Fast Numerical Method Research Articles

Performance Analysis of Non-uniform Sparse Segmentation Integral Method Based on Gauss Integral in EM Forward of Electrical Antenna Under Stratified Ocean

In order to solve the problem that it is difficult to take into account the computational efficiency and accuracy of electromagnetic field froward modeling of electrical antenna under the stratified ocean, evaluate the performance of two non-uniform sparse segmentation integral calculation methods based on Gauss integrals and improve the efficiency and accuracy of data interpretation in engineering applications, firstly two non-uniform sparse segmentation calculation methods based on Gauss integral nodes (Gauss-Legendre integral nodes and Gauss-Chebyshev integral nodes) are introduced. Then, two fast numerical integration methods corresponding of the time and frequency domain of the electrical antenna electromagnetic field under stratified ocean is established. Finally, by comparing and analyzing the results of the traditional uniform dense segmentation numerical integral calculation from the two aspects of computational accuracy and efficiency, the results show that two non-uniform sparse segmentation integral methods can improve the computational efficiency under the condition that the calculation accuracy is comparable, and the non-uniform sparse integral calculation method based on the Gauss-Chebyshev integral nodes is more significant than the calculation efficiency performance of the method based on the Gauss-Legendre integral node. At the same time, the general parameter setting of the non-uniform sparse segmentation integral method is given in this paper.

Simulation of Anharmonic Bloch Oscillations: Numerical Problems and Nonlinear Effects

A stable and fast numerical method for calculating complex energies and wave functions of carriers in one-dimensional electrically biased periodic structures is presented. Optical transitions in a superlattice (Bloch oscillations) have been investigated using this method. It is shown that the transition probabilities depend nonlinearly on the applied (sufficiently strong) field. In this case, the transitions at the double and triple Bloch frequencies can be more intense than that at the fundamental frequency. A similar situation is observed in the superlattice with a split miniband (the nonlinearity is more pronounced at strong splitting).

A fast numerical method for fractional partial integro-differential equations with spatial-time delays

This study aims to efficiently solve the space-time fractional partial integro-differential equations with spatial-time delays, employing a fast numerical methodology dependent upon the matching polynomial of complete graph and matrix-collocation procedure. This methodology provides a sustainable approach for each computation limit since it arises from the durable graph structure of complete graph and fractional matrix relations. The convergence analysis is established using the residual function of mean value theorem for double integrals. An error estimation is also implemented. All computations are performed with the aid of a unique computer program, which returns the desired results in seconds. Some specific numerical problems are tested to discuss the applicability of the method in tables and figures. It is stated that the method stands for fast, simple and highly accurate computation.

Simple computation of ignition voltage of self-sustaining gas discharges

A robust, fast, and accurate numerical method is proposed for finding the voltage of the ignition of DC self-sustaining gas discharges in a wide range of conditions. The method is based on physical grounds and builds up from the idea that the ignition of a self-sustaining gas discharge should be associated with a resonance that would occur in a non-self-sustained discharge in the same electrode configuration. Examples of the application of the method are shown for various configurations: parallel-plate discharge, coaxial and wire-to-plane corona discharges, and a discharge along a dielectric surface. The results conform to the conventional Townsend breakdown condition for the parallel-plate configuration and are in good agreement with existing experimental data for the other configurations. The method has the potential of providing a reference point for optimization of the hold-off capability of high-power switchgear operating in low-frequency fields.

Fast high-order method for multi-dimensional space-fractional reaction–diffusion equations with general boundary conditions

To achieve the efficient and accurate long-time integration, we propose a fast and stable high-order numerical method for solving fractional-in-space reaction–diffusion equations. The proposed method is explicit in nature and utilizes the fourth-order compact finite difference scheme and matrix transfer technique (MTT) in space with FFT-based implementation. Time integration is done through the modified fourth-order exponential time differencing Runge–Kutta scheme. The linear stability analysis and various numerical experiments including two-dimensional (2D) Fitzhugh–Nagumo, Allen–Cahn, Gierer–Meinhardt, Gray–Scott and three-dimensional (3D) Schnakenberg models are presented to demonstrate the accuracy, efficiency, and stability of the proposed method.

Моделирование ангармонических блоховских осцилляций: численные проблемы и нелинейные эффекты

The work presents robust and fast numerical method for calculation of complex-valued energies and wave functions of carriers in one-dimensional electrically biased periodic structures. Using this method optical transitions were studied in a superlattice (Bloch oscillations). Transition probabilities were shown to disobey linear dependence from applied field when it is large enough; in this case transitional with double and triple Bloch frequency can be more intensive than on single frequency. In a superlattice with a split miniband the same holds, with non-linearity being more pronounced in wider split case.

Electroencephalographic Source Reconstruction by the Finite-Element Approximation of the Elliptic Cauchy Problem.

This paper develops a novel approach for fast and reliable reconstruction of EEG sources in MRI-based head models. The inverse EEG problem is reduced to the Cauchy problem for an elliptic partial-derivative equation. The problem is transformed into a regularized minimax problem, which is directly approximated in a finite-element space. The resulting numerical method is efficient and easy to program. It eliminates the need to solve forward problems, which can be a tedious task. The method applies to complex anatomical head models, possibly containing holes in surfaces, anisotropic conductivity, and conductivity variations inside each tissue. The method has been verified on a spherical shell model and an MRI-based head. Numerical experiments indicate high accuracy of localization of brain activations (both cortical potential and current) and rapid execution time. This study demonstrates that the proposed approach is feasible for EEG source analysis and can serve as a rapid and reliable tool for EEG source analysis. The significance of this study is that it develops a fast, accurate, and simple numerical method of EEG source analysis, applicable to almost arbitrary complex head models.

An efficient numerical method for the valuation of American multi-asset options

In this paper, a fast and efficient numerical method which relies on the far field truncation technique, the finite element discretization, and the projection contraction method (PCM) is proposed for pricing American multi-asset options. It is well known that American multi-asset option satisfies a linear complementarity problem (LCP), which is a multi-dimensional variable coefficient parabolic model on an unbounded domain. First, we transform it into a constant coefficient parabolic LCP on a bounded domain by some skillful transformations and far-field boundary estimate. Then, the variational inequality (VI) corresponding to the truncated LCP is obtained. Further, it is discretized by the finite element method and the implicit difference method in spatial and temporal directions, respectively. Based on the symmetric positive definiteness of the full-discrete matrix, the discretized VI is solved by the PCM. Finally, numerical simulations are provided to verify the efficiency of the proposed method.

Solving differential Riccati equations: A nonlinear space-time method using tensor trains

Differential Riccati equations are at the heart of many applications in control theory. They are time-dependent, matrix-valued, and in particular nonlinear equations that require special methods for their solution. Low-rank methods have been used heavily for computing a low-rank solution at every step of a time-discretization. We propose the use of an all-at-once space-time solution leading to a large nonlinear space-time problem for which we propose the use of a Newton–Kleinman iteration. Approximating the space-time problem in a higher-dimensional low-rank tensor form requires fewer degrees of freedom in the solution and in the operator, and gives a faster numerical method. Numerical experiments demonstrate a storage reduction of up to a factor of 100.

Fast Numerical Power Loss Calculation for High-Frequency Litz Wires

This article presents a fast numerical calculation method of realistic power losses for high-frequency litz wires. Explicitly, the imperfect structure of litz wires is considered when calculating losses due to an excitation current (skin losses) and external magnetic fields (proximity losses). Calculations of litz wires with more than 1000 strands were performed on a personal computer and have been validated by measurements up to 10 MHz. In the calculation, the impact of the bundle structure on skin and proximity losses is examined. The method allows to select a suitable litz wire for a specific application or to design a litz wire considering realistic twisting structures.

Interior Ballistic Characteristics of Electromagnetic Rail Launcher Under Continuous Firing

According to the characteristics of an actual electromagnetic rail launcher (EMRL), this article points out defects in the traditional dynamic response model of EMRL, proposes an improved dynamic response model, and analyzes its dynamic and stopping response characteristics. On this basis, it presents a fast and accurate numerical calculation method for three-dimensional transient electromagnetic (EM) force of the EMRL and an order-reduced model of EMRL dynamic response. Furthermore, the second development of commercial software leads to the establishment of an interior ballistic simulation model with EM-structural coupling. The model is used to analyze the interior ballistic characteristics of the EMRL under sequential discharge and continuous firing. The finding is that the dynamic response of the first launching rail will cause a deformation in the second launching rail, in which a greater fluctuation happens. When the time interval between successive launches reaches 1.0 s, the deformation of the second launching rail will no longer be affected by the dynamic response of the first launching rail.

Sparse Harmonic Transforms: A New Class of Sublinear-Time Algorithms for Learning Functions of Many Variables

In this paper we develop fast and memory efficient numerical methods for learning functions of many variables that admit sparse representations in terms of general bounded orthonormal tensor product bases. Such functions appear in many applications including, e.g., various Uncertainty Quantification (UQ) problems involving the solution of parametric PDE that are approximately sparse in Chebyshev or Legendre product bases (Chkifa et al. in Polynomial approximation via compressed sensing of high-dimensional functions on lower sets. arXiv:1602.05823 , 2016; Rauhut and Schwab in Math Comput 86(304):661–700, 2017). We expect that our results provide a starting point for a new line of research on sublinear-time solution techniques for UQ applications of the type above which will eventually be able to scale to significantly higher-dimensional problems than what are currently computationally feasible. More concretely, let $${\mathcal {B}}$$ be a finite Bounded Orthonormal Product Basis (BOPB) of cardinality $$|{\mathcal {B}}| = N$$ . Herein we will develop methods that rapidly approximate any function f that is sparse in the BOPB, that is, $$f: {\mathcal {D}} \subset {\mathbb {R}}^D \rightarrow {\mathbb {C}}$$ of the form $$\begin{aligned} f(\varvec{x}) = \sum _{b \in {\mathcal {S}}} c_b \cdot b(\varvec{x}) \end{aligned}$$ with $${\mathcal {S}} \subset {\mathcal {B}}$$ of cardinality $$|{\mathcal {S}}| = s \ll N$$ . Our method adapts the CoSaMP algorithm (Needell and Tropp in Appl Comput Harmon Anal 26(3):301–321, 2009) to use additional function samples from f along a randomly constructed grid $${\mathcal {G}} \subset {\mathbb {R}}^D$$ with universal approximation properties in order to rapidly identify the multi-indices of the most dominant basis functions in $${\mathcal {S}}$$ component by component during each CoSaMP iteration. It has a runtime of just $$(s \log N)^{{\mathcal {O}}(1)}$$ , uses only $$(s \log N)^{{\mathcal {O}}(1)}$$ function evaluations on the fixed and nonadaptive grid $${\mathcal {G}}$$ , and requires not more than $$(s \log N)^{{\mathcal {O}}(1)}$$ bits of memory. We emphasize that nothing about $${\mathcal {S}}$$ or any of the coefficients $$c_b \in {\mathbb {C}}$$ is assumed in advance other than that $${\mathcal {S}} \subset {\mathcal {B}}$$ has $$|{\mathcal {S}}| \le s$$ . Both $${\mathcal {S}}$$ and its related coefficients $$c_b$$ will be learned from the given function evaluations by the developed method. For $$s\ll N$$ , the runtime $$(s \log N)^{{\mathcal {O}}(1)}$$ will be less than what is required to simply enumerate the elements of the basis $${\mathcal {B}}$$ ; thus our method is the first approach applicable in a general BOPB framework that falls into the class referred to as sublinear-time. This and the similarly reduced sample and memory requirements set our algorithm apart from previous works based on standard compressive sensing algorithms such as basis pursuit which typically store and utilize full intermediate basis representations of size $$\varOmega (N)$$ during the solution process.

Fast and High-Order Accuracy Numerical Methods for Time-Dependent Nonlocal Problems in $${\pmb {\mathbb {R}}}^2$$

In this paper, we study the Crank-Nicolson method for temporal dimension and the piecewise quadratic polynomial collocation method for spatial dimensions of time-dependent nonlocal problems. The new theoretical results of such discretization are that the proposed numerical method is unconditionally stable and its global truncation error is of $\mathcal{O}\left(\tau^2+h^{4-\gamma}\right)$ with $0<\gamma<1$, where $\tau$ and $h$ are the discretization sizes in the temporal and spatial dimensions respectively. Also we develop the conjugate gradient squared method to solving the resulting discretized nonsymmetric and indefinite systems arising from time-dependent nonlocal problems including two-dimensional cases. By using additive and multiplicative Cauchy kernels in non-local problems, structured coefficient matrix-vector multiplication can be performed efficiently in the conjugate gradient squared iteration. Numerical examples are given to illustrate our theoretical results and demonstrate that the computational cost of the proposed method is of $O(M \log M)$ operations where $M$ is the number of collocation points.

Fast and robust numerical method for inverse kinematics with prioritized multiple targets for redundant robots

In this paper, a new numerical method for inverse kinematics with prioritized multiple targets is proposed. The proposed method is constructed based on the virtual spring model and joint-based damping control. The targets are prioritized by adjusting the effect of the virtual springs. The proposed method has the following three features. First, it does not require complex calculations such as a Jacobian matrix projection into the null space. Second, it can solve prioritized inverse kinematics problems in the position level without integrating the joint velocity. Third, it is robust to parameter variations and singular configurations. The second feature is motivated by the background that most industrial robots in factories are used as position-controlled robots. Simulation experiments using a 9-DOF redundant robot show that the proposed method is faster and more robust than the conventional method. The proposed method is expected to be useful for helping to avoid collisions between links and obstacles using the redundancy.

Experimental and theoretical investigations about the nonlinear vibrations of rectangular atomic force microscope cantilevers immersed in different liquids

Nonlinear flexural vibrations of rectangular atomic force microscope cantilever have been investigated by both the theoretical model and experimental works. As for the theoretical model, the Timoshenko beam theory which takes the rotatory inertia and shear deformation effects into consideration has been adopted. To increase the accuracy of the theoretical model, all necessary details for cantilever and sample surface have been taken into account. Differential quadrature method as a simple and fast numerical method has been used for solving the differential equations. During the investigation, the softening behavior was observed for all cases. It was also seen that raising the amplitude of vibrations led to a decrease in the nonlinear resonant frequency to linear resonant frequency ratio. The effects of different parameters such as normal and lateral contact stiffness, cantilever thickness, the angle between cantilever and sample surface and tip height in the presence of air as environment on the softening behavior were also examined. It was also demonstrated that increasing the lateral and normal contact stiffness, but decreasing the Timoshenko beam parameter would lead to an increase in the amplitude of vibrations for the first and second modes. The vibrational behavior of cantilever immersed in different liquids including water, methanol, acetone and carbon tetrachloride has been studied. Results show that increasing the liquid density reduces the nonlinear frequency. Furthermore, experimental works were compared with theoretical model for water and air environments. Results show good agreement.

Stability of the inverse source problem for the Helmholtz equation in R3

We consider the reconstruction of a compactly supported source term in the constant-coefficient Helmholtz equation in R3, from the measurement of the outgoing solution at a source-enclosing sphere. The measurement is taken at a finite number of frequencies. We explicitly characterize certain finite-dimensional spaces of sources that can be stably reconstructed from such measurements. The characterization involves only the measurement frequencies and the problem geometry parameters. We derive a singular value decomposition of the measurement operator, and prove a lower bound for the spectral bandwidth of this operator. By relating the singular value decomposition and the eigenvalue problem for the Dirichlet–Laplacian on the source support, we devise a fast and stable numerical method for the source reconstruction. We do numerical experiments to validate the stability and efficiency of the numerical method.

A Fast and Accurate Method for Computing the Microwave Heating of Moving Objects

In this paper, we show a fast and accurate numerical method for simulating the microwave heating of moving objects, which is still a challenge because of its complicated mathematical model simultaneously coupling electromagnetic field, thermal field, and temperature-dependent moving objects. By contrast with most discrete methods whose dielectric parameters of the heated samples are updated only when they move to a new position or even turn a circle, in our simulations a real-time procedure is added to renew the parameters during the whole heating process. Furthermore, to avoid the mesh-mismatch induced by remeshing the moving objects, we move the cavity instead of samples. To verify the efficiency and accuracy, we compared our method with the arbitrary Lagrangian–Eulerian method, one of the most accurate methods for computing this process until now. For the same computation model, our method helps in decreasing the computing time by about 90% with almost the same accuracy. Moreover, the influence of the rotational speed on the microwave heating is systematically investigated by using this method. The results show the widely used speed in domestic microwave ovens, 5 rpm, is indeed a good choice for improving the temperature uniformity with high energy efficiency.

Reduced order modeling of time-dependent incompressible Navier–Stokes equation with variable density based on a local radial basis functions-finite difference (LRBF-FD) technique and the POD/DEIM method

The main propose of this investigation is to introduce a rapid and impressive numerical procedure to simulate the time dependent incompressible Navier–Stokes equation with variable density. The developed formulation is constructed by using the meshfree RBF-FD technique. Also, to improve the numerical results, an extra diffusion term has been added to the density equation with a small coefficient. On the other hand, a reduced order technique e.g. proper orthogonal decomposition (POD) has been employed to get a fast numerical method and to decrease the elapsed CPU time. On the other hand, since the considered problem is fully nonlinear, thus in the reduced order model based on the POD idea, there are some nonlinear terms that they take more computational time. Hence, we employ the discrete empirical interpolation method (DEIM) to overcome the nonlinear terms. Four test problems have been proposed to show the ability of the numerical simulations.

A high-order perturbation of envelopes (HOPE) method for scattering by periodic inhomogeneous media

The interaction of linear waves with periodic structures arises in a broad range of scientific and engineering applications. For such problems it is often mandatory that numerical simulations be rapid, robust, and highly accurate. With such qualities in mind High-Order Spectral methods are often utilized, and in this paper we describe and test a perturbative method which fits into this class. Here we view the inhomogeneous (but laterally periodic) permittivity as a perturbation of a constant value and pursue (regular) perturbation theory. We demonstrate that not only does this lead to a fast and accurate numerical method, but also that the expansion of the field in this geometric parameter is valid for large deformations (up to topological obstruction). Finally, we show that, if the permittivity deformation is spatially analytic, then so is the field scattered by it.

A numerical method for pricing discrete double barrier option by Chebyshev polynomials

In this article, a fast numerical method based on orthogonal Chebyshev polynomials for pricing discrete double barrier option is illustrated. At first, a recursive formula for computing price of discrete double barrier option is obtained. Then, these recursive formulas are estimated by Chebyshev polynomials and expressed in operational matrix form that reduce CPU time of algorithm. Finally, the effectiveness and validity of the presented method is demonstrated by comparison with the obtained numerical results with some other algorithms.