asymptotic optimality of cl and generalized cross validation in ridge regression with application to spline smoothing the asymptotic optimality of mallows c l and generalized cross validation is demonstrated in the setting of ridge regression an application is made to spline smoothing in nonparametric regression a counterexample is given to help understand why sometimes gcv may not be asymptotically optimal the coefficient of variation for the eigenvalues of the information matrix must be large in order to guarantee the optimality of gcv the proff is based on the connection between gcv and stein s unbiased risk estimate

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

a discontinuous galerkin finite element method for time dependent partial differential equations with higher order derivatives in this paper we develop a new discontinuous galerkin dg finite element method for solving time dependent partial differential equations pdes with higher order spatial derivatives unlike the traditional local discontinuous galerkin ldg method the method in this paper can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system stability is ensured by a careful choice of interface numerical fluxes the method can be designed for quite general nonlinear pdes and we prove stability and give error estimates for a few representative classes of pdes up to fifth order numerical examples show that our scheme attains the optimal k 1 th order of accuracy when using piecewise k th degree polynomials under the condition that k 1 is greater than or equal to the order of the equation

the subspace information criterion for infinite dimensional hypothesis spaces a central problem in learning is selection of an appropriate model this is typically done by estimating the unknown generalization errors of a set of models to be selected from and then choosing the model with minimal generalization error estimate in this article we discuss the problem of model selection and generalization error estimation in the context of kernel regression models e g kernel ridge regression kernel subset regression or gaussian process regression previously a non asymptotic generalization error estimator called the subspace information criterion sic was proposed that could be successfully applied to finite dimensional subspace models sic is an unbiased estimator of the generalization error for the finite sample case under the conditions that the learning target function belongs to a specified reproducing kernel hilbert space rkhs h and the reproducing kernels centered on training sample points span the whole space h these conditions hold only if dim h l where l is the number of training examples therefore sic could be applied only to finite dimensional rkhss in this paper we extend the range of applicability of sic and show that even if the reproducing kernels centered on training sample points do not span the whole space h sic is an unbiased estimator of an essential part of the generalization error our extension allows the use of any rkhss including infinite dimensional ones i e richer function classes commonly used in gaussian processes support vector machines or boosting we further show that when the kernel matrix is invertible sic can be expressed in a much simpler form making its computation highly efficient in computer simulations on ridge parameter selection with real and artificial data sets sic is compared favorably with other standard model selection techniques for instance leave one out cross validation or an empirical bayesian method

ridge regression biased estimation for nonorthogonal problems in multiple regression it is shown that parameter estimates based on minimum residual sum of squares have a high probability of being unsatisfactory if not incorrect if the prediction vectors are not orthogonal proposed is an estimation procedure based on adding small positive quantities to the diagonal of x x introduced is the ridge trace a method for showing in two dimensions the effects of nonorthogonality it is then shown how to augment x x to obtain biased estimates with smaller mean square error

maximum principle satisfying second order discontinuous galerkin schemes for convection diffusion equations on triangular meshes we propose second order accurate discontinuous galerkin dg schemes which satisfy a strict maximum principle for general nonlinear convection diffusion equations on unstructured triangular meshes motivated by genuinely high order maximum principle satisfying dg schemes for hyperbolic conservation laws perthame 1996 and zhang 2010 14 26 we prove that under suitable time step restriction for forward euler time stepping for general nonlinear convection diffusion equations the same scaling limiter coupled with second order dg methods preserves the physical bounds indicated by the initial condition while maintaining uniform second order accuracy similar to the purely convection cases the limiters are mass conservative and easy to implement strong stability preserving ssp high order time discretizations will keep the maximum principle following the idea in zhang 2012 30 we extend the schemes to two dimensional convection diffusion equations on triangular meshes there are no geometric constraints on the mesh such as angle acuteness numerical results including incompressible navier stokes equations are presented to validate and demonstrate the effectiveness of the numerical methods

maximum principle satisfying high order finite volume weighted essentially nonoscillatory schemes for convection diffusion equations to easily generalize the maximum principle satisfying schemes for scalar conservation laws in x zhang and c w shu j comput phys 229 2010 pp 3091 3120 to convection diffusion equations we propose a nonconventional high order finite volume weighted essentially nonoscillatory weno scheme which can be proved maximum principle satisfying two dimensional extensions are straightforward we also show that the same idea can be used to construct high order schemes preserving the maximum principle for two dimensional incompressible navier stokes equations in the vorticity stream function formulation numerical tests for the fifth order weno schemes are reported

third order maximum principle satisfying dg schemes for convection diffusion problems with anisotropic diffusivity abstract for a class of convection diffusion equations with variable diffusivity we construct third order accurate discontinuous galerkin dg schemes on both one and two dimensional rectangular meshes the dg method with an explicit time stepping can well be applied to nonlinear convection diffusion equations it is shown that under suitable time step restrictions the scaling limiter proposed in liu and yu 2014 23 when coupled with the present dg schemes preserves the solution bounds indicated by the initial data i e the maximum principle while maintaining uniform third order accuracy these schemes can be extended to rectangular meshes in three dimension the crucial for all model scenarios is that an effective test set can be identified to verify the desired bounds of numerical solutions this is achieved mainly by taking advantage of the flexible form of the diffusive flux and the adaptable decomposition of weighted cell averages numerical results are presented to validate the numerical methods and demonstrate their effectiveness

the finite volume scheme preserving maximum principle for diffusion equations with discontinuous coefficient abstract we construct a new scheme whose primary unknowns include both cell centered unknowns and edge unknowns on discontinuous line to solve diffusion equations with discontinuous coefficient first two linear fluxes are given on both sides of the cell edge respectively in order to deal with the defect of the existing scheme preserving maximum principle for solving diffusion problem with discontinuous coefficient in addition to cell centered unknowns we also introduce cell edge unknowns on discontinuous line as basic unknowns second the conservative flux is constructed by using nonlinear weighted combination of these two linear fluxes for the cell edge unknowns on discontinuous line we add an equation by using the continuity of normal flux compared to the classical cell centered nonlinear finite volume scheme the introduction of cell edge unknowns on discontinuous edge is the key point for our scheme to solve diffusion equations with discontinuous coefficient then we prove that the scheme satisfies the discrete maximum principle based on this the existence of a solution for the scheme is also obtained numerical results are presented to show that our scheme obtains almost second order accuracy for solution on random meshes preserves discrete maximum principle and is superior to the existing scheme in sheng and yuan 2011 preserving the maximum principle on dealing with the problems with strong anisotropic discontinuous coefficient on distorted meshes

positivity preserving discontinuous galerkin schemes for linear vlasov boltzmann transport equations we develop a high order positivity preserving discontinuous galerkin dg scheme for linear vlasov boltzmann transport equations vlasov bte under the action of quadratically confined electrostatic potentials the solutions of such btes are positive probability distribution functions and it is very challenging to have a mass conservative high order accurate scheme that preserves positivity of the numerical solutions in high dimensions our work extends the maximum principle satisfying scheme for scalar conservation laws in a recent work by x zhang and c w shu to include the linear boltzmann collision term the dg schemes we developed conserve mass and preserve the positivity of the solution without sacrificing accuracy a discussion of the standard semi discrete dg schemes for the bte are included as a foundation for the stability and error estimates for this new scheme numerical results of the relaxation models are provided to validate the method

on positivity preserving high order discontinuous galerkin schemes for compressible navier stokes equations we construct a local lax friedrichs type positivity preserving flux for compressible navier stokes equations which can be easily extended to multiple dimensions for generic forms of equations of state shear stress tensor and heat flux with this positivity preserving flux any finite volume type schemes including discontinuous galerkin dg schemes with strong stability preserving runge kutta time discretizations satisfy a weak positivity property with a simple and efficient positivity preserving limiter high order explicit runge kutta dg schemes are rendered preserving the positivity of density and internal energy without losing local conservation or high order accuracy numerical tests suggest that the positivity preserving flux and the positivity preserving limiter do not induce excessive artificial viscosity and the high order positivity preserving dg schemes without other limiters can produce satisfying non oscillatory solutions when the nonlinear diffusion in compressible navier stokes equations is accurately resolved

discrete maximum principle for the p 1 p 0 weak galerkin finite element approximations abstract this paper presents two discrete maximum principles dmp for the numerical solution of second order elliptic equations arising from the weak galerkin finite element method the results are established by assuming an h acute angle condition for the underlying finite element triangulations the mathematical theory is based on the well known de giorgi technique adapted in the finite element context some numerical results are reported to validate the theory of dmp

a maximum principle preserving finite element method for scalar conservation equations this paper introduces a first order viscosity method for the explicit approximation of scalar conservation equations with lipschitz fluxes using continuous finite elements on arbitrary grids in any space dimension provided the lumped mass matrix is positive definite the method is shown to satisfy the local maximum principle under a usual cfl condition the method is independent of the cell type for instance the mesh can be a combination of tetrahedra hexahedra and prisms in three space dimensions

a high order discontinuous galerkin method for 2d incompressible flows in this paper we introduce a high order discontinuous galerkin method for two dimensional incompressible flow in the vorticity stream function formulation the momentum equation is treated explicitly utilizing the efficiency of the discontinuous galerkin method the stream function is obtained by a standard poisson solver using continuous finite elements there is a natural matching between these two finite element spaces since the normal component of the velocity field is continuous across element boundaries this allows for a correct upwinding gluing in the discontinuous galerkin framework while still maintaining total energy conservation with no numerical dissipation and total enstrophy stability the method is efficient for inviscid or high reynolds number flows optimal error estimates are proved and verified by numerical experiments

third order maximum principle satisfying direct discontinuous galerkin methods for time dependent convection diffusion equations on unstructured triangular meshes we develop 3rd order maximum principle satisfying direct discontinuous galerkin methods 8 9 19 21 for convection diffusion equations on unstructured triangular mesh we carefully calculate the normal derivative numerical flux across element edges and prove that with proper choice of parameter pair 0 1 in the numerical flux formula the quadratic polynomial solution satisfies strict maximum principle the polynomial solution is bounded within the given range and third order accuracy is maintained there is no geometric restriction on the meshes and obtuse triangles are allowed in the partition a sequence of numerical examples are carried out to demonstrate the accuracy and capability of the maximum principle satisfying limiter

an equivalence between sparse approximation and support vector machines in the first part of this paper we show a similarity between the principle of structural risk minimization principle srm vapnik 1982 and the idea of sparse approximation as defined in chen donoho and saunders 1995 and olshausen and field 1996 then we focus on two specific approximate implementations of srm and sparse approximation which have been used to solve the problem of function approximation for srm we consider the support vector machine technique proposed by v vapnik and his team at at t bell labs and for sparse approximation we consider a modification of the basis pursuit de noising algorithm proposed by chen donoho and saunders 1995 we show that under certain conditions these two techniques are equivalent they give the same solution and they require the solution of the same quadratic programming problem

high order bound preserving discontinuous galerkin methods for wormhole propagation on triangular meshes abstract wormhole propagation arising in petroleum engineering is used to describe the distribution of acid and the increase of porosity in carbonate reservoir under dissolution of injected acid the important physical features of porosity and acid concentration include their boundedness between 0 and 1 as well as the monotone increasing for porosity how to keep these properties in the simulation is crucial to the robustness of the numerical algorithm in this paper we propose high order bound preserving discontinuous galerkin methods to keep these important physical properties the main technique is to introduce a new variable r to replace the original acid concentration and use a consistent flux pair to deduce a ghost equation such that the positive preserving technique can be applied on both original and deduced equations a high order slope limiter is used to keep a polynomial upper bound which changes over time for r moreover the high order accuracy is attained by the flux limiter numerical examples are given to demonstrate the high order accuracy and bound preserving property of the numerical technique

subspace information criterion for nonquadratic regularizers model selection for sparse regressors nonquadratic regularizers in particular the l sub 1 norm regularizer can yield sparse solutions that generalize well in this work we propose the generalized subspace information criterion gsic that allows to predict the generalization error for this useful family of regularizers we show that under some technical assumptions gsic is an asymptotically unbiased estimator of the generalization error gsic is demonstrated to have a good performance in experiments with the l sub 1 norm regularizer as we compare with the network information criterion nic and cross validation in relatively large sample cases however in the small sample case gsic tends to fail to capture the optimal model due to its large variance therefore also a biased version of gsic is introduced which achieves reliable model selection in the relevant and challenging scenario of high dimensional data and few samples

a high order scheme for unsteady heat conduction equations abstract in this work a high order scheme is proposed for solving the heat conduction problems in this scheme the governing equation of heat conduction is firstly written into a system of first order partial differential equations which are integrated over the control volumes around each node point and a finite time step the resulting time integrals are approximated by numerical quadrature with an undetermined weighting parameter the error analysis allows us to determine the parameter and the ratio of the time step to the square of grid spacing by making the error of the scheme as small as possible both theoretical and numerical results show that the proposed high order scheme can achieve at least sixth order accuracy at the expense of the same computing cost as the traditional finite volume method and the high order scheme demonstrates a better performance than the high order compact finite difference method and the runge kutta discontinuous galerkin method

a second order maximum principle preserving lagrange finite element technique for nonlinear scalar conservation equations this paper proposes an explicit at least second order maximum principle sat isfying lagrange finite element method for solving nonlinear scalar conservation equations the technique is based on a new viscous bilinear form introduced in guermond and nazarov com put methods appl mech engrg 272 2014 pp 198 213 a high order entropy viscosity method and the boris book zalesak flux correction technique the algorithm works for arbitrary meshes in any space dimension and for all lipschitz fluxes the formal second order accuracy of the method and its convergence properties are tested on a series of linear and nonlinear benchmark problems

positivity preserving high order local discontinuous galerkin method for parabolic equations with blow up solutions in this paper we apply positivity preserving local discontinuous galerkin ldg methods to solve parabolic equations with blow up solutions this model is commonly used in combustion problems however previous numerical methods are mainly based on a second order finite difference method this is because the positivity preserving property can hardly be satisfied for high order ones leading to incorrect blow up time and blow up sets recently we have applied discontinuous galerkin methods to linear hyperbolic equations involving singularities and obtained good approximations for nonlinear problems some special limiters are constructed to capture the singularities precisely we will continue this approach and study parabolic equations with blow up solutions we will construct special limiters to keep the positivity of the numerical approximations due to the dirichlet boundary conditions we have to modify the numerical fluxes and the limiters used in the schemes numerical experiments demonstrate that our schemes can capture the blow up sets and high order approximations yield better numerical blow up time

new support vector algorithms we propose a new class of support vector algorithms for regression and classification in these algorithms a parameter lets one effectively control the number of support vectors while this can be useful in its own right the parameterization has the additional benefit of enabling us to eliminate one of the other free parameters of the algorithm the accuracy parameter epsilon in the regression case and the regularization constant c in the classification case we describe the algorithms give some theoretical results concerning the meaning and the choice of and report experimental results

stability and generalization we define notions of stability for learning algorithms and show how to use these notions to derive generalization error bounds based on the empirical error and the leave one out error the methods we use can be applied in the regression framework as well as in the classification one when the classifier is obtained by thresholding a real valued function we study the stability properties of large classes of learning algorithms such as regularization based algorithms in particular we focus on hilbert space regularization and kullback leibler regularization we demonstrate how to apply the results to svm for regression and classification

some comments on cp i study the typical configuration of a cp plot when the number of variables in the regression problem is large and there are many weak effects i show that a particular configuration that is very commonly seen can arise in a simple way i give a formula by means of which the risk incurred by the minimum cp rule can be estimated

generalised information criteria in model selection summary the problem of evaluating the goodness of statistical models is investigated from an information theoretic point of view information criteria are proposed for evaluating models constructed by various estimation procedures when the specified family of probability distributions does not contain the distribution generating the data the proposed criteria are applied to the evaluation of models estimated by maximum likelihood robust penalised likelihood bayes procedures etc we also discuss the use of the bootstrap in model evaluation problems and present a variance reduction technique in the bootstrap simulation

a comparison of gcv and gml for choosing the smoothing parameter in the generalized spline smoothing problem the partially improper prior behind the smoothing spline model is used to obtain a generalization of the maximum likelihood gml estimate for the smoothing parameter then this estimate is compared with the generalized cross validation gcv estimate both analytically and by monte carlo methods the comparison is based on a predictive mean square error criteria it is shown that if the true unknown function being estimated is smooth in a sense to be defined then the gml estimate undersmooths relative to the gcv estimate and the predictive mean square error using the gml estimate goes to zero at a slower rate than the mean square error using the gcv estimate if the true function is rough then the gcv and gml estimates have asymptotically similar behavior a monte carlo experiment was designed to see if the asymptotic results in the smooth case were evident in small sample sizes mixed results were obtained for n 32 gcv was somewhat better than gml for n 64 and gcv was decidedly superior for n 128 in the n 32 case gcv was better for smaller sigma 2 and the comparison close for larger sigma 2 the theoretical results are shown to extend to the generalized spline smoothing model which includes the estimate of functions given noisy values of various integrals of them

maximum principle preserving third order local discontinuous galerkin method for convection diffusion equations on overlapping meshes abstract local discontinuous galerkin ldg methods are popular for convection diffusion equations in ldg methods we introduce an auxiliary variable p to represent the derivative of the primary variable u and solve them on the same mesh it is well known that the maximum principle preserving mpp ldg method is only available up to second order accuracy recently we introduced a new algorithm and solve u and p on different meshes and obtained stability and optimal error estimates in this paper we will continue this approach and construct mpp third order ldg methods for convection diffusion equations on overlapping meshes the new algorithm is more flexible and does not increase any computational cost numerical evidence will be given to demonstrate the accuracy and good performance of the third order mpp ldg method

trading variance reduction with unbiasedness the regularized subspace information criterion for robust model selection in kernel regression a well known result by stein 1956 shows that in particular situations biased estimators can yield better parameter estimates than their generally preferred unbiased counterparts this letter follows the same spirit as we will stabilize the unbiased generalization error estimates by regularization and finally obtain more robust model selection criteria for learning we trade a small bias against a larger variance reduction which has the beneficial effect of being more precise on a single training set we focus on the subspace information criterion sic which is an unbiased estimator of the expected generalization error measured by the reproducing kernel hilbert space norm sic can be applied to the kernel regression and it was shown in earlier experiments that a small regularization of sic has a stabilization effect however it remained open how to appropriately determine the degree of regularization in sic in this article we derive an unbiased estimator of the expected squared error between sic and the expected generalization error and propose determining the degree of regularization of sic such that the estimator of the expected squared error is minimized computer simulations with artificial and real data sets illustrate that the proposed method works effectively for improving the precision of sic especially in the high noise level cases we furthermore compare the proposed method to the original sic the cross validation and an empirical bayesian method in ridge parameter selection with good results

bias variance decompositions for likelihood based estimators the bias variance decomposition of mean squared error is well understood and relatively straightforward in this note a similar simple decomposition is derived valid for any kind of error measure that when using the appropriate probability model can be derived from a kullback leibler divergence or log likelihood

maximum principle satisfying third order discontinuous galerkin schemes for fokker planck equations we design and analyze up to third order accurate discontinuous galerkin dg methods satisfying a strict maximum principle for fokker planck equations a procedure is established to identify an effective test set in each computational cell to ensure the desired bounds of numerical averages during time evolution this is achievable by taking advantage of the two parameters in the numerical flux and a novel decomposition of weighted cell averages based on this result a scaling limiter for the dg method with first order euler forward time discretization is proposed to solve the one dimensional fokker planck equations strong stability preserving high order time discretizations will keep the maximum principle it is straightforward to extend the method to two and higher dimensions on rectangular meshes we also show that a modified limiter can preserve the strict maximum principle for dg schemes solving fokker planck equations as a consequence the present schemes preserve steady states numerical tests

multi symplectic quasi interpolation method for hamiltonian partial differential equations abstract in this paper we propose a multi symplectic quasi interpolation method for solving multi symplectic hamiltonian partial differential equations based on the method of lines we first discretize the multi symplectic pdes using quasi interpolation method and then employ appropriate time integrators to obtain the full discrete system the local conservation properties including multi symplectic conservation laws energy conservation laws and momentum conservation laws are discussed in detail for illustration we provide two concrete examples the nonlinear wave equation and the nonlinear schrodinger equation the salient feature of our multi symplectic quasi interpolation method is that it is valid both on uniform grids and nonuniform grids the numerical results show the good accuracy and excellent conservation properties of the proposed method

subspace information criterion for model selection the problem of model selection is considerably important for acquiring higher levels of generalization capability in supervised learning in this article we propose a new criterion for model selection the subspace information criterion sic which is a generalization of mallows s cl it is assumed that the learning target function belongs to a specified functional hilbert space and the generalization error is defined as the hilbert space squared norm of the difference between the learning result function and target function sic gives an unbiased estimate of the generalization error so defined sic assumes the availability of an unbiased estimate of the target function and the noise covariance matrix which are generally unknown a practical calculation method of sic for least mean squares learning is provided under the assumption that the dimension of the hilbert space is less than the number of training examples finally computer simulations in two examples show that sic works well even when the number of training examples is small

efficient accurate and flexible finite element solvers for chemotaxis problems in the framework of finite element discretizations we introduce a fully nonlinear newton like method and a linearized second order approach in time applied to certain partial differential equations for chemotactic processes incorporating two entities a chemical agent and the reacting population of certain biological organisms species we investigate the benefit of a corresponding monolithic approach and the decoupled variant in particular we analyze accuracy efficiency and stability of different methods and their dependences on certain parameters in order to identify a well suited finite element solver for chemotaxis problems

discrete maximum principle and adequate discretizations of linear parabolic problems in this paper we analyze the connections between the different qualitative properties of numerical solutions of linear parabolic problems with dirichlet type boundary condition first we formulate the qualitative properties for the differential equations and shed light on their relations then we show how the well known discretization schemes can be written in the form of a one step iterative process we give necessary and sufficient conditions of the main qualitative properties of these iterations we apply the results to the finite difference and galerkin finite element solutions of linear parabolic problems in our main result we show that the nonnegativity preservation property is equivalent to the maximum minimum principle and they imply the maximum norm contractivity in one two and three dimensions we list sufficient a priori conditions that ensure the required qualitative properties finally we demonstrate the above results on numerical examples

bound preserving modified exponential runge kutta discontinuous galerkin methods for scalar hyperbolic equations with stiff source terms abstract in this paper we develop bound preserving modified exponential runge kutta rk discontinuous galerkin dg schemes to solve scalar hyperbolic equations with stiff source terms by extending the idea in zhang and shu 43 exponential strong stability preserving ssp high order time discretizations are constructed and then modified to overcome the stiffness and preserve the bound of the numerical solutions it is also straightforward to extend the method to two dimensions on rectangular and triangular meshes even though we only discuss the bound preserving limiter for dg schemes it can also be applied to high order finite volume schemes such as weighted essentially non oscillatory weno finite volume schemes as well

a positivity preserving high order semi lagrangian discontinuous galerkin scheme for the vlasov poisson equations the vlasov poisson equations describe the evolution of a collisionless plasma represented through a probability density function pdf that self interacts via an electrostatic force one of the main difficulties in numerically solving this system is the severe time step restriction that arises from parts of the pdf associated with moderate to large velocities the dominant approach in the plasma physics community for removing these time step restrictions is the so called particle in cell pic method which discretizes the distribution function into a set of macro particles while the electric field is represented on a mesh several alternatives to this approach exist including fully lagrangian fully eulerian and so called semi lagrangian methods the focus of this work is the semi lagrangian approach which begins with a grid based eulerian representation of both the pdf and the electric field then evolves the pdf via lagrangian dynamics and finally projects this evolved field back onto the original eulerian mesh in particular we develop in this work a method that discretizes the 1 1 vlasov poisson system via a high order discontinuous galerkin dg method in phase space and an operator split semi lagrangian method in time second order accuracy in time is relatively easy to achieve via strang operator splitting with additional work using higher order splitting and a higher order method of characteristics we also demonstrate how to push this scheme to fourth order accuracy in time we show how to resolve all of the lagrangian dynamics in such a way that mass is exactly conserved positivity is maintained and high order accuracy is achieved the poisson equation is solved to high order via the smallest stencil local discontinuous galerkin ldg approach we test the proposed scheme on several standard test cases

on positivity preserving finite volume schemes for euler equations we consider the positivity preserving property of first and higher order finite volume schemes for one and two dimensional euler equations of gas dynamics a general framework is established which shows the positivity of density and pressure whenever the underlying one dimensional first order building block based on an exact or approximate riemann solver and the reconstruction are both positivity preserving appropriate limitation to achieve a high order positivity preserving reconstruction is described

high order local maximum principle preserving mpp discontinuous galerkin finite element method for the transport equation abstract in this work we present a fct like maximum principle preserving mpp method to solve the transport equation we use high order polynomial spaces in particular we consider up to 5th order spaces in two and three dimensions and 23rd order spaces in one dimension the method combines the concepts of positive basis functions for discontinuous galerkin finite element spatial discretization locally defined solution bounds element based flux correction and non linear local mass redistribution we consider a simple 1d problem with non smooth initial data to explain and understand the behavior of different parts of the method convergence tests in space indicate that high order accuracy is achieved numerical results from several benchmarks in two and three dimensions are also reported

an introduction to kernel based learning algorithms this paper provides an introduction to support vector machines kernel fisher discriminant analysis and kernel principal component analysis as examples for successful kernel based learning methods we first give a short background about vapnik chervonenkis theory and kernel feature spaces and then proceed to kernel based learning in supervised and unsupervised scenarios including practical and algorithmic considerations we illustrate the usefulness of kernel algorithms by discussing applications such as optical character recognition and dna analysis

a study on moving mesh finite element solution of the porous medium equation an adaptive moving mesh finite element method is studied for the numerical solution of the porous medium equation with and without variable exponents and absorption the method is based on the moving mesh partial differential equation approach and employs its newly developed implementation the implementation has several improvements over the traditional one including its explicit compact form of the mesh velocities ease to program and less likelihood of producing singular meshes three types of metric tensor that correspond to uniform and arclength based and hessian based adaptive meshes are considered the method shows first order convergence for uniform and arclength based adaptive meshes and second order convergence for hessian based adaptive meshes it is also shown that the method can be used for situations with complex free boundaries emerging and splitting of free boundaries and the porous medium equation with variable exponents and absorption two dimensional numerical results are presented

bound preserving discontinuous galerkin methods for conservative phase space advection in curvilinear coordinates we extend the positivity preserving method of zhang and shu 49 to simulate the advection of neutral particles in phase space using curvilinear coordinates the ability to utilize these coordinates is important for non equilibrium transport problems in general relativity and also in science and engineering applications with specific geometries the method achieves high order accuracy using discontinuous galerkin dg discretization of phase space and strong stability preserving runge kutta ssp rk time integration special care is taken to ensure that the method preserves strict bounds for the phase space distribution function f i e f 0 1 the combination of suitable cfl conditions and the use of the high order limiter proposed in 49 is sufficient to ensure positivity of the distribution function however to ensure that the distribution function satisfies the upper bound the discretization must in addition preserve the divergence free property of the phase space flow proofs that highlight the necessary conditions are presented for general curvilinear coordinates and the details of these conditions are worked out for some commonly used coordinate systems i e spherical polar spatial coordinates in spherical symmetry and cylindrical spatial coordinates in axial symmetry both with spherical momentum coordinates results from numerical experiments including one example in spherical symmetry adopting the schwarzschild metric demonstrate that the method achieves high order accuracy and that the distribution function satisfies the maximum principle

a positivity preserving high order discontinuous galerkin scheme for convection diffusion equations abstract for constructing high order accurate positivity preserving schemes for convection diffusion equations we construct a simple positivity preserving diffusion flux discontinuous galerkin dg schemes with such a positivity preserving diffusion flux are nonlinear schemes which can be regarded as a reduction of the high order positivity preserving dg schemes for compressible navier stokes equations in 1 to scalar diffusion operators in this paper we focus on the local dg method to discuss how to apply such a flux a limiter on the auxiliary variable for approximating the gradient of the solution must be used so that the diffusion flux is positivity preserving in the sense that dg schemes with this flux satisfies a weak positivity property together with a positivity preserving limiter high order dg schemes with strong stability preserving time discretizations can be rendered positivity preserving without losing conservation or high order accuracy for convection diffusion problems with periodic boundary conditions or a special class of dirichlet or neumann boundary conditions numerical tests on a few parabolic equations and an application to modeling electrical discharges are shown to demonstrate the performance of this scheme

preserving nonnegativity of an affine finite element approximation for a convection diffusion reaction problem an affine finite element scheme approximation of a time dependent linear convection diffusion reaction problem in 2d and 3d is presented for these equations which do not satisfy an underlying maximum principle sufficient conditions are given in terms of the coefficient functions the computational grid and the discretization parameters to ensure that the nonnegativity property of the true solution is also satisfied by its approximation numerical examples are given which confirm the necessity and sufficiency of the discretization conditions to guarantee the nonnegativity of the approximation we present a monotone fem scheme for a convection diffusion reaction problem in 2 3d the considered equation does not possess an underlying maximum principle sufficient conditions are given to ensure the nonnegativity of the approximations numerical examples confirm the necessity and sufficiency of the conditions

on the existence of maximum principles in parabolic finite element equations in 1973 h fujii investigated discrete versions of the maximum principle for the model heat equation using piecewise linear finite elements in space in particular he showed that the lumped mass method allows a maximum principle when the simplices of the triangulation are acute and this is known to generalize in two space dimensions to triangulations of delauney type in this note we consider more general parabolic equations and first show that a maximum principle cannot hold for the standard spatially semidiscrete problem we then show that for the lumped mass method the above conditions on the triangulation are essentially sharp this is in contrast to the elliptic case in which the requirements are weaker we also study conditions for the solution operator acting on the discrete initial data with homogeneous lateral boundary conditions to be a contraction or a positive operator

sign preserving of principal eigenfunctions in p1 finite element approximation of eigenvalue problems of second order elliptic operators this paper is concerned with the p1 finite element approximation of the eigenvalue problem of second order elliptic operators subject to the dirichlet boundary condition the focus is on the preservation of the basic properties of the principal eigenvalue and eigenfunctions of the continuous problem it is shown that when the stiffness matrix is an irreducible m matrix the discrete eigenvalue problem maintains almost all of the basic properties such as the smallest eigenvalue being real and simple and the corresponding eigenfunctions being either positive or negative inside the physical domain mesh conditions leading to such a stiffness matrix are also studied a sufficient condition is that the mesh is simplicial interiorly connected and acute when measured in the metric specified by the inverse of the diffusion matrix the acute requirement can be replaced by the delaunay condition in two dimensions numerical results show that when the stiffness matrix is not an m matrix a finite element approximation can be structurally different from the continuous eigenvalue problem the eigenfunctions corresponding to the smallest eigenvalue can change sign over the physical domain and the smallest eigenvalue in modulus can even be complex for the case with nonsymmetric operators

high order bound preserving discontinuous galerkin methods for compressible miscible displacements in porous media on triangular meshes abstract in this paper we develop high order bound preserving bp discontinuous galerkin dg methods for the coupled system of compressible miscible displacements on triangular meshes we consider the problem with multi component fluid mixture and the volumetric concentration of the jth component c j should be between 0 and 1 there are three main difficulties firstly c j does not satisfy a maximum principle therefore the numerical techniques introduced in zhang and shu 2010 44 cannot be applied directly the main idea is to apply the positivity preserving techniques to all c j s and enforce j c j 1 simultaneously to obtain physically relevant approximations by doing so we have to treat the time derivative of the pressure d p d t as a source in the concentration equation and choose suitable fluxes in the pressure and concentration equations secondly it is not easy to construct first order numerical fluxes for interior penalty dg methods on triangular meshes one of the key points in the high order bp technique applied in this paper is the combination of high order and lower order numerical fluxes we will construct second order bp schemes and use the second order numerical fluxes as the lower order one finally the classical slope limiter cannot be applied to c j to construct the bp technique we will not approximate c j directly therefore a new limiter will be introduced numerical experiments will be given to demonstrate the high order accuracy and good performance of the numerical technique

more comments on c p we discuss the interpretation of c p plots and show how they can be calibrated in several ways we comment on the practice of using the display as a basis for formal selection of a subset regression model and extend the range of application of the device to encompass arbitrary linear estimates of the regression coefficients for example ridge estimates