Exponential Integral Research Articles

Stirling's approximation and a hidden link between two of Ramanujan's approximations

A conjectured relation between Ramanujan's asymptotic approximations to the exponential function and the exponential integral is established. The proof involves Stirling numbers, second-order Eulerian numbers, modifications of both of these, and Stirling's approximation to the gamma function. Our work provides new information about the coefficients in Stirling's approximation and their connection to Ramanujan's approximation coefficients. A more analytic second proof of the main result is also included in an appendix.

L 2 hypocoercivity, deviation bounds, hitting times and Lyapunov functions

We establish that, for a Markov semi-group, $L^2$ hypocoercivity, i.e. contractivity for a modified $L^2$ norm, implies quantitative deviation bounds for additive functionals of the associated Markov process and exponential integrability of the hitting time of sets with positive measure. Moreover, in the case of diffusion processes and under a strong hypoellipticity assumption, we prove that $L^2$ hypocoercivity implies the existence of a Lyapunov function for the generator.

Study of Forward and Backward Scattering in Two-Region Milne Problem for Non-absorbing Medium Using a Synthetic Kernel

In this work, the effect of an extremely anisotropic scattering function on the two-region Milne problem is studied. This scattering function divides the neutrons resulting from the collisions into three parts; part ( which moves backward, part (m) which moves forward, and part (n) which emerge isotropically from the collisions, where + m + n = 1. The integral version of the transport equation is solved using trial functions based on Case’s eigenmodes and exponential integral function. Hence, the solution to the Milne problem is formulated in terms of characteristic quantities such as the extrapolation length and the fractional scalar flux discontinuity. Numerical results for the analytically evaluated quantities are presented. Some of our numerical results are compared with the available published results.

An exponential integration generalized multiscale finite element method for parabolic problems

We consider linear and semilinear parabolic problems posed in high-contrast multiscale media in two dimensions. The presence of high-contrast multiscale media adversely affects the accuracy, stability, and overall efficiency of numerical approximations such as finite elements in space combined with some time integrator. In many cases, implementing time discretizations such as finite differences or exponential integrators may be impractical because each time iteration needs the computation of matrix operators involving very large and ill-conditioned sparse matrices. Here, we propose an efficient Generalized Multiscale Finite Element Method (GMsFEM) that is robust against the high-contrast diffusion coefficient. We combine GMsFEM with exponential integration in time to obtain a good approximation of the final time solution. Our approach is efficient and practical because it computes matrix functions of small matrices given by the GMsFEM method. We present representative numerical experiments that show the advantages of combining exponential integration and GMsFEM approximations. The constructions and methods developed here can be easily adapted to three-dimensional domains.

Scalable matrix-free solver for 3D transfer of polarized radiation in stellar atmospheres

We present an efficient and massively parallel solution strategy for the transfer problem of polarized radiation, for a 3D stationary medium out of local thermodynamic equilibrium. Scattering processes are included accounting for partial frequency redistribution effects. Such a setting is one of the most challenging ones in radiative transfer modeling. The problem is formulated for a two-level atomic model, which allows linearization. The discrete ordinate method alongside an exponential integrator are used for discretization. Efficient solution is obtained with a Krylov method equipped with a tailored physics-based preconditioner. A matrix-free approach results in a lightweight implementation, suited for tackling large problems. Near-optimal strong and weak scalability are obtained with two complementary decompositions of the computational domain. The presented approach made it possible to perform simulations for the Ca i line at 4227Å with more than 109 degrees of freedom in less than half an hour on massively parallel machines, always converging in a few iterations for the proposed tests.

Jacobian-free High Order Local Linearization methods for large systems of initial value problems

In this paper, the class of Jacobian-free High Order Local Linearization (HOLL) methods is introduced for integrating large systems of initial value problems. Unlike other high order exponential integrators, this new class of methods involves the approximation of just a single phi-function times vector and does not require the evaluation and storing of Jacobian matrices, which result more efficient in terms of flops operations and computer memory. Specifically, the new Jacobian-free integrators are constructed by approximating the products of Jacobian matrix times vector that appear in the ordinary HOLL methods. A general result on the convergence rate of the new methods is derived in terms of the convergence rate of the mentioned approximations, resulting in a simple order condition that preserves the order of the ordinary HOLL methods. To design effective integrators of the new class, a novel Matrix-free Krylov-Padé approximation to the product of phi-function times vector is also proposed as well as an adaptive strategy for the automatic selection of the Krylov dimension and Padé order. As a particular instance, a class of Jacobian-free Locally Linearized Runge-Kutta schemes is developed, and schemes of third to fifth order explicitly constructed. Numerical simulations are provided illustrating the theoretical findings concerning the convergence rate of the introduced methods as well as their performance in comparison with other Jacobian-free integrators.

Some results on derangement polynomials

We study moments of the difference $D_n(x)-x^n n! {\rm e}^{-1/x}$ concerning derangement polynomials $D_n(x)$. For the first moment, we obtain an explicit formula in terms of the exponential integral function and we show that it is always negative for $x>0$. For the higher moments, we obtain a multiple integral representation of the order of the moment under computation.

Exploiting Kronecker structure in exponential integrators: Fast approximation of the action of [formula omitted]-functions of matrices via quadrature

In this article, we propose an algorithm for approximating the action of φ−functions of matrices against vectors, which is a key operation in exponential time integrators. In particular, we consider matrices with Kronecker sum structure, which arise from problems admitting a tensor product representation. The method is based on quadrature approximations of the integral form of the φ−functions combined with a scaling and modified squaring method. Owing to the Kronecker sum representation, only actions of 1D matrix exponentials are needed at each quadrature node and assembly of the full matrix can be avoided. Additionally, we derive a priori bounds for the quadrature error, which show that, as expected by classical theory, the rate of convergence of our method is supergeometric. Guided by our analysis, we construct a fast and robust method for estimating the optimal scaling factor and number of quadrature nodes that minimizes the total cost for a prescribed error tolerance. We investigate the performance of our algorithm by solving several linear and semilinear time-dependent problems in 2D and 3D. The results show that our method is accurate and orders of magnitude faster than the current state-of-the-art.

Numerical solutions of the time‐dependent Schrödinger equation with position‐dependent effective mass

AbstractNumerical solution of the time‐dependent Schrödinger equation with a position‐dependent effective mass is challenging to compute due to the presence of the non‐constant effective mass. To tackle the problem we present operator splitting‐based numerical methods. The wavefunction will be propagated either by the Krylov subspace method‐based exponential integration or by an asymptotic Green's function‐based time propagator. For the former, the wavefunction is given by a matrix exponential whose associated matrix–vector product can be approximated by the Krylov subspace method; and for the latter, the wavefunction is propagated by an integral with retarded Green's function that is approximated asymptotically. The methods have complexity per step with appropriate algebraic manipulations and fast Fourier transform, where is the number of spatial points. Numerical experiments are presented to demonstrate the accuracy, efficiency, and stability of the methods.

A necessary and sufficient condition for the convergence of the derivative martingale in a branching Lévy process

A continuous-time particle system on the real line satisfying the branching property and an exponential integrability condition is called a branching L\'evy process, and its law is characterized by a triplet $(\sigma^2,a,\Lambda)$. We obtain a necessary and sufficient condition for the convergence of the derivative martingale of such a process to a non-trivial limit in terms of $(\sigma^2,a,\Lambda)$. This extends previously known results on branching Brownian motions and branching random walks. To obtain this result, we rely on the spinal decomposition and establish a novel zero-one law on the perpetual integrals of centred L\'evy processes conditioned to stay positive.

The Galerkin Analysis for the Random Periodic Solution of Semilinear Stochastic Evolution Equations

In this paper, we study the numerical method for approximating the random periodic solution of semilinear stochastic evolution equations. The main challenge lies in proving a convergence over an infinite time horizon while simulating infinite-dimensional objects. We first show the existence and uniqueness of the random periodic solution to the equation as the limit of the pull-back flows of the equation, and observe that its mild form is well defined in the intersection of a family of decreasing Hilbert spaces. Then, we propose a Galerkin-type exponential integrator scheme and establish its convergence rate of the strong error to the mild solution, where the order of convergence directly depends on the space (among the family of Hilbert spaces) for the initial point to live. We finally conclude with a best order of convergence that is arbitrarily close to 0.5.

An Application of Elastic-Net Regularized Linear Inverse Problem in Seismic Data Inversion

In exploration geophysics, seismic impedance is a physical characteristic parameter of underground formations. It can mark rock characteristics and help stratigraphic analysis. Hence, seismic data inversion for impedance is a key technology in oil and gas reservoir prediction. To invert impedance from seismic data, one can perform reflectivity series inversion first. Then, under a simple exponential integration transformation, the inverted reflectivity series can give the final inverted impedance. The quality of the inverted reflectivity series directly affects the quality of impedance. Sparse-spike inversion is the most common method to obtain reflectivity series with high resolution. It adopts a sparse regularization to impose sparsity on the inverted reflectivity series. However, the high resolution of sparse-spike-like reflectivity series is obtained at the cost of sacrificing small reflectivity. This is the inherent problem of sparse regularization. In fact, the reflectivity series from the actual impedance well log is not strictly sparse. It contains not only the sparse major large reflectivity, but also small reflectivity between major reflectivity. That is to say, the large reflectivity is sparse, but the small reflectivity is dense. To combat this issue, we adopt elastic-net regularization to replace sparse regularization in seismic impedance inversion. The elastic net is a hybrid regularization that combines sparse regularization and dense regularization. The proposed inversion method was performed on a synthetic seismic trace, which is created from an actual well log. Then, a real seismic data profile was used to test the practice application. The inversion results showed that it provides an effective new alternative method to invert impedance.

Plurisubharmonic functions and real submanifolds of [formula omitted

We give an estimate for the volume of an analytic variety (or more generally the mass of a positive closed current) close to a real submanifold M. Applications are given to the Hausdorff measure of the intersection of the variety with M and the exponential integrability of plurisubharmonic functions on M.

Error estimates for a splitting integrator for abstract semilinear boundary coupled systems

Abstract We derive a numerical method, based on operator splitting, to abstract parabolic semilinear boundary coupled systems. The method decouples the linear components that describe the coupling and the dynamics in the abstract bulk- and surface-spaces, and treats the nonlinear terms similarly to an exponential integrator. The convergence proof is based on estimates for a recursive formulation of the error, using the parabolic smoothing property of analytic semigroups, and a careful comparison of the exact and approximate flows. This analysis also requires a deep understanding of the effects of the Dirichlet operator (the abstract version of the harmonic extension operator), which is essential for the stable coupling in our method. Numerical experiments, including problems with dynamic boundary conditions, reporting on convergence rates are presented.

Extensive novel waves evolution of three-dimensional Yu–Toda–Sasa–Fukuyama equation compatible with plasma and electromagnetic applications

There is no doubt that the investigation of the interaction and propagation of plasma and electromagnetic waves play an important role in understanding these phenomena. The three-dimensional Yu–Toda–Sasa–Fukuyama equation (YTSFE) is a competent mathematical model of waves in plasma, electromagnetics, or fluids. An optimal system of infinitesimal symmetries is constructed to discover extensive and astonishingly exact solutions to the YTSFE. The outstanding solutions include periodic, polynomials, fractional, logarithmic, exponential, hyperbolic, exponential integral, Airy and complex functions. These solutions are significant because they help understand how plasma and electromagnetic applications work at different boundary or initial conditions.

Mechanical formation damage management and sealed boundaries identification in pressure-sensitive oil reservoirs with source effects

Sealed zones identification is essential for a successful exploratory and production campaign during the development of an oil field. This work develops a new unsteady two-dimensional (2-D) integro-differential solution for permeability loss monitoring in a well near a sealant fault. The model presented in this study allows solving the nonlinear hydraulic diffusivity equation (NHDE) with an oil source term. The appropriate Green’s Function (GF) for a sealed barrier represents the instantaneous oil point-source in the well. Based on the well-known image method, the pressure field for the constant permeability solution is given by the sum of two exponential integral functions Ei(tD). Although, this solution does not consider the nonlinear effect caused by the permeability loss. Therewith, a new deviation factor is derived and coupled to a first-order series expansion to approach this phenomenon. The calibration of the proposed solution was carried out by a finite-difference-based simulator named IMEX® broadly used in the reservoir engineering works, and the results were accurate. The results clearly present the instantaneous permeability decay effect by a deviation when compared to the linear solution in a semi-log plot. The main advantages of the proposed solution are the accuracy, availability of a wide table of the GFs, ease of implementation, and computational cost savings. It constitutes a useful and attractive mathematical tool to calibrate new models and support the well-reservoir performance management.

Generalized Newton-Leibniz formula and the embedding of the Sobolev functions with dominating mixed smoothness into Hölder spaces

<abstract><p>It is well-known that the embedding of the Sobolev space of weakly differentiable functions into Hölder spaces holds if the integrability exponent is higher than the space dimension. In this paper, the embedding of the Sobolev functions into the Hölder spaces is expressed in terms of the minimal weak differentiability requirement independent of the integrability exponent. The proof is based on the generalization of the Newton-Leibniz formula to the $ n $-dimensional rectangle and the inductive application of the Sobolev trace embedding results. The new method is applied to prove the embedding of the Sobolev spaces with dominating mixed smoothness into Hölder spaces. Counterexamples demonstrate that in terms of minimal weak regularity degree the Sobolev spaces with dominating mixed smoothness present the largest class of weakly differentiable functions with the upgrade of pointwise regularity to continuity. Remarkably, it also presents the largest class of weakly differentiable functions where the generalized Newton-Leibniz formula holds.</p></abstract>

A new exponential type Kernel integral transform: Khalouta transform and its applications

In this paper, we suggest a new integral transform called the Khalouta transform, which is a generalization of many integral transforms having exponential type kernel. We discuss certain results on the inverse and the existence of this integral transform. We present useful properties of the Khalouta transform and their applications to solve differential equations. Furthermore, we prove the duality between the Khalouta transform and other transforms such as the Laplace-Carson transform, Sumudu transform, ZZ transform, ZMA transform, Elzaki transform, Aboodh transform, Natural transform and Shehu transform. Finally, we ensure the efficiency and accuracy of the Khalouta transform by solving various examples of both ordinary, integro and partial differential equations.

Laser-Based Dynamic Optical Wireless Power Transfer for Electrified Vehicles Covered with Solar Cells

This research proposes and theoretically analyzes a dynamic optical wireless power transfer (OWPT) system for wireless charging of aerial and ground electric vehicles (EVs). In the proposed system, an overhead facility is utilized to host laser transmitters, while solar arrays are attached on EVs to be energy receivers. Moreover, laser transmitters are able to rotate, point upward and downward to wirelessly charge aerial and ground EVs. Owing to the movement of EVs, the wirelessly transmitted power and energy to EVs are varying with time and distance, hence are not as simple as in the case of static wireless charging, i.e. when EVs are stopped or hovering. This paper therefore proposes an analytical analysis for the dependence of wirelessly transmitted laser power and energy to moving EVs on the transmitting distance, utilizing the exponential integral function. The derived results reveal that there exist a unique maximum power point and a unique maximum energy point, due to the monotonic increase and decrease of the transmitted power and energy. These give rise to determining the optimal distances at which the wirelessly transmitted laser power and energy are maximum. Numerical simulations are then carried out to validate and illustrate theoretical results.

Optimal proportional reinsurance and pairs trading under exponential utility criterion for the insurer

<p style='text-indent:20px;'>This paper studies the optimal proportional reinsurance and investment strategy for an insurer who invests one paired assets, where their price spread is described by Ornstein-Uhlenbeck (O-U) processes. The insurer's objective is to maximize the expected exponential utility of the terminal wealth in a finite time horizon under two risk models: a classical risk model and a diffusion model. Using the classical stochastic control approach based on the Hamilton-Jacobi-Bellman equation, we characterize the optimal strategies and provide a verification result for the value function via the exponential integrability of the square of an O-U process. Finally, numerical examples are performed to obtain sensitivity analysis.</p>