disjoint pattern database heuristics we describe a new technique for designing more accurate admissible heuristic evaluation functions based on pattern databases j culberson j schaeffer comput intelligence 14 3 1998 318 334 while many heuristics such as manhattan distance compute the cost of solving individual subgoals independently pattern databases consider the cost of solving multiple subgoals simultaneously existing work on pattern databases allows combining values from different pattern databases by taking their maximum if the subgoals can be divided into disjoint subsets so that each operator only affects subgoals in one subset then we can add the pattern database values for each subset resulting in a more accurate admissible heuristic function we used this technique to improve performance on the fifteen puzzle by a factor of over 2000 and to find optimal solutions to 50 random instances of the twenty four puzzle

decay estimates of functions through singular extensions of vector valued laplace transforms abstract let x be a banach space and let f l r x whose laplace transform extends analytically to some region containing i r 0 possibly having a pole at the origin in this paper we give estimates of the decay of certain slight suitable modification of f in terms of the growth of its laplace transform along the imaginary axis this technique is applied to obtain decay estimates of smooth orbits of bounded c 0 semigroups whose infinitesimal generators have an arbitrary finite boundary spectrum these results are close to those given recently by c j k batty and t duyckaerts

stability theory for semigroups using l p l q fourier multipliers abstract we study polynomial and exponential stability for c 0 semigroups using the recently developed theory of operator valued l p l q fourier multipliers we characterize polynomial decay of orbits of a c 0 semigroup in terms of the l p l q fourier multiplier properties of its resolvent using this characterization we derive new polynomial decay rates which depend on the geometry of the underlying space we do not assume that the semigroup is uniformly bounded our results depend only on spectral properties of the generator as a corollary of our work on polynomial stability we reprove and unify various existing results on exponential stability and we also obtain a new theorem on exponential stability for positive semigroups

the multidimensional wave equation with generalized acoustic boundary conditions i strong stability we study the wave equation on a domain of mathbb r d d geq 2 with dynamical boundary control applied on a part of the boundary and a dirichlet boundary condition on the remaining part we prove the asymptotic stability and nonuniform stability of the associated semigroup

heat wave interaction in 2 3 dimensions optimal rational decay rate abstract in this paper we consider a simplified version of a fluid structure pde model which has been of longstanding interest within the mathematical and biological sciences in it a n dimensional heat equation replaces the original stokes system so as to ultimately have a vector valued heat equation and vector valued wave equation compose the coupled pde system under study the coupling between the two disparate pde components occurs across a boundary interface as such the entire pde dynamics manifests features of both hyperbolicity and parabolicity for this heat structure system our main result of uniform stability is as follows given smooth initial data i e data in the domain of the underlying semigroup generator of the coupled pde system the corresponding solutions decay at the rate of o t 1 this establishes the long conjectured optimal rate the problem of obtaining sharp rational decay rates for the heat wave pde under present consideration has been a much considered problem with the modus operandi of earlier efforts taking place within the time domain by contrast we adopt here a frequency domain approach which is based upon a recent resolvent criterion and which was initiated in our prior effort wherein we obtained the rate of decay o 1 t the present optimal improvement o t 1 is achieved by employing an additional tool in our analysis a microlocal analysis argument to estimate a critical term involving two problematic boundary traces

a numerical method for solving incompressible flow problems with a surface of discontinuity a numerical method for solving problems in which a moving surface of discontinuity separates regions of incompressible flow is presented the method developed is notable in that it does not introduce any artificial smoothing of the change in fluid properties across the surface of discontinuity this results in an increase in accuracy relative to methods which introduce smoothing effects the method was also shown to be fairly versatile problems describing a free surface an immiscible fluid interface and a premixed flame discontinuity were solved there is a limitation however in that the method appears to be most suitable for application to inviscid problems the reason for this limitation and possible approaches toward resolving it are discussed

direct integrals of strongly continuous operator semigroups abstract the goal of this article is to develop a theory for direct integrals of c 0 semigroups on hilbert spaces parallel to the recent approach by lachowicz and moszynski for direct sums of banach spaces diagonal operators and semigroups in it we deal with the existence and characterisation of semigroups asymptotic rates and questions of decomposability

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

rates of convergence to equilibrium for collisionless kinetic equations in slab geometry abstract this work deals with free transport equations with partly diffuse stochastic boundary operators in slab geometry such equations are governed by stochastic semigroups in l 1 spaces we prove convergence to equilibrium at the rate o t k 2 k 1 1 t for l 1 initial data g in a suitable subspace of the domain of the generator t where k n depends on the properties of the boundary operators near the tangential velocities to the slab this result is derived from a quantified version of ingham s tauberian theorem by showing that f g s lim e 0 i s e t 1 g exists as a c k function on r 0 such that d j d s j f g s c s 2 j 1 near s 0 and bounded as s 0 j k various preliminary results of independent interest are given and some related open problems are pointed out

optimal rates of decay for operator semigroups on hilbert spaces abstract we investigate rates of decay for c 0 semigroups on hilbert spaces under assumptions on the resolvent growth of the semigroup generator our main results show that one obtains the best possible estimate on the rate of decay that is to say an upper bound which is also known to be a lower bound under a comparatively mild assumption on the growth behaviour this extends several statements obtained by batty et al 2016 6 in fact for a large class of semigroups our condition is not only sufficient but also necessary for this optimal estimate to hold even without this assumption we obtain a new quantified asymptotic result which in many cases of interest gives a sharper estimate for the rate of decay than was previously available and for semigroups of normal operators we are able to describe the asymptotic behaviour exactly we illustrate the strength of our theoretical results by using them to obtain sharp estimates on the rate of energy decay for a wave equation subject to viscoelastic damping at the boundary

optimal energy decay in a one dimensional wave heat system with infinite heat part abstract using recent results in the theory of c 0 semigroups due to batty et al 2016 5 we study energy decay in a one dimensional coupled wave heat system with finite wave part and infinite heat part our main result provides a sharp estimate for the rate of energy decay of a certain class of classical solutions the present paper can be thought of as a natural sequel to a recent work by batty et al 2016 6 which studied a similar wave heat system with finite wave and heat parts using a celebrated result due to borichev and tomilov

a level set formulation of eulerian interface capturing methods for incompressible fluid flows a level set formulation is derived for incompressible immiscible navier stokes equations separated by a free surface the interface is identified as the zero level set of a smooth function eulerian finite difference methods based on this level set formulation are proposed these methods are robust and efficient and are capable of computing interface singularities such as merging and reconnection numerical experiments are presented to demonstrate the effectiveness of the methods

regularity analysis for an abstract system of coupled hyperbolic and parabolic equations abstract in this paper we provide a complete regularity analysis for the following abstract system of coupled hyperbolic and parabolic equations u t t a u a w w t a u t k a w u 0 u 0 u t 0 v 0 w 0 w 0 where a is a self adjoint positive definite operator on a complex hilbert space h and 0 1 0 1 we are able to decompose the unit square of the parameter into three parts where the semigroup associated with the system is analytic of specific order gevrey classes and non smoothing respectively moreover we will show that the orders of gevrey class is sharp under proper conditions

asymptotics for infinite systems of differential equations this paper investigates the asymptotic behavior of solutions to certain infinite systems of ordinary differential equations in particular we use results from ergodic theory and the asymptotic theory of c 0 semigroups to obtain a characterization in terms of convergence of certain cesaro averages of those initial values which lead to convergent solutions moreover we obtain estimates on the rate of convergence for solutions whose initial values satisfy a stronger ergodic condition these results rely on a detailed spectral analysis of the operator describing the system which is made possible by certain structural assumptions on the operator the resulting class of systems is sufficiently broad to cover a number of important applications including in particular both the so called robot rendezvous problem and an important class of platoon systems arising in control theory our method leads to new results in both cases

asymptotics and approximation of large systems of ordinary differential equations abstract in this paper we continue our earlier investigations into the asymptotic behaviour of infinite systems of coupled differential equations under the mild assumption that the so called characteristic function of our system is completely monotonic we obtain a drastically simplified condition which ensures boundedness of the associates semigroup if the characteristic function satisfies certain additional conditions we deduce sharp rates of convergence to equilibrium we moreover address the important and delicate issue of the role of the infinite system in understanding the asymptotic behaviour of large but finite systems and we provide a precise way of obtaining size independent rates of convergence for families of finite dimensional systems finally we illustrate our abstract results in the setting of the well known platoon problem

on the decay rate for the wave equation with viscoelastic boundary damping abstract we consider the wave equation with a boundary condition of memory type under natural conditions on the acoustic impedance of the boundary a corresponding semigroup of contractions is known to exist with the help of quantified tauberian theorems we establish energy decay rates via resolvent estimates on the generator of the semigroup using a variational approach we reduce resolvent estimates to estimates for a sesquilinear form induced by an operator characteristic function arising form the matrix representation of the generator under not too strict additional assumptions on the acoustic impedance we establish an upper bound on the resolvent for the wave equation on the interval or the disc we prove our estimates to be sharp

a boundary condition capturing method for poisson s equation on irregular domains interfaces have a variety of boundary conditions or jump conditions that need to be enforced the ghost fluid method gfm was developed to capture the boundary conditions at a contact discontinuity in the inviscid euler equations and has been extended to treat more general discontinuities such as shocks detonations and deflagrations and compressible viscous flows in this paper a similar boundary condition capturing approach is used to develop a new numerical method for the variable coefficient poisson equation in the presence of interfaces where both the variable coefficients and the solution itself may be discontinuous this new method is robust and easy to implement even in three spatial dimensions furthermore the coefficient matrix of the associated linear system is the standard symmetric matrix for the variable coefficient poisson equation in the absence of interfaces allowing for straightforward application of standard black box solvers

a boundary condition capturing method for incompressible flame discontinuities abstract in this paper we propose a new numerical method for treating two phase incompressible flow where one phase is being converted into the other e g the vaporization of liquid water we consider this numerical method in the context of treating discontinuously thin flame fronts for incompressible flow this method was designed as an extension of the ghost fluid method 1999 j comput phys 152 457 and relies heavily on the boundary condition capturing technology developed in liu et al 2000 j comput phys 154 15 for the variable coefficient poisson equation and in kang et al in press j comput phys for multiphase incompressible flow our new numerical method admits a sharp interface representation similar to the method proposed in helenbrook et al 1999 j comput phys 148 366 since the interface boundary conditions are handled in a simple and straightforward fashion the code is very robust e g no special treatment is required to treat the merging of flame fronts the method is presented in three spatial dimensions with numerical examples in one two and three spatial dimensions

a front tracking method for viscous incompressible multi fluid flows abstract a method to simulate unsteady multi fluid flows in which a sharp interface or a front separates incompressible fluids of different density and viscosity is described the flow field is discretized by a conservative finite difference approximation on a stationary grid and the interface is explicitly represented by a separate unstructured grid that moves through the stationary grid since the interface deforms continuously it is necessary to restructure its grid as the calculations proceed in addition to keeping the density and viscosity stratification sharp the tracked interface provides a natural way to include surface tension effects both two and three dimensional full numerical simulations of bubble motion are presented

optimal polynomial decay of functions and operator semigroups we characterize the polynomial decay of orbits of hilbert space c 0 semigroups in resolvent terms we also show that results of the same type for general banach space semigroups and functions obtained recently in batty and duyckaerts j evol eq 8 765 780 2008 are sharp this settles a conjecture posed in batty and duyckaerts 2008

improved volume conservation in the computation of flows with immersed elastic boundaries this paper introduces a recipe for the construction of a finite difference divergence operator the coefficients of which are completely determined once an interpolation scheme has been chosen substitution of this divergence operator for the previously used divergence based on central differences makes a dramatic improvement in the overall volume conservation that is observed in an immersed boundary computation this improvement is particularly important for computations in which an elastic boundary separates chambers containing fluid at substantially different pressures a situation which is prominent in cardiac fluid dynamics during the contraction of the ventricles

the non analytic growth bound of a c0 semigroup and inhomogeneous cauchy problems abstract the non analytic growth bound t of a c0 semigroup t measures the extent to which t can be approximated by a holomorphic function and it is related to spectral properties of the generator a in regions of c far from the real axis we show that t can be characterised by means of fourier multiplier properties of the resolvent of a far from the real axis and also by existence and uniqueness of mild solutions of inhomogeneous cauchy problems of the form u t au t f t on r where the carleman spectra of f and u are far from the origin the corresponding results for the exponential growth bound 0 t have been established earlier by other authors

optimal rates of decay in the katznelson tzafriri theorem for operators on hilbert spaces abstract the katznelson tzafriri theorem is a central result in the asymptotic theory of discrete operator semigroups it states that for a power bounded operator t on a banach space we have t n i t 0 if and only if t t 1 the main result of the present paper gives a sharp estimate for the rate at which this decay occurs for operators on hilbert space assuming the growth of the resolvent norms r e i t as 0 satisfies a mild regularity condition this significantly extends an earlier result by the second author which covered the important case of polynomial resolvent growth we further show that under a natural additional assumption our condition on the resolvent growth is not only sufficient but also necessary for the conclusion of our main result to hold by considering a suitable class of toeplitz operators we show that our theory has natural applications even beyond the setting of normal operators for which we in addition obtain a more general result