understanding of complex curves from the UIUC airfoil and superformula datasets, demonstrating a deeper dive into simulations. This study signifies a pivotal juncture wherein machine-learning-based complex geometrical formulations are synergistically combined with actual isogeometric analysis.

Numerical solution of two-dimensional stochastic Fredholm integral equations on hypercube domains via meshfree approach

The paper introduces a local reproducing kernel method based on spatial trial space spanned by the Newton basis functions in the “native” Hilbert space of the reproducing kernel. It is a truly meshless approach which uses the local sub clusters of domain nodes, named local domains of influence, for the calculation of the spatial partial derivatives. With the selected domain of influence, an approximation function is introduced as a sum of weighted local Newton basis functions. Then the collocation approach is used to determine weights. After the successful approximation function creation, all the needed differential operators can be constructed by applying an arbitrary operator on the approximation function. The main advantage of using the local method is that the overlapping domains of influence results in many small matrices with the dimension of the number of nodes included in the domain of influence for each center node, instead of a large collocation matrix, and hence the sparse global derivative matrices from local contributions. So the method needs less computer storage and flops. The method leads to a system of ordinary differential equations (ODEs) for the time-dependent partial differential equations (PDEs). This method can be implemented on a set of uniform or random nodes, without any a priori knowledge of node to node connectivity. We have chosen uniform nodal arrangement due to their suitability and better accuracy. To decrease accuracy degradations near boundaries, some different edge improvement techniques are also used. The method is tested on four benchmark problems on Burgers’ equation with mixed boundary conditions. Five nodded domains of influence are used in the local support. The numerical results for different value of Reynolds numbers (Re) is compared with analytical solution as well as some other numerical methods. It is shown that the proposed method is efficient, accurate and stable for flow with reasonably high values of Re.

A unified numerical method for computing the quadrature weights, integration matrix, and differentiation matrix is newly developed in this study. For this purpose, a spline-like interpolation using piecewise continuous polynomials is converted into a global spline interpolation formula, with which the quadrature formulas can be derived from integration and differentiation of the transformed function in an exact manner. To prove the usefulness of the suggested approach, both the Lagrange and tension spline interpolations are represented in exactly the same form as global spline interpolation. The applicability of the proposed method on arbitrary nodes is illustrated using two different sets of nodes. A series of validations using three test functions is conducted to show the flexibility in selecting computational nodes with the present method.

This paper addresses the problem of deriving a policy from the value function in the context of reinforcement learning in continuous state and input spaces. We propose a novel method based on genetic programming to construct a symbolic function, which serves as a proxy to the value function and from which a continuous policy is derived. The symbolic proxy function is constructed such that it maximizes the number of correct choices of the control input for a set of selected states. Maximization methods can then be used to derive a control policy that performs better than the policy derived from the original approximate value function. The method was experimentally evaluated on two control problems with continuous spaces, pendulum swing-up and magnetic manipulation, and compared to a standard policy derivation method using the value function approximation. The results show that the proposed method and its variants outperform the standard method.