Asymptotic behavior of degenerate linear transport equations

We study in this paper a few simple examples of hypocoercive systems in which the coercive part is degenerate. We prove that the (completely explicit) speed of convergence is at least of inverse power type (the power depending on the features of the considered system).     

Stabilized spectral element approximation for the Navier–Stokes equations

The conforming spectral element methods are applied to solve the linearized Navier–Stokes equations by the help of stabilization techniques like those applied for finite elements. The stability and convergence analysis is carried out and essential numerical results are presented demonstrating the high accuracy of the method as well as its robustness. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 115–141, 1998     

Adaptive stochastic weak approximation of degenerate parabolic equations of Kolmogorov type

Degenerate parabolic equations of Kolmogorov type occur in many areas of analysis and applied mathematics. In their simplest form these equations were introduced by Kolmogorov in 1934 to describe the probability density of the positions and velocities of particles but the equations are also used as prototypes for evolution equations arising in the kinetic theory of gases. More recently equations of Kolmogorov type have also turned out to be relevant in option pricing in the setting of certain models for stochastic volatility and in the pricing of Asian options. The purpose of this paper is to numerically solve the Cauchy problem, for a general class of second order degenerate parabolic differential operators of Kolmogorov type with variable coefficients, using a posteriori error estimates and an algorithm for adaptive weak approximation of stochastic differential equations. Furthermore, we show how to apply these results in the context of mathematical finance and option pricing. The approach outlined in this paper circumvents many of the problems confronted by any deterministic approach based on, for example, a finite-difference discretization of the partial differential equation in itself. These problems are caused by the fact that the natural setting for degenerate parabolic differential operators of Kolmogorov type is that of a Lie group much more involved than the standard Euclidean Lie group of translations, the latter being relevant in the case of uniformly elliptic parabolic operators.     

Solutions to degenerate complex Hessian equations

Let (X,ω)$(X,\omega )$ be an n-dimensional compact Kähler manifold and fix an integer m   such that 1?m?n$1?m?n$. We study degenerate complex Hessian equations of the form (ω+dd^cφ)^m∧ω^n−m=F(x,φ)ωⁿ${(\omega +d{d}^c\phi )}^m\wedge {\omega }^{n-m}=F(x,\phi ){\omega }^n$. Under some natural conditions on F, this equation has a unique continuous solution. When X is homogeneous and ω is invariant under the Lie group action, we further show that the solution is Hölder continuous.     

A numerical approach to degenerate parabolic equations

Summary. In this work we propose a numerical approach to solve some kind of degenerate parabolic equations. The underlying idea is based on the maximum principle. More precisely, we locally perturb the (initial and boundary) data instead of the nonlinear diffusion coefficients, so that the resulting problem is not degenerate. The efficiency of this method is shown analytically as well as numerically. The numerical experiments show that this new approach is comparable with the existing ones. Received January 20, 1999 / Revised version received February 28, 2000 / Published online July 25, 2001     

Weak solutions to the stochastic porous media equation via Kolmogorov equations: The degenerate case

A stochastic version of the porous medium equation with coloured noise is studied. The corresponding Kolmogorov equation is solved in the space L²(H,ν) where ν is an infinitesimally excessive measure. Then a weak solution is constructed.     

Deterministic particle approximation for nonlocal transport equations with nonlinear mobility

We construct a deterministic, Lagrangian many-particle approximation to a class of nonlocal transport PDEs with nonlinear mobility arising in many contexts in biology and social sciences. The approximating particle system is a nonlocal version of the follow-the-leader scheme. We rigorously prove that a suitable discrete piece-wise density reconstructed from the particle scheme converges strongly in $L_{loc}^1$ towards the unique entropy solution to the target PDE as the number of particles tends to infinity. The proof is based on uniform BV estimates on the approximating sequence and on the verification of an approximated version of the entropy condition for large number of particles. As part of the proof, we also prove uniqueness of entropy solutions. We further provide a specific example of non-uniqueness of weak solutions and discuss the interplay of the entropy condition with the steady states. Finally, we produce numerical simulations supporting the need of a concept of entropy solution in order to get a well-posed semigroup in the continuum limit, and showing the behaviour of solutions for large times.     

A relaxation approximation for transport equations in the diffusive limit

A stable relaxation approximation for a transport equation with the diffusive scaling is developed. The relaxation approximation leads in the small mean free path limit to the higher-order diffusion equation obtained from the asymptotic analysis of the transport equation.     

求解阵列天线自适应滤波问题的一种调比随机逼近算法

提出了求解阵列天线自适应滤波问题的一种调比随机逼近算法.每一步迭代中,算法选取调比的带噪负梯度方向作为新的迭代方向.相比已有的其他随机逼近算法,这个算法不需要调整稳定性常数,在一定程度上解决了稳定性常数选取难的问题.数值仿真实验表明,算法优于已有的滤波算法,且比经典Robbins-Monro （RM）算法具有更好的稳定性.     

Critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux

We establish the critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux and then determine the blow-up rates and the blow-up sets for the nonglobal solutions.     

Persistence of bounded solutions to degenerate Sobolev-Galpern equations

In this paper the persistence of bounded solutions to degenerate evolution equations of Sobolev-Galpern type is discussed. In order to define the evolution operator well, we study the existence and uniqueness of solutions to its linear form. On this basis we discuss exponential dichotomies of the evolution operator and give the Fredholm alternative result for bounded solutions of nonhomogeneous linear degenerate equations. This result enables us to give a condition for the persistence of bounded solutions of a general degenerate nonlinear autonomous equation under a nonautonomous perturbation.     

Asymptotic approximation of degenerate fiber integrals

We study asymptotics of fiber integrals depending on a large parameter. When the critical fiber is singular, full-asymptotic expansions are established in two different cases: local extremum and isolated real principal type singularities. The main coefficients are computed and invariantly expressed. In the most singular cases, it is shown that the leading term of the expansion is related to invariant measures on the spherical blow-up of the singularity. The results can be applied to certain degenerate oscillatory integrals which occur in spectral analysis and quantum mechanics.     

Rational approximation to the Thomas-Fermi equations

We show that a simple and straightforward rational approximation to the Thomas-Fermi equation provides the slope at origin with unprecedented accuracy and that Padé approximants of relatively low order are far more accurate than more elaborate approaches proposed recently by other authors. We consider both the Thomas-Fermi equation for isolated atoms and for atoms in strong magnetic fields.     

Homogenization of degenerate elliptic equations

We consider the divergent elliptic equations whose weight function and its inverse are assumed locally integrable. The equations of this type exhibit the Lavrentiev phenomenon, the nonuniqueness of weak solutions, as well as other surprising consequences. We classify the weak solutions of degenerate elliptic equations and show the attainability of the so-called W-solutions. Investigating the homogenization of arbitrary attainable solutions, we find their different asymptotic behavior. Under the assumption of the higher integrability of the weight function we estimate the difference between the exact solution and certain special approximations.     

Implicit Milstein method for stochastic differential equations via the Wong-Zakai approximation

The aim of this paper is to derive a numerical scheme for solving stochastic differential equations (SDEs) via Wong-Zakai approximation. One of the most important methods for solving SDEs is Milstein method, but this method is not so popular because the cost of simulating the double stochastic integrals is high. For overcoming this complexity, we present an implicit Milstein scheme based on Wong-Zakai approximation by approximating the Brownian motion with its truncated Haar expansion. The main advantages of this method lie in the fact that it preserves the convergence order and also stability region of the Milstein method while its simulation is much easier than Milstein scheme. We show the convergence rate of the method by some numerical examples.     

On the continuity of solutions to degenerate elliptic equations

The local behavior of solutions to a degenerate elliptic equation

     

Relaxation approximation to bed-load sediment transport

In this work we propose and apply a numerical method based on finite volume relaxation approximation for computing the bed-load sediment transport in shallow water flows, in one and two space dimensions. The water flow is modeled by the well-known nonlinear shallow water equations which are coupled with a bed updating equation. Using a relaxation approximation, the nonlinear set of equations (and for two different formulations) is transformed to a semilinear diagonalizable problem with linear characteristic variables. A second order MUSCL-TVD method is used for the advection stage while an implicit–explicit Runge–Kutta scheme solves the relaxation stage. The main advantages of this approach are that neither Riemann problem solvers nor nonlinear iterations are required during the solution process. For the two different formulations, the applicability and effectiveness of the presented scheme is verified by comparing numerical results obtained for several benchmark test problems.