improved axisymmetric lattice boltzmann scheme this paper proposes an improved lattice boltzmann scheme for incompressible axisymmetric flows the scheme has the following features first it is still within the framework of the standard lattice boltzmann method using the single particle density distribution function and consistent with the philosophy of the lattice boltzmann method second the source term of the scheme is simple and contains no velocity gradient terms owing to this feature the scheme is easy to implement in addition the singularity problem at the axis can be appropriately handled without affecting an important advantage of the lattice boltzmann method the easy treatment of boundary conditions the scheme is tested by simulating hagen poiseuille flow three dimensional womersley flow wheeler benchmark problem in crystal growth and lid driven rotational flow in cylindrical cavities it is found that the numerical results agree well with the analytical solutions and or the results reported in previous studies

an axisymmetric multiple relaxation time lattice boltzmann scheme a multiple relaxation time mrt lattice boltzmann lb scheme developed for axisymmetric flows recovers the complete continuity and navier stokes equations this scheme follows the strategy of the standard d2q9 model by using a single particle distribution function and a simple collision streaming updating rule the extra terms related to axisymmetry in the macroscopic equations are recovered by adding source terms into the lb equation which are simple and involve no gradients the compressible effect retained in the navier stokes equations is recovered by introducing a term related to the reversed transformation matrix for mrt collision operator so as to produce a correct bulk viscosity making it suitable for compressible flows with high frequency and low mach number the validity of the scheme is demonstrated by testing the hagen poiseuille flow and 3d womersley flow as well as the standing acoustic waves in a closed cylindrical chamber the numerical experiments show desirable stability at low viscosities enabling to simulate a standing ultrasound field in centimeters space

linear quadratic fractional gaussian control in this paper a control problem for a linear stochastic system driven by a noise process that is an arbitrary zero mean square integrable stochastic process with continuous sample paths and a cost functional that is quadratic in the system state and the control is solved an optimal control is given explicitly as the sum of the well known linear feedback control for the associated deterministic linear quadratic control problem and the prediction of the response of a system to the future noise process the optimal cost is also given the special case of a noise process that is an arbitrary standard fractional brownian motion is noted explicitly with an explicit expression for the prediction of the future response of a system to the noise process that is used the optimal control

linear quadratic mean field type games jump diffusion process with regime switching in this paper we study a linear quadratic mean field type game problem under jump diffusion regime switching state dynamics that include mean field terms in the states and in the cost function in the cost functional we consider conditional variance of both the states and control actions mean of both states and control actions and covariance terms the underlying problem is solved in a semiexplicit way by using the direct method

axisymmetric lattice boltzmann method a lattice boltzmann method is developed for incompressible axisymmetric flows both force and source or sink terms are incorporated into the lattice boltzmann equation in a natural way which is consistent in dimension with the lattice boltzmann equation the correct macroscopic equations for incompressible axisymmetric flows are recovered through the chapman enskog expansion it turns out that the added terms are nothing but the additional in the governing equations for the axisymmetric flows compared with the navier stokes equations resulting in a simple and efficient model this provides an additional unique advantage that the proposed scheme is naturally suitable for general axisymmetric flows involving more physical phenomena two numerical simulations have been presented to verify the method

solvable stochastic differential games in rank one compact symmetric spaces some nonlinear stochastic differential games are formulated in the family of complex and quaternion projective spaces that are among the rank one compact symmetric spaces that consist of the spheres the projective spaces over r c and h and one arising from an exceptional lie algebra called the cayley plane the payoff functionals for the differential games are obtained from some eigenfunctions of the radial part of the laplacians for these riemannian manifolds and these payoffs induce symmetries for the game problems that reduce the required analysis to a radial direction in these manifolds these projective spaces are given a natural riemannian metric from the killing forms of the compact lie groups for these spaces that are the special unitary groups su n and the symplectic groups sp n explicit optimal control strategies are obtained for these differential games and the explicit payoffs are given a countable family of distinct solvable stochastic differential games can be obtained for

linear exponential quadratic gaussian control in this technical note an optimal control problem for a linear stochastic system with brownian motion and a cost that is an exponential of a quadratic functional of the state and the control is solved by obtaining explicitly an optimal control and the optimal cost while this solution has been previously obtained the approach given here is direct and elementary and does not use the well known solution methods of the hamilton jacobi bellman equation or the stochastic maximum principle the approach given here presents a basic insight in the solution by providing a simple explanation for the additional term in the riccati equation for the optimal control as compared to the riccati equation for the linear quadratic gaussian control problem

lattice boltzmann equation hydrodynamics by inserting position and time dependent source or forcing terms into the microscopic evolution equation of a lattice boltzmann fluid and treating the generalized scheme within the usual chapman enskog methodology we show that the emergent dynamics of the lattice fluid may be usefully transformed our method of adjustment is demonstrated by implementing the cylindrical polar coordinate form of the continuity and momentum equations on a rectangular lattice and generating results for pipe flow with straightforward systematic adjustment of the simulation our approach produces results in excellent agreement with theory

discrete lattice effects on the forcing term in the lattice boltzmann method we show that discrete lattice effects must be considered in the introduction of a force into the lattice boltzmann equation a representation of the forcing term is then proposed with the representation the navier stokes equation is derived from the lattice boltzmann equation through the chapman enskog expansion several other existing force treatments are also examined

risk sensitive zero sum differential games we consider two player risk sensitive zero sum differential games rszsdgs in our problem setup both the drift term and the diffusion term in the controlled stochastic differential equation are dependent on the state and controls of both players and the objective functional is of the risk sensitive type first a stochastic maximum principle type necessary condition for an open loop saddle point of the rszsdg is established via nonlinear transformations of the adjoint processes of the equivalent risk neutral stochastic zero sum differential game in particular we obtain two variational inequalities namely the pair of saddle point inequalities of the rszsdg next we obtain the hamilton jacobi isaacs partial differential equation for the rszsdg which provides a sufficient condition for a feedback saddle point of the rszsdg using a logarithmic transformation of the associated value function finally we study the extended linear quadratic rszsdg lq rszsdg we show intractability of the extended lq rszsdg with the state and or controls of both players appearing in the diffusion term this unexpected intractability could lead to nonlinear open loop and feedback saddle points even if the problem itself is essentially lq and the isaacs condition holds

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

stochastic differential games and viscosity solutions of hamilton jacobi bellman isaacs equations in this paper we study zero sum two player stochastic differential games with the help of the theory of backward stochastic differential equations bsdes more precisely we generalize the results of the pioneering work of fleming and souganidis indiana univ math j 38 1989 pp 293 314 by considering cost functionals defined by controlled bsdes and by allowing the admissible control processes to depend on events occurring before the beginning of the game this extension of the class of admissible control processes has the consequence that the cost functionals become random variables however by making use of a girsanov transformation argument which is new in this context we prove that the upper and the lower value functions of the game remain deterministic apart from the fact that this extension of the class of admissible control processes is quite natural and reflects the behavior of the players who always use the maximum of available information its combination with bsde methods in particular that of the notion of stochastic backward semigroups introduced by peng bsde and stochastic optimizations in topics in stochastic analysis science press beijing 1997 allows us then to prove a dynamic programming principle for both the upper and the lower value functions of the game in a straightforward way the upper and the lower value functions are then shown to be the unique viscosity solutions of the upper and the lower hamilton jacobi bellman isaacs equations respectively for this peng s bsde method is extended from the framework of stochastic control theory into that of stochastic differential games

schematic models for fragmentation of brittle solids in one and two dimensions abstract stochastic models for the development of cracks in 1d and 2d objects are presented in one dimension we focus on particular scenarios for interacting fragments if1 and non interacting fragments nif1 during the breakup process for 2d objects we consider only nif1 but analyze isotropic and anisotropic development of fissures analytical results are given for many observables power law size distributions are predicted for some of the fragmentation pictures considered

switching games of stochastic differential systems a two player zero sum switching game is formulated for general stochastic differential systems and is studied using a combined dynamic programming and viscosity solution approach the existence of the game value is proved for the proof of the related dynamic programming principle ddp for the lower and upper value functions the measurability problem of the same kind as mentioned in the paper of fleming and souganidis is also encountered and we are able to get around it via a delicate adaptation of their technique moreover the traditional direct method to prove the time continuity of lower and upper value functions also gives rise to a serious measurability problem to get around the new difficulty a subtle dynamic programming argument is developed to obtain the time continuity which in return is used to derive the ddp for random intermediate times from the ddp with deterministic intermediate times

scaling and multiscaling in models of fragmentation we introduce a simple geometric model which describes the kinetics of fragmentation of d dimensional objects in one dimension our model coincides with the randon scission model and shows a simple scaling behavior in the long time limit for dg1 the volume of the fragments is characterized by a single scale 1 t while other geometric properties such as the length are characterized by an infinite number of length scales and thus exhibit multiscaling

numerical investigation of flows in czochralski crystal growth by an axisymmetric lattice boltzmann method an alternative new method called lattice boltzmann method lbm is applied in this work to simulate the flows in czochralski crystal growth which is one of the widely used prototypical systems for melt crystal growth the standard lbm can only be used in cartesian coordinate system and we extend it to be applicable to this axisymmetric thermal flow problem avoiding the use of three dimensional lbm on cartesian coordinate system the extension is based on the following idea by inserting position and time dependent source terms into the evolution equation of standard lbm the continuity and ns equations on the cylindrical coordinate system 1 can be recovered our extension is validated by its application to the benchmark problem suggested by wheeler 2

multiscaling in fragmentation abstract we review recent results on random fragmentation of multidimensional objects in addition to the obvious volume conservation these processes exhibit an infinite number of hidden conservation laws the fragment size distribution is characterized by an infinite number of scales and thus exhibits multiscaling nevertheless the volume distribution function shows ordinary scaling i e it is characterized by a single scale

an alternative implementation of the kinetic theory based axisymmetric lattice boltzmann model abstract the semi implicit centered scheme as an approximated implementation of the trapezium rule based scheme has proved to be capable of avoiding the discrete lattice effects in our recent forcing scheme analysis for the axisymmetric lb method and is applied here for an alternative implementation of the kinetic theory based axisymmetric model by guo et al in particular the external force terms and the first order part of the source terms accounting for the axisymmetric effect are discretized with the semi implicit centered scheme while the rest are represented with the trapezium rule the proposed axisymmetric lb scheme has the advantage of simpler calculations of the macroscopic variables since the zeroth and the first order moments of the distribution function are independent of the complicated source terms chapman enskog analysis in the appendix demonstrates that axisymmetric navier stokes n s equations are recovered from the proposed model and the error terms caused by the semi implicit centered scheme do not affect the overall accuracy of the present model moreover the accuracy and applicability of the proposed axisymmetric lb model are verified by the adopted numerical tests and the complicated macroscopic variables computations in the guo et al model and its derivative models are avoided by the present model

optimal control of partially observable stochastic systems with an exponential of integral performance index the stochastic control problem with linear stochastic differential equations driven by brownian motion processes and as cost functional the exponential of a quadratic form is considered the soluti

on the optimal control of stochastic systems with an exponential of integral performance index abstract in this paper a theory of optimal control is developed for stochastic systems whose performance is measured by the exponential of an integral form such a formulation of the cost function is shown to be not only general and useful but also analytically tractable starting with very general classes of stochastic systems optimality conditions are obtained which exploit the multiplicative decomposability of the exponential of integral form specializing to partially observed systems of stochastic differential equations with brownian motion disturbances optimality conditions are obtained which parallel those for systems with integral costs also treated are the special cases of linear systems with exponential of quadratic costs for which explicit optimal controls are obtainable in addition several general results of independent interest are obtained which concern optimality of stochastic systems

modified lattice boltzmann model for axisymmetric flows a modified lattice boltzmann model based on the two dimensional nine velocity lattice bhatnagar gross krook fluid is presented for axisymmetric flows a spatially and temporally varying source term is incorporated into the evolution equation for the momentum distribution function on a two dimensional cartesian lattice the precise form of the source term is derived through a chapman enskog analysis so that the additional axisymmetric contributions in the navier stokes equations are furnished when written in the cylindrical polar coordinate system

two dimensional model for binary fragmentation process with random system of forces random stopping and material resistance abstract this work presents the numerical results obtained from large scale parallel distributed simulations of a self similar model for two dimensional discrete in time and continuous in space binary fragmentation its main characteristics are 1 continuous material 2 uniform and independent random distribution of the net forces denoted by f x and f y that produce the fracture 3 these net forces act at random positions of the fragments and generate the fracture following a maximum criterion 4 the fragmentation process has the property that every fragment fracture stops at each time step with an uniform probability p 5 the material presents an uniform resistance r to the fracture process through a numerical study was obtained an approximate power law behavior for the small fragments size distribution for a wide range of the main parameters of the model the stopping probability p and the resistance r the visualizations of the model resemble real systems the approximate power law distribution is a non trivial result which reproduces empirical results of some highly energetic fracture processes

axisymmetric multiphase lattice boltzmann method a lattice boltzmann method for axisymmetric multiphase flows is presented and validated the method is capable of accurately modeling flows with variable density we develop the classic shan chen multiphase model phys rev e 47 1815 1993 for axisymmetric flows the model can be used to efficiently simulate single and multiphase flows the convergence to the axisymmetric navier stokes equations is demonstrated analytically by means of a chapmann enskog expansion and numerically through several test cases in particular the model is benchmarked for its accuracy in reproducing the dynamics of the oscillations of an axially symmetric droplet and on the capillary breakup of a viscous liquid thread very good quantitative agreement between the numerical solutions and the analytical results is observed

diffuse reflection boundary condition for high order lattice boltzmann models with streaming collision mechanism for lattice boltzmann lb models their particle feature is ensured by the streaming collision mechanism however this mechanism is often discarded for high order lb models due to multi speed lattices here we propose a new way of implementing the diffuse reflection boundary condition that can maintain the characteristic streaming collision mechanism for high order models it is then tested on four lattice models namely d2q16 d2q17 d2q37 and d3q121 for isothermal and thermal couette flows and lid driven cavity flows our implementation is shown to be able to achieve second order accuracy globally

linear exponential quadratic stochastic differential games in this technical note a two person noncooperative stochastic differential game described by a linear stochastic equation with a brownian motion and a payoff that is the exponential of a quadratic functional in the state and the control strategies of the two players is explicitly solved the optimal strategies form a nash equilibrium the value of the game is also determined explicitly the solution of this problem is naturally related to a linear quadratic stochastic differential game and a linear exponential quadratic control problem

discrete time linear quadratic mean field type repeated games perfect incomplete and imperfect information in this paper we study linear quadratic mean field type repeated games with arbitrary number of interacting decision makers different information structures are examined i perfect knowledge of the model ii imperfect knowledge of the model under perfect state observation we provide semi explicit solutions for both non cooperative and cooperative cases the state dynamics is given by a stochastic difference equation that includes mean field terms i e the expected value for both the state and control inputs in addition the cost function incorporates not only the expected value of the states but also the variance of both system state and control inputs the effect of the information is quantified using price of anarchy and price of imperfect information finally some numerical examples are presented

alternative extrapolation based symmetry boundary implementations for the axisymmetric lattice boltzmann method in this study alternative symmetry boundary implementations for the axisymmetric lattice boltzmann lb method are proposed based on the nonequilibrium extrapolation and the direct extrapolation schemes the proposed boundary schemes are directly implemented on the symmetry axis and the postcollision distribution function and the macroscopic variables at the boundary nodes are extrapolated from the inner fluid nodes thereby the singularities arising at the symmetry axis r 0 during the collision and the macroscopic variable calculations are completely avoided the accuracy of the present schemes is consistent with the well established axisymmetric lb model moreover in comparison with previous symmetry boundary schemes the present implementations are slightly more accurate than the symmetry scheme by guo et al phys rev e 79 046708 2009 10 1103 physreve 79 046708 and numerically more stable than the specular reflection based schemes

lattice boltzmann method for 3 d flows with curved boundary in this work we investigate two issues that are important to computational efficiency and reliability in fluid dynamic applications of the lattice boltzmann equation lbe 1 computational stability and accuracy of different lattice boltzmann models and 2 the treatment of the boundary conditions on curved solid boundaries and their 3 d implementations three athermal 3 d lbe models q15d3 q19d3 and q27d3 are studied and compared in terms of efficiency accuracy and robustness the boundary treatment recently developed by filippova and hanel 1998 j comp phys 147 219 and mei et al 1999 j comp phys 155 307 in 2 d is extended to and implemented for 3 d the convergence stability and computational efficiency of the 3 d lbe models with the boundary treatment for curved boundaries were tested in simulations of four 3 d flows 1 fully developed flows in a square duct 2 flow in a 3 d lid driven cavity 3 fully developed flows in a circular pipe and 4 a uniform flow over a sphere we found that while the 15 velocity 3 d q15d3 model is more prone to numerical instability and the q27d3 is more computationally intensive the q19d3 model provides a balance between computational reliability and efficiency through numerical simulations we demonstrated that the boundary treatment for 3 d arbitrary curved geometry has second order accuracy and possesses satisfactory stability characteristics

accuracy analysis of high order lattice boltzmann models for rarefied gas flows in this work we have theoretically analyzed and numerically evaluated the accuracy of high order lattice boltzmann lb models for capturing non equilibrium effects in rarefied gas flows in the incompressible limit the lb equation is shown to be able to reduce to the linearized bhatnagar gross krook bgk equation therefore when the same gauss hermite quadrature is used lb method closely resembles the discrete velocity method dvm in addition the order of hermite expansion for the equilibrium distribution function is found not to be directly correlated with the approximation order in terms of the knudsen number to the bgk equation for incompressible flows meanwhile we have numerically evaluated the lb models for a standing shear wave problem which is designed specifically for assessing model accuracy by excluding the influence of gas molecule surface interactions at wall boundaries the numerical simulation results confirm that the high order terms in the discrete equilibrium distribution function play a negligible role in capturing non equilibrium effect for low speed flows by contrast appropriate gauss hermite quadrature has the most significant effect on whether lb models can describe the essential flow physics of rarefied gas accurately our simulation results where the effect of wall gas interactions is excluded can lead to conclusion on the lb modeling capability that the models with higher order quadratures provide more accurate results for the same order gauss hermite quadrature the exact abscissae will also modestly influence numerical accuracy using the same gauss hermite quadrature the numerical results of both lb and dvm methods are in excellent agreement for flows across a broad range of the knudsen numbers which confirms that the lb simulation is similar to the dvm process therefore lb method can offer flexible models suitable for simulating continuum flows at the navier stokes level and rarefied gas flows at the linearized boltzmann model equation level

linear exponential quadratic control for mean field stochastic systems in this technical note we consider linear exponential quadratic leq control for mean field stochastic differential equations mfsdes the mfsde includes the expectation value of state and control and the objective functional is exponential of a quadratic functional in state control and their expectations we obtain the explicit optimal solution as well as the optimal cost the corresponding optimal solution is linear in state and its expectation which is characterized by the riccati differential equations rdes the results are obtained by showing that after applying the completion of squares method the remaining exponentiated stochastic integral and additional rde terms can be eliminated together by taking expectation since they constitute the associated radon nikodym derivative as an extension of the problem the leq zero sum differential game is considered for which we obtain the explicit optimal solution saddle point equilibrium as well as the optimal cost