Backward Problem Research Articles

Finite element method based computational time reversal in elastodynamics: Refocusing, reconstruction and its numerical sensitivity

In this paper, we study the accuracy and robustness of the computational time reversal approach based on the explicit finite element method for application in nondestructive testing in solids. The main goal of this paper is to find a methodology for correct and accurate reconstruction of the original source time history. For numerical modeling of frontal (forward) and reverse (backward) problems of elastic wave propagation, we use the finite element method and explicit time integration with the lumped mass matrix. The suggested methodology is applicable in each finite element open source or commercial software. A special attention is paid to prescription of boundary conditions/loading for the reverse problem for accurate reconstruction of time history of the original source. For evaluation of the reconstruction quality, we suggest certain cost functions. Based on several numerical tests, we show effects of prescription of boundary conditions/loading in time reversal, effect of mesh size and time step size, an unknown obstacle, a number of sources, and environmental disturbance (noise) on the correctness of reconstruction of the original source.

A modified quasi-boundary value method for a backward problem for the inhomogeneous time conformable fractional heat equation in a cylinder

ABSTRACT The time conformable heat equation is a generalization of classical heat equation involved local and limit-based derivative, which is called conformable fractional derivative. In this paper, we study a backward problem for the time conformable fractional heat equation defined in cylindrical coordinates for the axis-symmetric case which is a severely ill-posed problem. By using a modified quasi-boundary value method, the problem is regularized and some new error estimates are obtained. The numerical experiment shows that the method is feasible and effective.

A comparative analysis of multi-machine learning algorithms for data-driven RANS turbulence modelling

To improve the speed or accuracy of Reynolds-averaged Navier-Stokes in numerical simulations of turbulence, five machine learning algorithms including Random Forests, eXtreme Gradient Boosting, K-Nearest Neighbours, Support Vector Machine and Artificial Neural Network are introduced in this paper. We establish the nonlinear mapping relationship between the average flow field and the steady-state eddy viscosity. The machine learning surrogate models for the Spalart-Allmaras turbulence model are constructed and used to solve the backward facing step problem. It is demonstrated that the machine learning surrogate models constructed by the five machine learning algorithms can effectively express the complex mapping relationship between the average turbulent flow field and the steady-state eddy viscosity. They can not only obtain reasonable simulation accuracy but also significantly accelerate the solution time of the flow field. The algorithms of eXtreme Gradient Boosting, K-Nearest Neighbours and Artificial Neural Network have better performance when considering the calculation accuracy, time cost and memory cost. It shows the great potential of applying machine learning algorithms to RANS turbulence model and also provides a new idea for industrial simulations of turbulent flows.

Well-posedness for the backward problems in time for general time-fractional diffusion equation

In this article, we consider an evolution partial differential equation with Caputo time-derivative with the zero Dirichlet boundary condition: \partial^{\alpha}_tu + Au = F where 0 < \alpha < 1 and the principal part -A , is a non-symmetric elliptic operator of the second order. Given a source F , we prove the well-posedness for the backward problem in time and our result generalizes the existing results assuming that -A is symmetric. The key is a perturbation argument and the completeness of the generalized eigenfunctions of the elliptic operator A .

Graphene-based microwave coaxial antenna for microwave ablation: thermal analysis

AbstractIn this study, the problem of backward heating in microwave ablation technique is examined and an electromagnetic solution based on the use of high impedance graphene material is presented for its mitigation. In this context, a one-atom-thick graphene layer is added on the coaxial double slot antenna. In addition to the electromagnetic behavior, thermal effects caused by the graphene-covered antenna are emphasized. The graphene's conductivity being highly dependent on its chemical potential and the relaxation time, a parametric study is performed to determine a range of tolerances within which the graphene-coated antenna outperform a typical graphene-free antenna. The range of values is found to be 0 < μc < 0.5 eV and τ < 0.4 ps, for the chemical potential and the relaxation time, respectively. The backward heating problem being prevented, the ablation region is ensured to be spherical around the tip of the antenna. Effects of the graphene layer to the heat dissipation in the tissue, the necrotic tissue ratio (damage to the cancerous tissue of the caused by electromagnetic energy), and the treatment time using the coaxial double slot antenna were examined. The results show that the heat dissipation is concentrated around the slots (region of cancerous tissue) and a higher necrotic tissue ratio can be achieved with a graphene-covered double slot antenna in a shorter time.

Iterated fractional Tikhonov regularization method for solving the spherically symmetric backward time-fractional diffusion equation

In this paper, we focus on a backward problem for an inhomogeneous time-fractional diffusion equation on a spherical symmetric domain. This problem is ill-posed in the sense of Hadamard. We propose an iterated fractional Tikhonov regularization method in both cases: the deterministic case and random noise case. The a-priori and the a-posteriori choice rules for regularization parameters are discussed and both rules yield the corresponding convergence rates. The iterated fractional Tikhonov regularization method goes beyond the saturation results of classical Tikhonov method, and iterative fractional Tikhonov regularization method is superior to classical iterative Tikhonov regularization method under the a-priori parameter choice rule. A numerical example is conducted for showing the validity of the proposed method.

Some features of solving an inverse backward problem for a generalized Burgers’ equation

Abstract In this paper, we consider an inverse backward problem for a nonlinear singularly perturbed parabolic equation of the Burgers’ type. We demonstrate how a method of asymptotic analysis of the direct problem allows developing a rather simple algorithm for solving the inverse problem in comparison with minimization of the cost functional. Numerical experiments demonstrate the effectiveness of this approach.

Numerical analysis of backward subdiffusion problems

The aim of this paper is to develop and analyze numerical schemes for approximately solving the backward problem of subdiffusion equation involving a fractional derivative in time with order α ∈ (0, 1). After using quasi-boundary value method to regularize the ‘mildly’ ill-posed problem, we propose a fully discrete scheme by applying finite element method (FEM) in space and convolution quadrature (CQ) in time. We provide a thorough error analysis of the resulting discrete system in both cases of smooth and nonsmooth data. The analysis relies heavily on smoothing properties of (discrete) solution operators, and nonstandard error estimate for the direct problem in terms of problem data regularity. The theoretical results are useful to balance discretization parameters, regularization parameter and noise level. Numerical examples are presented to illustrate the theoretical results.

Backward Reachability Analysis for Nonlinear Dynamical Systems via Pseudospectral Method

In this paper, we propose a new approach to solving the backward reachability problem for nonlinear dynamical systems. Previously, this class of problems has been studied within frameworks of optimal control and zero-sum differential games, where a backward reachable set can be expressed as the zero sublevel set of the value function that can be characterized by solving the Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE). In many cases, however, a high computational cost is incurred in numerically solving such HJB PDEs due to the curse of dimensionality. We use the pseudospectral method to convert the associated optimal control problem into nonlinear programs (NLPs). We then show that the zero sublevel set obtained by the optimal cost of the NLP is the corresponding backward reachable set. Note that our approach does not require solving complex HJB PDEs. Therefore, it can reduce computation time and handle high-dimensional dynamical systems, compared with the numerical software package developed by I. Mitchell, which has been used widely in the literature to obtain backward reachable sets by solving HJB equations. We provide several examples to validate the effectiveness of the proposed approach.

Regularization of the fractional Rayleigh\u2013Stokes equation using a fractional Landweber method

In this paper, we consider a time-fractional backward problem for the fractional Rayleigh–Stokes equation in a general bounded domain. We propose a fractional Landweber regularization method for solving this problem. Error estimates between the regularized solution and the sought solution are also obtained under some choice rules for both a-priori and a-posterior regularization parameters.

In search of civil society: From peasant populism to postpeasant illiberalism in provincial Hungary

The need to rebuild civil society was a prominent theme in dissident writings in East-Central Europe in late socialism, but the revival of this concept deserves close scrutiny and local contextualization. This article identifies two currents in Hungarian debates, one focused on addressing problems of backwardness by opening up paths of material embourgeoisement and the other on abstract liberal notions of associational freedom. It then outlines successive transformations of economic and social life in a small Hungarian town where no industry existed prior to socialism and the dominant political forces were populist in the sense of ‘peasantism’. The agrarian and industrial transformations of the socialist decades were undone in the 1990s. In the 2010s, under governments led by Viktor Orbán, it is argued that norms of civility have been threatened by postpeasant illiberalism. If civil society was the gauntlet laid down to social theorists by East-Central Europe in 1989, the challenge posed by this region nowadays is the theorization of incivility and a new brand of populism. It is suggested that these political processes are driven by the collapse of socialist embourgeoisement and the emergence of a new national bourgeoisie under peripheral capitalism, and that some of the moral responsibility for these developments lies with the unwavering intellectual enthusiasts of abstract liberalism.

On a final value problem for parabolic equation on the sphere with linear and nonlinear source

Parabolic equation on the unit sphere arise naturally in geophysics and oceanography when we model a physical quantity on large scales. In this paper, we consider a problem of finding the initial state for backward parabolic problem on the sphere. This backward parabolic problem is ill-posed in the sense of Hadamard. The solutions may be not exists and if they exists then the solution does not continuous depends on the given observation. The backward problem for homogeneous parabolic problem was recently considered in the paper Q.T. L. Gia, N.H. Tuan, T. Tran. However, there are very few results on the backward problem of nonlinear parabolic equation on the sphere. In this paper, we do not consider the its existence, we only study the stability of the solution if it exists. By applying some regularized method and some techniques on the spherical harmonics, we approximate the problem and then obtain the convalescence rate between the regularized solution and the exact solution.

A modified fractional Landweber method for a backward problem for the inhomogeneous time-fractional diffusion equation in a cylinder

In this paper, we consider a backward problem for the inhomogeneous time-fractional diffusion equation in a cylinder. Such a problem is ill-posed. Based on the solution given by the separation of variables, we apply a modified fractional Landweber method to solve it. Error estimates are presented under the a-priori and the a-posteriori choice rules for regularization parameters, respectively. Relative to the classical Landweber method, the modified fractional Landweber method can reduce the total number of iterations to achieve the same accuracy. Numerical example is given to show the validity of the scheme.

Solving the Riesz\u2013Feller space-fractional backward diffusion problem by a generalized Tikhonov method

The article investigates a Riesz–Feller space-fractional backward diffusion problem. We develop a generalized Tikhonov regularization method to overcome the ill-posedness of this problem, and then based on the result of conditional stability, we derive the convergence estimates of logarithmic and double logarithmic types for the regularized method by adopting a-posteriori choice rules of regularization parameter. Finally, by using the finite difference method, we solve a direct problem to construct the data, and some corresponding results of numerical simulations are presented to verify the convergence and stability for this method.

Spectral graph wavelet regularization and adaptive wavelet for the backward heat conduction problem

This paper proposes a new regularization technique using spectral graph wavelet for backward heat conduction problem (BHCP) on the graph. The method uses the fourth-order compact difference for an approximation of differential operators. Meanwhile, the error estimate between the exact and wavelet regularized solution is derived. Adaptive node arrangement is obtained using spectral graph wavelet. The essence of the method is that the same operator can be used for the construction of the spectral graph wavelet and the approximation of the differential operator. The proposed method is applied to solve BHCP on graph arising from regular one-dimensional and two-dimensional mesh. Numerical results show that we can obtain a stable solution using spectral graph wavelet regularization. Moreover, spectral graph wavelet adaptivity and fourth-order compact difference scheme lead to a very efficient solution.

A two-level variational multiscale meshless local Petrov-Galerkin (VMS-MLPG) method for incompressible Navier-Stokes equations

A two-level variational multiscale meshless local Petrov-Galerkin (VMS-MLPG) method is presented for incompressible Navier-Stokes equations based on two local Gauss integrations which effectively replace a unit operator (first level) and an orthogonal project operator (second level). The present VMS-MLPG method allows arbitrary combinations of interpolation functions for the velocity and pressure fields, specifically the equal-order interpolations that are easy to implement and satisfy the Babuska-Breezi (B-B) condition. The prediction accuracy and the numerical stability of the proposed method for the lid-driven cavity flow and the backward facing step flow problems are analyzed and validated by comparing with the SUMLPG method and the benchmark solutions. It is shown that the present VMS-MLPG method can guarantee the numerical stability and obtain the reasonable solutions for incompressible Navier-Stokes equation.

A backward problem for distributed order diffusion equation: uniqueness and numerical solution

In this paper we consider the identification of the initial condition for a distributed order diffusion equation. We first prove the unique existence and the regularity properties of the strong solution on the bounded temporal-spacial domain based on the eigenfunction expansions. The ill-posedness of the backward problem is interpreted by the compactness of the observation operator. Next the Laplace transformation technique and analytic continuation method are adopted to prove the uniqueness of the backward problem. Then for stabilizing the ill-posed problem, the backward problem is formulated as a Tikhonov type optimization problems, and the conjugate gradient method is adopted to solve the optimization problem with the help of the variational adjoint technique. Finally four numerical examples are given to show the efficiency and stability of the proposed method.

An extension of the landweber regularization for a backward time fractional wave problem

In this paper, we investigate numerical methods for a backward problem of the time-fractional wave equation in bounded domains. We propose two fractional filter regularization methods, which can be regarded as an extension of the classical Landweber iteration for the time-fractional wave backward problem. The idea is first to transform the ill-posed backward problem into a weighted normal operator equation, then construct the regularization methods for the operator equation by introducing suitable fractional filters. Both a priori and a posteriori regularization parameter choice rules are investigated, together with an estimate for the smallest regularization parameter according to a discrepancy principle. Furthermore, an error analysis is carried out to derive the convergence rates of the regularized solutions generated by the proposed methods. The theoretical estimate shows that the proposed fractional regularizations efficiently overcome the well-known over-smoothing drawback caused by the classical regularizations. Some numerical examples are provided to confirm the theoretical results. In particular, our numerical tests demonstrate that the fractional regularization is actually more efficient than the classical methods for problems having low regularity.

The Influence of Working Backward Problem Solving Strategy Towards Mathematical Reasoning Ability in Terms of Students’ Emotional Intelligence

The study aimed to know the description of student’s mathematical reasoning ability and student’s emotional intelligence, to find out whether there were differences in the mathematical reasoning ability between students who used and those who did not use working backward problem solving strategy, identify whether there were differences in mathematical reasoning ability between students who used and those who did not use working backward problem solving strategy in terms of the emotional intelligence, and to find out whether there was an influence of interaction on the working backward problem solving strategy towards the mathematical reasoning ability in terms of the emotional intelligence of the students of grade VIII of MTsN 1 Makassar. The study used the quantitative approach with 3×2 factorial design. The population of the study were the students of grade VIII of MTsN 1 Makassar consisting of 440 students with 80 students as the samples (2 classes). The data analysis technique used was descriptive statistical and inferential statistical analysis. The findings of the study showed that (1) the mathematical reasoning ability of the students who used and those who did not use the working backward problem solving strategy were in the high category, meanwhile the students’ emotional intelligence was in the medium category, (2)there were differences in the mathematical reasoning abilities between students who used and those who did not use working backward problem solving strategy, (3)there were differences in mathematical reasoning ability between students who used and those who did not use working backward problem solving strategy in terms of student’s emotional intelligence, and (4)there was no interaction effect of working backward problem solving strategy towards the mathematical reasoning ability in terms of the emotional intelligence of the students of grade VIII of MTsN 1 Makassar

SPECIAL ECONOMIC ZONES AS AN INSTRUMENT OF ECONOMIC GROWTH AND DEVELOPMENT OF INDUSTRIAL POTENTIAL OF RUSSIA

The article considers an instrument of influence on the economic development of the Russian Federation in the form of creating special economic zones. The current state of Russian enterprises is described in terms of the theory of techno-economic paradigm. The problem of technological backwardness of Russia is revealed. In this regard, the authors consider special economic zones as an engine for the development of the Russian Federation. The principles of functioning of these zones are described, privileges for enterprises registered in the territory of special economic zones are listed. The types of zones are also characterized, among which four are distinguished by the legislation of Russia: industrial-production, technical-innovative, tourist-recreational and port. Examples are given for each type of special economic zone. The authors examined the stages of construction of these zones, and the legislation that determines their functioning. The article also provides an example of the Innopolis special economic zone: its specialization, what benefits it gives residents and what is located on its territory. The article’s conclusion provides a list of benefits that an enterprise can count on if it is registered as a resident in the territory of a special economic zone. It also proves the existence of a direct relationship between the effective functioning of special economic zones and the acceleration of the economic development of the state, which allows it to enter the next techno-economic paradigm.