rigidity and tolerance in point processes gaussian zeros and ginibre eigenvalues let be a translation invariant point process on the complex plane c and let d c be a bounded open set we ask the following what does the point configuration out obtained by taking the points of outside d tell us about the point configuration in of inside d we show that for the ginibre ensemble out determines the number of points in in for the translation invariant zero process of a planar gaussian analytic function we show that out determines the number as well as the center of mass of the points in in further in both models we prove that the outside says nothing more about the inside in the sense that the conditional distribution of the inside points given the outside is mutually absolutely continuous with respect to the lebesgue measure on its supporting submanifold

some filter regularization methods for a backward heat conduction problem in this paper a one dimensional backward heat conduction problem is investigated it is well known that such problem is ill posed some filter regularization methods are used to solve it convergence estimates under two a priori bound assumptions for the exact solution are given based on the conditional stabilities finally numerical examples are given to show that our used numerical methods are effective and stable

on fractional powers of generators of fractional resolvent families abstract we show that if a generates a bounded times resolvent family for some 0 2 then a generates an analytic times resolvent family for 0 2 2 and 0 2 and a generalized subordination principle is derived in particular if a generates a bounded times resolvent family for some 1 2 then a 1 generates an analytic c 0 semigroup such relations are applied to study the solutions of cauchy problems of fractional order and first order

a novel multigrid based preconditioner for heterogeneous helmholtz problems an iterative solution method in the form of a preconditioner for a krylov subspace method is presented for the helmholtz equation the preconditioner is based on a helmholtz type differential operator with a complex term a multigrid iteration is used for approximately inverting the preconditioner the choice of multigrid components for the corresponding preconditioning matrix with a complex diagonal is validated with fourier analysis multigrid analysis results are verified by numerical experiments high wavenumber helmholtz problems in heterogeneous media are solved indicating the performance of the preconditioner

cauchy problems for fractional differential equations with riemann liouville fractional derivatives abstract in this paper we are concerned with cauchy problems of fractional differential equations with riemann liouville fractional derivatives in infinite dimensional banach spaces we introduce the notion of fractional resolvent obtain some its properties and present a generation theorem for exponentially bounded fractional resolvents moreover we prove that a homogeneous order cauchy problem is well posed if and only if its coefficient operator is the generator of an order fractional resolvent and we give sufficient conditions to guarantee the existence and uniqueness of weak solutions and strong solutions of an inhomogeneous order cauchy problem

on a backward parabolic problem with local lipschitz source abstract we consider the regularization of the backward in time problem for a nonlinear parabolic equation in the form u t a u t f u t t u 1 where a is a positive self adjoint unbounded operator and f is a local lipschitz function as known it is ill posed and occurs in applied mathematics e g in neurophysiological modeling of large nerve cell systems with action potential f in mathematical biology a new version of quasi reversibility method is described we show that the regularized problem with a regularization parameter 0 is well posed and that its solution u t converges on 0 1 to the exact solution u t as 0 these results extend some earlier works on the nonlinear backward problem

a comparison of regularizations for an ill posed problem we consider numerical methods for a quasi boundary value regularization of the backward parabolic problem given by u t au 0 0 t t u t f where a is positive self adjoint and unbounded the regularization due to clark and oppenheimer perturbs the final value u t by adding u 0 where is a small parameter we show how this leads very naturally to a reformulation of the problem as a second kind fredholm integral equation which can be very easily approximated using methods previously developed by ames and epperson error estimates and examples are provided we also compare the regularization used here with that from ames and epperson

the fractional d alembert s formulas abstract in this paper we develop generalized d alembert s formulas for abstract fractional integro differential equations and fractional differential equations on banach spaces some examples are given to illustrate our abstract results and the probability interpretations of these fractional d alembert s formulas are also given moreover we also provide d alembert s formulas for abstract fractional telegraph equations

fractional calculus and continuous time finance in this paper we present a rather general phenomenological theory of tick by tick dynamics in financial markets many well known aspects such as the levy scaling form follow as particular cases of the theory the theory fully takes into account the non markovian and non local character of financial time series predictions on the long time behaviour of the waiting time probability density are presented finally a general scaling form is given based on the solution of the fractional diffusion equation

a fractional landweber method for solving backward time fractional diffusion problem abstract in this paper we propose a fractional landweber regularization method for solving a backward time fractional diffusion problem choice rules for both a priori and a posterior regularization parameters are discussed in deriving the corresponding convergence proofs for the backward problem for verification on the theoretical estimates several numerical examples are constructed and compared with some existing works

a new iterative solver for the time harmonic wave equation the time harmonic wave equation also known as the helmholtz equation is obtained if the constant density acoustic wave equation is transformed from the time domain to the frequency domain its discretization results in a large sparse linear system of equations in two dimensions this system can be solved efficiently by a direct method in three dimensions direct methods cannot be used for problems of practical sizes because the computational time and the amount of memory required become too large iterative methods are an alternative these methods are often based on a conjugate gradient iterative scheme with a preconditioner that accelerates its convergence the iterative solution of the time harmonic wave equation has long been a notoriously difficult problem in numerical analysis recently a new preconditioner based on a strongly damped wave equation has heralded a breakthrough the solution of the linear system associated with the preconditioner is approximated by another iterative method the multigrid method the multigrid method fails for the original wave equation but performs well on the damped version the performance of the new iterative solver is investigated on a number of 2d test problems the results suggest that the number of required iterations increases linearly with frequency even for a strongly heterogeneous model where earlier iterative schemes fail to converge complexity analysis shows that the new iterative solver is still slower than a time domain solver to generate a full time series we compare the time domain numeric results obtained using the new iterative solver with those using the direct solver and conclude that they agree very well quantitatively the new iterative solver can be applied straightforwardly to 3d problems

the fractional order governing equation of levy motion a governing equation of stable random walks is developed in one dimension this fokker planck equation is similar to and contains as a subset the second order advection dispersion equation ade except that the order a of the highest derivative is fractional e g the 1 65th derivative fundamental solutions are levy s a stable densities that resemble the gaussian except that they spread proportional to time 1 a have heavier tails and incorporate any degree of skewness the measured variance of a plume undergoing levy motion would grow faster than fickian plume at a rate of time 2 a where 0 a 2 the equation is parsimonious since the parameters are not functions of time or distance the scaling behavior of plumes that undergo levy motion is accounted for by the fractional derivatives which are appropriate measures of fractal functions in real space the fractional derivatives are integrodifferential operators so the fractional ade describes a spatially nonlocal process that is ergodic and has analytic solutions for all time and space

determinantal point processes associated with hilbert spaces of holomorphic functions we study determinantal point processes on mathbb c induced by the reproducing kernels of generalized fock spaces as well as those on the unit disc mathbb d induced by the reproducing kernels of generalized bergman spaces in the first case we show that all reduced palm measures of the same order are equivalent the radon nikodym derivatives are computed explicitly using regularized multiplicative functionals we also show that these determinantal point processes are rigid in the sense of ghosh and peres hence reduced palm measures of different orders are singular in the second case we show that all reduced palm measures of all orders are equivalent the radon nikodym derivatives are computed using regularized multiplicative functionals associated with certain blaschke products the quasi invariance of these determinantal point processes under the group of diffeomorphisms with compact supports follows as a corollary

the pseudospectral method comparisons with finite differences for the elastic wave equation the pseudospectral or fourier method has been used recently by several investigators for forward seismic modeling the method is introduced here in two different ways as a limit of finite differences of increasing orders and by trigonometric interpolation an argument based on spectral analysis of a model equation shows that the pseudospectral method for the accuracies and integration times typical of forward elastic seismic modeling may require in each space dimension as little as a quarter the number of grid points compared to a fourth order finite difference scheme and one sixteenth the number of points as a second order finite difference scheme for the total number of points in two dimensions these factors become 1 16 and 1 256 respectively in three dimensions they become 1 64 and 1 4 096 respectively in a series of test calculations on the two dimensional elastic wave equation only minor degradations are found in cases with variable coefficients and discontinuous interfaces

regularized solutions for abstract volterra equations abstract we introduce a two kernel dependent family of strong continuous operators defined in a banach space which allows us to consider in an unified treatment the notions of among others c 0 semigroups of operators cosine families n times integrated semigroups resolvent families and k generalized solutions the results are applied to the study of existence and uniqueness of solutions for the volterra equation of convolution type u f a au in the case a is not necessarily densely defined examples for equations defined in l p spaces are also given

regularization and convergence for ill posed backward evolution equations in banach spaces abstract this work is concerned with a mathematical study of ill posed backward evolution equations associated with densely defined linear differential operators in banach spaces a general approach is presented to investigate the convergence and stability of a class of regularized solutions for ill posed backward evolution equations associated with sectorial or half strip operators generalized concepts of qualification pairs and index functions are introduced to characterize the explicit convergence rates of the concerned regularized solutions applications of our results to general backward evolution equations are also investigated

spectral methods in time for hyperbolic equations a pseudospectral numerical scheme for solving linear periodic hyperbolic problems is described it has infinite accuracy both in time and in space the high accuracy in time is achieved without increasing the computational work and memory space which is needed for a regular one step explicit scheme the algorithm is shown to be optimal in the sense that among all the explicit algorithms of a certain class it requires the least amount of work to achieve a certain given resolution the class of algorithms referred to consists of all explicit schemes which may be represented as a polynomial in the spatial operator

characterization of compactness for resolvents and its applications abstract in this paper we derive some characterizations of compactness for resolvents as well as the subordination principle associated with the compactness as applications we obtain the existence of fractional evolution equations by using schauder s fixed point theorem a simple example about fractional heat equation is also given to illustrate our theory

three dimensional elastic modeling by the fourier method earlier work on three dimensional forward modeling is extended to elastic waves using the equations of con servation of momentum and the stress strain relations for an isotropic elastic medium undergoing infinitesimal deformation in addition to arbitrary compressional or p wave velocity and density variation in lateral and vertical directions elastic modeling permits shear or s wave velocity variation as well the elastic wave equation is solved using a generalization of the method for the acoustic case computation of each time step begins by computing six strain components by per forming nine spatial partial differentiation operations on the three displacement components from the pre vious time step the six strains and two lame constants are linearly combined to yield six stress components

a nonlinear parabolic equation backward in time regularization with new error estimates abstract consider a nonlinear backward parabolic problem in the form u t a u t f t u t u t g where a is a positive self adjoint unbounded operator based on the fundamental solution to the parabolic equation we propose to solve this problem by the fourier truncated method which generates a well posed integral equation then the well posedness of the proposed regularizing problem and convergence property of the regularizing solution to the exact one are proven our regularizing scheme can be considered a new regularization with the advantage of a relatively small amount of computation compared with the quasi reversibility or quasi boundary value regularizations error estimates for this method are provided together with a selection rule for the regularization parameter these errors show that our method works effectively

the local and global existence of solutions for a time fractional complex ginzburg landau equation abstract we study the following time fractional complex nonlinear ginzburg landau equation e i 0 c d t u u e i u p 1 u x r n t 0 u 0 x u 0 x x r n where 0 1 r 2 2 p 1 u 0 l q r n q q c n p 1 2 and q 1 is a complex valued function and d t 0 c u t 0 i t 1 u t x u 0 x where i t 1 0 denotes a left riemann liouville fractional integral of order 1 by defining two operators and establishing some estimates of them we prove the well posedness of the mild solution for this problem in c 0 t l q r n and l 2 r q n r q 0 t l r r n where r satisfies 1 q 1 r 2 n moreover we also obtain the existence of global solutions when u 0 l q c r n is sufficiently small

a new regularized method for two dimensional nonhomogeneous backward heat problem we consider the problem of finding from the final data u x y t g x y the initial data u x y 0 of the temperature function u x y t x y i 0 p x 0 p t 0 t satisfying the following systemu t u x x u y y f x y t x y t ix 0 t u 0 y t u p y t u x 0 t u x p t 0 x y t ix 0 t the problem is severely ill posed in this paper a simple and convenient new regularization method for solving this problem is considered meanwhile some quite sharp error estimates between the approximate solution and exact solution are provided a numerical example also shows that the method works effectively

brownian time processes the pde connection and the half derivative generator we introduce a class of interesting stochastic processes based on brownian time processes these are obtained by taking markov processes and replacing the time parameter with the modulus of brownian motion they generalize the iterated brownian motion ibm of burdzy and the markov snake of le gall and they introduce new interesting examples after defining brownian time processes we relate them to fourth order parabolic partial differential equations pde s we then study their exit problem as they exit nice domains in r d and connect it to elliptic pde s we show that these processes have the peculiar property that they solve fourth order parabolic pde s but their exit distribution at least in the standard brownian time process case solves the usual second order dirichlet problem we recover fourth order pde s in the elliptic setting by encoding the iterative nature of the brownian time process through its exit time in a standard brownian motion we also show that it is possible to assign a formal generator to these non markovian processes by giving such a generator in the half derivative sense

time evolution of the wave equation using rapid expansion method forward modeling of seismic data and reverse time migration are based on the time evolution of wavefields for the case of spatially varying velocity we have worked on two approaches to evaluate the time evolution of seismic wavefields an exact solution for the constant velocity acoustic wave equation can be used to simulate the pressure response at any time for a spatially varying velocity a one step method can be developed where no intermediate time responses are required using this approach we have solved for the pressure response at intermediate times and have developed a recursive solution the solution has a very high degree of accuracy and can be reduced to various finite difference time derivative methods depending on the approximations used although the two approaches are closely related each has advantages depending on the problem being solved

time step n tupling for wave equations abstractwe have constructed novel temporal discretizations for wave equations we first select an explicit time integrator that is of second order leading to classic time marching schemes in which the next value of the wavefield at the discrete time t t is computed from current values known at time t and the previous time t t then we determine how the time step can be doubled tripled or generally n tupled producing a new time stepping method in which the next value of the wavefield at the discrete time t n t is computed from current values known at time t and the previous time t n t in between time values of the wavefield are eliminated using the fourier method to calculate space derivatives the new time integrators allow larger stable time steps than traditional time integrators however like the lax wendroff procedure they require more computational effort per time step because the new schemes are developed from the classic second order time stepping scheme they will have the same proper

a helmholtz iterative solver for 3d seismic imaging problems a preconditioned iterative solver for the 3d frequency domain wave equation applied to seismic problems is evaluated the preconditioner corresponds to an approximate inverse of a heavily damped wave equation deduced from the undamped wave equation the approximate inverse is computed with one multigrid cycle numerical results show that the method is robust and that the number of iterations increases roughly linearly with frequency when the grid spacing is adapted to keep a constant number of discretization points per wavelength to evaluate the relevance of this iterative solver the number of floating point operations required for two imaging problems are roughly evaluated this rough estimate indicates that the time domain migration approach is more than one order of magnitude faster the full wave form tomography based on a least squares formulation and a scale separation approach has the same complexity in both domains

conditional measures of generalized ginibre point processes the main result of this paper is that conditional measures of generalized ginibre point processes with respect to the configuration in the complement of a bounded open subset on c are orthogonal polynomial ensembles with weights found explicitly an especially simple formula for conditional measures is obtained in the particular case of radially symmetric determinantal point processes including the classical ginibre point process

a new numerical method for backward parabolic problems in the maximum norm setting a new method for solving numerically backward parabolic problems is proposed as usual for this kind of ill posed problems it is assumed that an a priori bound for the solution is available the algorithm consists of two basic steps first a standard forward integration is performed in order to approximate the solution at suitable future time levels second a holomorphic recovery procedure is carried out providing the required approximations for the preceding times the analysis is valid in the maximum norm setting and rigorous estimates are derived among other advantages the method can also be applied to nonlinear problems and it produces a continuous output some numerical illustrations are presented

a parallel method for backward parabolic problems based on the laplace transformation a parallel method for time discretization of backward parabolic problems is proposed the problem is reformulated to a set of helmholtz type problems with a parameter on a suitably chosen contour in the complex plane after solving the resulting elliptic equations which can be solved in parallel we obtain a regularized solution with high frequency terms cut off by the inverse laplace transforms without requiring the knowledge of the eigenfunctions of the differential operator since the regularized solution is obtained without artificial perturbation and high frequency components of the noise are suppressed the quality of the solution is improved significantly compared to those obtained by other methods two different numerical inversions of laplace transforms with an arbitrary high order of accuracy and spectral accuracy respectively are used error estimates and numerical examples are presented

absolute continuity and singularity of palm measures of the ginibre point process we prove a dichotomy between absolute continuity and singularity of the ginibre point process mathsf g and its reduced palm measures mathsf g mathbf x mathbf x in mathbb c ell ell 0 1 2 ldots namely reduced palm measures mathsf g mathbf x and mathsf g mathbf y for mathbf x in mathbb c ell and mathbf y in mathbb c n are mutually absolutely continuous if and only if ell n they are singular each other if and only if ell not n furthermore we give an explicit expression of the radon nikodym density d mathsf g mathbf x d mathsf g mathbf y for mathbf x mathbf y in mathbb c ell

modified quasi reversibility method for final value problems in banach spaces this paper is concerned with the final value problem associated with a linear operator a in a banach space where a is the generator of a uniformly bounded analytic semigroup based on the delaubenfels functional calculus we use new quasi reversibility method introduced by boussetila and rebbani recently to form an approximate problem we obtain some results in a banach space similar to those in a hilbert space

characterizations of domains of fractional powers for non negative operators abstract it is known that a closed linear operator a defined densely on a banach space x is non negative if and only if a generates a bounded analytic times resolvent family for some 0 2 in this paper by using such resolvent families as well as the komatsu representations we characterize systematically the domains of fractional powers of non negative operators on banach spaces gaps between the subset 0 d a and the whole space x are discussed as well

a spectral scheme for wave propagation simulation in 3 d elastic anisotropic media this work presents a new scheme for wave propagation simulation in three dimensional 3 d elastic anisotropic media the algorithm is based on the rapid expansion method rem as a time integration algorithm and the fourier pseudospectral method for computation of the spatial derivatives the rem expands the evolution operator of the second order wave equation in terms of chebychev polynomials constituting an optimal series expansion with exponential convergence the modeling allows arbitrary elastic coefficients and density in lateral and vertical directions numerical methods which are based on finite difference techniques in time and space are not efficient when applied to realistic 3 d models since they require considerable computer memory and time to obtain accurate results on the other hand the fourier method permits a significant reduction of the working space and the rem algorithm gives machine accuracy with half the computational effort as the usual second order temporal differencing sch

mild solutions to integro differential equations in banach spaces abstract in this paper we study the existence and uniqueness of mild solutions to integro differential equations in terms of a resolvent operator on the interval 0 2 and on the real line moreover we characterize the spectrum of the resolvent family that solves the volterra equation u a u a a u f in terms of their mild periodic solutions

a modified method of stochastic regularization for an ill posed cauchy problem abstract this paper is concerned with the ill posed cauchy problem associated with a densely defined linear operator in a banach space we investigate a method of stochastic regularization for this ill posed cauchy problem by using fractional powers and functional calculus for the sectorial operators

using the rapid expansion method for accurate time stepping in modeling and reverse time migration reverse time migration is based on seismic forward modeling algorithms where spatial derivatives usually are calculated by finite differences or by the fourier method time integration in general is done by finite difference time stepping of low orders if the spatial derivatives are calculated by high order methods and time stepping is based on low order methods there is an imbalance that might require that the time step size needs to be very small to avoid numerical dispersion as a result computing times increase using the rapid expansion method rem avoids numerical dispersion if the number of expansion terms is chosen properly comparisons with analytical solutions show that the rem is preferable especially at larger propagation times for reverse time migration the rem needs to be applied in a time stepping manner this is necessary because the original implementation based on very large time spans requires that the source term is separable in space and time this is not appropriate for reverse time migration where the sources have different time histories in reverse time migration it might be desirable to use the poynting vector information to estimate opening angles to improve the quality of the image in the solution of the wave equation this requires that one calculates not only the pressure wavefield but also its time derivative the rapid expansion method can be extended easily to provide this time derivative with negligible extra cost

on a backward heat problem with time dependent coefficient regularization and error estimates in this paper we consider a homogeneous backward heat conduction problem which appears in some applied subjects this problem is ill posed in the sense that the solution if it exists does not depend continuously on the final data a new regularization method is applied to formulate regularized solutions which are stably convergent to the exact ones with holder estimates a numerical example shows that the computational effect of the method is all satisfactory

logarithmic well posed approximation of the backward heat equation in banach space abstract in this paper we apply a logarithmic approximation introduced by boussetila and rebbani in order to study an ill posed cauchy problem in banach space the proposed method induces a well posed problem whose approximation parameter yields continuous dependence on modeling throughout properties of holomorphic semigroups are utilized culminating in the motivating application which is the backward heat equation in l p r 1 p

determinantal processes and completeness of random exponentials the critical case for a locally finite point set lambda subset mathbb r consider the collection of exponential functions given by mathcal e lambda e i lambda x lambda in lambda we examine the question whether mathcal e lambda spans the hilbert space l 2 pi pi when lambda is random for several point processes of interest this belongs to a certain critical case of the corresponding question for deterministic lambda about which little is known for lambda the continuum sine kernel process obtained as the bulk limit of gue eigenvalues we establish that mathcal e lambda is indeed complete almost surely we also answer an analogous question on mathbb c for the ginibre ensemble arising as weak limits of the spectra of certain non hermitian gaussian random matrices in fact we establish completeness for any rigid determinantal point process in a general setting in addition we partially answer two questions of lyons and steif about stationary determinantal processes on mathbb z d

ill posed problems associated with sectorial operators in banach space abstract ill posed problems of the form u t a u h 0 t t u 0 are studied in banach space under two related scenarios both of which consider a well posed logarithmic approximation in the first scenario a generates a bounded holomorphic semigroup of angle 2 and so a is sectorial of angle 0 in the second where may be relaxed to any value in 0 2 we impose an additional assumption that e a is sectorial regularization results for the original ill posed problem are then established in the linear case where h h t and subsequently extended to nonlinear problems where h is replaced by h t u t we compare the two cases in terms of regularization convergence rates and apply our results to backward heat equations both linear and nonlinear in l p r n 1 p

future sequential regularization methods for iii posed volterra equations applications to the inverse heat conduction problem abstract we develop a theoretical context in which to study the future sequential regularization method developed by beck for the inverse heat conduction problem in the process we generalize beck s ideas and view that method as one in a large class of regularization methods in which the solution of an ill posed first kind volterra equation is seen to be the limit of a sequence of solutions of well posed second kind volterra equations such techniques are important because standard regularization methods such as tikhonov regularization tend to transform a naturally sequential volterra problem into a full domain fredholm problem destroying the underlying causal nature of the volterra model and leading to inefficient global approximation strategies in contrast the ideas we present here preserve the original volterra structure of the problem and thus can lead to easily implemented localized approximation strategies theoretical properties of these methods are discussed and proofs of convergence are given

a kernel based method for the approximate solution of backward parabolic problems a method for approximating the solution of self adjoint parabolic equations backwards in time is presented the technique uses the green s function of the parabolic equation to formulate the problem as a first kind fredholm integral equation and then does a regularized least squares solution of the resulting discrete system error estimates and examples are presented direct knowledge of the kernel of the differential operator is not required

a multigrid method enhanced by krylov subspace iteration for discrete helmholtz equations standard multigrid algorithms have proven ineffective for the solution of discretizations of helmholtz equations in this work we modify the standard algorithm by adding gmres iterations at coarse levels and as an outer iteration we demonstrate the algorithm s effectiveness through theoretical analysis of a model problem and experimental results in particular we show that the combined use of gmres as a smoother and outer iteration produces an algorithm whose performance depends relatively mildly on wave number and is robust for normalized wave numbers as large as 200 for fixed wave numbers it displays grid independent convergence rates and has costs proportional to the number of unknowns

time discretization of fractional subdiffusion equations via fractional resolvent operators abstract in this work we study time discretization of subdiffusion equations that is fractional differential equations of order 0 1 assuming that a is the generator of a fractional resolvent family s t t 0 which allows to write the solution to the subdiffusion equation t u t a u t f t as a variation of constants formula we find an interesting connection between s t t 0 and a discrete resolvent family s n n n and then by using the properties of s t t 0 we study the existence of solutions to the discrete subdiffusion equation c u n a u n f n n n where based on the backward euler method for a 0 given c u n is an approximation of t u t at time t n n we study simultaneously the fractional derivative in the caputo and riemann liouville sense we also provide error estimates and some experiments to illustrate the results

the quasi invariance property for the gamma kernel determinantal measure abstract the gamma kernel is a projection kernel of the form a x b y b x a y x y where a and b are certain functions on the one dimensional lattice expressed through euler s function the gamma kernel depends on two continuous parameters its principal minors serve as the correlation functions of a determinantal probability measure p defined on the space of infinite point configurations on the lattice as was shown earlier a borodin g olshanski adv math 194 2005 141 202 arxiv math ph 0305043 p describes the asymptotics of certain ensembles of random partitions in a limit regime theorem the determinantal measure p is quasi invariant with respect to finitary permutations of the nodes of the lattice this result is motivated by an application to a model of infinite particle stochastic dynamics

regularization for ill posed cauchy problems associated with generators of analytic semigroups this paper is concerned with the ill posed cauchy problem associated with a densely defined linear operator a in a banach space our main result is that if a is the generator of an analytic semigroup then there exists a family of regularizing operators for such an ill posed cauchy problem by using the quasi reversibility method fractional powers and semigroups of linear operators the applications to ill posed partial differential equations are also given

continuous dependence on modeling for nonlinear ill posed problems abstract the nonlinear ill posed cauchy problem d u d t a u t h t u t u 0 where a is a positive self adjoint operator on a hilbert space h h and h 0 t h h is a uniformly lipschitz function is studied in order to establish continuous dependence results for solutions to approximate well posed problems the authors show here that solutions of the problem if they exist depend continuously on solutions to corresponding approximate well posed problems if certain stabilizing conditions are imposed the approximate problem is given by d v d t f a v t h t v t v 0 for suitable functions f the main result is that u t v t c 1 t t m t t where c and m are computable constants independent of and 0 1 this work extends to the nonlinear case earlier results by the authors and by ames and hughes

efficient 3d frequency response modeling with spectral accuracy by the rapid expansion method frequency responses of seismic wave propagation can be obtained either by directly solving the frequency domain wave equations or by transforming the time domain wavefields using the fourier transform the former approach requires solving systems of linear equations which becomes progressively difficult to tackle for larger scale models and for higher frequency components on the contrary the latter approach can be efficiently implemented using explicit time integration methods in conjunction with running summations as the computation progresses commonly used explicit time integration methods correspond to the truncated taylor series approximations that can cause significant errors for large time steps the rapid expansion method rem uses the chebyshev expansion and offers an optimal solution to the second order in time wave equations when applying the fourier transform to the time domain wavefield solution computed by the rem we can derive a frequency response modeling formula that has the same form as the original time domain rem equation but with different summation coefficients in particular the summation coefficients for the frequency response modeling formula corresponds to the fourier transform of those for the time domain modeling equation as a result we can directly compute frequency responses from the chebyshev expansion polynomials rather than the time domain wavefield snapshots as do other time domain frequency response modeling methods when combined with the pseudospectral method in space this new frequency response modeling method can produce spectrally accurate results with high efficiency

brownian subordinators and fractional cauchy problems a brownian time process is a markov process subordinated to the absolute value of an independent one dimensional brownian motion its transition densities solve an initial value problem involving the square of the generator of the original markov process an apparently unrelated class of processes emerging as the scaling limits of continuous time random walks involves subordination to the inverse or hitting time process of a classical stable subordinator the resulting densities solve fractional cauchy problems an extension that involves fractional derivatives in time in this paper we will show a close and unexpected connection between these two classes of processes and consequently an equivalence between these two families of partial differential equations

random point fields associated with certain fredholm determinants i fermion poisson and boson point processes we introduce certain classes of random point fields including fermion and boson point processes which are associated with fredholm determinants of certain integral operators and study some of their basic properties limit theorems correlation functions palm measures etc also we propose a conjecture on an analogue of the determinant and permanent

optimal error bound and simplified tikhonov regularization method for a backward problem for the time fractional diffusion equation in this paper we consider a backward problem for a time fractional diffusion equation such a problem is ill posed the optimal error bound for the problem under a source condition is analyzed a simplified tikhonov regularization method is utilized to solve the problem and its convergence rates are analyzed under an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule respectively numerical examples show that the proposed regularization method is effective and stable and both parameter choice rules work well

a backward parabolic equation with a time dependent coefficient regularization and error estimates we consider the problem of determining the temperature u x t for x t 0 p x 0 t in the parabolic equation with a time dependent coefficient this problem is severely ill posed i e the solution if it exists does not depend continuously on the given data in this paper we use a modified method for regularizing the problem and derive an optimal stability estimation a numerical experiment is presented for illustrating the estimate

3d finite difference frequency domain modeling of visco acoustic wave propagation using a massively parallel direct solver a feasibility study we present a finite difference frequency domain method for 3d visco acoustic wave propagation modeling in the frequency domain the underlying numerical problem is the resolution of a large sparse system of linear equations whose right hand side term is the source this system is solved with a massively parallel direct solver we first present an optimal 3d finite difference stencil for frequency domain modeling the method is based on a parsimonious staggered grid method differential operators are discretized with second order accurate staggered grid stencils on different rotated coordinate systems to mitigate numerical anisotropy an antilumped mass strategy is implemented to minimize numerical dispersion the stencil incorporates 27 grid points and spans two grid intervals dispersion analysis shows that four grid points per wavelength provide accurate simulations in the 3d domain to assess the feasibility of the method for frequency domain full waveform inversion we computed simulations in the 3d seg eage overthrust model for frequencies 5 7 and 10 hz results confirm the huge memory requirement of the factorization several hundred figabytes but also the cpu efficiency of the resolution phase few seconds per shot heuristic scalability analysis suggests that the memory complexity of the factorization is o 35n 4 for a n 3 grid our method may provide a suitable tool to perform frequency domain full waveform inversion using a large distributed memory platform further investigation is still necessary to assess more quantitatively the respective merits and drawbacks of time and frequency domain modeling of wave propagation to perform 3d full waveform inversion

random point fields associated with certain fredholm determinants ii fermion shifts and their ergodic and gibbs properties we construct and study a family of probability measures on the configuration space over countable discrete space associated with nonnegative definite symmetric operators via determinants under a mild condition they turn out unique gibbs measures also some ergodic properties including the entropy positivity are discussed in the lattice case