Convergence analysis of a proximal newton method 1

Recently Fukushima and Qi proposed a proximal Newton method for minimizating a nonsmooth convex function. An alternative global convergence proof for that method is presented in this paper. Global convergence was established without any additional assumption on the objective function. We also show that the infimum of a convex function is always equal to the infimun of its Moreau—Yosida regularization     

Convergence to periodic solutions

Open Problems and Conjectures Edited by Gerry Ladas In this section we present some open problems and conjectures about some interesting types of difference equations. Please submit your problems and conjectures with all relevant information to G. Ladas     

Convergence of solutions of stochastic differential equations to the Arratia flow

We consider the solution x _ε of the equation

Convergence to self-similar solutions for a coagulation equation

Convergence to self-similar profiles is shown for solutions to the Oort-Hulst-Safronov coagulation equation with constant coagulation kernel. A dynamical systems approach is used on the equation written in self-similar variables, for which two Liapunov functionals are identified. For initial data decaying sufficiently rapidly at infinity, decay rates are also obtained.Received: October 17, 2002     

Convergence of numerical solutions to stochastic age-dependent population equations

In this paper, stochastic age-dependent population equations, one of the important classes of hybrid systems, are studied. In general, most of stochastic age-dependent population equations do not have explicit solutions, thus numerical approximation schemes are invaluable tools for exploring their properties. The main purpose of this paper is to develop a numerical scheme and show the convergence of the numerical approximation solution to the true solution.     

Convergence rate of numerical solutions to SFDEs with jumps

In this paper, we are interested in numerical solutions of stochastic functional differential equations with jumps. Under a global Lipschitz condition, we show that the pth-moment convergence of Euler-Maruyama numerical solutions to stochastic functional differential equations with jumps has order 1/p for any p≥2. This is significantly different from the case of stochastic functional differential equations without jumps, where the order is 1/2 for any p≥2. It is therefore best to use the mean-square convergence for stochastic functional differential equations with jumps. Moreover, under a local Lipschitz condition, we reveal that the order of mean-square convergence is close to 1/2, provided that local Lipschitz constants, valid on balls of radius j, do not grow faster than logj.     

Convergence of difference solutions to generalized solutions of the Dirichlet problem for a second-order elliptic equation

Exact difference scheme operators are applied to construct a difference scheme for the Dirichlet problem for a secondorder elliptic equation with variable coefficients in a rectangle. If the solution belongs to the class W ₂ ² (), the scheme is of first-order accuracy in the grid norm of W ₂ ¹ ().Translated from Vychislitel'naya i Prikladnaya Matematika, No. 57, pp. 21–26, 1985.     

Convergence to steady states of solutions to nonlinear integral evolution equations

We prove that any global bounded solution of the nonlinear evolutionary integral equation $$\dot{u}(t) + \int\limits_0^t a(t-s)\mathcal{E}'(u(s))ds =f(t), \quad t >0 $$ tends to a single equilibrium state for long time (i.e., ${\mathcal{E}'(\vartheta)=0}$ where ${\vartheta= \lim_{t \rightarrow \infty} u(t)}$ on a real Hilbert space), where ${\mathcal{E}'}$ is the Fréchet derivative of a functional ${\mathcal{E}}$ , which satisfies the ?ojasiewicz?CSimon inequality near ${\vartheta}$ . The vector-valued function f and the scalar kernel a satisfy suitable conditions.     

Convergence to steady state of solutions of the euler equations,I

In this paper we analyze the convergence to steady state of solutions of the compressible and the incompressible isentropic Euler equations in two space dimensions. In the compressible case, the original equations do not converge. We replace the equation of continuity with an elliptic equation for the density, obtaining a new set of equations, which have the same steady solution. In the incompressible case, the equation of continuity is replaced by a Poisson equation for the pressure. In both cases, we linearize the equations around a steady solution and show that the unsteady solution of the linearized equations converges to the steady solution, if the steady solution is sufficiently smooth. In the proof we consider how the energy of the time dependent part developes with time, and find that it decrease exponentially.     

Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line

We study a large time behavior of a solution to the initial boundary value problem for an isentropic and compressible viscous fluid in a one-dimensional half space. The unique existence and the asymptotic stability of a stationary solution are proved by S. Kawashima, S. Nishibata and P. Zhu for an outflow problem where the fluid blows out through the boundary. The main concern of the present paper is to investigate a convergence rate of a solution toward the stationary solution. For the supersonic flow at spatial infinity, we obtain an algebraic or an exponential decay rate. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in the spatial asymptotic point, the solution converges to the corresponding stationary solution with the same rate in time as time tends to infinity. An algebraic convergence rate is also obtained for the transonic flow. These results are proved by the weighted energy method.     

Convergence of solutions of von Karman evolution equations to equilibria

We consider von Karman evolution equations with nonlinear interior dissipation and with clamped boundary conditions. Under some conditions we prove that every energy solution converges to a stationary solution and establish a rate of convergence. Earlier this result was known in the case when the set of equilibria was finite and hyperbolic. In our argument we use the fact that the von Karman nonlinearity is analytic on an appropriate space and apply the Lojasiewicz–Simon method in the form suggested by A. Haraux and M. Jendoubi.     

Random A-permutations: Convergence to a Poisson process

Suppose that S _n is the permutation group of degree n, A is a subset of the set of natural numbers ?, and T _n(A) is the set of all permutations from S _n whose cycle lengths belong to the set A. Permutations from T _n are usually called A-permutations. We consider a wide class of sets A of positive asymptotic density. Suppose that ζ _mn is the number of cycles of length m of a random permutation uniformly distributed on T _n. It is shown in this paper that the finite-dimensional distributions of the random process {tz _mn, m ε A} weakly converge as n → ∞ to the finite-dimensional distributions of a Poisson process on A.     

Convergence of solutions to nonlinear dispersive equations without convexity conditions

This paper gets a series of results about the convergence of solutions {u_δ ^c} for partial differential equations of the form u_t + f_x(u) + δu_χχχ ≡ εu_χχ and u_t + f_χ(u) + δu_χχχ ≡ εu_χχ as ε and δ approach zero. Where the flux functions need no convexity conditions     

Convergence rate of approximate solutions to weakly coupled nonlinear systems

We study the convergence rate of approximate solutions to nonlinear hyperbolic systems which are weakly coupled through linear source terms. Such weakly coupled systems appear, for example, in the context of resonant waves in gas dynamics equations.

This work is an extension of our previous scalar analysis. This analysis asserts that a One Sided Lipschitz Condition (OSLC, or -stability) together with -consistency imply convergence to the unique entropy solution. Moreover, it provides sharp convergence rate estimates, both global (quantified in terms of the -norms) and local.

We focus our attention on the -stability of the viscosity regularization associated with such weakly coupled systems. We derive sufficient conditions, interesting for their own sake, under which the viscosity (and hence the entropy) solutions are -stable in an appropriate sense. Equipped with this, we may apply the abovementioned convergence rate analysis to approximate solutions that share this type of -stability.

     

Convergence conditions for the Seidel process

The convergence condition of the successive approximation process based on the Seidel method is derived for a system of two transcendental equations with allowance for specific functional dependences.Ukrainian External Polytechnical Institute, Khar'kov. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 74, pp. 19–20, 1992;     

The projected newton method for solving inverse eigenvalue problems - the case of multiple eigenvaues

This paper deals with the optimal solution of ill-posed linear problems, i.e..linear problems for which the solution operator is unbounded. We consider worst-case ar,and averagecase settings. Our main result is that algorithms having finite error (for a given setting) exist if and only if the solution operator is bounded (in that setting). In the worst-case setting, this means that there is no algorithm for solving ill-posed problems having finite error. In the average-case setting, this means that algorithms having finite error exist if and only lf the solution operator is bounded on the average. If the solution operator is bounded on the average, we find average-case optimal information of cardinality n and optimal algorithms using this information, and show that the average error of these algorithms tends to zero as n→∞. These results are then used to determine the [euro]-complexity, i.e., the minimal costof finding an [euro]-accurate approximation. In the worst-case setting, the [euro]comp1exity of an illposed problem is infinite for all [euro]>0; that is, we cannot find an approximation having finite error and finite cost. In the average-case setting, the [euro]-complexity of an ill-posed problem is infinite for all [euro]>0 iff the solution operator is not bounded on the average, moreover, if the the solutionoperator is bounded on the average, then the [euro]-complexity is finite for all [euro]>0.     

Convergence analysis of a minimax method for finding multiple solutions of hemivariational inequality in Hilbert space

In 2013, a minimax method for finding saddle points of locally Lipschitz continuous functional was designed (Yao Math. Comp. 82 2087–2136 2013). The method can be applied to numerically solve hemivariational inequality for multiple solutions. Its subsequence and sequence convergence results in functional analysis were established in the same paper. But, since these convergence results do not consider discretization, they are not convergence results in numerical analysis. In this paper, we point out what approximation problem is, when this minimax method is used to solve hemivariational inequality and the finite element method is used in discretization. Computation of the approximation problem is discussed, numerical experiment is carried out and its global convergence is verified. Finally, as element size goes to zero, convergence of solutions of the approximation problem to solutions of hemivariational inequality is proved.