adaptive moving mesh computations for reaction diffusion systems in this paper we describe an adaptive moving mesh technique and its application to reaction diffusion models from chemistry the method is based on a coordinate transformation between physical and computational coordinates the transformation can be viewed as a solution of adaptive mesh partial differential equations pdes which is derived from the minimization of a mesh energy integral for an efficient implementation we have used an approach in which the numerical solution of the physical pdes and the adaptive pdes are decoupled further to avoid solving large nonlinear systems a second order implicit explicit time integration method in combination with the iterative method bi cgstab is applied in the method of lines procedure numerical examples are given in one and two space dimensions

singular perturbation analysis of the pore creation transient electroporation in which electric pulses create transient pores in the cell membrane is an important technique for drug and dna delivery electroporation kinetics is mathematically described by an advection diffusion boundary value problem this study uses singular perturbation to derive a reduced description of the pore creation transient in the form of a single integrodifferential equation for the transmembrane voltage v t the number of pores and the distribution of their radii are computed from v t the analysis contains two nonstandard features the use of the voltage deviation to peel away the strong exponential dependence of pore creation upon the transmembrane potential and the autonomous approximation of the pore evolution comparing the predictions of the reduced equation with the simulations of the original problem demonstrates that this analysis allows one to predict with good accuracy the number and distribution of pores as a function of the electric pulse strength

invariant pattern recognition using radon dual tree complex wavelet and fourier transforms an invariant pattern recognition descriptor is proposed in this paper by using the radon transform the dual tree complex wavelet transform and the fourier transform the radon transform can capture the directional features of the pattern image by projecting the pattern onto different orientation slices the dual tree complex wavelet transform can select shift invariant features in a multiresolution way the fourier transform can extract features that are invariant to rotation of the patterns standard normalization techniques are used to normalize the input pattern image so that it is translation and scale invariant experiments conducted in this paper show that the proposed descriptor achieves high recognition rates for different combinations of rotation angles and noise levels the descriptor is very robust to gaussian white noise even when the noise level is very high

contour based handwritten numeral recognition using multiwavelets and neural networks abstract in this paper we develop a handwritten numeral recognition descriptor using multiwavelets and neural networks we first trace the contour of the numeral then normalize and resample the contour so that it is translation and scale invariant we then perform multiwavelet orthonormal shell expansion on the contour to get several resolution levels and the average finally we use the shell coefficients as features to input into a feed forward neural network to recognize the handwritten numerals the main advantage of the orthonormal shell decomposition is that it decomposes a signal into multiresolution levels but without down sampling wavelet transforms with down sampling can give very different coefficients when the input signal is shifted this is the main limitation of wavelet transforms in pattern recognition for the shell expansion we prefer multiwavelets to scalar wavelets because we have two coordinates x and y for each point on the contour if we extract features from x and y separately just as wunsch et al did pattern recognition 28 1995 1237 then we may not get the best features in addition we know that multiwavelets have advantages over scalar wavelets such as short support orthogonality symmetry and higher order of vanishing moments these properties allow multiwavelets to outperform scalar wavelets in some applications e g signal denoising ieee trans signal process 46 12 1998 3414 we conducted experiments and found that it is feasible to use multiwavelet features in handwritten numeral recognition

a fully adaptive reaction diffusion integration scheme with applications to systems biology adaptive integration schemes for ode systems typically function by adjusting the time step size so as to keep the truncation error below some desired value for adaptive integration of pde systems involving coupled kinetic reaction and diffusion operations truncation error arises not only from the individual propagators but also from their method of coupling a common second order accurate method for coupling operators is strang s method of operator splitting we derive an expression for the truncation error resulting from strang splitting reaction and diffusion operators for an arbitrary number of spatial dimensions and demonstrate its use in adaptive time step algorithms in addition we present explanations of the second order implicit reaction and diffusion operators and their individual error calculations used in our implementation of the scheme finally using example simulations we discuss the use of this calculation for problems in systems biology

a real distinct poles exponential time differencing scheme for reaction diffusion systems a second order exponential time differencing etd method for reaction diffusion systems which uses a real distinct poles discretization method for the underlying matrix exponentials is developed the method is established to be stable and second order convergent it is demonstrated to be robust for problems involving non smooth initial and boundary conditions and steep solution gradients we discuss several advantages over competing second order etd schemes

invariant fourier wavelet descriptor for pattern recognition abstract we present a novel set of descriptors for recognizing complex patterns such as roadsigns keys aircrafts characters etc given a pattern we first transform it to polar coordinate r using the centre of mass of the pattern as origin we then apply the fourier transform along the axis of polar angle and the wavelet transform along the axis of radius r the features thus obtained are invariant to translation rotation and scaling as an example we apply the method to a database of 85 printed chinese characters the result shows that the fourier wavelet descriptor is an efficient representation which can provide for reliable recognition

wavelet descriptors for multiresolution recognition of handprinted characters we present a novel set of shape descriptors that represents a digitized pattern in a concise way and that is particularly well suited for the recognition of handprinted characters the descriptor set is derived from the wavelet transform of a pattern s contour the approach is closely related to feature extraction methods by fourier series expansion the motivation to use an orthonormal wavelet basis rather than the fourier basis is that wavelet coefficients provide localized frequency information and that wavelets allow us to decompose a function into a multiresolution hierarchy of localized frequency bands we describe a character recognition system that relies upon wavelet descriptors to simultaneously analyze character shape at multiple levels of resolution the system was trained and tested on a large database of more than 6000 samples of handprinted alphanumeric characters the results show that wavelet descriptors are an efficient representation that can provide for reliable recognition in problems with large input variability

tensor product adaptive grids based on coordinate transformations in this paper we discuss a two dimensional adaptive grid method that is based on a tensor product approach adaptive grids are a commonly used tool for increasing the accuracy and reducing computational costs when solving both partial differential equations pdes and ordinary differential equations a traditional and widely used form of adaptivity is the concept of equidistribution which is well defined and well understood in one space dimension the extension of the equidistribution principle to two or three space dimensions however is far from trivial and has been the subject of investigation of many researchers during the last decade besides the nonsingularity of the transformation that defines the nonuniform adaptive grid the smoothness of the grid or transformation plays an important role as well we will analyse these properties and illustrate their importance with numerical experiments for a set of time dependent pde models with steep moving pulses fronts and boundary layers

the fast discrete radon transform i theory an inversion scheme for reconstruction of images from projections based on the slope intercept form of the discrete radon transform is presented a seminal algorithm for the forward and the inverse transforms proposed by g beylkin 1987 demonstrated poor dispersion characteristics for steep slopes and could not invert transforms based on nonlinear slope variations by formulating the computation of a discrete computation of the continuous radon transform formula the authors explicitly derive fast generalized inversion methods that overcome the original shortcomings the generalized forward frt and inverse ifrt algorithms proposed are fast eliminate interpolation calculations and convert directly between a raster scan grid and a rectangular polar grid in one step

efficient approximation of the solution of certain nonlinear reaction diffusion equations with large absorption we study the positive stationary solutions of a standard finite difference discretization of the semilinear heat equation with nonlinear neumann boundary conditions we prove that if the absorption is large enough compared with the flux in the boundary there exists a unique solution of such a discretization which approximates the unique positive stationary solution of the continuous equation furthermore we exhibit an algorithm computing an e approximation of such a solution by means of a homotopy continuation method the cost of our algorithm is linear in the number of nodes involved in the discretization and the logarithm of the number of digits of approximation required

adaptive mesh methods for one and two dimensional hyperbolic conservation laws we develop efficient moving mesh algorithms for one and two dimensional hyperbolic systems of conservation laws the algorithms are formed by two independent parts pde evolution and mesh redistribution the first part can be any appropriate high resolution scheme and the second part is based on an iterative procedure in each iteration meshes are first redistributed by an equidistribution principle and then on the resulting new grids the underlying numerical solutions are updated by a conservative interpolation formula proposed in this work the iteration for the mesh redistribution at a given time step is complete when the meshes governed by a nonlinear equation reach the equilibrium state the main idea of the proposed method is to keep the mass conservation of the underlying numerical solution at each redistribution step in one dimension we can show that the underlying numerical approximation obtained in the mesh redistribution part satisfies the desired tvd property which guarantees that the numerical solution at any time level is tvd provided that the pde solver in the first part satisfies such a property several test problems in one and two dimensions are computed using the proposed moving mesh algorithm the computations demonstrate that our methods are efficient for solving problems with shock discontinuities obtaining the same resolution with a much smaller number of grid points than the uniform mesh approach

a fast discrete approximation algorithm for the radon transform this paper addresses fast parallel methods for the computation of the radon or hough transform the radon transform of an image is a set of projections of the image taken at different angles its computation is important in image processing and computer vision for problems such as pattern recognition and reconstruction of medical images a unique new method for combining partial results is presented from which an algorithm is constructed that computes a provably good approximation to the discrete radon transform the approximate discrete radon transform adrt algorithm computes 4 n 4 projections through an n times n image in time o n 2 mathrm lg n the majority of previous algorithms are o n 3 the method is quite simple and easy to parallelize a parallel version of the adrt requires only o mathrm lg n parallel steps on o n 2 processors ignoring communication time an additional property of the algorithm is that it can be applied directly to compute the backprojection step of the inverse rt

on the invertibility of mappings arising in 2d grid generation problems in adapting a grid for a computational fluid dynamics problem one uses a mapping from the unit square onto itself that is the solution of an elliptic partial differential equation with rapidly varying coefficients for a regular discretization this mapping has to be invertible we will show that such result holds for general elliptic operators in two dimensions the carleman hartman wintner theorem will be fundamental in our proof we will also explain why such a general result cannot be expected to hold for the three dimensional cube

image representation via a finite radon transform a model of finite radon transforms composed of radon projections is presented the model generalizes to finite group projections in the classical radon transform theory the radon projector averages a function on a group over cosets of a subgroup reconstruction formulae that were formally similar to the convolved backprojection ones are derived and an iterative reconstruction technique is found to converge after a finite number of steps applying these results to the group z sub 2 sup p new computationally favorable image representations have been obtained a numerical study of the transform coding aspects is attached

moving mesh strategy based on a gradient flow equation for two dimensional problems in this paper we introduce a moving mesh method for solving pdes in two dimensions it can be viewed as a higher dimensional generalization of the moving mesh pde mmpde strategy developed in our previous work for one dimensional problems w huang y ren and r d russell siam j numer anal 31 1994 pp 709 730 the mmpde is derived from a gradient flow equation which arises using a mesh adaptation functional in turn motivated from the theory of harmonic maps geometrical interpretations are given for the gradient equation and functional and basic properties of this mmpde are discussed numerical examples are presented where the method is used both for mesh generation and for solving time dependent pdes the results demonstrate the potential of the mesh movement strategy to concentrate the mesh points so as to adapt to special problem features and to also preserve a suitable level of mesh orthogonality

design and application of a gradient weighted moving finite element code ii in two dimensions in part i the authors reported on the design of a robust and versatile gradient weighted moving finite element gwmfe code in one dimension and on its application to a variety of pdes and pde systems this companion paper does the same for the two dimensional 2d case these moving node methods are especially suited to problems which develop sharp moving fronts especially problems where one needs to resolve the fine scale structure of the fronts the many potential pitfalls in the design of gwmfe codes and the special features of the implicit one dimensional 1d and 2d codes which contribute to their robustness and efficiency are discussed at length in part i this paper concentrates on issues unique to the 2d case brief explanations are given of the variational interpretation of gwmfe the geometrical mechanical interpretation simplified regularization terms and the treatment of systems a catalog of inner products which occur in gwmfe is given with particular attention paid to those involving second order operators after presenting an example of the 2d phenomenon of grid collapse and discussing the need for long time regularization the paper reports on the application of the 2d code to several nontrivial problems nonlinear arsenic diffusion in the manufacture of semiconductors the drift diffusion equations for semiconductor device simulation the buckley leverett black oil equations for reservoir simulation and the motion of surfaces by mean curvature

modeling postshock evolution of large electropores the smoluchowski equation se which describes the evolution of pores created by electric shocks cannot be applied to modeling large and long lived pores for two reasons 1 it does not predict pores of radius above 20nm without also predicting membrane rupture 2 it does not predict postshock growth of pores this study proposes a model in which pores are coupled by membrane tension resulting in a nonlinear generalization of se the predictions of the model are explored using examples of homogeneous all pore radii r are equal and heterogeneous 0 r r m a x distributions of pores pores in a homogeneous population either shrink to zero or assume a stable radius corresponding to the minimum of the bilayer energy for a heterogeneous population such a stable radius does not exist all pores except r m a x shrink to zero and r m a x grows to infinity however the unbounded growth of r m a x is not physical because the number of pores per cell decreases in time and the continuum model loses validity when the continuum formulation is replaced by the discrete one the model predicts the coarsening process all pores except r m a x shrink to zero and r m a x assumes a stable radius thus the model with tension coupled pores does not predict membrane rupture and the predicted postshock growth of pores is consistent with experimental evidence

an adaptive grid method and its application to steady euler flow calculations an adaptive remeshing procedure based on a cell volume deformation method is presented starting with an initial grid this method offers direct cell volume control through the specification of the transformation jacobian grid points are moved with appropriate grid velocities so that the specified cell volume distribution can be achieved at the end of the grid movement without adding or removing grid points the grid velocities are determined by solving a scalar poisson equation this method is applied to solving the compressible euler equations computational test cases of transonic flow over an airfoil are presented and demonstrate the desired control of grid sizes across shock waves

a compact fourth order l stable scheme for reaction diffusion systems with nonsmooth data exponential time differencing runge kutta etdrk schemes based on diagonal pade approximations for the numerical solution of reaction diffusion systems containing nonsmooth data have the disadvantage of producing poor numerical results when the time steps are too large relative to the spatial steps on the other hand methods based on sub diagonal pade approximations do not suffer from this bottleneck in this paper we present a novel modified version of cox and matthews fourth order etdrk scheme based on sub diagonal pade approximation to matrix exponential functions in combination with compact fourth order finite difference scheme in space for direct integration of reaction diffusion problems with nonsmooth data for an efficient implementation of the scheme a partial fraction splitting technique is utilized in which it is required to solve several backward euler type linear systems at each time step moreover the design of the algorithm offers parallel implementation so we implement the proposed algorithm in parallel on two processors utilizing matlabmpi a parallel message passing version of matlab and obtain that the parallel version is computationally more efficient than the existing schemes considered in this paper we investigate the amplification factor of the scheme and plot its boundaries of stability regions which give an indication of the stability of the scheme calculation of the local truncation error and an empirical convergence analysis demonstrates the fourth order accuracy of the proposed scheme accuracy computational efficiency and reliability of the new scheme are demonstrated with numerical examples and comparing it with existing schemes

computational solution of two dimensional unsteady pdes using moving mesh methods numerical experiments are described which illustrate some important features of the performance of moving mesh methods for solving two dimensional partial differential equations pdes here we are concerned with algorithms based on moving mesh methods proposed by w huang and r d russell siam j sci comput 20 998 1999 we show that the accuracy of the computations is strongly influenced by the choice of monitor function and we present a monitor function which yields a higher rate of convergence than those that are commonly used in an earlier paper g beckett j a mackenzie a ramage and d m sloan j comput phys 167 372 2001 we demonstrated a robust and efficient algorithm for problems in one space dimension in which the mesh equation is decoupled from the physical pde and the time step is controlled automatically the present work extends this algorithm to deal with problems in two space dimensions

string gradient weighted moving finite elements in multiple dimensions with applications in two dimensions we formulate the string gradient weighted moving finite element method sgwmfe for systems of pdes in multiple dimensions then we illustrate implementation issues for the method using two dimensions the method is applied successfully to solve the highly nonlinear unsteady porous medium equation in two dimensions and the results are compared to exact solutions we proceed to present results for the method when applied to the gray scott chemical reaction diffusion model in two dimensions finally we apply the method to the shallow water equations in two dimensions and compare the meshes produced with those produced using gradient weighted finite element methods gwmfe our conclusion is that sgwmfe is an easily applied member of the group of moving finite element methods

a second order exponential time differencing scheme for non linear reaction diffusion systems with dimensional splitting abstract a second order l stable exponential time differencing etd method is developed by combining an etd scheme with approximating the matrix exponentials by rational functions having real distinct poles rdp together with a dimensional splitting integrating factor technique a variety of non linear reaction diffusion equations in two and three dimensions with either dirichlet neumann or periodic boundary conditions are solved with this scheme and shown to outperform a variety of other second order implicit explicit schemes an additional performance boost is gained through further use of basic parallelization techniques

moving mesh methods with locally varying time steps the time steps associated with moving mesh methods are proportional to the smallest mesh size in space and as a result they are very small at each time level for some practical problems the physical phenomena develop dynamically singular or nearly singular solutions in fairly localized regions and therefore the smallest time step at each time level occurs only in these localized regions in this work we will develop a local time stepping algorithm for the moving mesh methods the principal idea will be demonstrated by investigating the nonlinear hyperbolic conservation laws numerical experiments are carried out to demonstrate the efficiency and robustness of the proposed methods

asymptotic model of electroporation electroporation is described mathematically by a partial differential equation pde that governs the distribution of pores as a function of their radius and time this pde does not have an analytical solution and because of the presence of disparate spatial and temporal scales numerical solutions are hard to obtain these difficulties limit the application of the pde only to experimental setups with a uniformly polarized membrane this study performs a rigorous asymptotic reduction of the pde to an ordinary differential equation ode that describes the dynamics of the pore density n t given n t the precise distribution of the pores in the space of their radii can be determined by an asymptotic approximation thus the asymptotic ode represents most of the phenomenology contained in the pde it is easy to solve numerically which makes it a powerful tool to study electroporation in experimental setups with significant spatial dependence such vesicles or cells in an external field

dimension splitting data points redistribution for meshless approximation to better approximate nearly singular functions with meshless methods we propose a data points redistribution method extended from the well known one dimensional equidistribution principle with properly distributed data points nearly singular functions can be well approximated by linear combinations of global radial basis functions the proposed method is coupled with an adaptive trial subspace selection algorithm in order to reduce computational cost in our numerical examples clear exponential convergence with respect to the numbers of data points can be observed