Exponential Scheme Research Articles

Computing coherent sets using the Fokker-Planck equation

We perform a numerical approximation of coherent sets in finite-dimensional smooth dynamical systems by computing singular vectors of the transfer operator for a stochastically perturbed flow. This operator is obtained by solution of a discretized Fokker-Planck equation. For numerical implementation, we employ spectral collocation methods and an exponential time differentiation scheme. We experimentally compare our approach with the more classical method by Ulam that is based on discretization of the transfer operator of the unperturbed flow.

Adaptive Hierarchical Sliding Mode Control with Input Saturation for Attitude Regulation of Multi-satellite Tethered System

This paper proposes a novel adaptive hierarchical sliding mode control for the attitude regulation of the multi-satellite inline tethered system, where the input saturation is taken into account. The governing equations for the attitude dynamics of the three-satellite inline tethered system are derived firstly by utilizing Lagrangian mechanics theory. Considering the fact that the attitude of the central satellite can be adjusted by using the simple exponential stabilization scheme, the decoupling of the central satellite and the terminal ones is presented, and in addition, the new adaptive sliding mode control law is applied to stabilize the attitude dynamics of the two terminal satellites based on the synchronization and partial contraction theory. In the adaptive hierarchical sliding mode control design, the input is modeled as saturated input due to the fact that the flywheel torque is bounded, and meanwhile, an adaptive update rate is introduced to eliminate the effect of the saturated input and the external perturbation. The proposed control scheme can be applied on the two-satellite system to achieve fixed-point rotation. Numerical results validate the effectiveness of the proposed method.

Fourth-order two-stage explicit exponential integrators for time-dependent PDEs

Among the family of fourth-order time integration schemes, the two-stage Gauss–Legendre method, which is an implicit Runge–Kutta method based on collocation, is the only superconvergent. The computational cost of this implicit scheme for large systems, however, is very high since it requires solving a nonlinear system at every step. Surprisingly, in this work we show that one can construct and prove convergence results for exponential methods of order four which use two stages only. Specifically, we derive two new fourth-order two-stage exponential Rosenbrock schemes for solving large systems of differential equations. Moreover, since the newly schemes are not only superconvergent but also fully explicit, they turn out to be very competitive compared to the two-stage Gauss–Legendre method as well as other fourth-order time integration schemes. Numerical experiments are given to demonstrate the efficiency of the new integrators.

Efficient and Stable Exponential Runge-Kutta Methods for Parabolic Equations

AbstractIn this paper we develop explicit fast exponential Runge-Kutta methods for the numerical solutions of a class of parabolic equations. By incorporating the linear splitting technique into the explicit exponential Runge-Kutta schemes, we are able to greatly improve the numerical stability. The proposed numerical methods could be fast implemented through use of decompositions of compact spatial difference operators on a regular mesh together with discrete fast Fourier transform techniques. The exponential Runge-Kutta schemes are easy to be adopted in adaptive temporal approximations with variable time step sizes, as well as applied to stiff nonlinearity and boundary conditions of different types. Linear stabilities of the proposed schemes and their comparison with other schemes are presented. We also numerically demonstrate accuracy, stability and robustness of the proposed method through some typical model problems.

Complexity of Bovine Rhodopsin Activation Revealed at Low Temperature and Alkaline pH.

The late intermediates involved in the activation mechanism of bovine rhodopsin are investigated by time-resolved optical absorption spectroscopy. Measurements from 10 μs to 200 ms after photolysis were carried out on membrane suspensions of bovine rhodopsin at a temperature of 15 °C and at pH of 7.3, 8.0, and 8.7. The time-resolved absorption spectra in the 330-650 nm range were analyzed by global exponential and kinetic scheme fitting methods. The results indicate an activation mechanism that is more complex than suggested previously. It involves interconnected branched pathways with two metarhodopsin I480 and two metarhodopsin II intermediates. The intermediates involved in this more complex mechanism need to be considered in spectroscopic studies that vary sample temperature and pH in order to enhance the presence of specific rhodopsin intermediates.

Efficiently pricing continuously monitored barrier options under stochastic volatility model with jumps

ABSTRACTStochastic volatility model with jumps fits almost perfectly the empirical implied volatility surface. Under this model, this paper considers continuously monitored barrier options pricing by Monte Carlo simulation. Based on quadratic exponential scheme, this paper develops an algorithm for pricing barrier options and provides convergence of the algorithm by moment-matching techniques. Variance reduction technique based on control variates further improves the efficiency of the algorithm. The algorithm is also extended to stochastic volatility model with contemporaneous jumps in variance and stock price. Simulations show that the proposed algorithm is efficient and easy to implement. Compared to contemporaneous jumps in variance and stock price, only jumps in stock price produce more profound impact on barrier options prices.

Solution of stochastic nonlinear time fractional PDEs using polynomial chaos expansion combined with an exponential integrator

The aim of this paper is to introduce an efficient numerical algorithm for the solution of stochastic time fractional stiff partial differential equations (PDEs). The time fractional derivative is described in the Caputo sense. By applying polynomial chaos (PC) expansion to discretize the random variable, a coupled time fractional deterministic system of PDEs is obtained. For the resulting system of time fractional PDEs we apply the Fourier spectral method to discretize the spatial variable, and use the fast Fourier transform (FFT) during the computation. Then, we use an exponential integrator method to overcome the stability issues due to the stiffness in the resulting time fractional semi-discrete system. The considered models have two challenging parameters, fractional order of equations and noise amplitude. In addition to exponential integrator, we also implemented a predictor–corrector method of Adams–Bashforth–Moulton type. We include numerical results of applying the developed method to stochastic time fractional Burgers and Kuramoto–Sivashinsky (KS) equations, for various values of noise intensity. A comparison of performance of the proposed scheme with fractional Adams method is also reported which confirms the efficiency and applicability of the proposed method based on the exponential integrator scheme.

A generalized version of Banerjee and Rahim model in economic and economic statistical designs of multivariate control charts under generalized exponential shock model

ABSTRACTIn this article, a method for economic and economic-statistical design of multivariate (T-squared) control charts under a two-parameter generalized exponential shock model and uniform sampling scheme which is based on primary work of Banerjee and Rahim (1988) and Yang and Rahim (2006) was developed. A sensitivity analysis was conducted to study the effects of process, and cost parameters on design parameters and designs were compared. Comparison showed that the difference between sampling interval and minimum cost in these two designs is negligible and economic-statistical design provides smaller average run length (ARL) under control.

An efficient exponential time integration method for the numerical solution of the shallow water equations on the sphere

The exponential propagation methods were applied in the past for accurate integration of the shallow water equations on the sphere. Despite obvious advantages related to the exact solution of the linear part of the system, their use for the solution of practical problems in geophysics has been limited because efficiency of the traditional algorithm for evaluating the exponential of Jacobian matrix is inadequate. In order to circumvent this limitation, we modify the existing scheme by using the Incomplete Orthogonalization Method instead of the Arnoldi iteration. We also propose a simple strategy to determine the initial size of the Krylov space using information from previous time instants. This strategy is ideally suited for the integration of fluid equations where the structure of the system Jacobian does not change rapidly between the subsequent time steps. A series of standard numerical tests performed with the shallow water model on a geodesic icosahedral grid shows that the new scheme achieves efficiency comparable to the semi-implicit methods. This fact, combined with the accuracy and the mass conservation of the exponential propagation scheme, makes the presented method a good candidate for solving many practical problems, including numerical weather prediction.

A STOCHASTIC APPROXIMATION ALGORITHM FOR STOCHASTIC SEMIDEFINITE PROGRAMMING

Motivated by applications to multi-antenna wireless networks, we propose a distributed and asynchronous algorithm for stochastic semidefinite programming. This algorithm is a stochastic approximation of a continuous-time matrix exponential scheme which is further regularized by the addition of an entropy-like term to the problem's objective function. We show that the resulting algorithm converges almost surely to an ɛ-approximation of the optimal solution requiring only an unbiased estimate of the gradient of the problem's stochastic objective. When applied to throughput maximization in wireless systems, the proposed algorithm retains its convergence properties under a wide array of mobility impediments such as user update asynchronicities, random delays and/or ergodically changing channels. Our theoretical analysis is complemented by extensive numerical simulations, which illustrate the robustness and scalability of the proposed method in realistic network conditions.

Weak convergence for a stochastic exponential integrator and finite element discretization of stochastic partial differential equation with multiplicative & additive noise

We consider a finite element approximation of a general semi-linear stochastic partial differential equation (SPDE) driven by space-time multiplicative and additive noise. We examine the full weak convergence rate of the exponential Euler scheme when the linear operator is self adjoint and also provide the full weak convergence rate for non-self-adjoint linear operator with additive noise. Key part of the proof does not rely on Malliavin calculus. For non-self-adjoint operators, we analyse the optimal strong error for spatially semi-discrete approximations for both multiplicative and additive noise with truncated and non-truncated noise. Depending on the regularity of the noise and the initial solution, we found that in some cases the rate of weak convergence is twice the rate of the strong convergence. Our convergence rate is in agreement with some numerical results in two dimensions.

Conciliating efficiency and dynamical consistency in the simulation of the effects of proliferation and motility of transforming growth factor β on cancer cells

In this work, we provide some discretizations of a partial differential equation that generalizes the well-known Fisher’s equation from population dynamics. The mathematical model of interest is a nonlinear diffusion-reaction equation that appears in the investigation of the proliferation and motility effect of transforming growth factor β on cancer cells. Only positive and bounded solutions are physically relevant in this context, and the discretizations that we provide in this manuscript are able to preserve both properties. One of the techniques is an implicit linear method that is motivated by previous approaches of the author. On the other hand, the second method is a novel explicit exponential technique which has the advantage of requiring less computational resources and less computer time. Similar qualitative results are obtained with both methods, but the latter one is able to handle finer grid meshes. Some qualitative and quantitative comparisons are carried out in support of the advantages of the exponential scheme. It is worthwhile to note that the explicit technique used in the present manuscript has the advantage over other exponential methodologies that it yields no singularities. In addition, the preservation of the properties of non-negativity and boundedness of both the solution and the total mass are distinctive features which are established analytically in this work. The numerical simulations on cancer growth obtained with the exponential method are found to be in good agreement with the experimental results available in the literature.

Fourier spectral method for higher order space fractional reaction–diffusion equations

Evolution equations containing fractional derivatives can provide suitable mathematical models for describing important physical phenomena. In this paper, we propose a fast and accurate method for numerical solutions of space fractional reaction–diffusion equations. The proposed method is based on an exponential integrator scheme in time and the Fourier spectral method in space. The main advantages of this method are that it yields a fully diagonal representation of the fractional operator, with increased accuracy and efficiency, and a completely straightforward extension to high spatial dimensions. Although, in general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives, we introduce them to describe fractional hyper-diffusions in reaction diffusion. The scheme justified by a number of computational experiments, this includes two and three dimensional partial differential equations. Numerical experiments are provided to validate the effectiveness of the proposed approach.

Exponential synchronization for stochastic neural networks driven by fractional Brownian motion

In this paper, we investigate the exponential synchronization problem for a new class of stochastic neural networks driven by fractional Brownian motion (fBm). In order to deal with the problem of stability for the error system, we propose the mild solution of system equation with respect to fractional Brownian motion based on the space of Hilbert–Schmidt operator law. By using the infinitesimal generator on analytic semigroup principle and associating with the well-known Hölder inequality, Gronwall inequality, we obtain the exponential synchronization criteria for the drive system and response system driven by fBm. Finally, two numerical examples as well as some evolution figures are implemented to demonstrate the effectiveness and feasibility of the proposed exponential synchronization schemes.

Exponential pseudo-splines: Looking beyond exponential B-splines

Pseudo-splines are a rich family of functions that allows the user to meet various demands for balancing polynomial reproduction (i.e., approximation power), regularity and support size. Such a family includes, as special members, B-spline functions, universally known for their usefulness in different fields of application. When replacing polynomial reproduction by exponential polynomial reproduction, a new family of functions is obtained. This new family is here constructed and called the family of exponential pseudo-splines. It is the nonstationary counterpart of (polynomial) pseudo-splines and includes exponential B-splines as a special subclass. In this work we provide a computational strategy for deriving the explicit expression of the Laurent polynomial sequence that identifies the family of exponential pseudo-spline nonstationary subdivision schemes. For this family we study its symmetry properties and perform its convergence and regularity analysis. Finally, we also show that the family of primal exponential pseudo-splines fills in the gap between exponential B-splines and interpolatory cardinal functions. This extends the analogous property of primal pseudo-spline stationary subdivision schemes.

Efficient Simulation of the Multi Asset Heston Model

This paper describes a procedure for efficiently simulating a multi asset Heston model with an arbitrary correlation structure. Very little literature can be found on the topic (e.g. Wadman (2010) and Dimitroff et al. (2011)), the latter being very restrictive on correlation assumptions. The scheme proposed in this text is based on Andersen's Quadratic Exponential (QE) scheme (2008) and operates with an arbitrary input correlation structure, which is partially decorrelated via a Gaussian copula approach to fit the single asset QE prerequisites. Given a long term horizon, it is shown numerically that, in the multi asset QE (MQE) scheme, all combinations of terminal correlations converge quickly to the true terminal correlations for decreasing Monte Carlo time step size, if the input correlation matrix is interpreted as the system's instantaneous correlation matrix. Convergence of vanilla and spread option prices is investigated, in order to verify the appropriate behaviour for higher moments of the marginal and the joint distribution under MQE. Finally, the superiority of MQE vs. Taylor based schemes is shown by comparing convergence of the empirical PDF, calculated with Monte Carlo, to the exact function calculated via Fourier inversion.

Time exponential splitting technique for the Klein–Gordon equation with Hagstrom–Warburton high-order absorbing boundary conditions

Klein-Gordon equations on an unbounded domain are considered in one dimensional and two dimensional cases. Numerical computation is reduced to a finite domain by using the Hagstrom-Warburton (H-W) high-order absorbing boundary conditions (ABCs). Time integration is made by means of exponential splitting schemes that are efficient and easy to implement. In this way, it is possible to achieve a negligible error due to the time integration and to study the behavior of the absorption error. Numerical experiments displaying the accuracy of the numerical solution for the two dimensional case are provided. The influence of the dispersion coefficient on the error is also studied.

High-order splitting schemes for semilinear evolution equations

We first derive necessary and sufficient stiff order conditions, up to order four, for exponential splitting schemes applied to semilinear evolution equations. The main idea is to identify the local splitting error as a sum of quadrature errors. The order conditions of the quadrature rules then yield the stiff order conditions in an explicit fashion, similarly to that of Runge–Kutta schemes. Furthermore, the derived stiff conditions coincide with the classical non-stiff conditions. Secondly, we propose an abstract convergence analysis, where the linear part of the vector field is assumed to generate a group or a semigroup and the nonlinear part is assumed to be smooth and to satisfy a set of compatibility requirements. Concrete applications include nonlinear wave equations and diffusion-reaction processes. The convergence analysis also extends to the case where the nonlinear flows in the exponential splitting scheme are approximated by a sufficiently accurate one-step method.

Solving Two Point Boundary Value Problems for Ordinary Differential Equations Using Exponential Finite Difference Method

In this article, a new exponential finite difference scheme for the numerical solution of two point boundary value problems with Dirichlet's boundary conditions is proposed. The scheme is based on an exponential approximation of the Taylor expansion for the discretized derivative .The convergence of the scheme discussed under appropriate condition .The theoretical and numerical results show that this new scheme is efficient and at least fourth order accurate.

Solving nonlinear two point boundary value problems using exponential finite difference method

In this article, we present exponential finite difference scheme for solving nonlinear two point boundary value problems with Dirichlet's boundary conditions . The local truncation error and under appropriate condition we have discussed the convergence of the proposed method. Numerical experiments demonstrate the use and computational efficiency of the method. Numerical results show that this method is at least fourth order accurate, which is good agreement with the theoretically established order of the method.