a parameterized deteriorated pss preconditioner and its optimization for nonsymmetric saddle point problems abstract based on the deteriorated positive definite and skew hermitian splitting dpss preconditioner we develop a parameterized dpss pdpss preconditioner for the nonsymmetric saddle point linear systems the spectral properties of the pdpss preconditioned coefficient matrix are analyzed moreover an upper bound of the degree of the minimal polynomial of the pdpss preconditioned matrix is also obtained inasmuch as the efficiency of the pdpss preconditioner depends on the values of its parameters we further derive a fast and effective method to compute the optimal parameter values of the pdpss preconditioner finally numerical examples are employed to illustrate the feasibility and the efficiency of the pdpss preconditioner

preconditioning for pde constrained optimization with total variation regularization abstract the efficient solution of pde constrained optimization problems is of great significance in many scientific and engineering applications the key point of achieving this goal is to solve the linear systems arising from such problems more efficiently by preconditioning in this paper we consider preconditioning the linear systems arising from pde constrained optimization problems with total variation tv regularization based on the block structure of the coefficient matrix or its row and column transformed forms we propose three kinds of new preconditioners by approximating the schur complements in different ways we analyze the eigenvalue properties of the preconditioned coefficient matrices and provide the eigenvalue bounds independent of mesh size we also show the relations of eigenvalue bounds corresponding to different preconditioners numerical results illustrate the efficiency of the proposed preconditioners by comparison with other existing methods

an all at once approach for the optimal control of the unsteady burgers equation we apply an all at once method for the optimal control of the unsteady burgers equation the nonlinear burgers equation is discretized in time using the semi implicit discretization and the resulting first order optimality conditions are solved iteratively by newton s method the discretize then optimize approach is used because it leads to a symmetric indefinite saddle point problem numerical results for the distributed unconstrained and control constrained problems illustrate the performance of the all at once approach with semi implicit time discretization

some preconditioners for elliptic pde constrained optimization problems abstract for the structured systems of linear equations arising from the galerkin finite element discretizations of elliptic pde constrained optimization problems some preconditioners are proposed to accelerate the convergence rate of krylov subspace methods such as gmres for both cases of the tikhonov parameter not very small equal or greater than 1e 6 and sufficiently small less than 1e 6 respectively we derive the explicit expressions for the eigenvalues and eigenvectors of the corresponding preconditioned matrices numerical results show that the corresponding preconditioned gmres methods perform and match well with the theoretical results

a block alternating splitting iteration method for a class of block two by two complex linear systems optimization problems with partial differential equations as constraints arise widely in many areas of science and engineering in this paper we focus on solving a class of block two by two complex linear systems arising from the distributed optimal control with time periodic parabolic equations a new block alternating splitting bas iteration method is presented for solving the class of complex linear systems the convergence theory and the spectral properties of the bas iteration method are discussed numerical experiments are presented to illustrate the efficiency of the bas iteration as a solver as well as a preconditioner for krylov subspace methods

an adjoint based jacobi type iterative method for elastic full waveform inversion problem full waveform inversion fwi is a promising technique that is capable of creating high resolution subsurface images of the earth from seismic data however it is computationally expensive especially when 3d elastic wave equation is considered in this paper an adjoint based computational algorithm has been proposed to address the computational challenges the important strategy in this work is to decouple the two lame parameters so that two half sized subproblems are resulted and solved separately mathematically the inverse problem is formulated as a pde constrained optimization problem in which the objective functional is defined as the misfit between observational and synthetic data the parameters to be recovered are the spatially varying lame parameters which are of great interests to geophysicists for the purpose of hydrocarbon exploration and subsurface imaging the gradient of the misfit functional with respect to the lame parameters is calculated by solving the adjoint elastic wave equation whilst the quasi newton method l bfgs is used to minimize the misfit functional numerical experiments demonstrated that the new method is accurate efficient and robust in recovering lame parameters from seismic data

a class of accelerated parameterized inexact uzawa algorithms for complex symmetric linear systems abstract we establish a class of accelerated parameterized inexact uzawa apiu algorithms for solving the complex symmetric linear systems our main contribution is accelerating the convergence of the piu algorithm by making use of the extrapolation technique which is based on the eigenvalues of the iterative matrix these accelerated parameterized inexact uzawa algorithms involve two iteration parameters whose special choices can recover the parameterized inexact uzawa algorithm and some other methods first the accelerated model for the piu algorithm is established and the accelerated piu algorithm is presented then we study the convergence of the corrected piu algorithm moreover we present the optimal iteration parameter and the corresponding optimal convergence factor for the piu method we also consider acceleration of the piu iteraton by krylov subspace methods numerical experiments are presented to illustrate the theoretical results and examine the numerical effectiveness of the new method

multigrid methods for pde optimization research on multigrid methods for optimization problems is reviewed optimization problems considered include shape design parameter optimization and optimal control problems governed by partial differential equations of elliptic parabolic and hyperbolic type

a unified approach to infinitesimal loewner and geroch transformations and the ernst and einstein maxwell equations it is demonstrated that the dual ernst equation of general relativity constitutes a stationary loewner system in an analogous manner it is shown that the einstein maxwell equations for stationary axisymmetric space times and their extension to einstein s equations coupled with an arbitrary number of u 1 gauge fields may be interpreted as generalized loewner systems moreover it is recorded that the base geroch transformation for the dual ernst equation may be located in a class of infinitesimal backlund transformations introduced by loewner in 1952 a geroch type transformation for a generic class of 2 1 dimensional loewner systems is set down and it is shown how the base geroch and hoenselaers kinnersley xanthopoulos transformations are naturally retrieved

a note on preconditioning for indefinite linear systems preconditioners are often conceived as approximate inverses for nonsingular indefinite matrices of saddle point or kkt form we show how preconditioners incorporating an exact schur complement lead to preconditioned matrices with exactly two or exactly three distinct eigenvalues thus approximations of the schur complement lead to preconditioners which can be very effective even though they are in no sense approximate inverses

parallel lagrange newton krylov schur methods for pde constrained optimization part i the krylov schur solver large scale optimization of systems governed by partial differential equations pdes is a frontier problem in scientific computation reduced quasi newton sequential quadratic programming sqp methods are state of the art approaches for such problems these methods take full advantage of existing pde solver technology and parallelize well however their algorithmic scalability is questionable for certain problem classes they can be very slow to converge in this two part article we propose a new method for steady state pde constrained optimization based on the idea of using a full space newton solver combined with an approximate reduced space quasi newton sqp preconditioner the basic components of the method are newton solution of the first order optimality conditions that characterize stationarity of the lagrangian function krylov solution of the karush kuhn tucker kkt linear systems arising at each newton iteration using a symmetric quasi minimum residual method preconditioning of the kkt system using an approximate state decision variable decomposition that replaces the forward pde jacobians by their own preconditioners and the decision space schur complement the reduced hessian by a bfgs approximation initialized by a two step stationary method accordingly we term the new method it lagrange newton krylov schur lnks it is fully parallelizable exploits the structure of available parallel algorithms for the pde forward problem and is locally quadratically convergent in part i of this two part article we investigate the effectiveness of the kkt linear system solver we test our method on two optimal control problems in which the state constraints are described by the steady state stokes equations the objective is to minimize dissipation or the deviation from a given velocity field the control variables are the boundary velocities numerical experiments on up to 256 cray t3e processors and on an sgi origin 2000 include scalability and performance assessment of the lnks algorithm and comparisons with reduced sqp for up to 1 000 000 state and 50 000 decision variables in part ii of the article we address globalization and inexactness issues and apply lnks to the optimal control of the steady incompressible navier stokes equations

shrinkage tuning parameter selection with a diverging number of parameters contemporary statistical research frequently deals with problems involving a diverging number of parameters for those problems various shrinkage methods e g the lasso and smoothly clipped absolute deviation are found to be particularly useful for variable selection nevertheless the desirable performances of those shrinkage methods heavily hinge on an appropriate selection of the tuning parameters with a fixed predictor dimension wang and co worker have demonstrated that the tuning parameters selected by a bayesian information criterion type criterion can identify the true model consistently in this work similar results are further extended to the situation with a diverging number of parameters for both unpenalized and penalized estimators consequently our theoretical results further enlarge not only the scope of applicability of the traditional bayesian information criterion type criteria but also that of those shrinkage estimation methods

robust preconditioners for optimal control with time periodic parabolic equation abstract based on a new approximation of a schur complement matrix and some preconditioning techniques considered in earlier papers three robust preconditioners are presented for solving the kkt systems arising from discretizing optimal control problem with time periodic parabolic equation the eigenvalue bounds of the preconditioned matrices and some new theoretical results are obtained moreover selection of quasi optimal parameter is studied and an explicit expression of the quasi optimal parameter is given numerical examples are given to illustrate the effectiveness of the three proposed optimized preconditioners

a preconditioner for optimal control problems constrained by stokes equation with a time harmonic control in this article we construct an efficient preconditioner for solving the algebraic systems arising from discretized optimal control problems with time periodic stokes equations based on a preconditioning technique for stationary stokes constrained optimal control problems considered in an earlier paper by the authors a simplified analysis of the derivation of the preconditioner and its properties is presented the preconditioner is fully parameter independent and the condition number of the corresponding preconditioned matrix is bounded by 2 the so constructed preconditioner is favourably compared with another robust preconditioner for the same problem

classification of integrable equations on quad graphs the consistency approach a classification of discrete integrable systems on quad graphs i e on surface cell decompositions with quadrilateral faces is given the notion of integrability laid in the basis of the classification is the three dimensional consistency this property yields among other features the existence of the discrete zero curvature representation with a spectral parameter for all integrable systems of the obtained exhaustive list the so called three leg forms are found this establishes lagrangian and symplectic structures for these systems and the connection to discrete systems of the toda type on arbitrary graphs generalizations of these ideas to the three dimensional integrable systems and to the quantum context are also discussed

linear integral transformations and hierarchies of integrable nonlinear evolution equations abstract integrable hierarchies of nonlinear evolution equations are investigated on the basis of linear integral equations these are riemann hilbert type of integral transformations which leave invariant an infinite sequence of ordinary differential matrix equations of increasing order in an indefinite parameter k the potential matrices in these equations obey a set of nonlinear recursion relations leading to a heirarchy of nonlinear partial differential equations in decreasing order the same equations give rise to a reciprocal hierarchy associated with heisenberg ferromagnet type of equations central in the treatment is an embedding of the hierarchy into an infinite matrix structure which is constructed on the basis of the integral equations in terms of this infinite matrix structure the equations governing the hierarchies become quite simple furthermore it leads in a straightforward way to various generalizations such as to other types of linear spectral problems multicomponent system and lattice equations generalizations to equations associated with noncommuting flows follow as a direct consequence of the treatment finally some results on conserved densities and the hamiltonian structure are briefly discussed

model selection via bayesian information criterion for quantile regression models bayesian information criterion bic is known to identify the true model consistently as long as the predictor dimension is finite recently its moderate modifications have been shown to be consistent in model selection even when the number of variables diverges those works have been done mostly in mean regression but rarely in quantile regression the best known results about bic for quantile regression are for linear models with a fixed number of variables in this article we investigate how bic can be adapted to high dimensional linear quantile regression and show that a modified bic is consistent in model selection when the number of variables diverges as the sample size increases we also discuss how it can be used for choosing the regularization parameters of penalized approaches that are designed to conduct variable selection and shrinkage estimation simultaneously moreover we extend the results to structured nonparametric quantile models with a diverging number of covariates we illustrate o

a model selection approach for the identification of quantitative trait loci in experimental crosses summary we consider the problem of identifying the genetic loci called quantitative trait loci qtls contributing to variation in a quantitative trait with data on an experimental cross a large number of different statistical approaches to this problem have been described most make use of multiple tests of hypotheses and many consider models allowing only a single qtl we feel that the problem is best viewed as one of model selection we discuss the use of model selection ideas to identify qtls in experimental crosses we focus on a back cross experiment with strictly additive qtls and concentrate on identifying qtls considering the estimation of their effects and precise locations of secondary importance we present the results of a simulation study to compare the performances of the more prominent methods

optimal control with stochastic pde constraints and uncertain controls abstract the optimal control of problems that are constrained by partial differential equations with uncertainties and with uncertain controls is addressed the lagrangian that defines the problem is postulated in terms of stochastic functions with the control function possibly decomposed into an unknown deterministic component and a known zero mean stochastic component the extra freedom provided by the stochastic dimension in defining cost functionals is explored demonstrating the scope for controlling statistical aspects of the system response one shot stochastic finite element methods are used to find approximate solutions to control problems it is shown that applying the stochastic collocation finite element method to the formulated problem leads to a coupling between stochastic collocation points when a deterministic optimal control is considered or when moments are included in the cost functional thereby forgoing the primary advantage of the collocation method over the stochastic galerkin method for the considered problem the application of the presented methods is demonstrated through a number of numerical examples the presented framework is sufficiently general to also consider a class of inverse problems and numerical examples of this type are also presented

extended bayesian information criteria for model selection with large model spaces summary the ordinary bayesian information criterion is too liberal for model selection when the model space is large in this paper we re examine the bayesian paradigm for model selection and propose an extended family of bayesian information criteria which take into account both the number of unknown parameters and the complexity of the model space their consistency is established in particular allowing the number of covariates to increase to infinity with the sample size their performance in various situations is evaluated by simulation studies it is demonstrated that the extended bayesian information criteria incur a small loss in the positive selection rate but tightly control the false discovery rate a desirable property in many applications the extended bayesian information criteria are extremely useful for variable selection in problems with a moderate sample size but with a huge number of covariates especially in genome wide association studies which are now an active area in genetics research

neumann neumann domain decomposition preconditioners for linear quadratic elliptic optimal control problems we present a class of domain decomposition dd preconditioners for the solution of elliptic linear quadratic optimal control problems our dd preconditioners are extensions of neumann neumann dd preconditioners which have been successfully applied to the solution of single pdes the dd preconditioners are based on a decomposition of the optimality conditions for the elliptic linear quadratic optimal control problem into smaller subdomain optimality conditions with dirichlet boundary conditions for the states and the adjoints on the subdomain interfaces these subdomain optimality conditions are coupled through neumann interface conditions for the states and the adjoints this decomposition leads to a schur complement system in which the unknowns are the state and adjoint variables on the subdomain interfaces the schur complement operator is the sum of subdomain schur complement operators the application of which is shown to correspond to the solution of subdomain elliptic linear quadratic optimal control problems which are essentially smaller copies of the original optimal control problem we show that under suitable conditions the application of the inverse of the subdomain schur complement operators requires the solution of a subdomain elliptic linear quadratic optimal control problem with neumann interface conditions for the state the subdomain schur complement operators are analyzed in the variational setting of the problem as well as the algebraic setting obtained after a finite element discretization of the problem definiteness properties of the algebraic form of the subdomain schur complement operator s are studied numerical tests show that the dependence of these preconditioners on mesh size and subdomain size is comparable to its counterpart applied to elliptic equations only these tests also show that the preconditioners are insensitive to the size of the control regularization parameter

nonconcave penalized likelihood with a diverging number of parameters a class of variable selection procedures for parametric models via nonconcave penalized likelihood was proposed by fan and li to simultaneously estimate parameters and select important variables they demonstrated that this class of procedures has an oracle property when the number of parameters is finite however in most model selection problems the number of parameters should be large and grow with the sample size in this paper some asymptotic properties of the nonconcave penalized likelihood are established for situations in which the number of parameters tends to as the sample size increases under regularity conditions we have established an oracle property and the asymptotic normality of the penalized likelihood estimators furthermore the consistency of the sandwich formula of the covariance matrix is demonstrated nonconcave penalized likelihood ratio statistics are discussed and their asymptotic distributions under the null hypothesis are obtained by imposing some mild conditions on the penalty functions the asymptotic results are augmented by a simulation study and the newly developed methodology is illustrated by an analysis of a court case on the sexual discrimination of salary

augmented lagrangian sqp methods for nonlinear optimalcontrol problems of tracking type an augmented lagrangian method with second order update is developed and its relationship to the sequential quadratic programming method is described the rate of convergence proof depends on a second order sufficient optimality condition which is shown to be satisfied for a class of nonlinear optimal control problems of tracking type numerical examples are included which demonstrate the globalizing effect of the augmented lagrangian method

fast iterative solution of reaction diffusion control problems arising from chemical processes pde constrained optimization problems and the development of preconditioned iterative methods for the efficient solution of the arising matrix systems is a field of numerical analysis that has recently been attracting much attention in this paper we analyze and develop preconditioners for matrix systems that arise from the optimal control of reaction diffusion equations which themselves result from chemical processes important aspects of our solvers are saddle point theory mass matrix representation and effective schur complement approximation as well as the incorporation of control constraints and application of the outer newton iteration to take into account the nonlinearity of the underlying pdes

analysis of the minimal residual method applied to ill posed optimality systems we analyze the performance of the minimal residual minres method applied to linear karush kuhn tucker systems arising in connection with inverse problems such optimality systems typically have a saddle point structure and have unique solutions for all alpha 0 where alpha is the parameter employed in the tikhonov regularization unfortunately the associated spectral condition number is very large for small values of alpha which strongly indicates that their numerical treatment is difficult our main result shows that a broad range of linear ill posed optimality systems can be solved efficiently with the minres method this result is obtained by carefully analyzing the spectrum of the associated saddle point operator except for a few isolated eigenvalues the spectrum consists of three bounded intervals krylov subspace methods handle such problems very well for severely ill posed cases techniques based on chebyshev polynomials are applied to prove that the number of iterations needed by

prolongation structure and linear eigenvalue equations for einstein maxwell fields the einstein maxwell equations for stationary axisymmetric exterior fields are shown to be the integrability conditions of a set of linear eigenvalue equations for pseudopotentials the prolongation structure in the spirit of wahlquist and estabrook 1975 of the einstein maxwell equations contains the su 2 1 lie algebra a new mapping of known solutions to other solutions has been found

lagrangian multiform structure for the lattice gel fand dikii hierarchy the lattice gel fand dikii hierarchy was introduced by nijhoff et al 1992 inverse problems 8 597 621 as the family of partial difference equations generalizing to higher rank the lattice korteweg de vries systems and includes in particular the lattice boussinesq system we present a lagrangian for the generic member of the lattice gel fand dikii hierarchy and show that it can be considered as a lagrangian 2 form when embedded in a higher dimensional lattice obeying a closure relation thus the multiform structure proposed by lobb and nijhoff 2009 j phys a math theor 42 454013 is extended to a multi component system

a parallel two level domain decomposition based one shot method for shape optimization problems summary a two level domain decomposition method is introduced for general shape optimization problems constrained by the incompressible navier stokes equations the optimization problem is first discretized with a finite element method on an unstructured moving mesh that is implicitly defined without assuming that the computational domain is known and then solved by some one shot lagrange newton krylov schwarz algorithms in this approach the shape of the domain its corresponding finite element mesh the flow fields and their corresponding lagrange multipliers are all obtained computationally in a single solve of a nonlinear system of equations highly scalable parallel algorithms are absolutely necessary to solve such an expensive system the one level domain decomposition method works reasonably well when the number of processors is not large aiming for machines with a large number of processors and robust nonlinear convergence we introduce a two level inexact newton method with a hybrid two level overlapping schwarz preconditioner as applications we consider the shape optimization of a cannula problem and an artery bypass problem in 2d numerical experiments show that our algorithm performs well on a supercomputer with over 1000 processors for problems with millions of unknowns copyright 2014 john wiley sons ltd

a robust multigrid method for elliptic optimal control problems we consider the discretized optimality system of a special class of elliptic optimal control problems and propose an all at once multigrid method for solving this discretized system under standard assumptions the convergence of the multigrid method and the robustness of the convergence rates with respect to the involved parameter are shown numerical experiments are presented for illustrating the theoretical results

on nonsingularity of block two by two matrices national natural science foundation 11271308 91118001 natural science foundation of fujian province for distinguished young scholars 2010j06002 program for new century excellent talents in universities pr china hundred talent project of chinese academy of sciences national basic research program 2011cb309703 national natural science foundation for creative research groups pr china 11021101

on the degrees of freedom of the lasso we study the effective degrees of freedom of the lasso in the framework of stein s unbiased risk estimation sure we show that the number of nonzero coefficients is an unbiased estimate forthe degrees of freedom of the lasso a conclusion that requires no special assumption on the predictors in addition the unbiased estimator is shown to be asymptotically consistent with these results on hand various model selection criteria cp aic and bic are available which along with the lars algorithm provide a principled and efficient approach to obtaining the optimal lasso fit with the computational effort of a single ordinary least squares fit

a new multigrid method for unconstrained parabolic optimal control problems abstract a second order leapfrog finite difference scheme in time is proposed and developed for solving the first order necessary optimality system of the distributed parabolic optimal control problems different from available approaches the proposed leapfrog scheme for the two point boundary optimality system is shown to be unconditionally stable and provides a second order accuracy though the classical leapfrog scheme usually is unstable moreover the proposed leapfrog scheme provides a feasible structure that leads to an effective implementation of a fast solver under the multigrid framework a detailed mathematical proof for the stability of the proposed scheme is provided in terms of a new norm that is more suitable and stronger to characterize the convergence than the l 2 norm often used in literature numerical experiments show that the proposed scheme significantly outperforms the widely used second order backward time differentiation approach and the resultant fast solver demonstrates a mesh independent convergence as well as a linear time complexity

piecewise linear regularized solution paths we consider the generic regularized optimization problem argmin l y x j efron hastie johnstone and tibshirani ann statist 32 2004 407 499 have shown that for the lasso that is if l is squared error loss and j 1 is the l 1 norm of the optimal coefficient path is piecewise linear that is is piecewise constant we derive a general characterization of the properties of loss l penalty j pairs which give piecewise linear coefficient paths such pairs allow for efficient generation of the full regularized coefficient paths we investigate the nature of efficient path following algorithms which arise we use our results to suggest robust versions of the lasso for regression and classification and to develop new efficient algorithms for existing problems in the literature including mammen and van de geer s locally adaptive regression splines

asymptotic behavior of m estimators of p regression parameters when p 2 n is large ii normal approximation in a general linear model y x beta r with y and r n dimensional x a n times p matrix and beta p dimensional let hat beta be an m estimator of beta satisfying 0 sum x i psi y i x i beta let p rightarrow infty such that p log n 3 2 n rightarrow 0 then max i x i hat beta beta rightarrow p 0 and it is possible to find a uniform normal approximation for the distribution of hat beta under which arbitrary linear combinations a n hat beta beta are asymptotically normal when appropriately normalized and hat beta beta x x hat beta beta is approximately chi 2 p

the risk inflation criterion for multiple regression a new criterion is proposed for the evaluation of variable selection procedures in multiple regression this criterion which we call the risk inflation is based on an adjustment to the risk essentially the risk inflation is the maximum increase in risk due to selecting rather than knowing the correct predictors a new variable selection procedure is obtained which in the case of orthogonal predictors substantially improves on aic c p and bic and is close to optimal in contrast to aic c p and bic which use dimensionality penalties of 2 2 and log n respectively this new procedure uses a penalty 2 log p where p is the number of available predictors for the case of nonorthogonal predictors bounds for the optimal penalty are obtained

parameter modified versions of preconditioning and iterative inner product free refinement methods for two by two block matrices abstract a special two by two block matrix form arises in many important applications extending earlier results it is shown that parameter modified versions of a very efficient preconditioner does not improve its rate of convergence this holds also for iterative refinement methods corresponding to a few fixed steps of the chebyshev accelerated method the parameter version can improve the defect correction method but the convergence of this method is slower than an iterative refinement method with an optimal parameter the paper includes also a discussion of how one can save computer elapsed times by avoiding use of global inner products such as by use of a chebyshev accelerated method instead of a krylov subspace method since accurate and even sharp eigenvalue bounds are available the chebyshev iteration method converges as fast as the krylov subspace method

preconditioning for allen cahn variational inequalities with non local constraints the solution of allen cahn variational inequalities with mass constraints is of interest in many applications this problem can be solved both in its scalar and vector valued form as a pde constrained optimization problem by means of a primal dual active set method at the heart of this method lies the solution of linear systems in saddle point form in this paper we propose the use of krylov subspace solvers and suitable preconditioners for the saddle point systems numerical results illustrate the competitiveness of this approach

forward regression for ultra high dimensional variable screening motivated by the seminal theory of sure independence screening fan and lv 2008 sis we investigate here another popular and classical variable screening method namely forward regression fr our theoretical analysis reveals that fr can identify all relevant predictors consistently even if the predictor dimension is substantially larger than the sample size in particular if the dimension of the true model is finite fr can discover all relevant predictors within a finite number of steps to practically select the best candidate from the models generated by fr the recently proposed bic criterion of chen and chen 2008 can be used the resulting model can then serve as an excellent starting point from where many existing variable selection methods e g scad and adaptive lasso can be applied directly fr s outstanding finite sample performances are confirmed by extensive numerical studies

generalized hyperbolic ernst equations for an einstein maxwell weyl field a new family of nonlinear partial differential equations is presented they represent a generalization of the hyperbolic ernst equations for an einstein mawxell weyl field in general relativity a backlund transformation for the system of equations under consideration is given and their direct relation to the complete boussinesq hierarchy of soliton equations is illustrated

splitting based block preconditioning methods for block two by two matrices of real square blocks recently bai proposed rotated block preconditioners for block two by two matrices of real square blocks these rotated block preconditioners have the product form of a scaled orthogonal matrix and a block two by two triangular matrix theoretical and numerical results have shown the superiority of these rotated block preconditioners see bai 2013 10 in this paper inspired by the efficiency and the special structure of the rotated block preconditioners we establish a new equivalent linear system to the original linear system in which an orthogonal matrix arises we construct block jacobi and block gauss seidel splitting iteration methods based on the coefficient matrix of the new linear system the convergence of these splitting iterations is also demonstrated then by utilizing the proposed block jacobi and block gauss seidel splittings we put forward block preconditioners which are of the product form of a scaled orthogonal matrix and a block two by two diagonal or a block two by two triangular matrix spectral distributions of these preconditioned matrices and numerical experiments show that the proposed splitting based block preconditioners can be quite competitive with the rotated block preconditioners when they are used to accelerate krylov subspace iteration methods such as gmres for solving the block two by two liner systems

can the strengths of aic and bic be shared a conflict between model indentification and regression estimation a traditional approach to statistical inference is to identify the true or best model first with little or no consideration of the specific goal of inference in the model identification stage can the pursuit of the true model also lead to optimal regression estimation in model selection it is well known that bic is consistent in selecting the true model and aic is minimax rate optimal for estimating the regression function a recent promising direction is adaptive model selection in which in contrast to aic and bic the penalty term is data dependent some theoretical and empirical results have been obtained in support of adaptive model selection but it is still not clear if it can really share the strengths of aic and bic model combining or averaging has attracted increasing attention as a means to overcome the model selection uncertainty can bayesian model averaging be optimal for estimating the regression function in a minimax sense we show that the answers to these questions are basically in the negative for any model selection criterion to be consistent it must behave suboptimally for estimating the regression function in terms of minimax rate of covergence and bayesian model averaging cannot be minimax rate optimal for regression estimation

smoothly clipped absolute deviation on high dimensions the smoothly clipped absolute deviation scad estimator proposed by fan and li has many desirable properties including continuity sparsity and unbiasedness the scad estimator also has the asymptotically oracle property when the dimension of covariates is fixed or diverges more slowly than the sample size in this article we study the scad estimator in high dimensional settings where the dimension of covariates can be much larger than the sample size first we develop an efficient optimization algorithm that is fast and always converges to a local minimum second we prove that the scad estimator still has the oracle property on high dimensional problems we perform numerical studies to compare the scad estimator with the lasso and sis scad estimators in terms of prediction accuracy and variable selectivity when the true model is sparse through the simulation we show that the variance estimator of fan and li still works well for some limited high dimensional cases where the true nonzero coeffic

smooth and robust solutions for dirichlet boundary control of fluid solid conjugate heat transfer problems we study a new optimization scheme that generates smooth and robust solutions for dirichlet velocity boundary control dvbc of conjugate heat transfer cht processes the solutions to the dvbc of the incompressible navier stokes equations are typically nonsmooth due to the regularity degradation of the boundary stress in the adjoint navier stokes equations this nonsmoothness is inherited by the solutions to the dvbc of cht processes since the cht process couples the navier stokes equations of fluid motion with the convection diffusion equations of fluid solid thermal interaction our objective in the cht boundary control problem is to select optimally the fluid inflow profile that minimizes an objective function that involves the sum of the mismatch between the temperature distribution in the fluid system and a prescribed temperature profile and the cost of the control our strategy to resolve the nonsmoothness of the boundary control solution is based on two features namely the objective function with a regularization term on the gradient of the control profile on both the continuous and the discrete levels and the optimization scheme with either explicit or implicit smoothing effects such as the smoothed steepest descent and the limited memory broyden fletcher goldfarb shanno l bfgs methods our strategy to achieve the robustness of the solution process is based on combining the smoothed optimization scheme with the numerical continuation technique on the regularization parameters in the objective function in the section of numerical studies we present two suites of experiments in the first one we demonstrate the feasibility and effectiveness of our numerical schemes in recovering the boundary control profile of the standard case of a poiseuille flow in the second one we illustrate the robustness of our optimization schemes via solving more challenging dvbc problems for both the channel flow and the flow past a square cylinder which use initial control profiles far from optimal and require the numerical continuation technique applied on regularization parameters we believe our solution strategy is general and can be applied to other large scale optimal control problems which involve multiphysics processes and require smooth approximations to the optimal control profile

fast parameterized inexact uzawa method for complex symmetric linear systems in previous years bai and wang presented a class of parameterized inexact uzawa piu methods for solving the generalized saddle point problems in this paper we consider the same method for iteratively solving the complex symmetric linear systems our main contribution is accelerating the convergence of the parameterized inexact uzawa method by correction technique first the corrected model for the piu method is established and the corrected piu method is presented then we study the convergence property of the corrected piu method in fact the corrected piu method can converge faster than some uzawa type and hss like methods finally numerical experiments on a few model problems are presented to illustrate the theoretical results and examine the numerical effectiveness of the new method

consistent model selection criteria on high dimensions asymptotic properties of model selection criteria for high dimensional regression models are studied where the dimension of covariates is much larger than the sample size several sufficient conditions for model selection consistency are provided non gaussian error distributions are considered and it is shown that the maximal number of covariates for model selection consistency depends on the tail behavior of the error distribution also sufficient conditions for model selection consistency are given when the variance of the noise is neither known nor estimated consistently results of simulation studies as well as real data analysis are given to illustrate that finite sample performances of consistent model selection criteria can be quite different

inexact interior point method for pde constrained nonlinear optimization starting from the inexact interior point framework from curtis schenk and wachter siam j sci comput 32 2012 pp 3447 3475 we propose an effective schur complement slack control preconditioner for the full lagrangian hessian matrix needed at each newton iteration together they yield a scalable robust and highly parallel method for the numerical solution of large scale nonconvex pde constrained optimization problems with inequality constraints because it uses the full hessian matrix modifying it whenever needed the method not only is globally convergent but also converges fast locally our preconditioner is not tailored to any particular class of pdes or constraints but instead judiciously exploits the sparsity structure of the hessian numerical examples from pde constrained optimal control parameter estimation and full waveform inversion demonstrate the robustness and efficiency of the method even in the presence of active inequality constraints

all at once solution of time dependent stokes control the solution of time dependent pde constrained optimization problems subject to unsteady flow equations presents a challenge to both algorithms and computing power in this paper we present an all at once approach where we solve for all time steps of the discretized unsteady stokes problem at once the most desirable feature of this approach is that for all steps of an iterative scheme we only need approximate solutions of the discretized stokes operator this leads to an efficient scheme which exhibits mesh independent behaviour

a multigrid method for constrained optimal control problems we consider the fast and efficient numerical solution of linear quadratic optimal control problems with additional constraints on the control discretization of the first order conditions leads to an indefinite linear system of saddle point type with additional complementarity conditions due to the control constraints the complementarity conditions are treated by a primal dual active set strategy that serves as outer iteration at each iteration step a kkt system has to be solved here we develop a multigrid method for its fast solution to this end we use a smoother which is based on an inexact constraint preconditioner we present numerical results which show that the proposed multigrid method possesses convergence rates of the same order as for the underlying elliptic pde problem furthermore when combined with a nested iteration the solver is of optimal complexity and achieves the solution of the optimization problem at only a small multiple of the cost for the pde solution

consistency of bayesian procedures for variable selection it has long been known that for the comparison of pairwise nested models a decision based on the bayes factor produces a consistent model selector in the frequentist sense here we go beyond the usual consistency for nested pairwise models and show that for a wide class of prior distributions including intrinsic priors the corresponding bayesian procedure for variable selection in normal regression is consistent in the entire class of normal linear models we find that the asymptotics of the bayes factors for intrinsic priors are equivalent to those of the schwarz bic criterion also recall that the jeffreys lindley paradox refers to the well known fact that a point null hypothesis on the normal mean parameter is always accepted when the variance of the conjugate prior goes to infinity this implies that some limiting forms of proper prior distributions are not necessarily suitable for testing problems intrinsic priors are limits of proper prior distributions and for finite sample sizes they have been proved to behave extremely well for variable selection in regression a consequence of our results is that for intrinsic priors lindley s paradox does not arise

global optimality of nonconvex penalized estimators nonconvex penalties such as the smoothly clipped absolute deviation or minimax concave penalties have desirable properties such as the oracle property even when the dimension of the predictive variables is large however checking whether a given local minimizer has such properties is not easy since there can be many local minimizers in this paper we give sufficient conditions under which a local minimizer is unique and show that the oracle estimator becomes the unique local minimizer with probability tending to one copyright 2012 oxford university press

consistent model selection criteria for quadratically supported risks in this paper we study asymptotic properties of model selection criteria for high dimensional regression models where the number of covariates is much larger than the sample size in particular we consider a class of loss functions called the class of quadratically supported risks which is large enough to include the quadratic loss huber loss quantile loss and logistic loss we provide sufficient conditions for the model selection criteria which are applicable to the class of quadratically supported risks our results extend most previous sufficient conditions for model selection consistency in addition sufficient conditions for pathconsistency of the lasso and nonconvex penalized estimators are presented here pathconsistency means that the probability of the solution path that includes the true model converges to 1 pathconsistency makes it practically feasible to apply consistent model selection criteria to high dimensional data the data adaptive model selection procedure is proposed which is selection consistent and performs well for finite samples results of simulation studies as well as real data analysis are presented to compare the finite sample performances of the proposed data adaptive model selection criterion with other competitors

a low rank in time approach to pde constrained optimization the solution of time dependent pde constrained optimization problems is a challenging task in numerical analysis and applied mathematics all at once discretizations and corresponding solvers provide efficient methods to robustly solve the arising discretized equations one of the drawbacks of this approach is the high storage demand for the vectors representing the discrete space time cylinder here we introduce a low rank in time technique that exploits the low rank nature of the solution the theoretical foundations for this approach originate in the numerical treatment of matrix equations and can be carried over to pde constrained optimization we illustrate how three different problems can be rewritten and used within a low rank krylov subspace solver with appropriate preconditioning

immersed finite elements for optimal control problems of elliptic pdes with interfaces this paper presents a numerical method and analysis based on the variational discretization concept for optimal control problems governed by elliptic pdes with interfaces the method uses a simple uniform mesh which is independent of the interface due to the jump of the coefficient across the interface the standard linear finite element method cannot achieve optimal convergence when the uniform mesh is used therefore the immersed finite element method ifem developed in li et al 20 is used to discretize the state equation required in the variational discretization approach optimal error estimates for the control state and adjoint state are derived numerical examples are provided to confirm the theoretical results

analysis on inexact block diagonal preconditioners for elliptic pde constrained optimization problems abstract by using the galerkin finite element method the distributed control problems can be discretized into a saddle point problem with a coefficient matrix of block three by three which can be reduced to a linear system with lower order we first introduce a class of inexact block diagonal preconditioners and estimate the lower and upper bounds of positive and negative eigenvalues of the preconditioned matrices respectively based on the cholesky decomposition of the known matrices we also analyze a lower triangular preconditioner to accelerate the minimal residual method for the reduced linear system and discuss its real and complex eigenvalues respectively moreover these bounds do not rely on the regularization parameter and the eigenvalues of the matrices in the discrete system numerical experiments are also presented to demonstrate the effectiveness and robustness of the two new preconditioners

a family of integrable nonlinear equations of hyperbolic type a new system of integrable nonlinear equations of hyperbolic type obtained by a two dimensional reduction of the anti self dual yang mills equations is presented it represents a generalization of the ernst weyl equation of general relativity related to colliding neutrino and gravitational waves as well as of the fourth order equation of schwarzian type related to the kdv hierarchies which was introduced by nijhoff hone and joshi recently an auto backlund transformation of the new system is constructed leading to a superposition principle remarkably similar to the one connecting four solutions of the kdv equation at the level of the ernst weyl equation this backlund transformation and the associated superposition principle yield directly a generalization of the single and double harrison transformations of the ernst equation respectively the very method of construction also allows for revealing in an essentially algorithmic fashion other integrability features of the main subsystems such as their reduction to the painleve transcendents

an asymptotic property of model selection criteria probability models are estimated by use of penalized log likelihood criteria related to akaike 1973 information criterion aic and minimum description length mdl the accuracies of the density estimators are shown to be related to the tradeoff between three terms the accuracy of approximation the model dimension and the descriptive complexity of the model classes the asymptotic risk is determined under conditions on the penalty term and is shown to be minimax optimal for some cases as an application we show that the optimal rate of convergence is simultaneously achieved for log densities in sobolev spaces w sub 2 sup s u without knowing the smoothness parameter s and norm parameter u in advance applications to neural network models and sparse density function estimation are also provided

optimal solvers for pde constrained optimization optimization problems with constraints which require the solution of a partial differential equation arise widely in many areas of the sciences and engineering particularly in problems of design the solution of such pde constrained optimization problems is usually a major computational task here we consider simple problems of this type distributed control problems in which the 2 and 3 dimensional poisson problem is the pde the large dimensional linear systems which result from discretization and which need to be solved are of saddle point type we introduce two optimal preconditioners for these systems which lead to convergence of symmetric krylov subspace iterative methods in a number of iterations which does not increase with the dimension of the discrete problem these preconditioners are block structured and involve standard multigrid cycles the optimality of the preconditioned iterative solver is proved theoretically and verified computationally in several test cases the theoretical proof indicates that these approaches may have much broader applicability for other pdes

a leapfrog multigrid algorithm for the optimal control of parabolic pdes with robin boundary conditions we develop a second order finite difference scheme for solving the first order necessary optimality systems arising from the optimal control of parabolic pdes with robin boundary conditions under the framework of matrix analysis the proposed leapfrog scheme is shown to be unconditionally stable and second order convergent for both time and spatial variables without the requirement of the classical courant friedrichs lewy cfl condition on the spatial and temporal mesh step sizes moreover the developed leapfrog scheme provides a well structured discrete algebraic system that allows us to establish an effective multigrid iterative fast solver the resultant multigrid solver demonstrates a mesh independent convergence rate and a linear time complexity numerical experiments are provided to illustrate the accuracy and efficiency of the proposed leapfrog scheme

balanced truncation model reduction for a class of descriptor systems with application to the oseen equations we discuss the computation of balanced truncation model reduction for a class of descriptor systems which include the semidiscrete oseen equations with time independent advection and the linearized navier stokes equations linearized around a steady state the purpose of this paper is twofold first we show how to apply standard balanced truncation model reduction techniques which apply to dynamical systems given by ordinary differential equations to this class of descriptor systems this is accomplished by eliminating the algebraic equation using a projection the second objective of this paper is to demonstrate how the important class of adi smith type methods for the approximate computation of reduced order models using balanced truncation can be applied without explicitly computing the aforementioned projection instead we utilize the solution of saddle point problems we demonstrate the effectiveness of the technique in the computation of reduced order models for semidiscrete oseen equations

tuning parameter selection in high dimensional penalized likelihood summary determining how to select the tuning parameter appropriately is essential in penalized likelihood methods for high dimensional data analysis we examine this problem in the setting of penalized likelihood methods for generalized linear models where the dimensionality of covariates p is allowed to increase exponentially with the sample size n we propose to select the tuning parameter by optimizing the generalized information criterion with an appropriate model complexity penalty to ensure that we consistently identify the true model a range for the model complexity penalty is identified in the generlized information criterion we find that this model complexity penalty should diverge at the rate of some power of log p depending on the tail probability behaviour of the response variables this reveals that using the akaike information criterion or bayes information criterion to select the tuning parameter may not be adequate for consistently identifying the true model on the basis of our theoretical study we propose a uniform choice of the model complexity penalty and show that the approach proposed consistently identifies the true model among candidate models with asymptotic probability 1 we justify the performance of the procedure proposed by numerical simulations and a gene expression data analysis

a discrete garnier type system from symmetry reduction on the lattice a symmetry reduction of the lattice modified boussinesq system is studied the full group of lie point symmetries of the relevant system is retrieved and certain group invariant solutions are considered by using an accessional generalized symmetry it is demonstrated that the symmetry reduction leads to a coupled set of second order nonlinear non autonomous ordinary difference equations involving six free parameters generalizing to higher order some of the known discrete analogues of the painleve vi equation the corresponding isomonodromic deformation problem is constructed through the symmetry reduction as well