global well posedness for the two dimensional nonlinear boussinesq equations with vertical dissipation abstract the present paper is devoted to the study of the global well posedness for the two dimensional nonlinear boussinesq equations with vertical dissipation in the absence of horizontal dissipation we establish a growth estimate on vertical component of velocity that is sup p 2 u 2 t l p p log p which is close to u 2 t l and is bounded via the low high decomposition technique this together with the smoothing effect in vertical direction enables us to obtain the h 1 estimate for velocity based on this we prove the existence and uniqueness of classical solution without smallness assumptions in addition we also discuss the global well posedness result for the rough initial data

wavelet like bases for the fast solutions of second kind integral equations a class of vector space bases is introduced for the sparse representation of discretizations of integral operators an operator with a smooth nonoscillatory kernel possessing a finite number of singularities in each row or column is represented in these bases as a sparse matrix to high precision a method is presented that employs these bases for the numerical solution of second kind integral equations in time bounded by o n log 2 n where n is the number of points in the discretization numerical results are given which demonstrate the effectiveness of the approach and several generalizations and applications of the method are discussed

global existence and asymptotic behavior for the 3d generalized hall mhd system abstract in this paper we consider the global existence for the 3d generalized hall mhd equations with fractional dissipative terms u and b under small initial data in the setting of sobolev norms with lower regularity for the global existence we enlarge the range of dissipative exponents from 1 7 6 to 1 3 2 which established in a recent work in addition the long time behavior and rates of decay for both the solutions and higher derivatives in different sobolev spaces are obtained by using the fourier splitting method which extends the previous work by chae and schonbek

denoising of dynamic contrast enhanced mr images using dynamic nonlocal means this paper presents a new algorithm for denoising dynamic contrast enhanced dce mr images it is a novel variation on the nonlocal means nlm algorithm the algorithm called dynamic nonlocal means dnlm exploits the redundancy of information in the temporal sequence of images empirical evaluations of the performance of the dnlm algorithm relative to seven other denoising methods simple gaussian filtering the original nlm algorithm a trivial extension of nlm to include the temporal dimension bilateral filtering anisotropic diffusion filtering wavelet adaptive multiscale products threshold and traditional wavelet thresholding are presented the evaluations include quantitative evaluations using simulated data and real data 20 dce mri data sets from routine clinical breast mri examinations as well as qualitative evaluations using the same real data 24 observers 14 image signal processing specialists 10 clinical breast mri radiographers the results of the quantitative evaluation using the simulated data show that the dnlm algorithm consistently yields the smallest mse between the denoised image and its corresponding original noiseless version the results of the quantitative evaluation using the real data provide evidence at the 0 05 level of significance that the dnlm algorithm yields the smallest mse between the denoised image and its corresponding original noiseless version the results of the qualitative evaluation provide evidence at the 0 05 level of significance that the dnlm algorithm performs visually better than all of the other algorithms collectively the qualitative and quantitative results suggest that the dnlm algorithm more effectively attenuates noise in dce mr images than any of the other algorithms

ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

global well posedness for a boussinesq navier stokes system with critical dissipation abstract in this paper we study a fractional diffusion boussinesq model which couples a navier stokes type equation with fractional diffusion for the velocity and a transport equation for the temperature we establish global well posedness results with rough initial data

on the numerical solution of two point boundary value problems abstract this paper presents a new numerical method for the solution of linear two point boundary value problems of ordinary differential equations after reducing the differential equation to a second kind integral equation the latter are discretized via a high order nystrom scheme a somewhat involved analytical apparatus is then constructed which allows for the solution of the discrete system using o n dot p2 operations where n is the number of nodes on the interval and p is the desired order of convergence thus the advantages of the integral equation formulation small condition number insensitivity to boundary layers insensitivity to end point singularities etc are retained while achieving a computational efficiency previously available only to finite difference or finite element methods keywords two point boundary value problems integral equations chebyshev polynomials approximation theory

scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

two denoising methods by wavelet transform to wavelet based noise reduction methods are discussed first we improve the traditional spatially selective noise filtration technique proposed by xu et al 1994 second we introduce a new threshold based denoising algorithm based on undecimated discrete wavelet transform simulations and comparisons are given

an efficient spectral method for ordinary differential equations with rational function coefficients the authors present some relations that allow the efficient approximate inversion of linear differential operators with rational function coefficients they employ expansions in terms of a large class of orthogonal polynomial families including all the classical orthogonal polynomials these families obey a simple 3 term recurrence relation for differentiation which implies that on an appropriately restricted domain the differentiation operator has a unique banded inverse the inverse is an integration operator for the family and it is simply the tridiagonal coefficient matrix for the recurrence since in these families convolution operators i e matrix representations of multiplication by a function are banded for polynomials the authors are able to obtain a banded representation for linear differential operators with rational coefficients this leads to a method of solution of initial or boundary value problems that besides having an operation count that scales linearly with the order of truncation n is computationally well conditioned among the applications considered is the use of rational maps for the resolution of sharp interior layers 19 refs 2 figs 3 tabs

numerical methods for stiff two point boundary value problems we consider the two point boundary value problem for stiff systems of ordinary differential equations for systems that can be transformed to essentially diagonally dominant form with appropriate smoothness conditions a priori estimates are obtained problems with turning points can be treated with this theory and we discuss this in detail we give robust difference approximations and present error estimates for these schemes in particular we give a detailed description of how to transform a general system to essentially diagonally dominant form and then stretch the independent variable so that the system will satisfy the correct smoothness conditions numerical examples are presented for both linear and nonlinear problems

spectral integration and two point boundary value problems a numerical method for two point boundary value problems with constant coefficients is developed which is based on integral equations and the spectral integration matrix for chebyshev nodes the method is stable achieves superalgebraic convergence and requires o n log n operations where n is the number of nodes in the discretization although stable spectral methods have been constructed in the past they have generally been based on reformulating the recurrence relations obtained through spectral differentiation in an attempt to avoid the ill conditioning introduced by that process

noise removal using fourth order partial differential equation with applications to medical magnetic resonance images in space and time we introduce a new method for image smoothing based on a fourth order pde model the method is tested on a broad range of real medical magnetic resonance images both in space and time as well as on nonmedical synthesized test images our algorithm demonstrates good noise suppression without destruction of important anatomical or functional detail even at poor signal to noise ratio we have also compared our method with related pde models

global regularity results for the 2d boussinesq equations with vertical dissipation abstract the two dimensional 2d incompressible boussinesq equations model geophysical fluids and play an important role in the study of the raleigh bernard convection mathematically this 2d system retains some key features of the 3d navier stokes and euler equations such as the vortex stretching mechanism the issue of whether the 2d boussinesq equations always possess global in time classical solutions can be difficult when there is only partial dissipation or no dissipation at all this paper obtains the global regularity for two partial dissipation cases and proves several global a priori bounds for two other prominent partial dissipation cases these results take us one step closer to a complete resolution of the global regularity issue for all the partial dissipation cases involving the 2d boussinesq equations

wavelet based rician noise removal for magnetic resonance imaging it is well known that magnetic resonance magnitude image data obey a rician distribution unlike additive gaussian noise rician noise is signal dependent and separating signal from noise is a difficult task rician noise is especially problematic in low signal to noise ratio snr regimes where it not only causes random fluctuations but also introduces a signal dependent bias to the data that reduces image contrast this paper studies wavelet domain filtering methods for rician noise removal we present a novel wavelet domain filter that adapts to variations in both the signal and the noise

a high order adaptive numerical method for recirculating flows at large reynolds number abstract a numerical method is constructed for two dimensional navier stokes flows in a circular domain adaptive computational grids are employed to resolve the boundary layer and the boundary vorticity values are obtained in high order of accuracy numerical results show a quite good agreement with the asymptotic expansion calculation kim siam j appl math 58 5 1998 1394 1413 extension to general shape domains is briefly addressed

global well posedness of the 2d boussinesq equations with vertical dissipation we prove the global well posedness of the two dimensional boussinesq equations with only vertical dissipation the initial data u 0 theta 0 are required to be only in the space x f in l 2 mathbb r 2 partial x f in l 2 mathbb r 2 and thus our result generalizes that of cao and wu arch rational mech anal 208 985 1004 2013 where the initial data are assumed to be in h 2 mathbb r 2 the assumption on the initial data is at the minimal level that is required to guarantee the uniqueness of the solutions a logarithmic type limiting sobolev embedding inequality for the l infty mathbb r 2 norm in terms of anisotropic sobolev norms and a logarithmic type gronwall inequality are established to obtain the global in time a priori estimates which guarantee the local solution to be a global one

global well posedness and asymptotic behavior for the 2d boussinesq system with variable viscosity abstract in this paper we investigate the global existence and uniqueness of strong solutions to 2d boussinesq system with variable kinematic viscosity depending on the temperature comparing with the previous results given by abidi and zhang 4 who considered the critical case 1 we weaken the dissipation effect in the temperature equation to the supercritical case 0 1 furthermore we obtain the algebraic decay estimate for u l 2

driven cavity flows by efficient numerical techniques abstract efficient and reliable numerical techniques of high order accuracy are presented for solving problems of steady viscous incompressible flow in the plane and are used to obtain accurate solutions for the driven cavity a solution is obtained at reynolds number 10 000 on a 180 180 grid the numerical methods combine an efficient linear system solver an adaptive newton like method for nonlinear systems and a continuation procedure for following a branch of solutions over a range of reynolds numbers

nonlocal image and movie denoising neighborhood filters are nonlocal image and movie filters which reduce the noise by averaging similar pixels the first object of the paper is to present a unified theory of these filters and reliable criteria to compare them to other filter classes a ccd noise model will be presented justifying the involvement of neighborhood filters a classification of neighborhood filters will be proposed including classical image and movie denoising methods and discussing further a recently introduced neighborhood filter nl means in order to compare denoising methods three principles will be discussed the first principle method noise specifies that only noise must be removed from an image a second principle will be introduced noise to noise according to which a denoising method must transform a white noise into a white noise contrarily to method noise this principle which characterizes artifact free methods eliminates any subjectivity and can be checked by mathematical arguments and fourier analysis noise to noise will be proven to rule out most denoising methods with the exception of neighborhood filters this is why a third and new comparison principle the statistical optimality is needed and will be introduced to compare the performance of all neighborhood filters the three principles will be applied to compare ten different image and movie denoising methods it will be first shown that only wavelet thresholding methods and nl means give an acceptable method noise second that neighborhood filters are the only ones to satisfy the noise to noise principle third that among them nl means is closest to statistical optimality a particular attention will be paid to the application of the statistical optimality criterion for movie denoising methods it will be pointed out that current movie denoising methods are motion compensated neighborhood filters this amounts to say that they are neighborhood filters and that the ideal neighborhood of a pixel is its trajectory unfortunately the aperture problem makes it impossible to estimate ground true trajectories it will be demonstrated that computing trajectories and restricting the neighborhood to them is harmful for denoising purposes and that space time nl means preserves more movie details

adaptive multiquadric collocation for boundary layer problems an adaptive collocation method based upon radial basis functions is presented for the solution of singularly perturbed two point boundary value problems using a multiquadric integral formulation the second derivative of the solution is approximated by multiquadric radial basis functions this approach is combined with a coordinate stretching technique the required variable transformation is accomplished by a conformal mapping an iterated sine transformation a new error indicator function accurately captures the regions of the interval with insufficient resolution this indicator is used to adaptively add data centres and collocation points the method resolves extremely thin layers accurately with fairly few basis functions the proposed adaptive scheme is very robust and reaches high accuracy even when parameters in our coordinate stretching technique are not chosen optimally the effectiveness of our new method is demonstrated on two examples with boundary layers and one example featuring an interior layer it is shown in detail how the adaptive method refines the resolution

a cure for noisy magnetic resonance images chi square unbiased risk estimation in this paper we derive an unbiased expression for the expected mean squared error associated with continuously differentiable estimators of the noncentrality parameter of a chi square random variable we then consider the task of denoising squared magnitude magnetic resonance mr image data which are well modeled as independent noncentral chi square random variables on two degrees of freedom we consider two broad classes of linearly parameterized shrinkage estimators that can be optimized using our risk estimate one in the general context of undecimated filterbank transforms and the other in the specific case of the unnormalized haar wavelet transform the resultant algorithms are computationally tractable and improve upon most state of the art methods for both simulated and actual mr image data

global regularity for the two dimensional anisotropic boussinesq equations with vertical dissipation this paper establishes the global in time existence of classical solutions to the two dimensional anisotropic boussinesq equations with vertical dissipation when only vertical dissipation is present there is no direct control on the horizontal derivatives and the global regularity problem is very challenging to solve this problem we bound the derivatives in terms of the l infty norm of the vertical velocity v and prove that v l r with 2 leqq r infty does not grow faster than sqrt r log r at any time as r increases a delicate interpolation inequality connecting v l infty and v l r then yields the desired global regularity

on the global well posedness of 2 d boussinesq system with variable viscosity abstract in this paper we investigate the global well posedness of 2 d boussinesq system which has variable kinematic viscosity and with thermal conductivity of d with general initial data provided that the viscosity coefficient is sufficiently close to some positive constant in l norm

a maximum principle applied to quasi geostrophic equations we study the initial value problem for dissipative 2d quasi geostrophic equations proving local existence global results for small initial data in the super critical case decay of lp norms and asymptotic behavior of viscosity solution in the critical case our proofs are based on a maximum principle valid for more general flows

on the numerical solution of two point boundary value problems ii abstract in a recent paper greengard and rokhlin introduce a numerical technique for the rapid solution of integral equations resulting from linear two point boundary value problems for second order ordinary differential equations in this paper we extend the method to systems of ordinary differential equations after reducing the system of differential equations to a system of second kind integral equations we discretize the latter via a high order nystrom scheme a somewhat involved analytical apparatus is then constructed which allows for the solution of the discrete system using o n p squared n cubed operations with n the number of nodes on the interval p the desired order of convergence and n the number of equations in the system thus the advantages of the integral equation formulation small condition number insensitivity to boundary layers insensitivity to end point singularities etc are retained while achieving a computational efficiency previously available only to finite difference of finite element methods we in addition present a newton method for solving boundary value problems for nonlinear first order systems in which each newton iterate is the solution of a second kind integral equation the analytical and numerical advantages of integral equations are thus obtained for nonlinear boundary value problems kr

vorticity boundary condition and related issues for finite difference schemes this paper discusses three basic issues related to the design of finite difference schemes forunsteadyviscous incompressible flows using vorticity formulations the boundary condition for vorticity an efficient time stepping procedure and the relation between these schemes and the ones based on velocity pressure formulation we show that many of the newly developed global vorticity boundary conditions can actually be written as some local formulas derived earlier we also show that if we couple a standard centered difference scheme with third or fourth order explicit runge kutta methods the resulting schemes havenocell reynolds number constraints for high reynolds number flows these schemes are stable under the cfl condition given by the convective terms finally we show that the classical mac scheme is the same as thom s formula coupled with second order centered differences in the interior in the sense that one can define discrete vorticity in a natural way for the mac scheme and get the same values as the ones computed from thom s formula we use this to derive an efficient fourth order runge kutta time discretization for the mac scheme from the one for thom s formula we present numerical results for driven cavity flow at high reynolds number 105

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

lower bounds for an integral involving fractional laplacians and the generalized navier stokes equations in besov spaces when estimating solutions of dissipative partial differential equations in l p related spaces we often need lower bounds for an integral involving the dissipative term if the dissipative term is given by the usual laplacian lower bounds can be derived through integration by parts and embedding inequalities however when the laplacian is replaced by the fractional laplacian the approach of integration by parts no longer applies in this paper we obtain lower bounds for the integral involving by combining pointwise inequalities for with bernstein s inequalities for fractional derivatives as an application of these lower bounds we establish the existence and uniqueness of solutions to the generalized navier stokes equations in besov spaces the generalized navier stokes equations are the equations resulting from replacing in the navier stokes equations by

global well posedness for the 2d boussinesq system with anisotropic viscosity and without heat diffusion abstract we establish global existence and uniqueness theorems for the two dimensional non diffusive boussinesq system with anisotropic viscosity acting only in the horizontal direction which arises in ocean dynamics models global well posedness for this system was proven by danchin and paicu however an additional smoothness assumption on the initial density was needed to prove uniqueness they stated that it is not clear whether uniqueness holds without this additional assumption the present work resolves this question and we establish uniqueness without this additional assumption furthermore the proof provided here is more elementary we use only tools available in the standard theory of sobolev spaces and without resorting to para product calculus we use a new approach by defining an auxiliary stream function associated with the density analogous to the stream function associated with the vorticity in 2d incompressible euler equations then we adapt some of the ideas of yudovich for proving uniqueness for 2d euler equations

numerical methods for singular perturbation problems consider the two point boundary value problem for a stiff system of ordinary differential equations an adaptive method to solve these problems even when turning points are present is discussed

a direct adaptive poisson solver of arbitrary order accuracy we present a direct adaptive solver for the poisson equation which can achieve any prescribed order of accuracy it is based on a domain decomposition approach using local spectral approximation as well as potential theory and the fast multipole method in two space dimensions the algorithm requireso nk work wherenis the number of discretization points andkis the desired order of accuracy

global well posedness for the 2 d boussinesq system with the temperature dependent viscosity and thermal diffusivity abstract we prove the global well posedness for the 2 d boussinesq system with the temperature dependent viscosity and thermal diffusivity

global regularity for the 2d boussinesq equations with partial viscosity terms abstract in this paper we prove the global in time regularity for the 2d boussinesq system with either the zero diffusivity or the zero viscosity we also prove that as diffusivity viscosity tends to zero the solutions of the fully viscous equations converge strongly to those of zero diffusion viscosity equations our result for the zero diffusion system in particular solves the problem no 3 posed by moffatt in r l ricca ed kluwer academic publishers dordrecht the netherlands 2001 pp 3 10

the 2d boussinesq equations with vertical viscosity and vertical diffusivity this paper aims at the global regularity of classical solutions to the 2d boussinesq equations with vertical dissipation and vertical thermal diffusion we prove that the l r norm of the vertical velocity v for any 1 0 would guarantee the global regularity of classical solutions

a fast adaptive numerical method for stiff two point boundary value problems we describe a robust adaptive algorithm for the solution of singularly perturbed two point boundary value problems many different phenomena can arise in such problems including boundary layers dense oscillations and complicated or ill conditioned internal transition regions working with an integral equation reformulation of the original differential equation we introduce a method for error analysis which can be used for mesh refinement even when the solution computed on the current mesh is underresolved based on this method we have constructed a black box code for stiff problems which automatically generates an adaptive mesh resolving all features of the solution the solver is direct and of arbitrarily high order accuracy and requires an amount of time proportional to the number of grid points

an integral equation approach to the incompressible navier stokes equations in two dimensions we present a collection of methods for solving the incompressible navier stokes equations in the plane that are based on a pure stream function formulation the advantages of this approach are twofold first the velocity is automatically divergence free and second complicated nonlocal boundary conditions for the vorticity are avoided the disadvantage is that the solution of a nonlinear fourth order partial differential equation is required by recasting this partial differential equation as an integral equation we avoid the ill conditioning which hampers finite difference and finite element methods in this environment by using fast algorithms for the evaluation of volume integrals we are able to solve the equations using o m or o m log m operations where m is the number of points in the discretization of the domain

a dictionary learning approach for poisson image deblurring the restoration of images corrupted by blur and poisson noise is a key issue in medical and biological image processing while most existing methods are based on variational models generally derived from a maximum a posteriori map formulation recently sparse representations of images have shown to be efficient approaches for image recovery following this idea we propose in this paper a model containing three terms a patch based sparse representation prior over a learned dictionary the pixel based total variation regularization term and a data fidelity term capturing the statistics of poisson noise the resulting optimization problem can be solved by an alternating minimization technique combined with variable splitting extensive experimental results suggest that in terms of visual quality peak signal to noise ratio value and the method noise the proposed algorithm outperforms state of the art methods

on the global well posedness for boussinesq system abstract in this paper we give a global well posedness result for the two dimensional boussinesq system with partial viscosity when the initial data v 0 b 1 1 r 2 l 2 r 2 and 0 b 2 1 0 r 2

multilevel matrix multiplication and fast solution of integral equations abstract a fast multigrid approach is described for the task of calculating k x y u y dy for each x r d discretizing by an equidistant grid with n points and meshsize h and approximating the integrations to o h2s accuracy it is shown that the complexity of this calculation can be reduced from o n2 to o sn provided the kernel k is sufficiently smooth for potential type kernels the complexity is reduced to o sn log n corresponding integral equations can be solved to a similar accuracy in basically the same amount of work using a special kind of distributed relaxation in a multigrid algorithm one and two dimensional numerical tests and theoretical derivations of optimal strategies are reported the method is applicable to the task of multiplying by any matrix with appropriate smoothness properties including most types of many body interactions

feature preserving mri denoising a nonparametric empirical bayes approach this paper presents a novel method for bayesian denoising of magnetic resonance mr images that bootstraps itself by inferring the prior i e the uncorrupted image statistics from the corrupted input data and the knowledge of the rician noise model the proposed method relies on principles from empirical bayes eb estimation it models the prior in a nonparametric markov random field mrf framework and estimates this prior by optimizing an information theoretic metric using the expectation maximization algorithm the generality and power of nonparametric modeling coupled with the eb approach for prior estimation avoids imposing ill fitting prior models for denoising the results demonstrate that unlike typical denoising methods the proposed method preserves most of the important features in brain mr images furthermore this paper presents a novel bayesian inference algorithm on mrfs namely iterated conditional entropy reduction icer this paper also extends the application of the proposed method for denoising diffusion weighted mr images validation results and quantitative comparisons with the state of the art in mr image denoising clearly depict the advantages of the proposed method