equilibrium points of a singular cooperative system with free boundary abstract in this paper we initiate the study of maps minimising the energy d u 2 2 u d x which due to lipschitz character of the integrand gives rise to the singular euler equations u u u u 0 u u 1 u m our primary goal in this paper is to set up a road map for future developments of the theory related to such energy minimising maps our results here concern regularity of the solution as well as that of the free boundary they are achieved by using monotonicity formulas and epiperimetric inequalities in combination with geometric analysis

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

optimal a priori estimates for interface problems we consider a priori estimates in weighted norms for interface problems with piecewise constant diffusion constants which do not depend on the ratio between the constants our result generalizes an estimate of lemrabet to arbitrary dimensions and includes curved boundaries furthermore we discuss criteria for the existence of a uniform poincare estimate in weighted norms in the affirmative case we obtain a robust finite element error bound in weighted norms finally we present numerical experiments including a case with no uniform poincare constant

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

determining the number of factors in the general dynamic factor model this article develops an information criterion for determining the number q of common shocks in the general dynamic factor model developed by forni et al as opposed to the restricted dynamic model considered by bai and ng and by amengual and watson our criterion is based on the fact that this number q is also the number of diverging eigenvalues of the spectral density matrix of the observations as the number n of series goes to infinity we provide sufficient conditions for consistency of the criterion for large n and t where t is the series length we show how the method can be implemented and provide simulations and empirics illustrating its very good finite sample performance application to real data adds a new empirical facet to an ongoing debate on the number of factors driving the u s economy

constrained factor models this article considers estimation and applications of constrained and partially constrained factor models when the dimension of explanatory variables is high both the classical and approximate factor models are investigated for estimation we employ both the maximum likelihood and least squares methods we show that the least squares estimation is based on constrained principal component analysis and provides consistent estimates for the model under certain conditions the normality condition is not used in the derivation we then propose likelihood ratio statistics to test the adequacy of factor constraints the test statistic is developed under the normality assumption but simulation results show that it continues to perform well even if the underlying distribution is student t the constraints are useful tools to incorporate prior information or substantive theory in applications of factor models in addition the constraints also serve as a statistical tool to obtain parsimonious econometric models

variational problems with two phases and their free boundaries the problem of minimizing vu 2 q2 x 2 v dx in an appropriate class of functions v is considered here q x 0 and a2 t x2 if v 0 any minimizer u is harmonic in u 0 and vu 2 has a jump a2 x 22 across the free boundary u 0 regularity and various properties are established for the minimizer u and for the free boundary introduction in this paper we consider the problem of minimizing j v f vv 2 q2 x x2 v dx vgk where q2 x 0 a22 ifu 0 and x2 u is lower semicontinuous at v 0 it is assumed that a2 0 and a a2 a22 0 the class k consists of all functions v in l loc s2 with vu g l2 i2 such that v u on a given open subset s of 3i2 and s is a domain in r the analogous problem for functions in k v k v 0 a e was studied in 1 in that paper it was proved that any local minimizer of j v in k is lipschitz continuous and if n 2 the free boundary 8 w 0 is analytic if q x is analytic the present variational problem is motivated by applications to the now of two liquids in models of jets and cavities these applications will be studied in other forthcoming papers 5 6 the present work is aimed at extending results of 1 in particular we shall establish nondegeneracy theorems the lipschitz continuity of the solution and some properties of the free boundary for n 2 the free boundary is proved to be continuously differentiable a new and rather powerful tool introduced in this paper is the monotonicity formula lemma 5 1 asserting that for a minimizer u if u x0 0 then

a theorem of real analysis and its application to free boundary problems in this note we show how comparison theorems for solutions to equations with non smooth coefficients yield in a very simple fashion some free boundary regularity theorems in cases where there is a priori geometrical information for instance when one knows that the solution u is monotone in some given direction

the two phase membrane problem an intersection comparison approach to the regularity at branch points for the two phase membrane problem delta u lambda 2 chi u 0 lambda 2 chi u 0 and lambda 0 we prove in two dimensions that the free boundary is in a neighborhood of each branch poin

convergence analysis of finite element methods for h curl ω elliptic interface problems in this article we investigate the analysis of a finite element method for solving h curl elliptic interface problems in general three dimensional polyhedral domains with smooth interfaces the continuous problems are discretized by means of the first family of lowest order nedelec h curl conforming finite elements on a family of tetrahedral meshes which resolve the smooth interface in the sense of sufficient approximation in terms of a parameter that quantifies the mismatch between the smooth interface and the triangulation optimal error estimates in the h curl norm are obtained for the first time the analysis is based on a strip argument a new extension theorem for h 1 curl functions across smooth interfaces a novel non standard interface aware interpolation operator and a perturbation argument for degrees of freedom for h curl conforming finite elements numerical tests are presented to verify the theoretical predictions and confirm the optimal order convergence of the numerical solution

the regularity theory for the parabolic double obstacle problem in this paper we study the regularity of the free boundaries of the parabolic double obstacle problem for the heat operator and fully nonlinear operator the result in this paper are generalizations of the theory for the elliptic problem in lee et al calc var partial differ equ 58 3 104 2019 and lee and park the regularity theory for the double obstacle problem for fully nonlinear operator 2018 to parabolic case and also the theory for the parabolic single obstacle problem in caffarelli et al j am math soc 17 4 827 869 2004 to double obstacle case new difficulties in the theory which are generated by the characteristic of parabolic pdes and the existence of the upper obstacle are discussed in detail furthermore the thickness assumptions to have the regularity of the free boundary are carefully considered

real interpolation of spaces of differential forms in this paper we study interpolation of hilbert spaces of differential forms using the real method of interpolation we show that the scale of fractional order sobolev spaces of differential l forms in hs with exterior derivative in hs can be obtained by real interpolation our proof heavily relies on the recent discovery of smoothed poincare lifting for differential forms m costabel and a mcintosh on bogovskii and regularized poincare integral operators for de rham complexes on lipschitz domains math z 265 2 297 320 2010 they enable the construction of universal extension operators for sobolev spaces of differential forms which in turns pave the way for a fourier transform based proof of equivalences of k functionals

forecasting using principal components from a large number of predictors this article considers forecasting a single time series when there are many predictors n and time series observations t when the data follow an approximate factor model the predictors can be summarized by a small number of indexes which we estimate using principal components feasible forecasts are shown to be asymptotically efficient in the sense that the difference between the feasible forecasts and the infeasible forecasts constructed using the actual values of the factors converges in probability to 0 as both n and t grow large the estimated factors are shown to be consistent even in the presence of time variation in the factor model

gradient estimates for variable coefficient parabolic equations and singular perturbation problems in this article we prove via monotonicity formulas interior and boundary gradient esti mates for solutions to second order parabolic equations in divergence form with dini top order coefficients we then prove uniform lipschitz estimates for solutions of singular perturbation prob lems using the previous results and two phase monotonicity formulas introduction the purpose of this paper is to study the regularity of spatial gradients of solutions to second order uniformly parabolic equations in divergence form with bounded lower order terms and dini continuous coefficients and also to establish uniform spatial lipschitz estimates for some singular perturbation problems in the same context in part 1 we obtain interior and boundary estimates for solutions to these linear equations as is well known the weakest assumption on the modulus of continuity of the top order terms to obtain interior continuity of the gradient of solutions is dini continuity see part 1 for the relevant definitions see for instance g w in the elliptic case the usual approach to such results is to use appropriate potentials here we use a monotonicity formula approach theo rem 1 2 9 which we believe is new the approach has the advantage that it allows us to show that dini continuity of the coefficients at a given point yields bound edness of the gradient at the same point using our monotonicity formula the fact that linear functions are almost solutions and a geometric argument from cl we also obtain under uniform dini continuity of the coefficients a uniform modulus of continuity estimate for gradients of solutions theorem 1 3 1 the fact that the monotonicity formula at a point depends only on the dini continuity at that same point enables us to use a simple reflection argument to extend these results up to the boundary theorem 1 4 5 and theorem 1 4 12 we thus obtain a modulus of continuity estimate up to the boundary for gradients of solutions with smooth dirichlet data under uniform dini continuity of the top order coeffi cients once this is established a compactness argument theorem 1 5 10 and a judicious application of harnack s principle allow us to obtain a hopf maximum principle theorem 1 5 10 corollary 1 5 13 and corollary 1 5 16 for our class of equations

regularity of a free boundary for viscosity solutions of nonlinear elliptic equations our objective in this paper is to analyze the regularity of the boundary of a set omega2 that admits a solution u to the following overdetermined problem f d 2 u chi omega in b u delu 0 in b omega here f is a convex uniformly elliptic operator of homogeneity one b is the unit ball in r n and the equation is satisfied in the viscosity sense we also consider the case of concave f in r 2

pca consistency in high dimension low sample size context principal component analysis pca is an important tool of dimension reduction especially when the dimension or the number of variables is very high asymptotic studies where the sample size is fixed and the dimension grows i e high dimension low sample size hdlss are becoming increasingly relevant we investigate the asymptotic behavior of the principal component pc directions hdlss asymptotics are used to study consistency strong inconsistency and subspace consistency we show that if the first few eigenvalues of a population covariance matrix are large enough compared to the others then the corresponding estimated pc directions are consistent or converge to the appropriate subspace subspace consistency and most other pc directions are strongly inconsistent broad sets of sucient conditions for each of these cases are specified and the main theorem gives a catalogue of possible combinations in preparation for these results we show that the geometric representation of hdlss data holds under general conditions which includes a mixing condition and a broad range of sphericity measures of the 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

extremal solutions of the two dimensionall problem of moments by using the theory of the principal function of a hyponormal operator with rank one self commutator one classifies some extremal points of the solutions of the truncatedl problem of moments in two real variables these extremal elements called degenerated solutions of thel problem are proved to be in a natural bijective correspondence with the bounded quadrature domains in the complex plane the main result of the paper characterizes these degenerated solutions from three distinct perspectives complex analysis moment problems and operator theory the interplay between the three points of view each with specific techniques constitutes the main body of the paper

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

arbitrage factor structure and mean variance analysis on large asset markets we examine the implications of arbitrage in a market with many assets the absence of arbitrage opportunities implies that the linear functionals that give the mean and cost of a portfolio are continuous hence there exist unique portfolios that represent these functionals these portfolios span the mean variance efficient set we resolve the question of when a market with many assets permits so much diversification that risk free investment opportunities are available ross 12 14 showed that if there is a factor structure then the mean returns are approximately linear functions of factor loadings we define an approximate factor structure and show that this weaker restriction is sufficient for ross result if the covariance matrix of the asset returns has only k unbounded eigenvalues then there is an approximate factor structure and it is unique the corresponding k eigenvectors converge and play the role of factor loadings hence only a principal component analysis is needed in empirical work

an unfitted finite element method based on nitsche s method for elliptic interface problems in this paper we propose a method for the finite element solution of elliptic interface problem using an approach due to nitsche the method allows for discontinuities internal to the elements in the approximation across the interface we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh further we derive a posteriori error estimates for the purpose of controlling functionals of the error and present some numerical examples

least squares estimation of large dimensional threshold factor models abstract this paper studies large dimensional factor models with threshold type regime shifts in the loadings we estimate the threshold by concentrated least squares and factors and loadings by principal components the estimator for the threshold is superconsistent with convergence rate that depends on the time and cross sectional dimensions of the panel and it does not affect the estimator for factors and loadings this has the same convergence rate as in linear factor models we propose model selection criteria and a linearity test empirical application of the model shows that connectedness in financial variables increases during periods of high economic policy uncertainty

computation of moments for maxwell s equations with random interfaces via pivoted low rank approximation abstract the aim of this paper is to compute the mean field and variance of solutions to three dimensional maxwell s equations with random interfaces via shape calculus and pivoted low rank approximation based on the perturbation theory and shape calculus we characterize the statistical moments of solutions to maxwell s equations with random interfaces in terms of the perturbation magnitude via the first order shape taylor expansion in order to capture oscillations with high resolution close to the interface an adaptive finite element method using nedelec s third order edge elements of the first kind is employed to solve the deterministic maxwell s equations with the mean interface to approximate the expectation of solutions for the second moment computation an efficient low rank approximation of the pivoted cholesky decomposition is proposed to compute the two point correlation function to approximate the variance of solutions numerical experiments are presented to demonstrate our theoretical results

shrinkage estimation of dynamic panel data models with interactive fixed effects abstract we consider the problem of determining the number of factors and selecting the proper regressors in linear dynamic panel data models with interactive fixed effects based on the preliminary estimates of the slope parameters and factors a la bai 2009 and moon and weidner 2015 we propose a method for simultaneous selection of regressors and factors and estimation through the method of adaptive group lasso least absolute shrinkage and selection operator we show that with probability approaching one our method can correctly select all relevant regressors and factors and shrink the coefficients of irrelevant regressors and redundant factors to zero further we demonstrate that our shrinkage estimators of the nonzero slope parameters exhibit some oracle property we conduct monte carlo simulations to demonstrate the superb finite sample performance of the proposed method we apply our method to study the determinants of economic growth and find that in addition to three common unobserved factors selected by our method government consumption share has negative effects whereas investment share and lagged economic growth have positive effects on economic growth

sparse estimation of a covariance matrix we suggest a method for estimating a covariance matrix on the basis of a sample of vectors drawn from a multivariate normal distribution in particular we penalize the likelihood with a lasso penalty on the entries of the covariance matrix this penalty plays two important roles it reduces the effective number of parameters which is important even when the dimension of the vectors is smaller than the sample size since the number of parameters grows quadratically in the number of variables and it produces an estimate which is sparse in contrast to sparse inverse covariance estimation our method s close relative the sparsity attained here is in the covariance matrix itself rather than in the inverse matrix zeros in the covariance matrix correspond to marginal independencies thus our method performs model selection while providing a positive definite estimate of the covariance the proposed penalized maximum likelihood problem is not convex so we use a majorize minimize approach in which we iteratively solve convex approximations to the original nonconvex problem we discuss tuning parameter selection and demonstrate on a flow cytometry dataset how our method produces an interpretable graphical display of the relationship between variables we perform simulations that suggest that simple elementwise thresholding of the empirical covariance matrix is competitive with our method for identifying the sparsity structure additionally we show how our method can be used to solve a previously studied special case in which a desired sparsity pattern is prespecified

an optimal partition problem related to nonlinear eigenvalues we extend to the case of many competing densities the results of the paper ann inst h poincare 6 2002 more precisely we are concerned with an optimal partition problem in n dimensional domains related to the method of nonlinear eigenvalues introduced by nehari acta math 105 1961 we prove existence of the minimal partition and some extremality conditions moreover in the case of two dimensional domains we give an asymptotic formula near the multiple intersection points finally we show some connections between the variational problem and the behavior of competing species systems with large interaction

boundary behaviour for a singular perturbation problem abstract in this paper we study the boundary behaviour of the family of solutions u e to singular perturbation problem u e e u e u e 1 in b 1 x n 0 x 1 where a smooth boundary data f is prescribed on the flat portion of b 1 here e 1 e e c 0 0 1 0 0 1 t m 0 is an approximation of identity if f z 0 whenever f z 0 then the level sets u e 0 approach the fixed boundary in tangential fashion with uniform speed the methods we employ here use delicate analysis of local solutions along with elaborated version of the so called monotonicity formulas and classification of global profiles

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

determining the number of factors in approximate factor models in this paper we develop some econometric theory for factor models of large dimensions the focus is the determination of the number of factors r which is an unresolved issue in the rapidly growing literature on multifactor models we first establish the convergence rate for the factor estimates that will allow for consistent estimation of r we then propose some panel criteria and show that the number of factors can be consistently estimated using the criteria the theory is developed under the framework of large cross sections n and large time dimensions t no restriction is imposed on the relation between n and t simulations show that the proposed criteria have good finite sample properties in many configurations of the panel data encountered in practice

topology of quadrature domains we address the problem of topology of quadrature domains namely we give upper bounds on the connectivity of the domain in terms of the number of nodes and their multiplicities in the quadrature identity

solutions of fully nonlinear elliptic equations with patches of zero gradient existence regularity and convexity of level curves in this paper we first construct viscosity solutions in the crandall lions sense of fully nonlinear elliptic equations of the form f d 2 u x g x u on u 0 in fact viscosity solutions are surprisingly weak since candidates for solutions are just continuous we only require that the test polynomials p those tangent from above or below to the graph of u at a point x 0 satisfy the correct inequality only if p x 0 0 that is we simply disregard those test polynomials for which p x 0 0 nevertheless this is enough by an appropriate use of the alexandroff bakelman technique to prove existence regularity and in two dimensions for f g cu c 0 and constant boundary conditions on a convex domains to prove that there is only one convex patch

weak galerkin methods for second order elliptic interface problems weak galerkin methods refer to general finite element methods for partial differential equations pdes in which differential operators are approximated by their weak forms as distributions such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions a weak galerkin finite element method wg fem is developed in this paper for solving elliptic pdes with discontinuous coefficients and interfaces theoretically it is proved that high order numerical schemes can be designed by using the wg fem with polynomials of high order on each element extensive numerical experiments have been carried out to validate the wg fem for solving second order elliptic interface problems high order of convergence is numerically confirmed in both l 2 and l norms for the piecewise linear wg fem special attention is paid to solve many interface problems in which the solution possesses a certain singularity due to the nonsmoothness of the interface a challenge in research is to design nearly second order numerical methods that work well for problems with low regularity in the solution the best known numerical scheme in the literature is of order o h to o h 1 5 for the solution itself in l norm it is demonstrated that the wg fem of the lowest order i e the piecewise constant wg fem is capable of delivering numerical approximations that are of order o h 1 75 to o h 2 in the l norm for c 1 or lipschitz continuous interfaces associated with a c 1 or h 2 continuous solution

numerical approximation of elliptic interface problems via isoparametric finite element methods this paper is concerned with the numerical approximation of elliptic interface problems via isoparametric finite element methods first a convergence analysis is carried which asserts that optimal rates of convergence are recovered in both the energy and the l 2 norms subsequently the efficiency of isoparametric finite elements for the problem at hand is also assessed via numerical experiments results both with smooth and piecewise regular interfaces are presented and discussed the numerical predictions corroborate the theoretical results and they also indicate that second order convergence is achieved in the l norm comparisons between isoparametric finite elements lagrangian finite elements and the immersed interface method are also performed

symmetry theorems for the overdetermined eigenvalue problems abstract the well known schiffer conjecture saying that for a smooth bounded domain r n if there exists a positive neumann eigenvalue such that the corresponding neumann eigenfunction u is constant on the boundary of then is a ball in this paper we shall prove that the schiffer conjecture holds if and only if the third order interior normal derivative of the corresponding neumann eigenfunction is constant on the boundary we also prove a similar result to the berenstein conjecture for the overdetermined dirichlet eigenvalue problem

obstacle problem for a non convex fully nonlinear operator abstract in this paper we study a priori c 2 estimate up to the boundary for f d 2 u 0 and the regularity of the free boundary of the obstacle problem for fully nonlinear operator under specific conditions for the operator and level sets of the operator the conditions are variations of conditions for the zero set of the operator in 7

a two step estimator for large approximate dynamic factor models based on kalman filtering this paper shows consistency of a two step estimation of the factors in a dynamic approximate factor model when the panel of time series is large n large in the first step the parameters of the model are estimated from an ols on principal components in the second step the factors are estimated via the kalman smoother the analysis develops the theory for the estimator considered in giannone et al 2004 and giannone et al 2008 and for the many empirical papers using this framework for nowcasting

nonlinear shrinkage estimation of large dimensional covariance matrices many statistical applications require an estimate of a covariance matrix and or its inverse when the matrix dimension is large compared to the sample size which happens frequently the sample covariance matrix is known to perform poorly and may suffer from ill conditioning there already exists an extensive literature concerning improved estimators in such situations in the absence of further knowledge about the structure of the true covariance matrix the most successful approach so far arguably has been shrinkage estimation shrinking the sample covariance matrix to a multiple of the identity by taking a weighted average of the two turns out to be equivalent to linearly shrinking the sample eigenvalues to their grand mean while retaining the sample eigenvectors our paper extends this approach by considering nonlinear transformations of the sample eigenvalues we show how to construct an estimator that is asymptotically equivalent to an oracle estimator suggested in previous work as demonstrated in extensive monte carlo simulations the resulting bona fide estimator can result in sizeable improvements over the sample covariance matrix and also over linear shrinkage

a matrix free chimera approach based on dirichlet dirichlet coupling for domain composition purposes abstract a novel approach using an algebraic based chimera type domain composition method in the context of the finite element method for non matching overlapping unstructured grids is proposed in this work the scheme is based on both the transfer of information across each grid interface via dirichlet boundary conditions and a high order interpolation algorithm to obtain one global solution of the system the solution can be obtained iteratively with a convergence rate that is similar to that obtained with an analogous conformal mesh and the matrix vector operator can be computed with completely decoupled operations on both meshes furthermore the scheme can be set as a linear operator that can be fed to a matrix free efficient iterative solver such as the biconjugate gradient stabilized method several numerical examples using non matching unstructured grids that are partially and completely overlapped with different element sizes are presented assessing the precision and convergence rate of the method

geometric approach to nonvariational singular elliptic equations in this work we develop a systematic geometric approach to study fully nonlinear elliptic equations with singular absorption terms as well as their related free boundary problems the magnitude of the singularity is measured by a negative parameter 1 for 0 0 in particular we show sharp hausdorff estimates which imply local finiteness of the perimeter of the region u 0 and the mathcal h n 1 almost everywhere weak differentiability property of mathfrak f

the structure of the free boundary in the fully nonlinear thin obstacle problem abstract we study the regularity of the free boundary in the fully nonlinear thin obstacle problem our main result establishes that the free boundary is c 1 near any regular point this extends to the fully nonlinear setting the celebrated result of athanasopoulos caffarelli salsa 1 the proofs we present here are completely independent from those in 1 and do not rely on any monotonicity formula furthermore an interesting and novel feature of our proofs is that we establish the regularity of the free boundary without classifying blow ups a priori they could be non homogeneous and or non unique we do not classify blow ups but only prove that they are 1d on x n 0

high dimensional covariance matrix estimation in approximate factor models the variance covariance matrix plays a central role in the inferential theories of high dimensional factor models in finance and economics popular regularization methods of directly exploiting sparsity are not directly applicable to many financial problems classical methods of estimating the covariance matrices are based on the strict factor models assuming independent idiosyncratic components this assumption however is restrictive in practical applications by assuming sparse error covariance matrix we allow the presence of the cross sectional correlation even after taking out common factors and it enables us to combine the merits of both methods we estimate the sparse covariance using the adaptive thresholding technique as in cai and liu j amer statist assoc 106 2011 672 684 taking into account the fact that direct observations of the idiosyncratic components are unavailable the impact of high dimensionality on the covariance matrix estimation based on the factor structure is then studied

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

positive definite l1 penalized estimation of large covariance matrices the thresholding covariance estimator has nice asymptotic properties for estimating sparse large covariance matrices but it often has negative eigenvalues when used in real data analysis to fix this drawback of thresholding estimation we develop a positive definite l1 penalized covariance estimator for estimating sparse large covariance matrices we derive an efficient alternating direction method to solve the challenging optimization problem and establish its convergence properties under weak regularity conditions nonasymptotic statistical theory is also established for the proposed estimator the competitive finite sample performance of our proposal is demonstrated by both simulation and real applications

regularity of a free boundary with application to the pompeiu problem in the unit ball b 0 1 let u and a domain in n solve the following overdetermined problem u in b 0 1 02 u jruj 0 inb 0 1 n where denotes the characteristic function and the equation is satisfled in the sense of distributions if the complement of does not develop cusp singularities at the origin then we prove is analytic in some small neighborhood of the origin the result can be modifled to yield for more general divergence form operators as an application of this then we obtain the regularity of the boundary of a domain without the pompeiu property provided its complement has no cusp singularities

regularity in the obstacle problem for parabolic non divergence operators of hormander type in this paper we continue the study initiated in 15 concerning the obstacle problem for a class of parabolic non divergence operators structured on a set of vector fields x x1 xq in rn with c coefficients satisfying hormander s finite rank condition i e the rank of lie x1 xq equals n at every point in rn in 15 we proved under appropriate assumptions on the operator and the obstacle the existence and uniqueness of strong solutions to a general obstacle problem the main result of this paper is that we establish further regularity in the interior as well as at the initial state of strong solutions compared to 15 we in this paper assume in addition that there exists a homogeneous lie group g rn ring operator such that x1 xq are left translation invariant on g and such that x1 xq are homogeneous of degree one

a recovery based a posteriori error estimator for h curl interface problems abstract this paper introduces a new recovery based a posteriori error estimator for the lowest order nedelec finite element approximation to the h curl interface problem the error estimator is analyzed by establishing both the reliability and the efficiency bounds and is supported by numerical results under certain assumptions it is proved that the reliability and efficiency constants are independent of the jumps of the coefficients

on the non tangential touch between the free and the fixed boundaries for the two phase obstacle like problem in this paper we consider the following two phase obstacle problem like equation in the unit half ball delta u lambda chi u 0 lambda chi u 0 we classify the angles of touch between the free boundary and the fixed one if the boundary data f satisfies certain conditions these results are connected to the results from andersson et al ark mat 44 1 1 15 2006

panel threshold models with interactive fixed effects abstract this paper studies estimation and inference in a panel threshold model in the presence of interactive fixed effects we study the asymptotic properties of the least squares estimators of the regression parameters in the shrinking threshold effect framework we find that under some regularity conditions the threshold parameter estimator possesses super consistency in the sense that its estimation error has an asymptotically negligible effect on the asymptotic properties of the slope coefficients the inference on the threshold parameter can be conducted based on a likelihood ratio test statistic as in the cross sectional or time series setup we also propose a test for the presence of the threshold effect monte carlo simulations suggest that our estimators and test statistics perform well in finite samples we apply our method to study the effect of financial development on economic growth and find that there is indeed a turning point in the effect for all three measures of financial development when the cross sectional dependence is properly accounted for

immersed finite element method for eigenvalue problem abstract we consider the approximation of elliptic eigenvalue problem with an interface the main aim of this paper is to prove the stability and convergence of an immersed finite element method ifem for eigenvalues using crouzeix raviart p 1 nonconforming approximation we show that spectral analysis for the classical eigenvalue problem can be easily applied to our model problem we analyze the ifem for elliptic eigenvalue problems with an interface and derive the optimal convergence of eigenvalues numerical experiments demonstrate our theoretical results

factor modeling for high dimensional time series inference for the number of factors this paper deals with the factor modeling for high dimensional time series based on a dimension reduction viewpoint under stationary settings the inference is simple in the sense that both the number of factors and the factor loadings are estimated in terms of an eigenanalysis for a nonnegative definite matrix and is therefore applicable when the dimension of time series is on the order of a few thousands asymptotic properties of the proposed method are investigated under two settings i the sample size goes to infinity while the dimension of time series is fixed and ii both the sample size and the dimension of time series go to infinity together in particular our estimators for zero eigenvalues enjoy faster convergence or slower divergence rates hence making the estimation for the number of factors easier in particular when the sample size and the dimension of time series go to infinity together the estimators for the eigenvalues are no longer consistent however our estimator for the number of the factors which is based on the ratios of the estimated eigenvalues still works fine furthermore this estimation shows the so called blessing of dimensionality property in the sense that the performance of the estimation may improve when the dimension of time series increases a two step procedure is investigated when the factors are of different degrees of strength numerical illustration with both simulated and real data is also reported

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

regularity properties of a free boundary near contact points with the fixed boundary in the upper half of the unit ball b x 0 let u and omega a domain in r n x is an element of r n x 1 0 solve the following overdetermined problem deltau chi omega in b u

on quadrature domains and the schwarz potential abstract let t be a distribution in r n whose support is compact a domain in r n is said to be a quadrature domain with respect to t if supp t and h h t for all functions h which are harmonic and integrable over we denote this by q t hl 1 this is equivalent to saying that there is a solution u called the modified schwarz potential to the following overdetermined cauchy problem u 1 t in and u u 0 on in this paper we investigate different properties of quadrature domains we show that if t is a fixed distribution then bounded domains in q t hl1 are uniformly bounded later we prove that if the modified schwarz potential of a domain is a polynomial then is a half space we also study the uniqueness problem namely we show that if d and are in q t hl1 v and u are the msps of d and respectively and sup d u 0 then meets x v x consequently if v 0 then d 0

a simple mesh generator in matlab creating a mesh is the first step in a wide range of applications including scientific computing and computer graphics an unstructured simplex mesh requires a choice of meshpoints vertex nodes

gls estimation of dynamic factor models in this article a simple two step estimation procedure of the dynamic factor model is proposed the estimator allows for heteroscedastic and serially correlated errors it turns out that the feasible two step estimator has the same limiting distribution as the generalized least squares gls estimator assuming that the covariance parameters are known in a monte carlo study of the small sample properties we find that the gls estimators may be substantially more efficient than the usual estimator based on principal components furthermore it turns out that the iterated version of the estimator may feature considerably improved properties in sample sizes usually encountered in practice

finite element methods and their convergence for elliptic and parabolic interface problems in this paper we consider the finite element methods for solving second order elliptic and parabolic interface problems in two dimensional convex polygonal domains nearly the same optimal l 2 norm and energy norm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth though the regularities of the solutions are low on the whole domain the assumptions on the finite element triangulation are reasonable and practical

positive definite estimators of large covariance matrices using convex optimization we construct a sparse estimator of the covariance matrix that is positive definite and performs well in high dimensional settings a lasso type penalty is used to encourage sparsity and a logarithmic barrier function is used to enforce positive definiteness consistency and convergence rate bounds are established as both the number of variables and sample size diverge an efficient computational algorithm is developed and the merits of the approach are illustrated with simulations and a speech signal classification example copyright 2012 oxford university press

analysis of a free boundary at contact points with lipschitz data in this paper we consider a minimization problem for the functional j u b 1 u 2 2 u 0 2 u 0 in the upper half ball b 1 n n 2 subject to a lipschitz continuous dirich

efficient estimation of approximate factor models via penalized maximum likelihood we study the estimation of a high dimensional approximate factor model in the presence of both cross sectional dependence and heteroskedasticity the classical method of principal components analysis pca does not efficiently estimate the factor loadings or common factors because it essentially treats the idiosyncratic error to be homoskedastic and cross sectionally uncorrelated for the efficient estimation it is essential to estimate a large error covariance matrix we assume the model to be conditionally sparse and propose two approaches to estimating the common factors and factor loadings both are based on maximizing a gaussian quasi likelihood and involve regularizing a large covariance sparse matrix in the first approach the factor loadings and the error covariance are estimated separately while in the second approach they are estimated jointly extensive asymptotic analysis has been carried out in particular we develop the inferential theory for the two step estimation because the proposed approaches take into account the large error covariance matrix they produce more efficient estimators than the classical pca methods or methods based on a strict factor model

spectrum estimation for large dimensional covariance matrices using random matrix theory estimating the eigenvalues of a population covariance matrix from a sample covariance matrix is a problem of fundamental importance in multivariate statistics the eigenvalues of covariance matrices play a key role in many widely used techniques in particular in principal component analysis pca in many modern data analysis problems statisticians are faced with large datasets where the sample size n is of the same order of magnitude as the number of variables p random matrix theory predicts that in this context the eigenvalues of the sample covariance matrix are not good estimators of the eigenvalues of the population covariance we propose to use a fundamental result in random matrix theory the marcenko pastur equation to better estimate the eigenvalues of large dimensional covariance matrices the marcenko pastur equation holds in very wide generality and under weak assumptions the estimator we obtain can be thought of as shrinking in a nonlinear fashion the eigenvalues of the sample covariance matrix to estimate the population eigenvalues inspired by ideas of random matrix theory we also suggest a change of point of view when thinking about estimation of high dimensional vectors we do not try to estimate directly the vectors but rather a probability measure that describes them we think this is a theoretically more fruitful way to think about these problems our estimator gives fast and good or very good results in extended simulations our algorithmic approach is based on convex optimization we also show that the proposed estimator is consistent

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

hausdorff measure and stability for the p obstacle problem 2 p in this paper we analyze the free boundary for the inhomogeneous obstacle problem with zero obstacle governed by the degenerate operator div a u f u 0 where f is a positive lipschitz function and a u is of the p laplacian type i e a u u p 2 u 2 p as a result of our analysis we obtain stability and finite n 1 hausdorff measure of the free boundary our technique is a modified version of that of l a caffarelli who obtained similar results for p 2

asymptotics of the principal components estimator of large factor models with weakly influential factors abstract this paper introduces a drifting parameter asymptotic framework to derive accurate approximations to the finite sample distribution of the principal components pc estimator in situations when the factors explanatory power does not strongly dominate the explanatory power of the cross sectionally and temporally correlated idiosyncratic terms under our asymptotics the pc estimator is inconsistent we find explicit formulae for the amount of the inconsistency and propose an estimator of the number of factors for which the pc estimator works reasonably well for the special case when the idiosyncratic terms are cross sectionally but not temporally correlated or vice versa we show that the coefficients in the ols regressions of the pc estimates of factors loadings on the true factors true loadings are asymptotically normal and find explicit formulae for the corresponding asymptotic covariance matrix we explain how to estimate the parameters of the derived asymptotic distributions our monte carlo analysis suggests that our asymptotic formulae and estimators work well even for relatively small n and t we apply our theoretical results to test a hypothesis about the factor content of the us stock return data

inferential theory for factor models of large dimensions this paper develops an inferential theory for factor models of large dimensions the principal components estimator is considered because it is easy to compute and is asymptotically equivalent to the maximum likelihood estimator if normality is assumed we derive the rate of convergence and the limiting distributions of the estimated factors factor loadings and common components the theory is developed within the framework of large cross sections n and a large time dimension t to which classical factor analysis does not apply we show that the estimated common components are asymptotically normal with a convergence rate equal to the minimum of the square roots of n and t the estimated factors and their loadings are generally normal although not always so the convergence rate of the estimated factors and factor loadings can be faster than that of the estimated common components these results are obtained under general conditions that allow for correlations and heteroskedasticities in both dimensions stronger results are obtained when the idiosyncratic errors are serially uncorrelated and homoskedastic a necessary and sufficient condition for consistency is derived for large n but fixed t

a defect corrected finite element approach for the accurate evaluation of magnetic fields on unstructured grids in electromagnetic simulations of magnets and machines one is often interested in a highly accurate and local evaluation of the magnetic field uniformity based on local post processing of the solution a defect correction scheme is proposed as an easy to realize alternative to higher order finite element or hybrid approaches radial basis functions rbfs are key for the generality of the method which in particular can handle unstructured grids also contrary to conventional finite element basis functions higher derivatives of the solution can be evaluated as required e g for deflection magnets defect correction is applied to obtain a solution with improved accuracy and adjoint techniques are used to estimate the remaining error for a specific quantity of interest significantly improved local convergence orders are obtained the scheme is also applied to the simulation of a sterngerlach magnet currently in operation