the primal dual active set strategy as a semismooth newton method this paper addresses complementarity problems motivated by constrained optimal control problems it is shown that the primal dual active set strategy which is known to be extremely efficient for this class of problems and a specific semismooth newton method lead to identical algorithms the notion of slant differentiability is recalled and it is argued that the max function is slantly differentiable in lp spaces when appropriately combined with a two norm concept this leads to new local convergence results of the primal dual active set strategy global unconditional convergence results are obtained by means of appropriate merit functions

tempered stable levy motion and transient super diffusion the space fractional diffusion equation models anomalous super diffusion its solutions are transition densities of a stable levy motion representing the accumulation of power law jumps the tempered stable levy motion uses exponential tempering to cool these jumps a tempered fractional diffusion equation governs the transition densities which progress from super diffusive early time to diffusive late time behavior this article provides finite difference and particle tracking methods for solving the tempered fractional diffusion equation with drift a temporal and spatial second order crank nicolson method is developed based on a finite difference formula for tempered fractional derivatives a new exponential rejection method for simulating tempered levy stables is presented to facilitate particle tracking codes

interior point methods in function space a primal dual interior point method for optimal control problems is considered the algorithm is directly applied to the infinite dimensional problem existence and convergence of the central path are analyzed and linear convergence of a short step path following method is established

semi smooth newton methods for state constrained optimal control problems abstract a regularized optimality system for state constrained optimal control problems is introduced and semi smooth newton methods for its solution are analyzed convergence of the regularized problems is proved numerical tests confirm the theoretical results and demonstrate the efficiency of the proposed methodology

a class of second order difference approximations for solving space fractional diffusion equations a class of second order approximations called the weighted and shifted grunwald difference wsgd operators are proposed for riemann liouville fractional derivatives with their effective applications to numerically solving space fractional diffusion equations in one and two dimensions the stability and convergence of our difference schemes for space fractional diffusion equations with constant coefficients in one and two dimensions are theoretically established several numerical examples are implemented to test the efficiency of the numerical schemes and confirm the convergence order and the numerical results for variable coefficients problem are also presented

numerical approximation of a time dependent nonlinear space fractional diffusion equation in this article we analyze a fully discrete numerical approximation to a time dependent fractional order diffusion equation which contains a nonlocal quadratic nonlinearity the analysis is performed for a general fractional order diffusion operator the nonlinear term studied is a product of the unknown function and a convolution operator of order 0 convergence of the approximation and a priori error estimates are given numerical computations are included which confirm the theoretical predictions

multigrid method for fractional diffusion equations the fractional diffusion equation is discretized by the implicit finite difference scheme with the shifted grunwald formula the scheme is unconditionally stable and the coefficient matrix possesses the toeplitz like structure a multigrid method is proposed to solve the resulting system meanwhile the fast toeplitz matrix vector multiplication is utilized to lower the computational cost with only o nlogn complexity where n is the number of the grid points numerical experiments are given to demonstrate the efficiency of the method

finite difference approximations for fractional advection dispersion flow equations fractional advection dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium in this paper we develop practical numerical methods to solve one dimensional fractional advection dispersion equations with variable coefficients on a finite domain the practical application of these results is illustrated by modeling a radial flow problem use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site

asymptotics of the solution to the conductivity equation in the presence of adjacent circular inclusions with finite conductivities abstract we consider the conductivity problem in the presence of adjacent circular inclusions with constant conductivities when two inclusions get closer and their conductivities degenerate to zero or infinity the gradient of the solution can be arbitrary large in this paper we derive an asymptotic formula of the solution which characterizes the gradient blow up of the solution in terms of conductivities of inclusions as well as the distance between inclusions the asymptotic formula is expressed in bipolar coordinates in terms of the lerch transcendent function and it is valid for inclusions with arbitrary constant conductivities we illustrate our results with numerical calculations

error estimates of crank nicolson type difference schemes for the subdiffusion equation a crank nicolson type difference scheme is proposed for solving the subdiffusion equation with fractional derivative and the truncation error is analyzed in detail at each temporal level only a tridiagonal linear system needs to be solved and the thomas algorithm may be used the solvability unconditional stability and h 1 norm convergence are proved the convergence order is rm min 2 gamma 2 1 gamma in the temporal direction and two in the spatial direction by the sobolev embedding inequality we obtain the maximum norm error estimate a spatial compact scheme based on the crank nicolson type difference scheme is also presented and similar results are given the convergence order is mathcal o tau rm min 2 gamma 2 1 gamma h 4 numerical experiments are included to support the theoretical results and comparisons with the related works are presented to show the effectiveness of our method

path following methods for a class of constrained minimization problems in function space path following methods for primal dual active set strategies requiring a regularization parameter are introduced existence of a primal dual path and its differentiability properties are analyzed monotonicity and convexity of the primal dual path value function are investigated both feasible and infeasible approximations are considered numerical path following strategies are developed and their efficiency is demonstrated by means of examples

a space time spectral method for the time fractional diffusion equation in this paper we consider the numerical solution of the time fractional diffusion equation essentially the time fractional diffusion equation differs from the standard diffusion equation in the time derivative term in the former case the first order time derivative is replaced by a fractional derivative making the problem global in time we propose a spectral method in both temporal and spatial discretizations for this equation the convergence of the method is proven by providing a priori error estimate numerical tests are carried out to confirm the theoretical results thanks to the spectral accuracy in both space and time of the proposed method the storage requirement due to the global time dependence can be considerably relaxed and therefore calculation of the long time solution becomes possible

a chebyshev pseudospectral method to solve the space time tempered fractional diffusion equation a chebyshev pseudospectral method to solve the space time tempered fractional diffusion equation

fluid limit of the continuous time random walk with general levy jump distribution functions the continuous time random walk ctrw is a natural generalization of the brownian random walk that allows the incorporation of waiting time distributions psi t and general jump distribution functions eta x there are two well known fluid limits of this model in the uncoupled case for exponential decaying waiting times and gaussian jump distribution functions the fluid limit leads to the diffusion equation on the other hand for algebraic decaying waiting times psi approximately t 1 beta and algebraic decaying jump distributions eta approximately x 1 alpha corresponding to levy stable processes the fluid limit leads to the fractional diffusion equation of order alpha in space and order beta in time however these are two special cases of a wider class of models here we consider the ctrw for the most general levy stochastic processes in the levy khintchine representation for the jump distribution function and obtain an integrodifferential equation describing the dynamics in the fluid limit the resulting equation contains as special cases the regular and the fractional diffusion equations as an application we consider the case of ctrws with exponentially truncated levy jump distribution functions in this case the fluid limit leads to a transport equation with exponentially truncated fractional derivatives which describes the interplay between memory long jumps and truncation effects in the intermediate asymptotic regime the dynamics exhibits a transition from superdiffusion to subdiffusion with the crossover time scaling as tauc approximately lambda alpha beta where 1 lambda is the truncation length scale the asymptotic behavior of the propagator green s function of the truncated fractional equation exhibits a transition from algebraic decay for t tauc

analysis of shape optimization for magnetic microswimmers we analyze an infinite dimensional geometrically constrained shape optimization problem for magnetically driven microswimmers locomotors in three dimensional 3 d stokes flow and give a well posed descent scheme for computing optimal shapes the problem is inspired by recent experimental work in this area we show the existence of a minimizer of the optimization problem using analytical tools for elastic rods that respect the excluded volume constraint we derive a variational gradient descent method for computing optimal locomotor shapes using the tools of shape differential calculus the descent direction is obtained by solving a saddle point system which we prove is well posed we also introduce a finite element approximation of the gradient descent method and prove its stability we present numerical results illustrating our method and the effect that finite aspect ratio and external cargo can have on the optimal shape the 3 d stokes equations are solved by a boundary integral method

new solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation a physical mathematical approach to anomalous diffusion is based on a generalized diffusion equation containing derivatives of fractional order in this paper an anomalous subdiffusion equation asub de is considered a new implicit numerical method inm and two solution techniques for improving the order of convergence of the inm for solving the asub de are proposed the stability and convergence of the inm are investigated by the energy method some numerical examples are given the numerical results demonstrate the effectiveness of theoretical analysis these methods and supporting theoretical results can also be applied to other fractional integro differential equations and higher dimensional problems

tempered fractional feynman kac equation theory and examples functionals of brownian and non brownian motions have diverse applications and attracted a lot of interest among scientists this paper focuses on deriving the forward and backward fractional feynman kac equations describing the distribution of the functionals of the space and time tempered anomalous diffusion belonging to the continuous time random walk class several examples of the functionals are explicitly treated including the occupation time in half space the first passage time the maximal displacement the fluctuations of the occupation fraction and the fluctuations of the time averaged position

tempered fractional calculus fractional derivatives and integrals are convolutions with a power law multiplying by an exponential factor leads to tempered fractional derivatives and integrals tempered fractional diffusion equations where the usual second derivative in space is replaced by a tempered fractional derivative govern the limits of random walk models with an exponentially tempered power law jump distribution the limiting tempered stable probability densities exhibit semi heavy tails which are commonly observed in finance tempered power law waiting times lead to tempered fractional time derivatives which have proven useful in geophysics the tempered fractional derivative or integral of a brownian motion called a tempered fractional brownian motion can exhibit semi long range dependence the increments of this process called tempered fractional gaussian noise provide a useful new stochastic model for wind speed data a tempered fractional difference forms the basis for numerical methods to solve tempered fractional diffusion equations and it also provides a useful new correlation model in time series

numerical schemes of the time tempered fractional feynmankac equation this paper focuses on providing the computation methods for the backward time tempered fractional feynmankac equation being one of the models recently proposed in wu etal 2016 the discretization for the tempered fractional substantial derivative is derived and the corresponding finite difference and finite element schemes are designed with well established stability and convergence the performed numerical experiments show the effectiveness of the presented schemes

nonsmooth data error estimates for approximations of an evolution equation with a positive type memory term we study the numerical approximation of an integro differential equation which is intermediate between the heat and wave equations the proposed discretization uses convolution quadrature based on the first and second order backward difference methods in time and piecewise linear finite elements in space optimal order error bounds in terms of the initial data and the inhomogeneity are shown for positive times without assumptions of spatial regularity of the data

high order algorithms for the fractional substantial diffusion equation with truncated levy flights the equation with the time fractional substantial derivative and space fractional derivative describes the distribution of the functionals of the levy flights and the equation is derived as the macroscopic limit of the continuous time random walk in unbounded domain and the levy flights have divergent second order moments however in more practical problems the physical domain is bounded and the involved observables have finite moments then the modified equation can be derived by tempering the levy measure of the levy flights and the corresponding tempered space fractional derivative is introduced this paper focuses on providing the high order algorithms for the modified equation i e the equation with the time fractional substantial derivative and space tempered fractional derivative more concretely the contributions of this paper are as follows 1 detailed numerical stability analysis and error estimates of the schemes with first order accuracy in time and second order in space are given in c

on distributions of functionals of anomalous diffusion paths functionals of brownian motion have diverse applications in physics mathematics and other fields the probability density function pdf of brownian functionals satisfies the feynman kac formula which is a schrodinger equation in imaginary time in recent years there is a growing interest in particular functionals of non brownian motion or anomalous diffusion but no equation existed for their pdf here we derive a fractional generalization of the feynman kac equation for functionals of anomalous paths based on sub diffusive continuous time random walk we also derive a backward equation and a generalization to levy flights solutions are presented for a wide number of applications including the occupation time in half space and in an interval the first passage time the maximal displacement and the hitting probability we briefly discuss other fractional schrodinger equations that recently appeared in the literature