advection diffusion reaction equations hyperbolization and high order ader discretizations the purpose of this paper is twofold first we extend the applicability of cattaneo s relaxation approach one of the currently known relaxation approaches to reformulate time dependent advection

pollutant transport by shallow water equations on unstructured meshes hyperbolization of the model and numerical solution via a novel flux splitting scheme abstract the purpose of this paper is twofold first using the cattaneo s relaxation approach we reformulate the system of governing equations for the pollutant transport by shallow water flows over non flat topography and anisotropic diffusion as hyperbolic balance laws with stiff source terms the proposed relaxation system circumvents the infinite wave speed paradox which is inherent in standard advection diffusion models this turns out to give a larger stability range for the choice of the time step second following a flux splitting approach we derive a novel numerical method to discretise the resulting problem in particular we propose a new flux splitting and study the associated two systems of differential equations called the hydrodynamic and the relaxed diffusive system respectively for the presented splitting we analyse the resulting two systems of differential equations and propose two discretisation schemes of the godunov type these schemes are simple to implement robust accurate and fast when compared with existing methods the resulting method is implemented on unstructured meshes and is systematically assessed for accuracy robustness and efficiency on a carefully selected suite of test problems including non flat topography and wetting and drying problems formal second order accuracy is assessed through convergence rates studies

uncertainty quantification methodology for hyperbolic systems with application to blood flow in arteries abstract we present a stochastic finite volume ader sfv ader methodology for uncertainty quantification uq in the general framework of systems of hyperbolic balance laws with uncertainty in parameters the resulting method is weakly intrusive meaning that deterministic solvers need minor modifications to include random contributions and has no theoretical accuracy barrier an illustration of the second order version for the viscous burgers equation with uncertain viscosity coefficient is first given in detail under different deterministic initial conditions the attainment of the theoretically expected convergence rate is demonstrated empirically results and features of our scheme are discussed we then extend the sfv ader method to a non linear hyperbolic system with source terms that models one dimensional blood flow in arteries assuming uncertainties in a set of parameters of the problem results are then compared with measurements in a well defined 1 1 experimental replica of the network of largest arteries in the human systemic circulation for a 99 confidence level severe influence of parameter fluctuations is seen in pressure profiles while secondary effects on flow rate profiles become visible particularly in terminal branches results suggest that the proposed methodology could successfully be applied to other physical problems involving hyperbolic balance laws

covariance regularization by thresholding this paper considers regularizing a covariance matrix of p variables estimated from n observations by hard thresholding we show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse in a suitable sense the variables are gaussian or sub gaussian and log p n 0 and obtain explicit rates the results are uniform over families of covariance matrices which satisfy a fairly natural notion of sparsity we discuss an intuitive resampling scheme for threshold selection and prove a general cross validation result that justifies this approach we also compare thresholding to other covariance estimators in simulations and on an example from climate data 1 introduction estimation of covariance matrices is important in a number of areas of statistical analysis including dimension reduction by principal component analysis pca classification by linear or quadratic discriminant analysis lda and qda establishing independence and conditional independence relations in the context of graphical models and setting confidence intervals on linear functions of the means of the components in recent years many application areas where these tools are used have been dealing with very high dimensional datasets and sample sizes can be very small relative to dimension examples include genetic data brain imaging spectroscopic imaging climate data and many others it is well known by now that the empirical covariance matrix for samples of size n from a p variate gaussian distribution np p is not a good estimator of the population covariance if p is large many results in random matrix theory illustrate this from the classical mary law 29 to the more recent work of johnstone and his students on the theory of the largest eigenvalues 12 23 30 and associated eigenvectors 24 however with the exception of a method for estimating the covariance spectrum 11 these probabilistic results do not offer alternatives to the sample covariance matrix alternative estimators for large covariance matrices have therefore attracted a lot of attention recently two broad classes of covariance estimators have emerged those that rely on a natural ordering among variables and assume that variables

on consistency and sparsity for principal components analysis in high dimensions principal components analysis pca is a classic method for the reduction of dimensionality of data in the form of n observations or cases of a vector with p variables contemporary datasets often have p comparable with or even much larger than n our main assertions in such settings are a that some initial reduction in dimensionality is desirable before applying any pca type search for principal modes and b the initial reduction in dimensionality is best achieved by working in a basis in which the signals have a sparse representation we describe a simple asymptotic model in which the estimate of the leading principal component vector via standard pca is consistent if and only if p n n 0 we provide a simple algorithm for selecting a subset of coordinates with largest sample variances and show that if pca is done on the selected subset then consistency is recovered even if p n n

covariance matrix selection and estimation via penalised normal likelihood summary we propose a nonparametric method for identifying parsimony and for producing a statistically efficient estimator of a large covariance matrix we reparameterise a covariance matrix through the modified cholesky decomposition of its inverse or the one step ahead predictive representation of the vector of responses and reduce the nonintuitive task of modelling covariance matrices to the familiar task of model selection and estimation for a sequence of regression models the cholesky factor containing these regression coefficients is likely to have many off diagonal elements that are zero or close to zero penalised normal likelihoods in this situation with l1 and l2 penalities are shown to be closely related to tibshirani s 1996 lasso approach and to ridge regression adding either penalty to the likelihood helps to produce more stable estimators by introducing shrinkage to the elements in the cholesky factor while because of its singularity the l1 penalty will set some elements to zero and produce interpretable models an algorithm is developed for computing the estimator and selecting the tuning parameter the proposed maximum penalised likelihood estimator is illustrated using simulation and a real dataset involving estimation of a 102 x 102 covariance matrix

optimal rates of convergence for sparse covariance matrix estimation this paper considers estimation of sparse covariance matrices and establishes the optimal rate of convergence under a range of matrix operator norm and bregman divergence losses a major focus is on the derivation of a rate sharp minimax lower bound the problem exhibits new features that are significantly different from those that occur in the conventional nonparametric function estimation problems standard techniques fail to yield good results and new tools are thus needed we first develop a lower bound technique that is particularly well suited for treating two directional problems such as estimating sparse covariance matrices the result can be viewed as a generalization of le cam s method in one direction and assouad s lemma in another this lower bound technique is of independent interest and can be used for other matrix estimation problems we then establish a rate sharp minimax lower bound for estimating sparse covariance matrices under the spectral norm by applying the general lower bound technique a thresholding estimator is shown to attain the optimal rate of convergence under the spectral norm the results are then extended to

high dimensional covariance matrix estimation using a factor model abstract high dimensionality comparable to sample size is common in many statistical problems we examine covariance matrix estimation in the asymptotic framework that the dimensionality p tends to as the sample size n increases motivated by the arbitrage pricing theory in finance a multi factor model is employed to reduce dimensionality and to estimate the covariance matrix the factors are observable and the number of factors k is allowed to grow with p we investigate the impact of p and k on the performance of the model based covariance matrix estimator under mild assumptions we have established convergence rates and asymptotic normality of the model based estimator its performance is compared with that of the sample covariance matrix we identify situations under which the factor approach increases performance substantially or marginally the impacts of covariance matrix estimation on optimal portfolio allocation and portfolio risk assessment are studied the asymptotic results are supported by a thorough simulation study

implicit semi analytical solution of the generalized riemann problem for stiff hyperbolic balance laws we present a semi analytical implicit solution to the generalized riemann problem grp for non linear systems of hyperbolic balance laws with stiff source terms the solution method is based on an implicit time taylor series expansion and the cauchy kowalewskaya procedure along with the solution of a sequence of classical riemann problems our new grp solver is then used to construct locally implicit ader methods of arbitrary accuracy in space and time for solving the general initial boundary value problem for non linear systems of hyperbolic balance laws with stiff source terms analysis of the method for model problems is carried out and empirical convergence rate studies for suitable tests problems are performed confirming the theoretically expected high order of accuracy

low dissipation centred schemes for hyperbolic equations in conservative and non conservative form abstract we propose a one parameter family of low dissipation centred numerical schemes for solving hyperbolic equations in conservative or non conservative form using finite volume or discontinuous galerkin finite element methods the new schemes spring out from the multi dimensional force method and are determined by a single parameter 1 given an increasing sequence of real numbers 1 1 2 k there corresponds a sequence of numerical schemes with stability restriction associated to a decreasing sequence of courant numbers 1 c 1 c 2 c k 0 and a decreasing sequence of corresponding numerical viscosity functions d 1 d 2 d k for a given courant number c k 1 there is a real number k 0 and a corresponding stable scheme with minimal numerical viscosity d k the proposed schemes suit very well the family of high order discontinuous galerkin finite element methods and the recently proposed class of ader tr schemes whose orders of accuracy define decreasing sequences of courant numbers as the order of accuracy increases the centred methods of this paper are stable in 2d and 3d in the frame of simultaneous updating formulae unlike other centred methods such as 1d force which are not furthermore the schemes are highly accurate for slowly moving waves which is precisely the kind of waves that traditional centred methods smear disastrously additional features of the proposed schemes include ease of implementation and applicability to any hyperbolic system either in conservative or non conservative form here the proposed schemes are analysed and computationally assessed through a suite of test problems for a linear model system for the euler equations in one and two space dimensions and for the baer nunziato equations for compressible two phase flow

regularized estimation of large covariance matrices this paper considers estimating a covariance matrix of p variables from n observations by either banding the sample covariance matrix or estimating a banded version of the inverse of the covariance we show that these estimates are consistent in the operator norm as long as logp 2 n 0 and obtain explicit rates the results are uniform over some fairly natural well conditioned families of covariance matrices we also introduce an analogue of the gaussian white noise model and show that if the population covariance is embeddable in that model and well conditioned then the banded approximations produce consistent estimates of the eigenvalues and associated eigenvectors of the covariance matrix the results can be extended to smooth versions of banding and to non gaussian distributions with su ciently short tails a resampling approach is proposed for choosing the banding parameter in practice this approach is illustrated numerically on both simulated and real data

hierarchical models of thin elastic structures overview and recent advances in error estimation and adaptivity abstract in the numerical approximation of hierarchical models of thin bodies there are two separate contributions to the total error a modeling error is the error due to dimensional reduction and b discretization error is the error due to the numerical approximation traditionally estimation of these errors has been performed in terms of abstract quantities like the global energy norm which does not give a clear representation of the error in the local features of the solution recently a new class of finite element error estimation techniques has emerged wherein errors in a simulation are measured not in norms but in quantities of interest to the analyst following a brief overview of the literature we focus on extending the theory of a posteriori estimation of the modeling error to local quantities of interest that have physical meaning independent estimates are obtained for the modeling and discretization errors in quantities of interest these local estimates are used to develop goal oriented adaptive modeling strategies in which the model is automatically adapted by controlling the two errors independently and effectively to obtain local quantities of interest to within a preset level of accuracy numerical examples are presented to demonstrate the accuracy of the error estimates and the adaptive strategy

an adaptive residual based splitting approach for the penalized navier stokes equations the interest on immersed boundary methods ibm is increasing in computational fluid dynamics as they simplify the mesh generation problem in this work we consider an approach based on the addition of a penalty term to the navier stokes equations to account for the wall boundary conditions to discretize the resulting equations we use a residual distribution approach previously developed by some of the authors to adapt the method to the ibm considered we developed a new formulation of residual distribution based on a strang splitting method in time coupling an implicit asymptotic integration procedure of the penalization ode with a simplified explicit residual distribution for the navier stokes equations the first method provides an operator which is exact up to orders 2 with the penalty parameter assuming values of the order of 10 10 a modification of the solution gradient reconstruction necessary for the evaluation of the viscous fluxes is also introduced in the paper this guarantees that correct physical values of the viscous stresses are recovered in vicinity of the solid we show formally and numerically that the approach proposed is second order accurate for smooth solutions we evaluate its potential for ibm by coupling the resulting method with unstructured mesh adaptation on wall boundaries several steady and time dependent tests are used to show the promising features of the method proposed

adaptive thresholding for sparse covariance matrix estimation in this article we consider estimation of sparse covariance matrices and propose a thresholding procedure that is adaptive to the variability of individual entries the estimators are fully data driven and demonstrate excellent performance both theoretically and numerically it is shown that the estimators adaptively achieve the optimal rate of convergence over a large class of sparse covariance matrices under the spectral norm in contrast the commonly used universal thresholding estimators are shown to be suboptimal over the same parameter spaces support recovery is discussed as well the adaptive thresholding estimators are easy to implement the numerical performance of the estimators is studied using both simulated and real data simulation results demonstrate that the adaptive thresholding estimators uniformly outperform the universal thresholding estimators the method is also illustrated in an analysis on a dataset from a small round blue cell tumor microarray experiment a supplement to this ar

high order ader schemes for a unified first order hyperbolic formulation of newtonian continuum mechanics coupled with electro dynamics abstract in this paper we propose a new unified first order hyperbolic model of newtonian continuum mechanics coupled with electro dynamics the model is able to describe the behavior of moving elasto plastic dielectric solids as well as viscous and inviscid fluids in the presence of electro magnetic fields it is actually a very peculiar feature of the proposed pde system that viscous fluids are treated just as a special case of elasto plastic solids this is achieved by introducing a strain relaxation mechanism in the evolution equations of the distortion matrix a which in the case of purely elastic solids maps the current configuration to the reference configuration the model also contains a hyperbolic formulation of heat conduction as well as a dissipative source term in the evolution equations for the electric field given by ohm s law via formal asymptotic analysis we show that in the stiff limit the governing first order hyperbolic pde system with relaxation source terms tends asymptotically to the well known viscous and resistive magnetohydrodynamics mhd equations furthermore a rigorous derivation of the model from variational principles is presented together with the transformation of the euler lagrange differential equations associated with the underlying variational problem from lagrangian coordinates to eulerian coordinates in a fixed laboratory frame the present paper hence extends the unified first order hyperbolic model of newtonian continuum mechanics recently proposed in 110 42 to the more general case where the continuum is coupled with electro magnetic fields the governing pde system is symmetric hyperbolic and satisfies the first and second principle of thermodynamics hence it belongs to the so called class of symmetric hyperbolic thermodynamically compatible systems shtc which have been studied for the first time by godunov in 1961 61 and later in a series of papers by godunov and romenski 67 69 119 an important feature of the proposed model is that the propagation speeds of all physical processes including dissipative processes are finite the model is discretized using high order accurate ader discontinuous galerkin dg finite element schemes with a posteriori subcell finite volume limiter and using high order ader weno finite volume schemes we show numerical test problems that explore a rather large parameter space of the model ranging from ideal mhd viscous and resistive mhd over pure electro dynamics to moving dielectric elastic solids in a magnetic field

model selection and estimation in the gaussian graphical model summary we propose penalized likelihood methods for estimating the concentration matrix in the gaussian graphical model the methods lead to a sparse and shrinkage estimator of the concentration matrix that is positive definite and thus conduct model selection and estimation simultaneously the implementation of the methods is nontrivial because of the positive definite constraint on the concentration matrix but we show that the computation can be done effectively by taking advantage of the efficient maxdet algorithm developed in convex optimization we propose a bic type criterion for the selection of the tuning parameter in the penalized likelihood methods the connection between our methods and existing methods is illustrated simulations and real examples demonstrate the competitive performance of the new methods

optimal rates of convergence for covariance matrix estimation covariance matrix plays a central role in multivariate statistical analysis significant advances have been made recently on developing both theory and methodology for estimating large covariance matrices however a minimax theory has yet been developed in this paper we establish the optimal rates of convergence for estimating the covariance matrix under both the operator norm and frobenius norm it is shown that optimal procedures under the two norms are different and consequently matrix estimation under the operator norm is fundamentally different from vector estimation the minimax upper bound is obtained by constructing a special class of tapering estimators and by studying their risk properties a key step in obtaining the optimal rate of convergence is the derivation of the minimax lower bound the technical analysis requires new ideas that are quite different from those used in the more conventional function sequence estimation problems

a constrained l1 minimization approach to sparse precision matrix estimation this article proposes a constrained l1 minimization method for estimating a sparse inverse covariance matrix based on a sample of n iid p variate random variables the resulting estimator is shown to have a number of desirable properties in particular the rate of convergence between the estimator and the true s sparse precision matrix under the spectral norm is when the population distribution has either exponential type tails or polynomial type tails we present convergence rates under the elementwise l norm and frobenius norm in addition we consider graphical model selection the procedure is easily implemented by linear programming numerical performance of the estimator is investigated using both simulated and real data in particular the procedure is applied to analyze a breast cancer dataset and is found to perform favorably compared with existing methods

improved accuracy for time splitting methods for the numerical solution of parabolic equations in this work we study time splitting strategies for the numerical approximation of evolutionary reaction diffusion problems in particular we formulate a family of domain decomposition splitting methods that overcomes some typical limitations of classical alternating direction implicit adi schemes the splitting error associated with such methods is observed to be o 2 in the time step in order to decrease the size of this splitting error to o 3 we add a correction term to the right hand side of the original formulation this procedure is based on the improved initialization technique proposed by douglas and kim in the framework of adi methods the resulting non iterative schemes reduce the global system to a collection of uncoupled subdomain problems that can be solved in parallel computational results comparing the newly derived algorithms with the crank nicolson scheme and certain adi methods are presented

sparsistency and rates of convergence in large covariance matrix estimation this paper studies the sparsistency and rates of convergence for estimating sparse covariance and precision matrices based on penalized likelihood with nonconvex penalty functions here sparsistency refers to the property that all parameters that are zero are actually estimated as zero with probability tending to one depending on the case of applications sparsity priori may occur on the covariance matrix its inverse or its cholesky decomposition we study these three sparsity exploration problems under a unified framework with a general penalty function we show that the rates of convergence for these problems under the frobenius norm are of order s n log p n n 1 2 where s n is the number of nonzero elements p n is the size of the covariance matrix and n is the sample size this explicitly spells out the contribution of high dimensionality is merely of a logarithmic factor the conditions on the rate with which the tuning parameter n goes to 0 have been made explicit and compared under different penalties as a result for the l 1 penalty to guarantee the sparsistency and optimal rate of convergence the number of nonzero elements should be small sn o pn at most among o pn2 parameters for estimating sparse covariance or correlation matrix sparse precision or inverse correlation matrix or sparse cholesky factor where sn is the number of the nonzero elements on the off diagonal entries on the other hand using the scad or hard thresholding penalty functions there is no such a restriction

random vectors in the isotropic position abstract letybe a random vector in r n satisfying e y y id letmbe a natural number and lety1 ymbe independent copies ofy we study the question of approximation of the identity operator by finite sums of the tensorsyi yi we prove that for some absolute constantc e 1 m i 1 m y i y i id c log n m e y log m 1 log m provided that the last expression is smaller than 1 we apply this estimate to improve a result of bourgain concerning the number of random points needed to bring a convex body into a nearly isotropic position

a five equation model for the simulation of miscible and viscous compressible fluids abstract typical multispecies compressible navier stokes computations employ conservative equations for mass fraction transport upwind discretisations of these governing equations produce spurious pressure oscillations at diffuse contact surfaces between gases of differing ratio of specific heat capacities which degrade the convergence rate of the algorithm adding quasi conservative equations for volume fraction can solve this error however this approach has been derived only for immiscible fluids here a five equation quasi conservative model is proposed that includes the effects of species diffusion viscosity and thermal conductivity the derivation of the model is presented along with a numerical method to solve the governing equations at second order accuracy in space and time formal convergence studies demonstrate the expected order of accuracy is achieved for three benchmark problems cross validated against two standard mass fraction models in these test cases the new model has between 2 and 10 times lower error for a given grid size simulations of a two dimensional air sf6 richtmyer meshkov instability demonstrate that the new model converges to the solution with four times fewer points in each direction when compared to the mass fraction model in an identical numerical framework this represents an 40 times lower computational cost for an equivalent error in two dimensional computations the proposed model is thus very suitable for direct numerical simulation and large eddy simulation of compressible mixing

brain venous haemodynamics neurological diseases and mathematical modelling a review behind medicine m is physiology p behind physiology is physics p and behind physics is always mathematics m for which i expect that the symmetry of the quadruplet mppm will be compatible with the characteristic bias of hyperbolic partial differential equations a theme of this paper i start with a description of several idiopathic brain pathologies that appear to have a strong vascular dimension of which the most prominent example considered here is multiple sclerosis the most common neurodegenerative disabling disease in young adults other pathologies surveyed here include retinal abnormalities transient global amnesia transient monocular blindness meniere s disease and idiopathic parkinson s disease it is the hypothesised vascular aspect of these conditions that links medicine to mathematics through fluid mechanics in very complex networks of moving boundary blood vessels the second part of this paper is about mathematical modelling of the human cardiovascular system with particular reference to the venous system and the brain a review of a recently proposed multi scale mathematical model then follows consisting of a one dimensional hyperbolic description of blood flow in major arteries and veins coupled to a lumped parameter description of the remaining main components of the human circulation derivation and analysis of the hyperbolic equations is carried out for blood vessels admitting variable material properties and with emphasis on the venous system a much neglected aspect of cardiovascular mathematics veins unlike their arterial counterparts are highly deformable even collapsible under mild physiological conditions we address mathematical and numerical challenges regarding the numerical analysis of the hyperbolic pdes we deploy a modern non linear finite volume method of arbitrarily high order of accuracy in both space and time the ader methodology in vivo validation examples and brain haemodynamics computations are shown we also point out two preliminary but important new findings through the use of mathematical models namely that extracranial venous strictures produce chronic intracranial venous hypertension and that augmented pressure increases the blood vessel wall permeability

generalized thresholding of large covariance matrices we propose a new class of generalized thresholding operators that combine thresholding with shrinkage and study generalized thresholding of the sample covariance matrix in high dimensions generalized thresholding of the covariance matrix has good theoretical properties and carries almost no computational burden we obtain an explicit convergence rate in the operator norm that shows the tradeoff between the sparsity of the true model dimension and the sample size and shows that generalized thresholding is consistent over a large class of models as long as the dimension p and the sample size n satisfy log p n 0 in addition we show that generalized thresholding has the sparsistency property meaning it estimates true zeros as zeros with probability tending to 1 and under an additional mild condition is sign consistent for nonzero elements we show that generalized thresholding covers as special cases hard and soft thresholding smoothly clipped absolute deviation and adaptive lasso and comp

reconstructed discontinuous galerkin methods for linear advection diffusion equations based on first order hyperbolic system abstract newly developed reconstructed discontinuous galerkin rdg methods are presented for solving linear advection diffusion equations on hybrid unstructured grids based on a first order hyperbolic system fohs formulation benefiting from both fohs and rdg methods the developed hyperbolic rdg methods are reliable accurate efficient and robust achieving higher orders of accuracy than conventional dg methods for the same number of degrees of freedom superior accuracy is achieved by reconstruction of higher order terms in the solution polynomial via gradient variables introduced to form a hyperbolic diffusion system and least squares variational reconstruction unsteady capability is demonstrated by an l stable implicit time integration scheme a number of advection diffusion test cases with a wide range of reynolds numbers including boundary layer type problems and unsteady cases are presented to assess accuracy and performance of the newly developed hyperbolic rdg methods numerical experiments demonstrate that the hyperbolic rdg methods are able to achieve the designed optimal order of accuracy for both solutions and their derivatives on regular irregular and heterogeneous grids indicating that the developed hyperbolic rdg methods provide an attractive and probably an even superior alternative for solving the linear advection diffusion equations

minimax and adaptive estimation of covariance operator for random variables observed on a lattice graph covariance structure plays an important role in high dimensional statistical inference in a range of applications including imaging analysis and fmri studies random variables are observed on a lattice graph in such a setting it is important to account for the lattice structure when estimating the covariance operator in this article we consider both minimax and adaptive estimation of the covariance operator over collections of polynomially decaying and exponentially decaying parameter spaces we first establish the minimax rates of convergence for estimating the covariance operator under the operator norm the results show that the dimension of the lattice graph significantly affects the optimal rates convergence often much more so than the dimension of the random variables we then consider adaptive estimation of the covariance operator a fully data driven block thresholding procedure is proposed and is shown to be adaptively rate optimal simultaneously over a wide range of polynomially decaying a