Method For Solution Of Problems Research Articles

A Tutorial Introduction to Inverse Problems in Magnetic Resonance.

There has been a tremendous increase in applications of the inverse problem framework to parameter estimation in magnetic resonance. Attempting to capture both the basics of this formalism and modern developments would require an article of inordinate length. Therefore, in the following, we provide basic material as a practical introduction to the topic and an entree to the literature. First, we describe the formulation of linear and nonlinear inverse problems, with an emphasis on signal equations arising in magnetic resonance. We then describe the Fredholm equation of the first kind as a paradigm for these problems. This is followed by much more detailed considerations for determining solutions in the linear case, including central concepts such as condition number, regularization, and stability. Solution methods for nonlinear inverse problems are described next, followed by a treatment of their stability and regularization. Finally, we provide an introduction to compressed sensing, with signal reconstruction formulated as the solution to an inverse problem, making use of much of the previous material. Throughout, the emphasis is on outlines of the theory and on numerical examples, rather than on mathematical rigor and completeness.

On the Problem of a Linear Function Localization on Permutations

Combinatorial optimization problems and methods of their solution have been a subject of numerous studies, since a large number of practical problems are described by combinatorial optimization models. Many studies consider approaches to and describe methods of solution for combinatorial optimization problems with linear or fractionally linear target functions on combinatorial sets such as permutations and arrangements. Studies consider solving combinatorial problems by means of well-known methods, as well as developing new methods and algorithms of searching a solution. We describe a method of solving a problem of a linear target function localization on a permutation set. The task is to find those locally admissible permutations on the permutation set, for which the linear function possesses a given value. In a general case, this problem may have no solutions at all. In the article, we propose a newly developed method that allows us to obtain a solution of such a problem (in the case that such solution exists) by the goal-oriented seeking for locally admissible permutations with a minimal enumeration that is much less than the number of all possible variants. Searching for the solution comes down to generating various permutations and evaluating them. Evaluation of each permutation includes two steps. The first step consists of function decreasing by transposing the numbers in the first n – 3 positions, and the second step is evaluation of the permutations for the remaining three numbers. Then we analyze the correlation (which is called balance) to define whether the considered permutation is the solution or not. In our article, we illustrate the localization method by solving the problem for n = 5. Keywords: localization, linear function, permutation, transposition, balance, position.

An explicit Tikhonov algorithm for nested variational inequalities

We consider nested variational inequalities consisting in a (upper-level) variational inequality whose feasible set is given by the solution set of another (lower-level) variational inequality. Purely hierarchical convex bilevel optimization problems and certain multi-follower games are particular instances of nested variational inequalities. We present an explicit and ready-to-implement Tikhonov-type solution method for such problems. We give conditions that guarantee the convergence of the proposed method. Moreover, inspired by recent works in the literature, we provide a convergence rate analysis. In particular, for the simple bilevel instance, we are able to obtain enhanced convergence results.

Iterative Method for the Numerical Solution of an Inverse Coefficient Problem for a System of Partial Differential Equations

For a system of partial differential equations, we consider the inverse problem of determining one of the coefficients based on additional information on one of the solution components. An iterative method is proposed for calculating the unknown coefficient based on the reduction of the inverse problem to a nonlinear operator equation. The convergence of the iterative method and the existence and uniqueness of a solution of the inverse problem are proved. Numerical results illustrating the convergence of the iterative numerical method are given.

Implementation of Differential Transform Method (DTM) for Large Deformation Analysis of Cantilever Beam

Large deformation of a cantilever beam is an eminent structural analysis and engineering problem. A non-linear differential equation governs the problem of large deformation of beams. It has many important applications scientific and engineering fields. It is quite intricate and complex to find a closed-form or an exact solution for such a problem. It is relatively easier to use analytic expressions for calculations than numerical or experimental analysis. Large deformation analysis has always been an active area of research for researchers. The main focus of this research is to implement a new method namely Differential Transformation Method (DTM) for non-linear beam problem analysis. DTM is quite reliable and efficient in giving an approximate analytical solution to beam problems. Moreover, it is a new application for DTM. Hence, the primary purpose of this work is to evaluate the utility of the method for solution of beam problem. DTM yields a 2-point boundary value problem. It is based Taylor’s series expansion. DTM helps to calculate the deflection angle and horizontal as well as vertical displacements of a cantilever beam, experiencing large deformation, in an explicit analytical form. Mathematical formulation code will be developed using MATLAB. Variation of rotation angle at free end will be analysed using different number of terms used and values of non-dimensional load “β”. Results of DTM prove the method to be quite effective and suitable for predicting solutions to non-linear beam problems. Therefore, DTM can be used for widespread applications for new, emerging and complex engineering problems.

On an iterative method for solution of direct problem for nonlinear hyperbolic differential equations

An iterative method for solution of Cauchy problem for one-dimensional nonlinear hyperbolic differential equation is proposed in this paper. The method is based on continuous method for solution of nonlinear operator equations. The keystone idea of the method consists in transition from the original problem to a nonlinear integral equation and its successive solution via construction of an auxiliary system of nonlinear differential equations that can be solved with the help of different numerical methods. The result is presented as a mesh function that consists of approximate values of the solution of stated problem and is constructed on a uniform mesh in a bounded domain of two-dimensional space. The advantages of the method are its simplicity and also its universality in the sense that the method can be applied for solving problems with a wide range of nonlinearities. Finally it should be mentioned that one of the important advantages of the proposed method is its stability to perturbations of initial data that is substantiated by methods for analysis of stability of solutions of systems of ordinary differential equations. Solving several model problems shows effectiveness of the proposed method.

On Global Dynamics in Duffing Equation with Quasiperiodic Perturbation

An iterative method for solution of Cauchy problem for one-dimensional nonlinear hyperbolic differential equation is proposed in this paper. The method is based on continuous method for solution of nonlinear operator equations. The keystone idea of the method consists in transition from the original problem to a nonlinear integral equation and its successive solution via construction of an auxiliary system of nonlinear differential equations that can be solved with the help of different numerical methods. The result is presented as a mesh function that consists of approximate values of the solution of stated problem and is constructed on a uniform mesh in a bounded domain of two-dimensional space. The advantages of the method are its simplicity and also its universality in the sense that the method can be applied for solving problems with a wide range of nonlinearities. Finally it should be mentioned that one of the important advantages of the proposed method is its stability to perturbations of initial data that is substantiated by methods for analysis of stability of solutions of systems of ordinary differential equations. Solving several model problems shows effectiveness of the proposed method.

A New Iterative Method for the Numerical Solution of High-Order Non-linear Fractional Boundary Value Problems

The boundary value problems (BVPs) have attracted the attention of many scientists from both practical and theoretical points of view, for these problems have remarkable applications in different branches of pure and applied sciences. Due to this important property, this research aims to develop an efficient numerical method for solving a class of nonlinear fractional BVPs. The proposed method is free from perturbation, discretization, linearization, or restrictive assumptions, and provides the exact solution in the form of a uniformly convergent series. Moreover, the exact solution is determined by solving only a sequence of linear BVPs of fractional-order. Hence, from practical viewpoint, the suggested technique is efficient and easy to implement. To achieve an approximate solution with enough accuracy, we provide an iterative algorithm that is also computationally efficient. Finally, four illustrative examples are given verifying the superiority of the new technique compared to the other existing results.

An Algorithm for Recovering the Characteristics of the Initial State of Supernova

An optimization method for the numerical solution of the inverse problem of recovering the initial state of a supernova is proposed. The gradient of the inverse problem objective functional is constructed. The solution of the direct and adjoint problems is based on a combination of the large particle method, Godunov’s method, and piecewise parabolic method on a local stencil. Results of the verification of the proposed numerical method and the results of numerical solution of the Evrard and Sedov collapse problems are presented.

The effect of problem-based learning model to students’ cognitive achievement on high and low students’ problem-solving abilities

This study aimed to determine the effect of PBL model on students’ cognitive achievement on high and low students’ problem solving abilities. This study was quasi experiment research by using posttest only non-equivalent control group design with 22 students of X(3) grader science class as control class (conventional model) and 21 students of X(4) as experiment class (using PBL model) at one of senior high school in Surakarta. Students’ problem solving abilities illustrated as abilities in four aspects: identification the problem, identification the cause of problem, create the method of problem solution, examine the result of problem solving by using 40 items of multiple choice question refers to Problem Solving Skill Test [12]. Students’ problem solving abilities measured before the research is done and divided into high and low categories. Students’ cognitive achievement illustrated as students’ concept understanding and measured by using essay test was given after the PBL model used. Instrument had been validated by expert judgment and students. Data analyze by using t-test. The result showed: 1) Sig. (0,00<0,05) Ho rejected, there was difference of students’ achievement between experiment class and control class; 2) Sig (0,683 >0,05) H0 accepted, there wasn’t difference students’ achievement with high problem solving abilities between low problem solving abilities.

CSG: A new stochastic gradient method for the efficient solution of structural optimization problems with infinitely many states

This paper presents a novel method for the solution of a particular class of structural optimzation problems: the continuous stochastic gradient method (CSG). In the simplest case, we assume that the objective function is given as an integral of a desired property over a continuous parameter set. The application of a quadrature rule for the approximation of the integral can give rise to artificial and undesired local minima. However, the CSG method does not rely on an approximation of the integral, instead utilizing gradient approximations from previous iterations in an optimal way. Although the CSG method does not require more than the solution of one state problem (of infinitely many) per optimization iteration, it is possible to prove in a mathematically rigorous way that the function value as well as the full gradient of the objective can be approximated with arbitrary precision in the course of the optimization process. Moreover, numerical experiments for a linear elastic problem with infinitely many load cases are described. For the chosen example, the CSG method proves to be clearly superior compared to the classic stochastic gradient (SG) and the stochastic average gradient (SAG) method.

Generalized Second Derivative Linear Multistep Methods Based on the Methods of Enright

The Adams-type second derivative multistep methods of W. H. Enright is generalized to a class of boundary value methods for the numerical solution of initial value problems in ordinary differential equations. The root distribution of the stability polynomials of the new class of methods are given for the purpose of their correct implementation as boundary value methods. Computational results on some stiff problems are compared with those from some existing methods, while ODE15s of MATLAB is used as a reference numerical solution.

A dynamical view of nonlinear conjugate gradient methods with applications to FFT-based computational micromechanics

For fast Fourier transform (FFT)-based computational micromechanics, solvers need to be fast, memory-efficient, and independent of tedious parameter calibration. In this work, we investigate the benefits of nonlinear conjugate gradient (CG) methods in the context of FFT-based computational micromechanics. Traditionally, nonlinear CG methods require dedicated line-search procedures to be efficient, rendering them not competitive in the FFT-based context. We contribute to nonlinear CG methods devoid of line searches by exploiting similarities between nonlinear CG methods and accelerated gradient methods. More precisely, by letting the step-size go to zero, we exhibit the Fletcher–Reeves nonlinear CG as a dynamical system with state-dependent nonlinear damping. We show how to implement nonlinear CG methods for FFT-based computational micromechanics, and demonstrate by numerical experiments that the Fletcher–Reeves nonlinear CG represents a competitive, memory-efficient and parameter-choice free solution method for linear and nonlinear homogenization problems, which, in addition, decreases the residual monotonically.

New Possibilities of Application of Artificial Intelligence Methods for High-Precision Solution of Boundary Value Problems

One of the main problems in modern mathematical modeling is to obtain high-precision solutions of boundary value problems. This study proposes a new approach that combines the methods of artificial intelligence and a classical analytical method. The use of the analytical method of fictitious canonic regions is proposed as the basis for obtaining reliable solutions of boundary value problems. The novelty of the approach is in the application of artificial intelligence methods, namely, genetic algorithms, to select the optimal location of fictitious canonic regions, ensuring maximum accuracy. A general genetic algorithm has been developed to solve the problem of determining the global minimum for the choice and location of fictitious canonic regions. For this genetic algorithm, several variants of the function of crossing individuals and mutations are proposed. The approach is applied to solve two test boundary value problems: the stationary heat conduction problem and the elasticity theory problem. The results of solving problems showed the effectiveness of the proposed approach. It took no more than a hundred generations to achieve high precision solutions in the work of the genetic algorithm. Moreover, the error in solving the stationary heat conduction problem was so insignificant that this solution can be considered as precise. Thus, the study showed that the proposed approach, combining the analytical method of fictitious canonic regions and the use of genetic optimization algorithms, allows solving complex boundary-value problems with high accuracy. This approach can be used in mathematical modeling of structures for responsible purposes, where the accuracy and reliability of the results is the main criterion for evaluating the solution. Further development of this approach will make it possible to solve with high accuracy of more complicated 3D problems, as well as problems of other types, for example, thermal elasticity, which are of great importance in the design of engineering structures.

Implicit Methods for Numerical Solution of Singular Initial Value Problems

Abstract Various order of implicit method has been formulated for solving initial value problems having an initial singular point. The method provides better result than those obtained by used implicit formulae developed based on Euler and Runge-Kutta methods. Romberg scheme has been used for obtaining more accurate result.

Integration of carrier selection and supplier selection problem in humanitarian logistics

Carrier selection is a crucial issue in humanitarian relief operations, and the shortage of transportation vehicles can considerably affect relief distribution. In this paper, the significance of commodity and vehicle suppliers and guaranteeing prevention of shortage via on-time supplying of each relief commodity by the suppliers is emphasized by suggesting the integration of carrier selection and supplier selection for developing contractual agreements in humanitarian logistics. For this close relationship to be described, a scenario-based two-stage stochastic programming model for the establishment of joint decision-making in preparedness and response phases is developed so that a contract can be created on the quantity of reservation of both commodities and vehicles by the suppliers, as well as the transmission of the reserved commodities in the form of commodity purchase and vehicle rent in the response phase. An efficient two-step solution technique is suggested as a solution method for real scale problems, the effectiveness of which can be seen through the results. To illustrate the applicability of the model, a case study is suggested. The application of the model is discussed via numerical studies as well as a sensitivity analysis. The results have provided an appropriate managerial insight for responding to disaster for relief agencies.

A Variational Perturbative Approach to Planning in Graph-Based Markov Decision Processes

Coordinating multiple interacting agents to achieve a common goal is a difficult task with huge applicability. This problem remains hard to solve, even when limiting interactions to be mediated via a static interaction-graph. We present a novel approximate solution method for multi-agent Markov decision problems on graphs, based on variational perturbation theory. We adopt the strategy of planning via inference, which has been explored in various prior works. We employ a non-trivial extension of a novel high-order variational method that allows for approximate inference in large networks and has been shown to surpass the accuracy of existing variational methods. To compare our method to two state-of-the-art methods for multi-agent planning on graphs, we apply the method different standard GMDP problems. We show that in cases, where the goal is encoded as a non-local cost function, our method performs well, while state-of-the-art methods approach the performance of random guess. In a final experiment, we demonstrate that our method brings significant improvement for synchronization tasks.

Fast implicit solvers for phase-field fracture problems on heterogeneous microstructures

We study fast and memory-efficient FFT-based implicit solution methods for small-strain phase-field crack problems for microstructured brittle materials. A fully implicit first order formulation of the problem coupling elasticity and damage permits using comparatively few, but large, time steps compared to semi-explicit schemes. We investigate memory-efficient FFT-based solution techniques, and identify the heavy ball scheme as particularly powerful. We discuss the memory-efficient implementation and present demonstrative numerical examples.

Polarimetric neutron tomography of magnetic fields: uniqueness of solution and reconstruction

We consider the problem of determination of a magnetic field from three dimensional polarimetric neutron tomography data. We see that this is an example of a non-Abelian ray transform and that the problem has a globally unique solution for smooth magnetic fields with compact support, and a locally unique solution for less smooth fields. We derive the linearization of the problem and note that the derivative is injective. We go on to show that the linearised problem about a zero magnetic field reduces to plane Radon transforms and suggest a modified Newton–Kantarovich method (MNKM) for the numerical solution of the non-linear problem, in which the forward problem is re-solved but the same derivative is used each time. Numerical experiments demonstrate that MNKM works for small enough fields (or large enough velocities) and we show an example where it fails to reconstruct a slice of the simulated data set. Lastly we show that, viewed as an optimization problem, the inverse problem is non-convex so we expect gradient based methods may fail.

Numerical solution of singular boundary value problems using advanced Adomian decomposition method

In this work, we present a powerful method for the numerical solution of non-linear singular boundary value problems, namely the advanced Adomian decomposition method which is the modification of the Adomian decomposition method. In this method, we use all the boundary conditions to obtain the coefficient of the approximate series solution. Moreover, convergence analysis and an error bound for the approximate solution are discussed. This method overcomes the singular behaviour of problems and illustrates the approximations of high precision with a large effective region of convergence. To prove the robustness and effectiveness of the proposed method, various examples are considered and the obtained results are compared with the other existing methods.