fast full search equivalent encoding algorithms for image compression using vector quantization three fast search routines to be used in the encoding phase of vector quantization vq image compression systems are presented these routines which are based on geometric considerations provide the same results as an exhaustive or full search examples show that the proposed algorithms need only 3 20 of the number of mathematical operations required by a full search and fewer than 50 of the operations required by recently proposed alternatives

hyperbolic conservation laws with relaxation the effect of relaxation is important in many physical situations it is present in the kinetic theory of gases elasticity with memory gas flow with thermo non equilibrium water waves etc the governing equations often take the form of hyperbolic conservation laws with lower order terms in this article we present and analyze a simple model of hyperbolic conservation laws with relaxation effects dynamic subcharacteristics governing the propagation of disturbances over strong wave forms are identified stability criteria for diffusion waves expansion waves and traveling waves are found and justified nonlinearly time asymptotic expansion and the energy method are used in the analysis for dissipative waves the expansion is similar in spirit to the chapman enskog expansion in the kinetic theory for shock waves however a different approach is needed

degenerate stochastic differential equations with holder continuous coefficients and super markov chains we consider the operator formula math acting on functions in c 2 b r d we prove uniqueness of the martingale problem for this degenerate operator under suitable nonnegativity and regularity conditions on ij and b i in contrast to previous work the b i need only be nonnegative on the boundary rather than strictly positive at the expense of the ij and b i being holder continuous applications to super markov chains are given the proof follows stroock and varadhan s perturbation argument but the underlying function space is now a weighted holder space and each component of the constant coefficient process being perturbed is the square of a bessel process

error bounds for the methods of glimm godunov and leveque the expected error in l 1 r at time t for glimm s scheme when applied to a scalar conservation law is bounded by left h frac 2 sqrt 3 left frac h delta t right 1 2 right left u 0 right bv bf r where h is the mesh size and delta t is the time step we show that the error in godunov s scheme is bounded by the same expression and use this result to analyze a variant of a large time step numerical method of leveque that may be viewed as a combination of godunov s scheme and a method of dafermos

the convergence rate of finite difference schemes in the presence of shocks finite difference approximations generically have cal o 1 pointwise errors close to a shock we show that this local error may effect the smooth part of the solution such that only first order is achieved even for formally higher order methods analytic and numerical examples of this form of accuracy are given we also show that a converging method will have the formal order of accuracy in domains where no characteristics have passed through a shock

schauder theorems for ornstein uhlenbeck equations in infinite dimension abstract we prove schauder type estimates for stationary and evolution equations driven by the classical ornstein uhlenbeck operator in a separable banach space endowed with a centered gaussian measure

fast fractal image block coding based on local variances in fractal image block coding most of the time is spent on finding a close match between a range block and a large pool of domain blocks for a large image this effect becomes aggravated as the domain pool increases exponentially we propose using the local variances of domain blocks to reduce the search space by sorting the contracted domain pool according to their local variances and defining an acceptance criterion for a close match we can confine all the potential close matches to a relatively small sized window to limit the search space the encoding time can hence be shortened with the decoded image quality as good as that using the full search method the speedup can be over ten times depending on the complexity of encoded images

local error estimates for discontinuous solutions of nonlinear hyperbolic equations let u x t be the possibly discontinuous entropy solution of a nonlinear scalar conservation law with smooth initial data suppose u varepsilon x t is the solution of an approximate viscosity regularization where varepsilon 0 is the small viscosity amplitude it is shown that by post processing the small viscosity approximation u varepsilon pointwise values of u and its derivatives with an error as close to e as desired can be recovered the analysis relies on the adjoint problem of the forward error equation which in this case amounts to a backward linear transport equation with discontinuous coefficients the novelty of our approach is to use a generalized e condition of the forward problem in order to deduce a w 1 infty energy estimate for the discontinuous backward transport equation this in turn leads to e uniform estimate on moments of the error u varepsilon u the approach presented does not follow the characteristics and therefore applies mutatis mutandis to o

the convergence rate of approximate solutions for nonlinear scalar conservation laws let v varepsilon x t varepsilon 0 be a family of approximate solutions for the nonlinear scalar conservation law u t f u x 0 with c 0 1 initial data assume that v varepsilon x t are textit lip stable in the sense that they satisfy oleinik s e entropy condition it is shown that if these approximate solutions are textit lip consistent i e if v varepsilon cdot 0 u cdot 0 textit lip x v t varepsilon f v varepsilon x textit lip x t mathcal o varepsilon then they converge to the entropy solution and the convergence rate estimate v varepsilon cdot t u cdot t textit lip x mathcal o varepsilon holds consequently the familiar l p type and new pointwise error estimates are derived these convergence rate results are demonstrated in the context of entropy satisfying finite difference and glimm s schemes

optimal l 1 rate of convergence for the viscosity method and monotone scheme to piecewise constant solutions with shocks we derive optimal error bounds for the viscosity method and monotone difference schemes to an initial value problem of scalar conservation laws with initial data being a finite number of piecewise constants subject to the initial discontinuities satisfying the entropy conditions it is known that the entropy solution of the problem is piecewise constant with a finite number of interacting shocks satisfying the entropy conditions a rigorous analysis shows that both the viscosity method and monotone schemes to approach the initial value problem have uniform l 1 error bounds of o epsilon and o delta x for the time infty t 0 respectively where epsilon and delta x are their corresponding viscosity coefficient and discrete mesh length the results are improvements over the half order rates of l 1 convergence numerical experiments for the lax friedrichs scheme are presented and numerical results justify the theoretical analysis

a fast fractal image coding based on kick out and zero contrast conditions a fast algorithm for fractal image coding based on a single kick out condition and the zero contrast prediction is proposed in this paper the single kick out condition can avoid a large number of range domain block matches when finding the best matched domain block an efficient method for zero contrast prediction is also proposed which can determine whether the contrast factor for a domain block is zero or not and compute the corresponding difference between the range block and the transformed domain block efficiently and exactly the proposed algorithm can achieve the same reconstructed image quality as the exhaustive search and can greatly reduce the required computation or runtime in addition this algorithm does not need any pre processing step or additional memory for its implementation and can combine with other fast fractal algorithms to further improve the speed experimental results show that the runtime is reduced by about 50 of that of the exhaustive search method when combined with the dct inner product algorithm the required runtime for the algorithm can be further reduced by about 50 the proposed algorithm was also compared to two other fast fractal algorithms experimental results also show that our algorithm achieves a better efficiency and requires a much smaller amount of memory for implementation

the lip stability and error estimates for a relaxation scheme we show the discrete lip stability for a relaxation scheme proposed by jin and xin comm pure appl math 48 1995 pp 235 277 to approximate convex conservation laws equipped with the lip stability we obtain global error estimates in the spaces w s p for 1 leq s leq 1 p 1 leq p leq infty and pointwise error estimates for the approximate solution obtained by the relaxation scheme the proof uses the framework introduced by nessyahu and tadmor siam j numer anal 29 1992 pp 1505 1519 we also show a maximum principle for the relaxation scheme when the initial data are in an equilibrium state

holder norm estimates for elliptic operators on finite and infinite dimensional spaces we introduce a new method for proving the estimate equation omitted where u solves the equation u u f the method can be applied to the laplacian on r it also allows us to obtain similar estimates when we replace the laplacian by an infinite dimensional ornstein uhlenbeck operator or other elliptic operators these operators arise naturally in martingale problems arising from measure valued branching diffusions and from stochastic partial differential equations

stability of systems of viscous conservation laws we consider systems of conservation laws and give conditions for nonlinear stability of viscous shock profiles the analysis applies to classical shocks of arbitrary strength 1998 john wiley sons inc

the sharpness of kuznetsov s o d x l 1 error estimate for monotone difference schemes we derive a lower error bound for monotone difference schemes to the solution of the linear advection equation with bv initial data a rigorous analysis shows that for any monotone difference scheme the lower l 1 error bound is o x where x is the spatial stepsize

a priori error estimates for numerical methods for scalar conservation laws part iii multidimensional flux splitting monotone schemes on non cartesian grids this paper is the third of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed in this paper we consider multidimensional flux splitting monotone schemes defined on non cartesian grids we identify those schemes which are consistent and prove that the l infty 0 t l 1 bbb r d norm of the error goes to zero as delta x 1 2 when the discretization parameter delta x goes to zero moreover we show that nonconsistent schemes can converge at optimal rates of delta x 1 2 because i the conservation form of the schemes and ii the so called consistency of the numerical fluxes allow the regularity properties of the approximate solution to compensate for their lack of consistency

viscous limits for piecewise smooth solutions to systems of conservation laws in this paper we study the zero dissipation problem for a general system of conservation laws with positive viscosity it is shown that if the solution of the problem with zero viscosity is piecewise smooth with a finite number of noninteracting shocks satisfying the entropy condition then there exist solutions to the corresponding system with viscosity that converge to the solutions of the system without viscosity away from shock discontinuities at a rate of order e as the viscosity coefficient e goes to zero the proof uses a matched asymptotic analysis and an energy estimate related to the stability theory for viscous shock profiles

a discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws abstract we present a new relaxation approximation to scalar conservation laws in several space variables by means of semilinear hyperbolic systems of equations with a finite number of velocities under a suitable multidimensional generalization of the whitham relaxation subcharacteristic condition we show the convergence of the approximated solutions to the unique entropy solution of the equilibrium cauchy problem

image coding based on a fractal theory of iterated contractive image transformations the author proposes an independent and novel approach to image coding based on a fractal theory of iterated transformations the main characteristics of this approach are that i it relies on the assumption that image redundancy can be efficiently exploited through self transformability on a block wise basis and ii it approximates an original image by a fractal image the author refers to the approach as fractal block coding the coding decoding system is based on the construction for an original image to encode of a specific image transformation a fractal code which when iterated on any initial image produces a sequence of images that converges to a fractal approximation of the original it is shown how to design such a system for the coding of monochrome digital images at rates in the range of 0 5 1 0 b pixel the fractal block coder has performance comparable to state of the art vector quantizers

a fast and efficient hybrid fractal wavelet image coder the excellent visual quality and compression rate of fractal image coding have limited applications due to exhaustive inherent encoding time this paper presents a new fast and efficient image coder that applies the speed of the wavelet transform to the image quality of the fractal compression fast fractal encoding using fisher s domain classification is applied to the lowpass subband of wavelet transformed image and a modified set partitioning in hierarchical trees spiht coding on the remaining coefficients furthermore image details and wavelet progressive transmission characteristics are maintained no blocking effects from fractal techniques are introduced and the encoding fidelity problem common in fractal wavelet hybrid coders is solved the proposed scheme promotes an average of 94 reduction in encoding decoding time comparing to the pure accelerated fractal coding results the simulations also compare the results to the spiht wavelet coding in both cases the new scheme improves the subjective quality of pictures for high medium low bitrates

discrete kinetic schemes for multidimensional systems of conservation laws we present here some numerical schemes for general multidimensional systems of conservation laws based on a class of discrete kinetic approximations which includes the relaxation schemes by s jin and z xin these schemes have a simple formulation even in the multidimensional case and do not need the solution of the local riemann problems for these approximations we give a suitable multidimensional generalization of the whitham s stability subcharacteristic condition in the scalar multidimensional case we establish the rigorous convergence of the approximated solutions to the unique entropy solution of the equilibrium cauchy problem

convergence rates for relaxation schemes approximating conservation laws in this paper we prove a global error estimate for a relaxation scheme approximating scalar conservation laws to this end we decompose the error into a relaxation error and a discretization error including an initial error omega ep we obtain the rate of convergence of sqrt ep in l1 for the relaxation step the estimate here is independent of the type of nonlinearity in the discretization step a convergence rate of sqrt del x in l1 is obtained these rates are independent of the choice of initial error omega ep thereby we obtain the order 1 2 for the total error

stability of rarefaction waves in viscous media we study the time asymptotic behavior of weak rarefaction waves of systems of conservation laws describing one dimensional viscous media with strictly hyperbolic flux functions our main result is to show that solutions of perturbed rarefaction data converge to an approximate burgers rarefaction wave for initial perturbations w 0 with small mass and localized as w 0 x mathcal o x 1 the proof proceeds by iteration of a pointwise ansatz for the error using integral representations of its various components based on green s functions we estimate the green s functions by careful use of the hopf cole transformation combined with a refined parametrix method as a consequence of our method we also obtain rates of decay and detailed pointwise estimates for the error this pointwise method has been used successfully in studying stability of shock and constant state solutions new features in the rarefaction case are time varying coefficients in the linearized equations and error waves of unbounded mass mathcal o log t these diffusion waves have amplitude mathcal o t 1 2logt in linear degenerate transversal fields and mathcal o t 1 2logt in genuinely nonlinear transversal fields a distinction which is critical in the stability proof

pointwise error estimates for scalar conservation laws with piecewise smooth solutions we introduce a new approach to obtain sharp pointwise error estimates for viscosity approximation and in fact more general approximations to scalar conservation laws with piecewise smooth solutions to this end we derive a transport inequality for an appropriately weighted error function the key ingredient in our approach is a one sided interpolation inequality between classical l1 error estimates and lip stability bounds the one sided interpolation interesting for its own sake enables us to convert a global l1 result into a nonoptimal local estimate this in turn provides the necessary bounds on the coefficients of the above mentioned transport inequality estimates on the weighted error then follow from the maximum principle and a bootstrap argument yields optimal pointwise error bound for the viscosity approximation unlike previous works in this direction our method can deal with finitely many waves with possible collisions moreover in our approach one does not follow the characteristics but instead makes use of the energy method and hence this approach could be extended to other types of approximate solutions

on the regularity of approximate solutions to conservation laws with piecewise smooth solutions in this paper we address the questions of the convergence rate for approximate solutions to conservation laws with piecewise smooth solutions in a weighted w1 1 space convergence rate for the derivative of the approximate solutions is established under the assumption that a weak pointwise error estimate is given in other words we are able to convert weak pointwise error estimates to optimal error bounds in a weighted w1 1 space for convex conservation laws the assumption of a weak pointwise error estimate is verified by tadmor siam j numer anal 28 1991 pp 891 906 therefore one immediate application of our w1 1 convergence theory is that for convex conservation laws we indeed have w1 1 error bounds for the approximate solutions to conservation laws furthermore the co en pointwise error estimates of tadmor and tang siam j numer anal 36 1999 pp 1739 1758 are recovered by the use of the w1 1 convergence result

the submartingale problem for a class of degenerate elliptic operators we consider the degenerate elliptic operator acting on c 2 b functions on 0 d mathcal l f x sum i 1 d a i x x i alpha i frac partial 2 f partial x i 2 x sum i 1 d b i x frac partial f partial x i x where the ai are continuous functions that are bounded above and below by positive constants the bi are bounded and measurable and the alpha i in 0 1 we impose neumann boundary conditions on the boundary of 0 d there will not be uniqueness for the submartingale problem corresponding to mathcal l if we consider however only those solutions to the submartingale problem for which the process spends 0 time on the boundary then existence and uniqueness for the submartingale problem for mathcal l holds within this class our result is equivalent to establishing weak uniqueness for the system of stochastic differential equations rm d x t i sqrt 2a i x t x t i alpha i 2 rm d w i t b i x t rm d t rm d l t x i quad x i t geq 0 where w t i are independent brownian motions and l x i t is a local time at 0 for xi

pointwise convergence to shock waves for viscous conservation laws we are interested in the pointwise behavior of the perturbations of shock waves for viscous conservation laws it is shown that besides a translation of the shock waves and of linear and nonlinear diffusion waves of heat and burgers equations a perturbation also gives rise to algebraically decaying terms which measure the coupling of waves of different characteristic families our technique is a combination of time asymptotic expansion construction of approximate green functions and analysis of nonlinear wave interactions the pointwise estimates yield optimal lp convergence of the perturbation to the shock and diffusion waves 1 p the new approach of obtaining pointwise estimates based on the green functions for the linearized system and the analysis of nonlinear wave interactions is also useful for studying the stability of waves of distinct types and nonclassical shocks these are being explored elsewhere 1997 john wiley sons inc

rarefactions and large time behavior for parabolic equations and monotone schemes we consider the large time behavior of monotone semigroups associated with degenerate parabolic equations and monotone difference schemes for an appropriate class of initial data the solution is shown to converge to rarefaction waves at a determined asymptotic rate

the optimal convergence rate of monotone finite difference methods for hyperbolic conservation laws we are interested in the rate of convergence in l1 of the approximate solution of a conservation law generated by a monotone finite difference scheme kuznetsov has proved that this rate is 1 2 ussr comput math math phys 16 1976 pp 105 119 and topics numer anal iii in proc roy irish acad conf dublin 1976 pp 183 197 and recently teng and zhang have proved this estimate to be sharp for a linear flux siam j numer anal 34 1997 pp 959 978 we prove by constructing appropriate initial data for the cauchy problem that kuznetsov s estimates are sharp for a nonlinear flux as well

viscosity methods for piecewise smooth solutions to scalar conservation laws it is proved that for scalar conservation laws if the flux function is strictly convex and if the entropy solution is piecewise smooth with finitely many discontinuities which includes initial central rarefaction waves initial shocks possible spontaneous formation of shocks in a future time and interactions of all these patterns then the error of viscosity solution to the inviscid solution is bounded by o log in the l 1 norm which is an improvement of the o upper bound if neither central rarefaction waves nor spontaneous shocks occur the error bound is improved to o

existence and uniqueness of traveling waves and error estimates for godunov schemes of conservation laws the existence and uniqueness of the lipschitz continuous traveling wave of godunov s scheme for scalar conservation laws are proved the structure of the traveling waves is studied the approximation error of godunov s scheme on single shock solutions is shown to be o 1 x

an analytic approach to existence and uniqueness for martingale problems in infinite dimensions we prove existence and uniqueness for a class of martingale problems in a hilbert space we solve the associated kolmogorov equation and prove that the corresponding semigroup is determined by a kernel of measures if a schauder type regularity is satisfied

convergence to equilibrium for the relaxation approximations of conservation laws we study the cauchy problem for 2 2 semilinear and quasilinear hyperbolic systems with a singular relaxation term special comparison and compactness properties are established by assuming the subcharacteristic condition therefore we can prove the convergence to equilibrium of the solutions of these problems as the singular perturbation parameter tends to 0 this research was strongly motivated by the recent numerical investigations of s jin and z xin on the relaxation schemes for conservation laws 1996 john wiley sons inc

the convergence rate of godunov type schemes godunov type schemes form a special class of transport projection methods for the approximate solution of nonlinear hyperbolic conservation laws the authors study the convergence rate of such sche