Reliable Approximate Solution Research Articles

Plug-and-Play Image Reconstruction Is a Convergent Regularization Method.

Non-uniqueness and instability are characteristic features of image reconstruction methods. As a result, it is necessary to develop regularization methods that can be used to compute reliable approximate solutions. A regularization method provides a family of stable reconstructions that converge to a specific solution of the noise-free problem as the noise level tends to zero. The standard regularization technique is defined by a variational image reconstruction that minimizes a data discrepancy augmented by a regularizer. The actual numerical implementation makes use of iterative methods, often involving proximal mappings of the regularizer. In recent years, Plug-and-Play (PnP) image reconstruction has been developed as a new powerful generalization of variational methods based on replacing proximal mappings by more general image denoisers. While PnP iterations yield excellent results, neither stability nor convergence in the sense of regularization have been studied so far. In this work, we extend the idea of PnP by considering families of PnP iterations, each accompanied by its own denoiser. As our main theoretical result, we show that such PnP reconstructions lead to stable and convergent regularization methods. This shows for the first time that PnP is as mathematically justified for robust image reconstruction as variational methods.

Optimizing option pricing: Exact and approximate solutions for the time-fractional Ivancevic model

This research investigates the time fractional Ivancevic option pricing model and presents two distinct solution methods: the invariant subspace method for obtaining exact solutions and the residual power series method for generating approximate solutions. In the invariant subspace method, we employ an algebraic approach to solve the time fractional Ivancevic option pricing model. By constructing an appropriate invariant subspace, we derive a system of fractional ordinary differential equations that characterizes the exact solution. This method provides a rigorous and efficient technique for generating various exact solutions. We also propose the residual power series method for obtaining approximate solutions. This method involves expressing the solution as an expansion of power functions and iteratively solving the residual equation to refine the approximation. By truncating the series expansion, we obtain an approximate solution that captures the essential features of the model while balancing computational efficiency and accuracy. Different graphs in 2-dimensions and 3-dimensions are presented to pioneer the obtained solutions. Our findings demonstrate the effectiveness of the invariant subspace method in generating exact solutions and shedding light on the model's dynamics. The residual power series method, on the other hand, provides a flexible and computationally efficient approach for obtaining reliable approximate solutions. The integration of exact and approximate methodologies presents a thorough structure for evaluating and valuing options in the time fractional Ivancevic option pricing model. This comprehensive approach yields valuable understandings for financial professionals and researchers alike.

An efficient technique based on Green’s function for solving two-point boundary value problems and its convergence analysis

This study proposes an accurate approximation to the solution of second-order nonlinear two-point boundary value problems, including the well-known Bratu problem, using an iterative technique based on Green’s function. The approach relies on constructing an equivalent integral representation of the problem incorporating Green’s function. The proposed methodology provides a reliable approximate solution and takes just a few iterations to achieve good accuracy. The mathematical formulation is further supported by discussing in detail the convergence analysis of this approach. Different numerical examples are used to check the robustness and effectiveness of the scheme. The numerical testing for nonlinear problems with nonlinear boundary conditions demonstrates that the proposed method outperforms other existing methods, including the finite element method, the finite volume method, the finite difference method, the B-spline method, and the Adomain’s decomposition method.

Approximate Analytical Solution of Fractional Lane-Emden Equation by Mittag-Leffler Function Method

The classical Lane-Emden differential equation, a nonlinear second-order differential equation, models the structure of an isothermal gas sphere in equilibrium under its own gravitation. In this paper, the Mittag-Leffler function expansion method is used to solve a class of fractional LaneEmden differential equation. In the proposed differential equation, the polytropic term f(y(x)) = ym(x) (where m = 0,1,2,... is the polytropic index; 0 < x <=1) is replaced with a linear combination f(y(x)) = a0 + a1y(x) + a2y2(x) + ··· + amym(x) + ··· + aNyN(x),0 <=m <=N,N <= N_0. Explicit solutions of the fractional equation, when f(y) are elementary functions are presented. In particular, we consider the special cases of the trigonometric, hyperbolic and exponential functions. Several examples are given to illustrate the method. Comparison of the Mittag-Leffler function method with other methods indicates that the method gives accurate and reliable approximate solutions of the fractional Lane-Emden differential equation.

Novel approach to the analysis of fifth-order weakly nonlocal fractional Schrödinger equation with Caputo derivative

The main goal of this study is to find solutions for the fractional model of the fifth-order weakly nonlocal Schrödinger equation incorporating nonlinearity of the parabolic law and external potential using a recent modification of the homotopy analysis method (HAM) called the q-homotopy analysis transform method (q-HATM). A mixture of q-HAM and Laplace transform is the projected solutions procedure. The method contributes approximate and exact (for some special cases) solutions such as the bright soliton, dark soliton, and exponential solutions. The simulation results using Mathematica package software, demonstrate that only a few terms are enough to achieve precise, effective, and reliable approximate solutions. In addition, in terms of plots for varying fractional order, the physical behavior of q-HATM solutions has been depicted and the numerical simulation is also exhibited. The results of q-HATM reveal that the projected method is competitive, reliable, and powerful for studying complex nonlinear models of fractional type.

IMPLEMENTING REPRODUCING KERNEL METHOD TO SOLVE SINGULARLY PERTURBED CONVECTION-DIFFUSION PARABOLIC PROBLEMS

In the present paper, reproducing kernel method (RKM) is introduced, which is employed to solve singularly perturbed convection-diffusion parabolic problems (SPCDPPs). It is noteworthy to mention that regarding very serve singularities, there are regular boundary layers in SPCDPPs. On the other hand, getting a reliable approximate solution could be difficult due to the layer behavior of SPCDPPs. The strategy developed in our method is dividing the problem region into two regions, so that one of them would contain a boundary layer behavior. For more illustrations of the method, certain linear and nonlinear SPCDPP are solved.

A new monotonically stable discrete model for the solution of differential equations emanating from the evaporating raindrop

Most of the models of the dynamics of evaporating raindrop have been created using Ordinary Differential Equations. Some models assumed that evaporation is affected by air resistance that is negligible and proportional to the velocity of transition others assumed that the air resistance is significant and proportional to the square of the velocity. Because of the significant differences in the basic assumption used by various modelers, the solution of the resulting equations produced differed values There are yet no work to confirm these dependencies. In this work we obtain a class of hybrid finite difference schemes that can be used to obtain a reliable approximate solution to some of these differential equation models. The schemes were found to possess the same monotonic properties as the analytic solution.

Seismic instability of free-standing statues atop multispondyle columns: A heuristic very stable system of ancient technology

The very complex rocking-sliding response of a free-standing heavy statue atop multi-rigid block assemblies under ground excitation, is extended and supplemented, to the case of multispondyle columns of circular cross-section. After the analytical derivation of the system of highly nonlinear differential equations governing the rocking – sliding response of a N-spondyle cantilever column, attention is focused on seeking the most reliable approximate solution due to the insuperable difficulties for evaluating the loss of energy between successive spondyles which affects decisively the final solution. This is achieved simulating the structural system of a “N-spondyle column – heavy statue” by an “equivalent” system of a “monolithic viscoelastic column – heavy statue” with viscosity coefficient experimentally obtained so that it matches the total loss of energy along the column height due to small slidings and impacts between successive spondyles. In this respect it was found that the accuracy of solution of the aforementioned structural system does not depend solely on the knowledge of the very complex response of a multispondyle column which can be avoided and thus simplifying the entire analysis. Moreover, some qualitatively obtained findings regarding the stability between monolithic and multispondyle columns as well as between geometrically similar columns where confirmed through simple experiments. Finally, it was determined a reduced total number of geometrical configuration patterns which may lead to overturning, and among these the critical one which provokes the first rocking initiation.

Making the Coupled Cluster Correlation Energy Machine-Learnable.

Calculating the electronic structure of molecules and solids has become an important pillar of modern research in diverse fields of research from biology and materials science to chemistry and physics. Unfortunately, increasingly accurate and thus reliable approximate solution schemes to the underlying Schrödinger equation scale steeply in computational cost, rendering most accurate approaches like "gold standard" coupled cluster theory, CC, quickly intractable for larger systems of interest. Here we show that this scaling can be significantly reduced by applying machine-learning to the CC correlation energy. We introduce a vector-based representation of CC wave functions and use potential energy surfaces of a small molecule test set to learn the correlation energy from this representation. Our results show that the CC correlation energy can be efficiently learned, even when the representation is constructed from approximate amplitudes provided by computationally less demanding Møller-Plesset (MP2) perturbation theory. Exploiting existing linear scaling MP2 implementations, this potentially opens the door to CC-quality molecular dynamics simulations.

A novel scheme for solving Caputo time-fractional nonlinear equations: theory and application

The current work contributes to find a supportive analytical solution of Caputo time-fractional nonlinear equations of the form $$\begin{aligned}&{\mathcal {D}}^{\alpha }_{t}\big [u(x,t)\big ]+\left( \sum _{k=1}^{N_1}\lambda _{k}u^k(x,t)\right) u_{x}(x,t)\nonumber \\&\quad +\sum _{k=2}^{N_2}\delta _{k}\frac{\partial ^ku(x,t)}{\partial x^k}=0,\ \ \ \ 0<\alpha \le 1. \end{aligned}$$ We proposed a modified version of the generalized Taylor power series method to extract a reliable approximate solution of this problem. Theorems regarding the convergence of the obtained solution are provided. The method is tested on two models of the proposed equation, namely the Caputo time-fractional Gardner and Kawahara equations. Finally, both graphical explanations and tabular analysis are performed to study the accuracy of our suggested scheme, and to study the effects of $$\alpha $$ on the behavior of the obtained solution.

Simulating Head Recovery and Intermittent Discharge in Free Flowing Wells.

Modeling of the head recovery (buildup) in and intermittent operation of flowing wells lacks rigorous analytical tools. Presently different methods are applied in discharging and recovery phases. The presented reliable approximate solution considers a continuous free flow process and time variant screen loss parameters. The latter are temporarily set to large value to prevent inflow into the well bore during the single recovery or in repeated shut down periods. This method is called as screen loss control or SLC technique and is verified by means of comparative analysis with the constant rate pumping-recovery test simulated in leaky aquifer. The evaluation of the free flow and recovery test in the free flowing well 28 (Artesia Heights) concluded with calibrated parameters close to those obtained in the earlier analyses based on split well flow process. The simulation of intermittent flowing well operation also underwent successful validation resulting in reasonable values of the Qw and s response functions.

The meshfree strong form methods for solving one dimensional inverse Cauchy-Stefan problem

In this paper, we extend the application of meshfree node based schemes for solving one-dimensional inverse Cauchy-Stefan problem. The aim is devoted to recover the initial and boundary conditions from some Cauchy data lying on the admissible curve s(t) as the extra overspecifications. To keep matters simple, the problem has been considered in one dimensional, however the physical domain of the problem is supposed as an irregular bounded domain in $$\mathbb {R}^2$$R2. The methods provide the space-time approximations for the heat temperature derived by expanding the required approximate solutions using collocation scheme based on radial point interpolation method (RPIM). The proposed method makes appropriate shape functions which possess the important Delta function property to satisfy the essential conditions automatically. In addition, to conquer the ill-posedness of the problem, particular optimization technique has been applied for solving the system of equations $$Ax=b$$Ax=b in which A is a nonsymmetric stiffness matrix. As the consequences, reliable approximate solutions are obtained which continuously depend on input data.

Intelligent computing to solve fifth-order boundary value problem arising in induction motor models

In this study, biologically inspired intelligent computing approached based on artificial neural networks (ANN) models optimized with efficient local search methods like sequential quadratic programming (SQP), interior point technique (IPT) and active set technique (AST) is designed to solve the higher order nonlinear boundary value problems arise in studies of induction motor. The mathematical modeling of the problem is formulated in an unsupervised manner with ANNs by using transfer function based on log-sigmoid, and the learning of parameters of ANNs is carried out with SQP, IPT and ASTs. The solutions obtained by proposed methods are compared with the reference state-of-the-art numerical results. Simulation studies show that the proposed methods are useful and effective for solving higher order stiff problem with boundary conditions. The strong motivation of this research work is to find the reliable approximate solution of fifth-order differential equation problems which are validated through strong statistical analysis.

Landau Subband and Landau Level Properties of AA-Stacked Graphene Superlattice

Magneto-electronic properties of AA-stacked graphene superlattice (simple hexagonal graphite) are studied by the tight-binding model with an exact diagonalization method. The Landau subbands (LSs) with strong energy dispersions exist along k(z) and each subband possesses two band-edge states. Density of states reflects main features of the LSs, such as asymmetric prominent peak structures and the semimetallic behavior. Under the AA-stacked configuration, the LS wave functions are characterized by two sublattices, a and b. The quantum number (n(c,v)) of each LS, which corresponds to the number of zero points in the dominating carrier distributions, is determined by a certain sublattice and independent of k(z). For each LS, the difference between the number of zero point of the a and b sublattices is fixed and equals one. Furthermore, a reliable approximate solution of the low-lying LS energy is obtained. Through observing this solution, the dependence of the LS energy on the field strength and quantum number, k(z)-dependent energy spacing between two LSs, and the values of atomic hopping integrals are reasonably determined. A comparison of the AA-stacked graphene superlattice and monolayer graphene demonstrates that they possess some similar magneto-electronic properties.

Enhanced Multistage Differential Transform Method: Application to the Population Models

We present an efficient computational algorithm, namely, the enhanced multistage differential transform method (E‐MsDTM) for solving prey‐predator systems. Since the differential transform method (DTM) is based on the Taylor series, it is difficult to obtain accurate approximate solutions in large domain. To overcome this difficulty, the multistage differential transform method (MsDTM) has been introduced and succeeded to have reliable approximate solutions for many problems. In MsDTM, it is the key to update an initial condition in each subdomain. The standard MsDTM utilizes the approximate solution directly to assign the new initial value. Because of local convergence of the Taylor series, the error is accumulated in a large domain. In E‐MsDTM, we propose the new technique to update an initial condition by using integral operator. To demonstrate efficiency of the proposed method, several numerical tests are performed and compared with ones obtained by other numerical methods such as MsDTM, multistage variational iteration method (MVIM), and fourth‐order Runge‐Kutta method (RK4).

Reliable Approximate Solution of Systems of Volterra Integro-Differential Equations with Time-Dependent Delays

Volterra integro-differential equations with time-dependent delay arguments can provide us with realistic models of many real-world phenomena. Delayed Lokta-Volterra predator-prey systems arise in Ecology and are well-known examples of delay Volterra integro-differential equations (DVIDEs) first introduced by Volterra in 1928. We investigate the numerical solution of systems of DVIDEs using an adaptive stepsize selection strategy. We will present a generic variable stepsize approach for solving systems of neutral DVIDEs based on an explicit continuous Runge-Kutta method using defect error control and study the convergence of the resulting numerical method for various kinds of delay arguments. We will show that the global error of the numerical solution can be effectively and reliably controlled by monitoring the size of the defect of the approximate solution and adjusting the stepsize on each step of the integration. Numerical results will be presented to demonstrate the effectiveness of this approach.

On the Brownian motion in a double-well potential in the overdamped limit

The Brownian motion of a particle in a double-well potential is considered and the position correlation function (excluding inertial effects) for the potential V( x)= ax 2/2+ bx 4/4 is evaluated first by averaging the governing Langevin equation over its realizations. The exact solution is then obtained via matrix continued fractions by using a representation, which symmetrizes the recurrence relations for the observables generated by the averaging procedure leading to convergence of these recurrence relations unlike the previous approaches to the problem. A reliable approximate solution based on the exponential separation of the time scales of the fast intrawell and low overbarrier relaxation processes associated with the bistable potential is also given. It is shown that a knowledge of the three characteristic relaxation times (the integral, effective and the longest relaxation times) of the position correlation function allows one to accurately predict the relaxation behavior of the system in the overdamped limit for all time scales of interest.

Incorporating load dependent servers in approximate mean value analysis

Queueing network performance modelling technology has made tremendous strides in recent years. Two of the most important developments in facilitating the modelling of large and complex systems are hierarchical modelling, in which a single load dependent server is used as a surrogate for a subsystem, and approximate mean value analysis, in which reliable approximate solutions of separable models are efficiently obtained. Unfortunately, there has been no successful marriage of these two developments; that is, existing algorithms for approximate mean value analysis do not accommodate load dependent servers reliably.This paper presents a successful technique for incorporating load dependent servers in approximate mean value analysis. We consider multiple class models in which the service rate of each load dependent server is a function of the queue length at that server. In other words, load dependent center k delivers “service units” at a total rate of f

Incorporating load dependent servers in approximate mean value analysis

Queueing network performance modelling technology has made tremendous strides in recent years. Two of the most important developments in facilitating the modelling of large and complex systems are hierarchical modelling , in which a single load dependent server is used as a surrogate for a subsystem, and approximate mean value analysis , in which reliable approximate solutions of separable models are efficiently obtained. Unfortunately, there has been no successful marriage of these two developments; that is, existing algorithms for approximate mean value analysis do not accommodate load dependent servers reliably. This paper presents a successful technique for incorporating load dependent servers in approximate mean value analysis. We consider multiple class models in which the service rate of each load dependent server is a function of the queue length at that server. In other words, load dependent center k delivers “service units” at a total rate of f @@@@ (n @@@@ ) when n @@@@ customers are present. We present extensive experimental validation which indicates that our algorithm contributes an average error in response times of less than 1% compared to the (much more expensive) exact solution. In addition to the practical value of our algorithm, several of the techniques that it employs are of independent interest.