Weak second-order stochastic Runge–Kutta methods for non-commutative stochastic differential equations

A new explicit stochastic Runge–Kutta scheme of weak order 2 is proposed for non-commutative stochastic differential equations (SDEs), which is derivative-free and which attains order 4 for ordinary differential equations. The scheme is directly applicable to Stratonovich SDEs and uses 2m-1$2m-1$ random variables for one step in the m-dimensional Wiener process case. It is compared with other derivative-free and weak second-order schemes in numerical experiments.     

Approximation of the soluticon of the stochastic navier-stokes equation ∗

We consider a stochastic equation of the Navier-Stokes type containing a noise part given by a stochastic Ito integral with respect to a Wiener process. The purpose of this paper is to approximate the solution of this nonlinear equation by the solutions of a sequence of linear equations with additive noise. We prove the convergence in mean squares     

Reciprocal diffusions and stochastic differential equations of second order ∗

We develop a theory of second order diffusion processes and associated stochastic differential equations of second order. We show that equations of evolution of the density, mean velocity and momentum flux are a family of first order conservation laws similar to those of continuum mechanics. We verify that the theory is satisfied for a large class of reciprocal Gaussian processes     

Asymptotic equivalence of linear stochastic Itô systems and oscillation of solutions of linear second-order equations

We obtain conditions for the asymptotic equivalence of linear stochastic and deterministic systems and analyze the oscillation of solutions of the Itô stochastic equation of the second order of the form \(\ddot x + (p(t) + q(t)\dot W(t))x = 0\) on the half-line.     

Designs of ϕ-optimal control for second-order processes

Consider a realization of the process on the intervalT=[0,1] for functionsf ₁(t),f ₂(t),...,f _n (t) inH(R), the reproducing kernel Hilbert space with reproducing kernelR(s,t) onT×T, whereR(s,t)=E[ξ(s)ξt)] is assumed to be continuous and known. Problems of the selection of functions {f _k (t)} _k=1 ⁿ to be ϕ-optimal design are given, and an unified approach to the solutions ofD-,A-,E- andD _s-optimal design problems are discussed.     

A note on stability of stochastic systems with unbounded perturbations ∗

In this note we consider the question of stabilizability on Hilbert space of systems governed by abstract stochastic differential equations containing randomly perturbed operators. It is shown that, under certain assumptions, a feedback control law can be found that makes the system exponentially stable in the mean square sense even in the presence of unbounded perturbations of the generator     

Multitime-scale singularly perturbed linear stochastic systems ∗

By applying diagonalization transformation, generalized variation of constants formula and theory of differential inequalities,the mean square convergence of solution process of a shingularly perturbed linear stochastic differential system of Itô-type is investigated. Moreover, slow and fast modes decomposition provides an auxiliary decoupled system whose solution processes are incorporated in approximating the solution processes of the original system     

A parallel version of a quasigradient methoid In stochastic control theory ∗

We solve an optimal control problem for controlled parabolic Ito equations by a stochastic quasigradient method. Because of high amounts of computation time required by numerical solution of such problems we investigate the parallelization of the algorithm. We distribute the computations of space stages over several processor nodes of a parallel computer. We obtain an efficient algorithm with low communication cost by using a ring topology     

Moreau–Yosida regularization of Lagrangian-dual functions for a class of convex optimization problems

In this paper, we consider the Lagrangian dual problem of a class of convex optimization problems, which originates from multi-stage stochastic convex nonlinear programs. We study the Moreau–Yosida regularization of the Lagrangian-dual function and prove that the regularized function η is piecewise C ², in addition to the known smoothness property. This property is then used to investigate the semismoothness of the gradient mapping of the regularized function. Finally, we show that the Clarke generalized Jacobian of the gradient mapping is BD-regular under some conditions.      

Singular limits for a parabolic–elliptic regularization of scalar conservation laws

We consider scalar hyperbolic conservation laws with a nonconvex flux, in one space dimension. Then, weak solutions of the associated initial value problems can contain undercompressive shock waves. We regularize the hyperbolic equation by a parabolic–elliptic system that produces undercompressive waves in the hyperbolic limit regime. Moreover we show that in another limit regime, called capillarity limit, we recover solutions of a diffusive–dispersive regularization, which is the standard regularization used to approximate undercompressive waves. In fact the new parabolic–elliptic system can be understood as a low-order approximation of the third-order diffusive–dispersive regularization, thus sharing some similarities with the relaxation approximations. A study of the traveling waves for the parabolic–elliptic system completes the paper.     

Self–regularization by projection for noisy pseudodifferential equations of negative order

It is well known, that pseudodifferential equations of negative order considered in Sobolev spaces with small smoothness indices are ill–posed. On the other hand, it is known that efficient discretization schemes with properly chosen discretization parameters allow to obtain a regularization effect for such equations. The main accomplishment of the present paper is the principle for the adaptive choice of the discretization parameters directly from noisy discrete data. We argue that the combination of this principle with wavelet–based matrix compression techniques leads to algorithms which are order–optimal in the sense of complexity.     

Schreier theory for singular extensions of categorical groups and homotopy classification∗ ∗

If G is a categorical group, a G-module is defined to be a braided categorical group (A c) together with an action of G on (A,c). In this work we define the notions of singular extension of G by the G-module (A,c) and of 1-cocycle of G with coefficients in (A,c) and we obtain, first, a bijection between the set of equivalence classes of singular extensions of G by (Ac) and the set of equivalence classes of 1-cocycles. Next, we associate to any G-module (Ac) a Kan fibration of simplicial sets ?: Ner(GAc)) → Ner(G)and then we show that there is a bijection between the set of equivalence classes of singular extensions of G by (A,c) and Γ[Ner(G,A,c/Ner(G)]the set of fibre homotopy classes of cross-sections of the fibration ?.     

Anticipated mean-field backward stochastic differential equations with jumps∗

Lithuanian Mathematical Journal - In this paper, we prove an existence and uniqueness theorem and a comparison theorem for a class of anticipated mean-field backward stochastic differential...     

Existence of periodic and subharmonic solutions for second-order superlinear difference equations

By critical point theory, a new approach is provided to study the existence and multiplicity results of periodic and subharmonic solutions for difference equations. For secord-order difference equations△2xn-1+f(n,xn)=0some new results are obtained for the above problems when f(t, z) has superlinear growth at zero and at infinityin z.     

Analysis of a Linear Sampling Method for Identification of Obstacles

Some difficulties are pointed out in the methods for identification of obstacles based on the numerical verification of tile inclusion of a function in the range of an operator. Numerical examples are given to illustrate theoretical conclusions. Alternative methods of identification of obstacles are mentioned： the Support Function Method （SFM） and the Modified Rayleigh Conjecture （MRC） method.     

Weak convergence of approximate solutions of stochastic equations with applications to random differential and integral equations ∗

In this paper, we considerably extend our earlier result about convergence in distribution of approximate solutions: of random operator equations, where the stochastic inputs and the underlying deterministic equation are simultaneously approximated. As a by-product, we obtain convergence results for approximate solutions of equations between spaces of probability measures. We apply our results to random Fredholm integral equations of the second kind and to a random [nbar]onlinear elliptic boundary value problem.     

The kernel of the bezoutian for operator Polynomials ∗

Based on two set partitions of the symmetric group S_n expansion theorems by diagonal elements for the permanent and the determinant are derived, for both the generic commuting and noncommuting cases. They are of the same type as the well-known Laplace expansions where either fixed rows or columns of a given matrix are chosen instead of diagonal elements.     

Strong convergence of a regularization method for Rockafellar’s proximal point algorithm

In this paper, for a monotone operator T, we shall show strong convergence of the regularization method for Rockafellar’s proximal point algorithm under more relaxed conditions on the sequences {r _k } and {t _k }, $$\lim\limits_{k\to\infty}t_k = 0;\quad \sum\limits_{k=0}^{+\infty}t_k = \infty;\quad\ \liminf\limits_{k\to\infty}r_k > 0.$$ Our results unify and improve some existing results.