self similar solutions to a density dependent reaction diffusion model in this paper we investigated a density dependent reaction diffusion equation u t u m xx u u m this equation is known as the extension of the fisher or kolmogoroff petrovsky piscounoff equation which is widely used in population dynamics combustion theory and plasma physics by employing a suitable transformation this equation was mapped to the anomalous diffusion equation where the nonlinear reaction term was eliminated due to its simpler form some exact self similar solutions with compact support have been obtained the solutions evolving from an initial state converge to the usual traveling wave at a certain transition time hence the connection between the self similar solution and the traveling wave solution is quite clear from these results moreover the solutions were found in a manner that propagates either to the right or to the left furthermore the two solutions form a symmetric solution expanding in both directions applications to spatiotemporal pattern formation in biological populations is discussed

a generalized proximal point algorithm and its convergence rate we propose a generalized proximal point algorithm ppa in the generic setting of finding a root of a maximal monotone operator in addition to the classical ppa a number of benchmark operator splitting methods in the pde and optimization literatures can be retrieved by this generalized ppa scheme we establish the convergence rate of this generalized ppa scheme under different conditions including estimating its worst case convergence rate measured by the iteration complexity under mild assumptions and deriving its linear convergence rate under certain stronger conditions throughout our discussion we pay particular attention to the special case where the operator is the sum of two maximal monotone operators and specify our theoretical results in the generic setting to this special case our result turns out to be a general and unified study on the convergence rate of a number of existing methods and subsumes some existing results in the literature

survival probability for the stadium billiard abstract we consider the open stadium billiard consisting of two semicircles joined by parallel straight sides with one hole situated somewhere on one of the sides due to the hyperbolic nature of the stadium billiard the initial decay of trajectories due to loss through the hole appears exponential however some trajectories bouncing ball orbits persist and survive for long times and therefore form the main contribution to the survival probability function at long times using both numerical and analytical methods we concur with previous studies that the long time survival probability for a reasonably small hole drops like c o n s t a n t t i m e 1 here we obtain an explicit expression for the constant

a strictly contractive peaceman rachford splitting method with logarithmic quadratic proximal regularization for convex programming recently a strictly contractive peaceman rachford splitting method prsm was proposed for a separable convex minimization model whose variables are subject to some linear constraints and two additional generic constraints in general the strictly contractive prsm requires to solve two constrained minimization subproblems at each iteration in this paper we consider the case where the additional constraints on variables are positive orthants and apply the well developed logarithmic quadratic proximal lqp regularization to regularize the subproblems of the strictly contractive prsm a new algorithm in combination with the strictly contractive prsm and the lqp regularization is thus proposed the new algorithm only needs to solve two unconstrained subproblems at each iteration an inexact version allowing the unconstrained subproblems to be solved approximately subject to certain inexactness criterion is also studied we prove the global convergence and establish a worst case convergence rate measured

convergence rate analysis of primal dual splitting schemes primal dual splitting schemes are a class of powerful algorithms that solve complicated monotone inclusions and convex optimization problems that are built from many simpler pieces they decompose problems that are built from sums linear compositions and infimal convolutions of simple functions so that each simple term is processed individually via proximal mappings gradient mappings and multiplications by the linear maps this leads to easily implementable and highly parallelizable or distributed algorithms which often obtain nearly state of the art performance in this paper we analyze a monotone inclusion problem that captures a large class of primal dual splittings as a special case we introduce a unifying scheme and use some abstract analysis of the algorithm to prove convergence rates of the proximal point algorithm forward backward splitting peaceman rachford splitting and forward backward forward splitting applied to the model problem our ergodic convergence rates are deduced under var

statistical properties of dynamical systems with some hyperbolicity this paper is about the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts though not necessarily all of their phase spaces the main results are for discrete time systems to put this work into context recall that for axiom a attractors the picture has been fairly complete since the 1970 s see s1 b r2 since then much progress has been made on two fronts there is a general nonuniform theory that deals with properties common to all diffeomorphisms with nonzero lyapunov exponents o p1 ka ly and there are detailed analyses of specific kinds of dynamical systems including for example billiards 1 dimensional and henon type maps s2 bsc hk j bc2 by1 statistical properties such as exponential decay of correlations are not enjoyed by all diffeomorphisms with nonzero lyapunov exponents the goal of this paper is a systematic understanding of these and other properties for a class of dynamical systems larger than axiom a this class will not be defined explicitly but it includes some of the much studied examples by looking at regular returns to sets with good hyperbolic properties one could give systems in this class a simple dynamical representation conditions for the existence of natural invariant measures exponential mixing and central limit theorems are given in terms of the return times these conditions can be checked in concrete situations giving a unified way of proving a number of results some new and some old among the new results are the exponential decay of correlations for a class of scattering billiards and for a positive measure set of henon type maps

ergodic billiards that are not quantum unique ergodic partially rectangular domains are compact two dimensional riemannian manifolds x either closed or with boundary that contain a flat rectangle or cylinder in this paper we are interested in partially rectangular domains with ergodic billiard flow examples are the bunimovich stadium the sinai billiard or donnelly surfaces we consider a one parameter family x t of such domains parametrized by the aspect ratio t of their rectangular part there is convincing theoretical and numerical evidence that the laplacian on x t with dirichlet or neumann boundary conditions is not quantum unique ergodic que we prove that this is true for all t in 1 2 excluding possibly a set of lebesgue measure zero this yields the first examples of ergodic billiard systems proven to be non que

escape exponent for transient chaos and chaotic scattering in non hyperbolic hamiltonian systems it is shown that in non hyperbolic hamiltonian systems where correlations decay as a power law transient chaos and chaotic scattering demonstrate different power laws numerical evidence of this effect for the stadium billiard is presented

on the convergence rate of douglas rachford operator splitting method this note provides a simple proof of a worst case convergence rate measured by the iteration complexity for the douglas rachford operator splitting method for finding a root of the sum of two maximal monotone set valued operators the accuracy of an iterate to the solution set is measured by the residual of a characterization of the original problem which is different from conventional measures such as the distance to the solution set

principles for the design of billiards with nonvanishing lyapunov exponents we introduce a large class of billiards with convex pieces of the boundary which have nonvanishing lyapunov exponents

inertial proximal strictly contractive peaceman rachford splitting method with an indefinite term for convex optimization abstract we consider an inertial proximal strictly contractive peaceman rachford splitting method abbreviated as ipscprsm with an indefinite proximal term for solving convex optimization problems with linear constraints with the aid of variational inequality proximal point method and fundamental inequality we prove global convergence of the proposed method and analyze iteration complexity in the best function value and feasibility residues the experimental results demonstrate the efficiency of the inertial extrapolation step and the indefinite proximal term even compared with the state of the art methods

on the o 1 n convergence rate of the douglas rachford alternating direction method alternating direction methods adms have been well studied in the literature and they have found many efficient applications in various fields in this note we focus on the douglas rachford adm scheme proposed by glowinski and marrocco and we aim at providing a simple approach to estimating its convergence rate in terms of the iteration number the linearized version of this adm scheme which is known as the split inexact uzawa method in the image processing literature is also discussed

local linear convergence of the alternating direction method of multipliers on quadratic or linear programs we introduce a novel matrix recurrence yielding a new spectral analysis of the local transient convergence behavior of the alternating direction method of multipliers admm for the particular case of a quadratic program or a linear program we identify a particular combination of vector iterates whose convergence can be analyzed via a spectral analysis the theory predicts that admm should go through up to four convergence regimes such as constant step convergence or linear convergence ending with the latter when close enough to the optimal solution if the optimal solution is unique and satisfies strict complementarity

models of collective cell spreading with variable cell aspect ratio a motivation for degenerate diffusion models continuum diffusion models are often used to represent the collective motion of cell populations most previous studies have simply used linear diffusion to represent collective cell spreading while others found that degenerate nonlinear diffusion provides a better match to experimental cell density profiles in the cell modeling literature there is no guidance available with regard to which approach is more appropriate for representing the spreading of cell populations furthermore there is no knowledge of particular experimental measurements that can be made to distinguish between situations where these two models are appropriate here we provide a link between individual based and continuum models using a multi scale approach in which we analyze the collective motion of a population of interacting agents in a generalized lattice based exclusion process for round agents that occupy a single lattice site we find that the relevant continuum description of the system is a linear diffusion equation whereas for elongated rod shaped agents that occupy l adjacent lattice sites we find that the relevant continuum description is connected to the porous media equation pme the exponent in the nonlinear diffusivity function is related to the aspect ratio of the agents our work provides a physical connection between modeling collective cell spreading and the use of either the linear diffusion equation or the pme to represent cell density profiles results suggest that when using continuum models to represent cell population spreading we should take care to account for variations in the cell aspect ratio because different aspect ratios lead to different continuum models

rate of convergence analysis of decomposition methods based on the proximal method of multipliers for convex minimization this paper presents two classes of decomposition algorithms based on the proximal method of multipliers pmm introduced in the mid 1970s by rockafellar for convex minimization we first show that

on non ergodic convergence rate of douglas rachford alternating direction method of multipliers this note proposes a novel approach to derive a worst case o 1 k o 1 k convergence rate measured by the iteration complexity in a non ergodic sense for the douglas rachford alternating direction method of multipliers proposed by glowinski and marrocco

local linear convergence of the alternating direction method of multipliers for quadratic programs the douglas rachford alternating direction method of multipliers admm has been widely used in various areas the global convergence of admm is well known while research on its convergence rate

dynamics of leaking hamiltonian systems in order to understand the dynamics in more detail in particular for visualizing the space filling unstable foliation of closed chaotic hamiltonian systems we propose to leak them up the cutting out of a finite region of their phase space the leak through which escape is possible leads to transient chaotic behavior of nearly all the trajectories the never escaping points belong to a chaotic saddle whose fractal unstable manifold can easily be determined numerically it is an approximant of the full hamiltonian foliation the better the smaller the leak is the escape rate depends sensitively on the orientation of the leak even if its area is fixed the applications for chaotic advection for chemical reactions superimposed on hydrodynamical flows and in other branches of physics are discussed

how chaotic is the stadium billiard a semiclassical analysis the impression gained from the literature published to date is that the spectrum of the stadium billiard can be adequately described semiclassically by the gutzwiller periodic orbit trace formula together with a modified treatment of the marginally stable family of bouncing ball orbits i show that this belief is erroneous the gutzwiller trace formula is not applicable for the phase space dynamics near the bouncing ball orbits unstable periodic orbits close to the marginally stable family in phase space cannot be treated as isolated stationary phase points when approximating the trace of the green s function semiclassical contributions to the trace show an dependent transition from hard chaos to integrable behaviour for trajectories approaching the bouncing ball orbits a whole region in phase space surrounding the marginal stable family acts semiclassically like a stable island with boundaries being explicitly dependent the localized bouncing ball states found in the billiard derive from this semiclassically stable island the bouncing ball orbits themselves however do not contribute to individual eigenvalues in the spectrum an ebk like quantization of the regular bouncing ball eigenstates in the stadium can be derived the stadium billiard is thus an ideal model for studying the influence of almost regular dynamics near marginally stable boundaries on quantum mechanics the behaviour is generically found at the border of classically stable islands in systems with a mixed phase space structure

on the linear convergence of the alternating direction method of multipliers we analyze the convergence rate of the alternating direction method of multipliers admm for minimizing the sum of two or more nonsmooth convex separable functions subject to linear constraints previous analysis of the admm typically assumes that the objective function is the sum of only two convex functions defined on two separable blocks of variables even though the algorithm works well in numerical experiments for three or more blocks moreover there has been no rate of convergence analysis for the admm without strong convexity in the objective function in this paper we establish the global r linear convergence of the admm for minimizing the sum of any number of convex separable functions assuming that a certain error bound condition holds true and the dual stepsize is sufficiently small such an error bound condition is satisfied for example when the feasible set is a compact polyhedron and the objective function consists of a smooth strictly convex function composed with a linear mapping and a nonsmooth ell 1 l1 regularizer this result implies the linear convergence of the admm for contemporary applications such as lasso without assuming strong convexity of the objective function

fisher equation with density dependent diffusion special solutions a special family of solutions explicit in the space and time variables is found for the fisher equation with density dependent diffusion the connection with the known travelling wave solution and the initial conditions from which that evolves is also shown