Sequence Of Approximate Solutions Research Articles

Ill-posed Boundary-value Problem for a System of Partial Differential Equations with Two Degenerate Lines

This paper is devoted to the investigation of ill-posed boundary-value problem for system of parabolic type equations with changing time direction with two degenerate lines. The problem under consideration is ill-posed in the sense of J. Hadamard, namely, there is no continuous dependence of the solution on the initial data. Such equations have many different applications, for example, describe the processes of heat propagation in inhomogeneous media, the interaction of filtration flows, mass transfer near the surface of an aircraft, and the description of complex viscous fluid flows. As possible applications should also indicate the task of calculating heat exchangers, in which the counter flow principle is used. Theorems on the uniqueness and conditional stability of a solution on a set of well-posedness are proved. We construct a sequence of approximate (regularized) solutions that are stable on the set of well-posedness

An iterative method to compute minimum norm solutions of ill-posed problems in Hilbert spaces

We study an algorithm to compute minimum norm solution of ill-posed problems in Hilbert spaces and investigate its regularizing properties with discrepancy principle stopping rule. This algorithm results from straightly applying the LSQR method to the main problem before discretizing. In fact, the proposed algorithm obtains a sequence of approximate solutions of the original problem. In order to test the new algorithm, it is implemented to solve system of linear integral equations of the first kind and some examples are given. Moreover, we compare the presented algorithm with the Tikhonov regularization method to compute the least norm solution when there are more than one solution.

Proximal bundle methods based on approximate subgradients for solving Lagrangian duals of minimax fractional programs

We are interested in this work to solve the Lagrangian dual of a generalized fractional program (GFP), which gives the minimal value of the primal problem, thanks to some duality results. With the help of a general minimax equality assumption, we give duality results under minimal assumptions. Since the associated parametric programs of the dual of GFP are always concave, we use a general approximating proximal scheme to these subproblems and construct bundle methods by means of approximate values and approximate subgradients of the dual parametric function. As for all dual algorithms, the proposed methods generate a sequence of values that converges from below to the minimal value of the GFP and a sequence of approximate solutions that converges to a solution of the Lagrangian dual of the GFP. For certain classes of problems, the convergence is at least linear.

Approximate Solution and its Convergence Analysis for Hypersingular Integral Equations

This paper propose a residual based Galerkin method with Legendre polynomial as a basis functions to find the approximate solution of hypersingular integral equations. These equations occur quite naturally in the field of aeronautics such as problem of aerodynamics of flight vehicles and during mathematical modeling of vortex wakes behind aircraft. The analytic solution of these kind of equations is known only for a particular case ( m(x,t) =0 in Eqn (1)). Also, in these singular integral equations which occur during the formulation of many boundary value problems, the known function m(x,t) in (Eqn (1)) is not always zero. Our proposed method find the approximate solution by converting the integral equations into a linear system of algebraic equations which is easy to solve. The convergence of sequence of approximate solutions is proved and error bound is obtained theoretically. The validation of derived theoretical results and implementation of method is also shown with the aid of numerical illustrations.

Simple non-perturbative resummation schemes beyond mean-field: case study for scalar \u03d54 theory in 1+1 dimensions

I present a sequence of non-perturbative approximate solutions for scalar ϕ4 theory for arbitrary interaction strength, which contains, but allows to systematically improve on, the familiar mean-field approximation. This sequence of approximate solutions is apparently well-behaved and numerically simple to calculate since it only requires the evaluation of (nested) one-loop integrals. To test this resummation scheme, the case of ϕ4 theory in 1+1 dimensions is considered, finding approximate agreement with known results for the vacuum energy and mass gap up to the critical point. Because it can be generalized to other dimensions, fermionic fields and finite temperature, the resummation scheme could potentially become a useful tool for calculating non-perturbative properties approximately in certain quantum field theories.

The zero relaxation limit for the Aw–Rascle–Zhang traffic flow model

We study the behavior of the Aw–Rascle–Zhang model when the relaxation parameter converges to zero. In a Lagrangian setting, we use the wavefront tracking method with splitting technique to construct a sequence of approximate solutions. We prove that this sequence converges to a weak entropy solution of the relaxed system associated to a given initial datum with bounded variation. Besides, we also provide an estimate on the decay of positive waves. We finally prove that the solutions of the Aw–Rascle–Zhang system with relaxation converge to a weak solution of the corresponding scalar conservation law when the relaxation parameter goes to zero.

О применении одного класса параметрических функций в качестве внешних штрафов при решении нелинейных задач с ограничениями

Kaplan’s one-parametric class of functions for solving nonlinear convex minimization problems with one-sided constraints is studied. The set defined by constraints is assumed to have a nonempty interior. In the monographs written by A. Fiacco, G. McCormick, E. Polak, and A. Kaplan, the penalty methods theory and classification of penalty functions are presented quite systematically. With these methods and approaches as backbones, in the present article, we establish that the class of functions under study belongs to the class of exterior penalty functions for problems of convex programming. Application of penalty methods to the solution of nonlinear extremal problems with constraints allows using the toolbox of unconstrained nonlinear optimization, including gradient methods. Kaplan’s penalty functions have fine differential properties. Therefore, they are suitable for use in iterative gradient methods of approximate solution of unconstrained extremal problems. After this, we prove the theorem on convergence of the sequence of approximate solutions of penalized unconstrained extremal problems to the exact solution of the original problem with constraints. As well, we establish a bound on the convergence rate for the penalty method with the one-parametric class of functions serving as penalty functions. This bound is derived provided that the exact resolving of the sequence of unconstrained extremal problems is fulfilled. With the help of these results, one may proceed further with a numerical analysis of the class of problems that are under discussion in the article.

Convergence of the backward Euler scheme for the operator-valued Riccati differential equation with semi-definite data

For initial value problems associated with operator-valued Riccati differential equations posed in the space of Hilbert–Schmidt operators existence of solutions is studied. An existence result known for algebraic Riccati equations is generalized and used to obtain the existence of a solution to the approximation of the problem via a backward Euler scheme. Weak and strong convergence of the sequence of approximate solutions is established permitting a large class of right-hand sides and initial data.

Positive control volume finite element scheme for a degenerate compressible two-phase flow in anisotropic porous media

In this paper, we are concerned with the convergence analysis of a positive control volume finite element scheme (CVFE) for a degenerate compressible two-phase flow model in anisotropic porous media. For this, we consider the global pressure saturation formulation. We next use an implicit Euler scheme in time and a CVFE discretization in space. This approach rests on a particular choice of the mean value of the gas density on the interfaces, a centered scheme of the total mobility, and the upwind approximation of fractional fluxes according to the total velocity. Thus, the maximum principle is fulfilled without any constraint on the stiffness coefficients. Moreover, uniform estimates on the discrete gradient of the global pressure and the dissipative term are derived. As the mesh size is sent to zero, we establish that the sequence of approximate solutions converges to a weak solution of the continuous problem. Numerical tests are presented in two dimensions to exhibit the behavior of the gas pressure and the water saturation through the medium.

On a doubly nonlinear PDE with stochastic perturbation

We consider a doubly nonlinear evolution equation with multiplicative noise and show existence and pathwise uniqueness of a strong solution. Using a semi-implicit time discretization we get approximate solutions with monotonicity arguments. We establish a-priori estimates for the approximate solutions and show tightness of the sequence of image measures induced by the sequence of approximate solutions. As a consequence of the theorems of Prokhorov and Skorokhod we get a.s. convergence of a subsequence on a new probability space which allows to show the existence of martingale solutions. Pathwise uniqueness is obtained by an $$L^1$$ -method. Using this result, we are able to show existence and uniqueness of strong solutions.

МЕТОД ОПРЕДЕЛЕНИЯ АПОСТЕРИОРНЫХ ОЦЕНОК ПОГРЕШНОСТЕЙ В РАСЧЕТАХ КОМПОЗИТНЫХ ОБОЛОЧЕК С ПРИМЕНЕНИЕМ МНОГОСЕТОЧНЫХ КОНЕЧНЫХ ЭЛЕМЕНТОВ

A numerical method of determining a posteriori error estimates of solutions created for composite cylindrical shells using multigrid finite elements (MFE) has been proposed. The suggested method is based on ZZ method proposed by O. C. Zienkiewicz and J. Z. Zhu for both energy norm and L 2 norm of solution errors estimates. In contrast to ZZ method, the suggested method uses MFE that takes into account complex shapes, heterogeneous and micro-heterogeneous body structures and forms small dimension discrete models for creating «precise» solutions. To give examples there was carried out analysis of error estimates for displacements and stresses in calculation of stress-strain state (SSS) of three-layer cylindrical shells with and without cutouts under local loading. It has been stated that analysis of SSS using MFE causes converging sequences of approximate solutions in norm L 2. Calculations that use mean square error for stresses in each finite element of the shell show that MFE allow to use arbitrarily small regular discretization grids all over the shell area without the necessity to tighten the grid in local areas for calculating SSS. This leads to simple algorithms of calculating SSS with the help of MFE and ensures considerable saving of computer resources. In the given examples the use of MFE decreases the dimension of the system of MFE algebraic equations and reduces computer memory volume by 1 500 and 8∙104 times respectively, compared to the finite elements base model that doesn’t use MFE.

Uniqueness of solutions of boundary value problems at resonance

In this paper, the method of upper and lower solutions is employed to obtain uniqueness of solutions for a boundary value problem at resonance. The shift method is applied to show the existence of solutions. A monotone iteration scheme is developed and sequences of approximate solutions are constructed that converge monotonically to the unique solution of the boundary value problem at resonance. Two examples are provided in which explicit upper and lower solutions are exhibited.

A Smoothing Approach for Minimizing A Linear Function Subject to Fuzzy Relation Inequalities with Addition–Min Composition

This paper mainly focuses on minimizing a linear function subject to fuzzy relation inequalities with addition–min composition. Although the problem has been proved to be equivalent to a linear programming, it is still difficult to efficiently solve when the numbers of constrains and variables come to about 200. In this paper, we devotes to constructing a smoothing approach for solving approximate solutions of the problem. Utilizing maximum entropy method, we approximate the constraints by continuously differentiable functions and prove that any cluster of an approximate solution sequence is an optimal point of the original problem. Numerical experiments show that the error of the approximate solutions is within a reasonable range. At the same time, compared to the linear programming approach, the smoothing approach costs much less computation time, especially for large-scale problems.

Dual algorithms based on the proximal bundle method for solving convex minimax fractional programs

In this work, we propose an approximating scheme based on the proximal point algorithm, for solving generalized fractional programs (GFP) by their continuous reformulation, also known to as partial dual counterparts of GFP. Bundle dual algorithms are then derived from this scheme. We prove the convergence and the rate of convergence of these algorithms. As for dual algorithms, the proposed methods generate a sequence of values that converges from below to the minimal value of \begin{document}$(P)$\end{document} , and a sequence of approximate solutions that converges to a solution of the dual problem. For certain classes of problems, the convergence is at least linear.

A New Smoothing Method for Mathematical Programs with Complementarity Constraints Based on Logarithm-Exponential Function

We present a new smoothing method based on a logarithm-exponential function for mathematical program with complementarity constraints (MPCC). Different from the existing smoothing methods available in the literature, we construct an approximate smooth problem of MPCC by partly smoothing the complementarity constraints. With this new method, it is proved that the Mangasarian-Fromovitz constraint qualification holds for the approximate smooth problem. Convergence of the approximate solution sequence, generated by solving a series of smooth perturbed subproblems, is investigated. Under the weaker constraint qualification MPCC-Cone-Continuity Property, it is proved that any accumulation point of the approximate solution sequence is a M-stationary point of the original MPCC. Preliminary numerical results indicate that the developed algorithm based on the partly smoothing method is efficient, particularly in comparison with the other similar ones.

Non-local multi-class traffic flow models

We prove the existence for small times of weak solutions for a class of non-local systems in one space dimension, arising in traffic modeling. We approximate the problem by a Godunov type numerical scheme and we provide uniform L ∞ and BV estimates for the sequence of approximate solutions. We finally present some numerical simulations illustrating the behavior of different classes of vehicles and we analyze two cost functionals measuring the dependence of congestion on traffic composition.

Existence of global solutions in a model of electrical activity of the monodomain type for a ventricle

Introduction: A monodomain model of electrical activity for an isolated ventricle is formulated. This model is written as a reaction diffusion PDE coupled to an ODE, The Rogers-Mculloch model is used to represent the electrical activity through the cell membrane. Method: We give a definition of weak and strong solution of the variational Cauchy problem associated to the monodomain model. A sequence of approximate solutions of Faedo-Galerkin type is proposed.Results: It is shown that the sequence of approximate solutions converge to a weak solution according to the proposed definition. Finally, we have that this weak solution is also a strong solution. Conclusion: The monodomain model of electrical activity in an isolated ventricle that is proposed has a weak solution in an appropriate sense. In addition, this weak solution is also a strong solution.

Existence of BV solutions for a non-conservative constrained Aw–Rascle–Zhang model for vehicular traffic

The main aim of this paper is to study the Aw–Rascle–Zhang (ARZ) model with non-conservative local point constraint on the density flux introduced in [10], its motivation being, for instance, the modeling of traffic across a toll gate. We prove the existence of weak solutions under assumptions that result to be more general than those required in [11]. More precisely, we do not require that the waves of the first characteristic family have strictly negative speeds of propagation. The result is achieved by showing the convergence of a sequence of approximate solutions constructed via the wave-front tracking algorithm. The case of solutions attaining values at the vacuum is considered. We also present an explicit numerical example to describe some qualitative features of the solutions.

An energy method for rough partial differential equations

We present a well-posedness and stability result for a class of nondegenerate linear parabolic equations driven by geometric rough paths. More precisely, we introduce a notion of weak solution that satisfies an intrinsic formulation of the equation in a suitable Sobolev space of negative order. Weak solutions are then shown to satisfy the corresponding energy estimates which are deduced directly from the equation. Existence is obtained by showing compactness of a suitable sequence of approximate solutions whereas uniqueness relies on a doubling of variables argument and a careful analysis of the passage to the diagonal. Our result is optimal in the sense that the assumptions on the deterministic part of the equation as well as the initial condition are the same as in the classical PDEs theory.

About the estimation of the convergence rate of projection-iteration processes of conditional minimization of a functional

We study projection-iterative processes based on the conditional gradient method to solve the problem of minimizing a functional in a real separable Hilbert space. To solve extremal problems, methods of approximate (projection) type are often used, which make it possible to replace the initial problem by a sequence of auxiliary approximating extremal problems. The work of many authors is devoted to the problems of approximating various classes of extremal problems. Investigations of projection and projection-iteration methods for solving extremal problems with constraints in Hilbert and reflexive Banach spaces were carried out, in particular, in the works of S.D. Balashova, in which the general conditions for approximation and convergence of sequences of exact and approximate solutions of approximating extremal problems considered both in subspaces of the original space and in certain spaces isomorphic to them were proposed. The projection-iterative approach to the approximate solution of an extremal problem is based on the possibility of applying iterative methods to the solution of approximating problems. Moreover, for each of the "approximate" extremal problems, only a few approximations are obtained with the help of a certain iteration method and the last of them as the initial approximation for the next "approximate" problem is used. This paper, in continuation of the author's past work to solve the problem of minimizing a functional on a convex set of Hilbert space, is devoted to obtaining theoretical estimates of the rate of convergence of the projection-iteration method based on the conditional gradient method (for different ways of specifying a step multiplier) of minimization of approximating functionals in certain spaces isomorphic to subspaces of the original space. We prove theorems on the convergence of a projection-iteration method and obtain estimates of error and convergence degree