least squares methods for stokes equations based on a discrete minus one inner product abstract the purpose of this paper is to develop and analyze least squares approximations for stokes and elasticity problems the major advantage of the least squares formulation is that it does not require that the classical ladyzhenskaya bab ska brezzi lbb condition be satisfied we provide two methods the first is posed in terms of the velocity pressure pair without the introduction of additional variables the second adds a vorticity variable in both cases we employ least squares functionals which involve a discrete inner product which is related to the inner product in h 1 the sobolev space of order minus one on the use of such inner products applied to second order problems was proposed in an earlier paper by bramble lazarov and pasciak 1994

type i and type ii bayesian methods for sparse signal recovery using scale mixtures in this paper we propose a generalized scale mixture family of distributions namely the power exponential scale mixture pesm family to model the sparsity inducing priors currently in use for sparse signal recovery ssr we show that the successful and popular methods such as lasso reweighted ell 1 and reweighted ell 2 methods can be formulated in an unified manner in a maximum a posteriori map or type i bayesian framework using an appropriate member of the pesm family as the sparsity inducing prior in addition exploiting the natural hierarchical framework induced by the pesm family we utilize these priors in a type ii framework and develop the corresponding em based estimation algorithms some insight into the differences between type i and type ii methods is provided and of particular interest in the algorithmic development is the type ii variant of the popular and successful reweighted ell 1 method extensive empirical results are provided and they show that the type ii methods exhibit better support recovery than the corresponding type i methods

complexity of partially separable convexly constrained optimization with non lipschitzian singularities an adaptive regularization algorithm using high order models is proposed for solving partially separable convexly constrained nonlinear optimization problems whose objective function contains non l

an interior point method for large scale l 1 regularized logistic regression logistic regression with l1 regularization has been proposed as a promising method for feature selection in classification problems in this paper we describe an efficient interior point method for solving large scale l1 regularized logistic regression problems small problems with up to a thousand or so features and examples can be solved in seconds on a pc medium sized problems with tens of thousands of features and examples can be solved in tens of seconds assuming some sparsity in the data a variation on the basic method that uses a preconditioned conjugate gradient method to compute the search step can solve very large problems with a million features and examples e g the 20 newsgroups data set in a few minutes on a pc using warm start techniques a good approximation of the entire regularization path can be computed much more efficiently than by solving a family of problems independently

optimality conditions and a smoothing trust region newton method for nonlipschitz optimization regularized minimization problems with nonconvex nonsmooth perhaps non lipschitz penalty functions have attracted considerable attention in recent years owing to their wide applications in image restoration signal reconstruction and variable selection in this paper we derive affine scaled second order necessary and sufficient conditions for local minimizers of such minimization problems moreover we propose a global convergent smoothing trust region newton method which can find a point satisfying the affine scaled second order necessary optimality condition from any starting point numerical examples are given to demonstrate the effectiveness of the smoothing trust region newton method

continuous groundwater velocity fields and path lines in linear bilinear and trilinear finite elements darcy velocity fields obtained from finite element solutions of heads in groundwater flow exhibit discontinuities at element boundaries thus giving rise to inaccuracies in path line construction several methods have been developed to resolve those inaccuracies including global postprocessing of velocity fields mixed hybrid finite element formulations and stream function formulations all of these techniques either lead to a considerable increase in the computational effort or are not general enough for all purposes the point of view taken here is that standard finite element flow models yield exact water balances which in the usual approach to path line computation are not fully used by starting out from patches with flux conserving boundaries a new postprocessing procedure is derived that allows construction of a continuous flux distribution over the whole model domain in this procedure only the head gradients of adjacent elements are used thus avoiding the solution of another global system of equations furthermore application of a stream function interpolation inside each patch allows the fast computation of path lines without any time stepping the path lines computed with the proposed procedure are compared to path lines computed by usual particle tracking and examples are presented showing that the new method is vastly superior to traditional particle tracking

accuracy of mixed and control volume finite element approximations to darcy velocity and related quantities the accuracy of the darcy velocity flux and stream function computed from lowest order triangle based control volume and mixed finite element approximations to the two dimensional pressure equation is considered the control volume finite element method similar to integrated finite difference methods and analogous to the interpolation of galerkin finite element results over control volumes is shown to yield a conservative velocity field and smooth streamlines the streamlines and fluxes through the system computed with the control volume finite element approach are compared to those computed from the mixed finite element method which approximates the pressure and velocity variables separately it is shown that for systems with only moderate degrees of heterogeneity the control volume finite element method is the more computationally efficient alternative i e it provides more accurate flow results for a given number of unknowns for more variable or discontinuous permeability fields by contrast such as sand shale systems the mixed finite element method is shown to approximate flow variables more accurately and more realistically than the control volume method with the same number of unknowns

on the truncated conjugate gradient method in this paper we consider the truncated conjugate gradient method for minimizing a convex quadratic function subject to a ball trust region constraint it is shown that the reduction in the objective function by the solution obtained by the truncated cg method is at least half of the reduction by the global minimizer in the trust region

a trust region method for solving generalized complementarity problems based on a semismooth equation reformulation using fischer s function a trust region algorithm is proposed for solving the generalized complementarity problem gcp the algorithm uses a generalized jacobian of the function involved in the semismooth equation and adopts the squared natural residual of the semismooth equation as a merit function the proposed algorithm is applicable to the nonlinear complementarity problem because the latter problem is a special case of the gcp global convergence and under a nonsingularity assumption local q superlinear or quadratic convergence of the algorithm are established moreover calculation of a generalized jacobian is discussed and numerical results are presented

multilevel iteration for mixed finite element systems with penalty in this paper the authors consider the solution of the discrete systems that arises when a mixed finite element approach is used to approximate the solution of second order elliptic boundary value problems by the introduction of a penalty parameter these equations can be approximated by the solution of a symmetric and positive definite penalty system on the velocity subspace iterative procedures are developed and analyzed for this penalty system based on the hierarchical basis approach as well as on the standard multigrid approach finally numerical experiments are presented that illustrate the convergence behavior suggested by the theory

the lowest order weak galerkin finite element method for the darcy equation on quadrilateral and hybrid meshes abstract this paper investigates the lowest order weak galerkin finite element method for solving the darcy equation on quadrilateral and hybrid meshes consisting of quadrilaterals and triangles in this approach the pressure is approximated by constants in element interiors and on edges the discrete weak gradients of these constant basis functions are specified in local raviart thomas spaces specifically r t 0 for triangles and unmapped r t 0 for quadrilaterals these discrete weak gradients are used to approximate the classical gradient when solving the darcy equation the method produces continuous normal fluxes and is locally mass conservative regardless of mesh quality and has optimal order convergence in pressure velocity and normal flux when the quadrilaterals are asymptotically parallelograms implementation is straightforward and results in symmetric positive definite discrete linear systems we present numerical experiments and comparisons with other existing methods

a trust region algorithm for nonsmooth optimization a trust region algorithm is proposed for minimizing the nonsmooth composite functionf x h f x wheref is smooth andh is convex the algorithm employs a smoothing function which is closely related to fletcher s exact differentiable penalty functions global and local convergence results are given considering convergence to a strongly unique minimizer and to a minimizer satisfying second order sufficiency conditions

smoothing technique and its applications in semidefinite optimization in this paper we extend the smoothing technique nesterov in math program 103 1 127 152 2005 nesterov in unconstrained convex mimimization in relative scale 2003 onto the problems of semidefinite optimization for that we develop a simple framework for estimating a lipschitz constant for the gradient of some symmetric functions of eigenvalues of symmetric matrices using this technique we can justify the lipschitz constants for some natural approximations of maximal eigenvalue and the spectral radius of symmetric matrices we analyze the efficiency of the special gradient type schemes on the problems of minimizing the maximal eigenvalue or the spectral radius of the matrix which depends linearly on the design variables we show that in the first case the number of iterations of the method is bounded by o 1 epsilon where epsilon is the required absolute accuracy of the problem in the second case the number of iterations is bounded by 4 delta sqrt 1 delta r ln r where is the required relative accuracy and r is the maximal rank of corresponding linear matrix inequality thus the latter method is a fully polynomial approximation scheme

reply to comment on application of the mixed hybrid finite element approximation in a groundwater flow model luxury or necessity by r mose p siegel p ackerer and g chavent

multicomponent fluid flow by discontinuous galerkin and mixed methods in unfractured and fractured media 1 a discrete fracture model for the flow of compressible multicomponent fluids in homogeneous heterogeneous and fractured media is presented in single phase in the numerical model we combine the mixed finite element mfe and the discontinuous galerkin dg methods we use the cross flow equilibrium concept to approximate the fractured matrix mass transfer the discrete fracture model is numerically superior to the single porosity model and overcomes limitations of the dual porosity models including the use of a shape factor the mfe method provides a direct and accurate approximation for the velocity field which is crucial for the convective terms in the flow equations the dg method associated with a slope limiter is used to approximate the species balance equations this method can capture the sharp moving fronts the calculation of the fracture fracture flux across three and higher intersecting fracture branches is a challenge in this work we provide an accurate approximation of these fluxes by using the mfe formulation numerical examples in unfractured and fractured media illustrate the efficiency and robustness of the proposed numerical model

a comparative study on the weak galerkin discontinuous galerkin and mixed finite element methods this paper presents a comparative study on the newly introduced weak galerkin finite element methods wgfems with the widely accepted discontinuous galerkin finite element methods dgfems and the classical mixed finite element methods mfems for solving second order elliptic boundary value problems we examine the differences similarities and connection among these methods in scheme formulations implementation strategies accuracy and computational cost the comparison and numerical experiments demonstrate that wgfems are viable alternatives to mfems and hold some advantages over dgfems due to their properties of local conservation normal flux continuity no need for penalty factor and definiteness of discrete linear systems

locally conservative coupling of stokes and darcy flows a locally conservative numerical method for solving the coupled stokes and darcy flows problem is formulated and analyzed the approach employs the mixed finite element method for the darcy region and the discontinuous galerkin method for the stokes region a discrete inf sup condition and optimal error estimates are derived

a mixed finite element technique based on implicit discretization of faults for permeability upscaling in fault damage zones 1 natural fault damage zones fdzs are often composed of complex arrays containing many thin faults to account for fdzs in large scale models it is necessary to represent the flow effects of these many small faults by some form of upscaling the explicit discretization of the geometry and topology of these thin faults and of the rock matrix typically leads to meshes comprising large numbers of elements making it difficult to solve the flow equations efficiently this paper proposes a practical technique based on the mixed finite element method mfem permitting the permeability upscaling in two dimensional fault damage zones that contain thin and isotropic low permeability faults this technique utilizes an implicit discretization scheme that is similar in principle to those used for discrete fracture networks the scheme described here treats flow across faults accurately but unlike the schemes used for fractures it neglects the flow along the faults because intrafault volumetric flow in thin faults is very small this approach allows for a simpler numerical discretization and a reduction in numerical calculation effort the scheme can be readily implemented in a standard mixed finite element code compared with the same mfem based on an explicit discretization scheme the two techniques are shown to produce results that agree closely the implicit technique is shown to be more efficient than the explicit technique for a typical fault damage zone model this technique is readily extendable to three dimensions

a triangle based mixed finite element finite volume technique for modeling two phase flow through porous media triangle based discretization techniques offer great advantages relative to standard finite difference methods for the modeling of flow through geometrically complex geological features the purpose of this paper is to develop and apply a triangle based method for the modeling of two phase flow through porous formations the formulation includes the effects of gravity compressibility and capillary pressure the technique entails a triangle based mixed finite element method for solution of the variable coefficient parabolic pressure equation and a second order tvd type total variation diminishing finite volume scheme for solution of the essentially hyperbolic saturation equation the method is applied to a variety of example problems and is shown to perform very well on problems involving geometric complexity coupled with heterogeneous generally anisotropic permeability descriptions

smoothing methods for nonsmooth nonconvex minimization we consider a class of smoothing methods for minimization problems where the feasible set is convex but the objective function is not convex not differentiable and perhaps not even locally lipschitz at the solutions such optimization problems arise from wide applications including image restoration signal reconstruction variable selection optimal control stochastic equilibrium and spherical approximations in this paper we focus on smoothing methods for solving such optimization problems which use the structure of the minimization problems and composition of smoothing functions for the plus function x many existing optimization algorithms and codes can be used in the inner iteration of the smoothing methods we present properties of the smoothing functions and the gradient consistency of subdifferential associated with a smoothing function moreover we describe how to update the smoothing parameter in the outer iteration of the smoothing methods to guarantee convergence of the smoothing methods to a stationary point of the original minimization problem

weak galerkin finite element methods for darcy flow anisotropy and heterogeneity this paper presents a family of weak galerkin finite element methods wgfems for darcy flow computation the wgfems are new numerical methods that rely on the novel concept of discrete weak gradients the wgfems solve for pressure unknowns both in element interiors and on the mesh skeleton the numerical velocity is then obtained from the discrete weak gradient of the numerical pressure the new methods are quite different than many existing numerical methods in that they are locally conservative by design the resulting discrete linear systems are symmetric and positive definite and there is no need for tuning problem dependent penalty factors we test the wgfems on benchmark problems to demonstrate the strong potential of these new methods in handling strong anisotropy and heterogeneity in darcy flow

adaptive cubic regularisation methods for unconstrained optimization part ii worst case function and derivative evaluation complexity an adaptive regularisation framework using cubics arc was proposed for unconstrained optimization and analysed in cartis gould and toint part i math program doi 10 1007 s10107 009 0286 5 2009 generalizing at the same time an unpublished method due to griewank technical report na 12 1981 damtp university of cambridge an algorithm by nesterov and polyak math program 108 1 177 205 2006 and a proposal by weiser deuflhard and erdmann optim methods softw 22 3 413 431 2007 in this companion paper we further the analysis by providing worst case global iteration complexity bounds for arc and a second order variant to achieve approximate first order and for the latter second order criticality of the iterates in particular the second order arc algorithm requires at most mathcal o epsilon 3 2 iterations or equivalently function and gradient evaluations to drive the norm of the gradient of the objective below the desired accuracy epsilon and mathcal o epsilon 3 iterations to reach approximate nonnegative curvature in a subspace the orders of these bounds match those proved for algorithm 3 3 of nesterov and polyak which minimizes the cubic model globally on each iteration our approach is more general in that it allows the cubic model to be solved only approximately and may employ approximate hessians

a trust region method for minimization of nonsmooth functions with linear constraints we introduce a trust region algorithm for minimization of nonsmooth functions with linear constraints at each iteration the objective function is approximated by a model function that satisfies a set of assumptions stated recently by qi and sun in the context of unconstrained nonsmooth optimization the trust region iteration begins with the resolution of an easy problem as in recent works of martinez and santos and friedlander martinez and santos for smooth constrained optimization in practical implementations we use the infinity norm for defining the trust region which fits well with the domain of the problem we prove global convergence and report numerical experiments related to a parameter estimation problem

on finite element methods for heterogeneous elliptic problems dealing with variational formulations of second order elliptic problems with discontinuous coefficients we recall a single field minimization problem of an extended functional presented by bevilacqua et al bevilacqua l feijoo r a rojas l f 1974 a variational principle for the laplace operator with application in the torsion of composite rods international journal of solids structuring 10 1091 1102 which we associate with the basic idea supporting discontinuous galerkin finite element methods we review residual based stabilized mixed methods applied to darcy flow in homogeneous porous media and extend them to heterogeneous media with an interface of discontinuity for smooth interfaces the proposed formulations preserve the continuity of the flux and exactly imposes the constraint between the tangent components of darcy velocity on the interface convergence studies for a heterogeneous and anisotropic porous medium confirm the same rates of convergence predicted for homogeneous problem with smooth solutions

cubic regularization of newton method and its global performance in this paper we provide theoretical analysis for a cubic regularization of newton method as applied to unconstrained minimization problem for this scheme we prove general local convergence results however the main contribution of the paper is related to global worst case complexity bounds for different problem classes including some nonconvex cases it is shown that the search direction can be computed by standard linear algebra technique

fosll for nonlinear partial differential equations in previous work the first order system ll fosll method was developed for linear partial differential equations this approach seeks to minimize the residual of the equations in a dual norm induced by the differential operator yielding approximations accurate in l 2 omega rather than h 1 omega or h div in this paper the general framework of fosll is extended to a wide range of nonlinear problems four approaches to propagating an inexact newton iteration based on a fosll approximation are presented and theory for robust convergence in l 2 omega is established numerical results are presented for two formulations of the steady incompressible navier stokes equations and for a diffusion equation with reduced regularity due to a discontinuous diffusion coefficient

first order system least squares for incompressible resistive magnetohydrodynamics magnetohydrodynamics mhd is a fluid theory that describes plasma physics by treating the plasma as a fluid of charged particles hence the equations that describe the plasma form a nonlinear system that couples navier stokes equations with maxwell s equations this paper shows that the first order system least squares fosls finite element method is a viable discretization for these large mhd systems to solve this system a nested iteration newton fosls amg approach is taken most of the work is done on the coarse grid including most of the linearizations we show that at most one newton step and a few v cycles are all that are needed on the finest grid here we describe how the fosls method can be applied to incompressible resistive mhd and how it can be used to solve these mhd problems efficiently a 3d steady state and a reduced 2d time dependent test problem are studied the latter equations can simulate a large aspect ratio tokamak the goal is to resolve as much physics from the test problems with the least amount of computational work we show that this is achieved in a few dozen work units or fine grid residual evaluations

necessary optimality conditions and exact penalization for non lipschitz nonlinear programs when the objective function is not locally lipschitz constraint qualifications are no longer sufficient for karush kuhn tucker kkt conditions to hold at a local minimizer let alone ensuring an exact penalization in this paper we extend quasi normality and relaxed constant positive linear dependence condition to allow the non lipschitzness of the objective function and show that they are sufficient for kkt conditions to be necessary for optimality moreover we derive exact penalization results for the following two special cases when the non lipschitz term in the objective function is the sum of a composite function of a separable lower semi continuous function with a continuous function and an indicator function of a closed subset we show that a local minimizer of our problem is also a local minimizer of an exact penalization problem under a local error bound condition for a restricted constraint region and a suitable assumption on the outer separable function when the non lipschitz term is the sum of a continuous function and an indicator function of a closed subset we also show that our problem admits an exact penalization under an extended quasi normality involving the coderivative of the continuous function

the maximum principle violations of the mixed hybrid finite element method applied to diffusion equations the affluent literature of finite element methods applied to linear parabolic problems generally produces numerical procedures with satisfactory properties from a numerical approximation point of view however some initial boundary conditions or abrupt sink source terms may cause large gradients at some points and consequently jumps in the solution that usually needs a certain period of time to become more and more smooth this intuitive fact of the diffusion process necessitates when applying numerical methods varying the mesh size in time and space according to the smoothness of the solution in this work the numerical behavior of the time dependent solutions for such problems during small time duration obtained by using the mixed hybrid finite element method mhfem is investigated numerical comparisons with the standard galerkin finite element fe as well as the finite difference fd methods are checked up on due to the fact that the mixed methods violate the discrete maximum principle some numerical experiments showed that the mhfem leads sometimes to non physical peaks in the solution a diffusivity criterion relating the mesh steps for an artificial initial boundary value problem will be presented one of the propositions given to avoid any non physical oscillations is to use the mass lumping techniques

locally conservative stabilized finite element methods for variably saturated flow standard galerkin finite element methods for variably saturated groundwater flow have several deficiencies for instance local oscillations can appear around sharp infiltration fronts without the use of mass lumping and velocity fields obtained from differentiation of pressure fields are discontinuous at element boundaries here we consider conforming finite element discretizations based on a multiscale formulation along with recently developed local postprocessing schemes the resulting approach maintains the basic flexibility and appeal of traditional finite element methods while controlling nonphysical oscillations and producing element wise mass conservative velocity fields accuracy and efficiency of the proposed schemes are evaluated through a series of steady state and transient variably saturated groundwater flow problems in homogeneous as well as heterogeneous domains the schemes are formulated for a generic nonlinear advection diffusion equation and are thus applicable to many other flow models

first order system least squares for the stokes equations with application to linear elasticity following our earlier work on general second order scalar equations here we develop a least squares functional for the two and three dimensional stokes equations generalized slightly by allowing a pressure term in the continuity equation by introducing a velocity flux variable and associated curl and trace equations we are able to establish ellipticity in an h1 product norm appropriately weighted by the reynolds number this immediately yields optimal discretization error estimates for finite element spaces in this norm and optimal algebraic convergence estimates for multiplicative and additive multigrid methods applied to the resulting discrete systems both estimates are naturally uniform in the reynolds number moreover our pressure perturbed form of the generalized stokes equations allows us to develop an analogous result for the dirichlet problem for linear elasticity where we obtain the more substantive result that the estimates are uniform in the poisson ratio

worst case complexity of smoothing quadratic regularization methods for non lipschitzian optimization in this paper we propose a smoothing quadratic regularization sqr algorithm for solving a class of nonsmooth nonconvex perhaps even non lipschitzian minimization problems which has wide applications in statistics and sparse reconstruction the proposed sqr algorithm is a first order method at each iteration the sqr algorithm solves a strongly convex quadratic minimization problem with a diagonal hessian matrix which has a simple closed form solution that is inexpensive to calculate we show that the worst case complexity of reaching an epsilon scaled stationary point is o epsilon 2 moreover if the objective function is locally lipschitz continuous the sqr algorithm with a slightly modified updating scheme for the smoothing parameter and iterate can obtain an epsilon clarke stationary point in at most o epsilon 3 iterations

on the superlinear convergence of a trust region algorithm for nonsmooth optimization it is proved that the second order correction trust region algorithm of fletcher 5 ensures superlinear convergence if some mild conditions are satisfied

a numerical study of a semi algebraic multilevel preconditioner for the local discontinuous galerkin method in this work we consider the local discontinuous galerkin ldg method applied to second order elliptic problems arising in the modeling of single phase flows in porous media it has been recently proven that the spectral condition number of the stiffness matrix exhibits an asymptotic behavior of o h 2 on structured and unstructured meshes where h is the mesh size thus efficient preconditioners are mandatory we present a semi algebraic multilevel preconditioner for the ldg method using local lagrange type interpolatory basis functions we show numerically that its performance does not degrade or at least the number of iterations increases very slowly as the number of unknowns augments the preconditioner is tested on problems with high jumps in the coefficients which is the typical scenario of problems arising in porous media

a trust region algorithm for minimization of locally lipschitzian functions the classical trust region algorithm for smooth nonlinear programs is extended to the nonsmooth case where the objective function is only locally lipschitzian at each iteration an objective function that carries both first and second order information is minimized over a trust region the term that carries the first order information is an iteration function that may not explicitly depend on subgradients or directional derivatives we prove that the algorithm is globally convergent this convergence result extends the result of powell for minimization of smooth functions the result of yuan for minimization of composite convex functions and the result of dennis li and tapia for minimization of regular functions in addition compared with the recent model of pang han and rangaraj for minimization of locally lipschitzian functions using a line search this algorithm has the same convergence property without assuming positive definiteness and uniform boundedness of the second order term applications of the algorithm to various nonsmooth optimization problems are discussed

an inexpensive method for the evaluation of the solution of the lowest order raviart thomas mixed method for the model problem operatorname div a nabla u f we show how the approximate solution produced by the raviart thomas lecture notes in mathematics 606 springer verlag berlin 1977 mixed method of lowest degree can be obtained from the solution of the p 1 nonconforming galerkin method modified in a virtually cost free manner

high order evaluation complexity for convexly constrained optimization with non lipschitzian group sparsity terms this paper studies high order evaluation complexity for partially separable convexly constrained optimization involving non lipschitzian group sparsity terms in a nonconvex objective function we propose a partially separable adaptive regularization algorithm using a pth order taylor model and show that the algorithm needs at most o epsilon p 1 p q 1 evaluations of the objective function and its first p derivatives whenever they exist to produce an epsilon delta approximate qth order stationary point our algorithm uses the underlying rotational symmetry of the euclidean norm function to build a lipschitzian approximation for the non lipschitzian group sparsity terms which are defined by the group ell 2 ell a norm with a in 0 1 the new result shows that the partially separable structure and non lipschitzian group sparsity terms in the objective function do not affect the worst case evaluation complexity order

a general mass conservative numerical solution for the unsaturated flow equation numerical approximations based on different forms of the governing partial differential equation can lead to significantly different results for unsaturated flow problems numerical solution based on the standard h based form of richards equation generally yields poor results characterized by large mass balance errors and erroneous estimates of infiltration depth conversely numerical solutions based on the mixed form of richards equation can be shown to possess the conservative property so that mass is perfectly conserved this leads to significant improvement in numerical solution performance while requiring no additional computational effort however use of the mass conservative method does not guarantee good solutions accurate solution of the unsaturated flow equation also requires use of a diagonal time or mass matrix only when diagonal time matrices are used can the solution be shown to obey a maximum principle which guarantees smooth nonoscillatory infiltration profiles this highlights the fact that proper treatment of the time derivative is critical in the numerical solution of unsaturated flow

first order system least squares for geometrically nonlinear elasticity we present a first order system least squares fosls method to approximate the solution to the equations of geometrically nonlinear elasticity in two dimensions with assumptions of regularity on the problem we show h 1 equivalence of the norm induced by the fosls functional in the case of pure displacement boundary conditions as well as local convergence of newton s method in a nested iteration setting theoretical results hold for deformations satisfying a small strain assumption a set we show to be largely coincident with the set of deformations allowed by the model numerical results confirm optimal multigrid performance and finite element approximation rates of the discrete functional with a total work bounded by about 25 fine grid relaxation sweeps

unconditionally stable mixed finite element methods for darcy flow unconditionally stable finite element methods for darcy flow are derived by adding least squares residual forms of the governing equations to the classical mixed formulations the proposed methods are free of mesh dependent stabilization parameters and allow the use of the classical continuous lagrangian finite element spaces of any order for the velocity and the potential stability convergence and error estimates are derived and numerical experiments are presented to demonstrate the flexibility of the proposed finite element formulations and to confirm the predicted rates of convergence

convergence of the gradient sampling algorithm for nonsmooth nonconvex optimization we study the gradient sampling algorithm of burke lewis and overton for minimizing a locally lipschitz function f on mathbb r n that is continuously differentiable on an open dense subset we strengthen the existing convergence results for this algorithm and introduce a slightly revised version for which stronger results are established without requiring compactness of the level sets of f in particular we show that with probability 1 the revised algorithm either drives the f values to infty or each of its cluster points is clarke stationary for f we also consider a simplified variant in which the differentiability check is skipped and the user can control the number of f evaluations per iteration

accelerating the cubic regularization of newton s method on convex problems in this paper we propose an accelerated version of the cubic regularization of newton s method nesterov and polyak in math program 108 1 177 205 2006 the original version used for minimizing a convex function with lipschitz continuous hessian guarantees a global rate of convergence of order o big 1 over k 2 big where k is the iteration counter our modified version converges for the same problem class with order o big 1 over k 3 big keeping the complexity of each iteration unchanged we study the complexity of both schemes on different classes of convex problems in particular we argue that for the second order schemes the class of non degenerate problems is different from the standard class

mixed finite elements and newton type linearizations for the solution of richards equation we present the development of a two dimensional mixed hybrid finite element mhfe model for the solution of the non linear equation of variably saturated flow in groundwater on unstructured triangular meshes by this approach the darcy velocity is approximated using lowest order raviart thomas rt0 elements and is exactly mass conserving hybridization is used to overcome the ill conditioning of the mixed system the scheme is globally first order in space nevertheless numerical results employing non uniform meshes show second order accuracy of the pressure head and normal fluxes on specific grid points the non linear systems of algebraic equations resulting from the mhfe discretization are solved using picard or newton iterations realistic sample tests show that the mhfe newton approach achieves fast convergence in many situations in particular when a good initial guess is provided by either the picard scheme or relaxation techniques copyright 1999 john wiley sons ltd

an impes scheme for a two phase flow in heterogeneous porous media using a structured grid abstract we develop a numerical scheme for a two phase immiscible flow in heterogeneous porous media using a structured grid finite element method which has been successfully used for the computation of various physical applications involving elliptic interface equations li et al 2003 2004 chou et al 2010 kwak et al 2010 chang and kwak 2011 the proposed method is based on the implicit pressure explicit saturation procedure to solve the pressure equation we use an ifem based on the rannacher turek rannacher and turek 1992 nonconforming space which is a modification of the work in kwak et al 2010 where broken p 1 nonconforming element of crouzeix raviart 1973 1 was developed for the darcy velocity we apply the mixed finite volume method studied in chou et al 2003 and kwak et al 2010 on the basis of immersed finite element method ifem in this way the darcy velocity of the flow can be computed cheaply locally after we solve the pressure equation the computed darcy velocity is used to solve the saturation equation explicitly the whole procedure can be implemented on a structured grid which is independent of the underlying heterogeneous porous media in fact we apply a new version of multigrid algorithm to solve the pressure equation for which the cpu time grows like o n numerical tests for analytic problems show that our method is almost optimal in all variables problem with sources sinks is also computed and it seems to work well

adaptive simulations of two phase flow by discontinuous galerkin methods in this paper we present and compare primal discontinuous galerkin formulations of the two phase flow equations the wetting phase pressure and saturation equations are decoupled and solved sequentially proposed adaptivity in space and time techniques yield accurate and efficient solutions slope limiters valid on non conforming meshes are also presented numerical examples of homogeneous and heterogeneous media are considered

a robust gradient sampling algorithm for nonsmooth nonconvex optimization let f be a continuous function on rl n and suppose f is continuously differentiable on an open dense subset such functions arise in many applications and very often minimizers are points at which f is not differentiable of particular interest is the case where f is not convex and perhaps not even locally lipschitz but is a function whose gradient is easily computed where it is defined we present a practical robust algorithm to locally minimize such functions based on gradient sampling no subgradient information is required by the algorithm when f is locally lipschitz and has bounded level sets and the sampling radius eps is fixed we show that with probability 1 the algorithm generates a sequence with a cluster point that is clarke eps stationary furthermore we show that if f has a unique clarke stationary point bar x then the set of all cluster points generated by the algorithm converges to bar x as eps is reduced to zero numerical results are presented demonstrating the robustness of the algorithm and its applicability in a wide variety of contexts including cases where f is not locally lipschitz at minimizers we report approximate local minimizers for functions in the applications literature which have not to our knowledge been obtained previously when the termination criteria of the algorithm are satisfied a precise statement about nearness to clarke eps stationarity is available a matlab implementation of the algorithm is posted at http www cs nyu edu overton papers gradsamp alg

optimal convergence of the original dg method for the transport reaction equation on special meshes we show that the approximation given by the original discontinuous galerkin method for the transport reaction equation in d space dimensions is optimal provided the meshes are suitably chosen the l 2 norm of the error is of order k 1 when the method uses polynomials of degree k these meshes are not necessarily conforming and do not satisfy any uniformity condition they are required only to be made of simplexes each of which has a unique outflow face we also find a new element by element postprocessing of the derivative in the direction of the flow which superconverges with order k 1

a group bridge approach for variable selection in multiple regression problems when covariates can be naturally grouped it is important to carry out feature selection at the group and within group individual variable levels simultaneously the existing methods including the lasso and group lasso are designed for either variable selection or group selection but not for both we propose a group bridge approach that is capable of simultaneous selection at both the group and within group individual variable levels the proposed approach is a penalized regularization method that uses a specially designed group bridge penalty it has the oracle group selection property in that it can correctly select important groups with probability converging to one in contrast the group lasso and group least angle regression methods in general do not possess such an oracle property in group selection simulation studies indicate that the group bridge has superior performance in group and individual variable selection relative to several existing methods

conditions for convergence of trust region algorithms for nonsmooth optimization this paper discusses some properties of trust region algorithms for nonsmooth optimization the problem is expressed as the minimization of a functionh f x whereh is convex andf is a continuously differentiable mapping from to bounds for the second order derivative approximation matrices are discussed it is shown that powel s 7 8 results hold for nonsmooth optimization

a new formulation of the mixed finite element method for solving elliptic and parabolic pde with triangular elements for the groundwater flow problem which corresponds to the darcy flow model we show how to produce a scheme with one unknown per element starting from a mixed formulation discretized with the raviart thomas triangular elements of lowest order the aim is here to obtain a new formulation with one unknown per element by elimination of the velocity variables q k grad p without any restriction concerning the computation of the velocity field in the first part we describe the triangular mixed finite element method used for solving darcy s and mass balance equations in the second part we study the elliptic parabolic problem we describe the new formulation of the problem in order to use mixed finite elements mfe with less unknowns without any specific numerical integration finally we compare the computational effort of the mfe method with the new formulation for different triangulations using numerical experiments in this work we show that the new formulation can be seen as a general formulation which can be equivalent to the finite volume or the finite difference methods in some particular cases

complexity analysis of interior point algorithms for non lipschitz and nonconvex minimization we propose a first order interior point algorithm for a class of non lipschitz and nonconvex minimization problems with box constraints which arise from applications in variable selection and regularized optimization the objective functions of these problems are continuously differentiable typically at interior points of the feasible set our first order algorithm is easy to implement and the objective function value is reduced monotonically along the iteration points we show that the worst case iteration complexity for finding an epsilon scaled first order stationary point is o epsilon 2 o 2 furthermore we develop a second order interior point algorithm using the hessian matrix and solve a quadratic program with a ball constraint at each iteration although the second order interior point algorithm costs more computational time than that of the first order algorithm in each iteration its worst case iteration complexity for finding an epsilon scaled second order stationary point is reduced to o epsilon 3 2 o 3 2 note that an epsilon scaled second order stationary point must also be an epsilon scaled first order stationary point

first order system least squares for second order partial differential equations part i this paper develops a least squares functional that arises from recasting general second order uniformly elliptic partial differential equations in n 2 or 3 dimensions as a system of first order equations in part i z cai r d lazarov t manteuffel and s mccormick siam j numer anal 31 1994 pp 1785 1799 a similar functional was developed and shown to be elliptic in the h divv times h 1 norm and to yield optimal convergence for finite element subspaces of h divv times h 1 in this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first order system the resulting functional is proved to be elliptic in the h 1 n 1 norm this immediately implies optimal error estimates for finite element approximation by standard subspaces of h 1 n 1 another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent as an alternative to perturbation based approaches the least squares approach developed here applies directly to convection diffusion reaction equations in a unified way and also admits a fast multigrid solver historically a missing ingredient in least squares methodology

superconvergent derivatives a taylor series analysis an analysis based on the use of taylor series expansions is developed to determine accuracy estimates for derivatives in one and two dimensions computed by differentiation of a finite element interpolant or approximation the analysis clarifies some issues concerning special points at which the derivatives are believed to be exceptionally accurate with higher convergence rates superconvergence moreover it leads directly to a class of post processing strategies for the derivatives and offers a more direct constructive approach to the subject

a unified approach to global convergence of trust region methods for nonsmooth optimization this paper investigates the global convergence of trust region tr methods for solving nonsmooth minimization problems for a class of nonsmooth objective functions called regular functions conditions are found on the tr local models that imply three fundamental convergence properties these conditions are shown to be satisfied by appropriate forms of fletcher s tr method for solving constrained optimization problems powell and yuan s tr method for solving nonlinear fitting problems zhang kim and lasdon s successive linear programming method for solving constrained problems duff nocedal and reid s tr method for solving systems of nonlinear equations and el hallabi and tapia s tr method for solving systems of nonlinear equations thus our results can be viewed as a unified convergence theory for tr methods for nonsmooth problems

hierarchical high order finite element spaces in h div ω h1 ω for a stabilized mixed formulation of darcy problem abstract the classical dual mixed finite element method for flow simulations is based on h div conforming approximation spaces for the flux which guarantees continuous normal components on element interfaces and discontinuous approximations in l 2 for the pressure however stability and convergence can only be obtained for compatible approximation spaces stabilized finite element methods may provide an alternative stable procedure to avoid this kind of delicate balance the main purpose of this paper is to present a high order finite element methodology to solve the darcy problem based on the combination of an unconditionally stable mixed finite element method with a hierarchical methodology for the construction of finite dimensional subspaces of h div and h 1 the chosen stabilized method is free of mesh dependent stabilization parameters and allows for the use of different high order finite element approximations for the flux and the pressure variables without requiring any compatibility constraint as required in mixed methods for these problems convergence studies are presented comparing the numerical solutions obtained for different approximation orders on quadrilateral elements with the ones given by classical mixed formulation with raviart thomas elements

smoothing neural network for constrained non lipschitz optimization with applications in this paper a smoothing neural network snn is proposed for a class of constrained non lipschitz optimization problems where the objective function is the sum of a nonsmooth nonconvex function and a non lipschitz function and the feasible set is a closed convex subset of using the smoothing approximate techniques the proposed neural network is modeled by a differential equation which can be implemented easily under the level bounded condition on the objective function in the feasible set we prove the global existence and uniform boundedness of the solutions of the snn with any initial point in the feasible set the uniqueness of the solution of the snn is provided under the lipschitz property of smoothing functions we show that any accumulation point of the solutions of the snn is a stationary point of the optimization problem numerical results including image restoration blind source separation variable selection and minimizing condition number are presented to illustrate the theoretical results and show the efficiency of the snn comparisons with some existing algorithms show the advantages of the snn

approximate static condensation algorithm for solving multi material diffusion problems on meshes non aligned with material interfaces abstract in this paper we describe a new algorithm for solving multi material diffusion problem when material interfaces are not aligned with the mesh in this case interface reconstruction methods are used to construct approximate representation of interfaces between materials they produce so called multi material cells in which materials are represented by material polygons that contain only one material the reconstructed interface is not continuous between cells we suggest the new method for solving multi material diffusion problems on such meshes and compare its performance with known homogenization methods

error analysis of multipoint flux domain decomposition methods for evolutionary diffusion problems abstract we study space and time discretizations for mixed formulations of parabolic problems the spatial approximation is based on the multipoint flux mixed finite element method which reduces to an efficient cell centered pressure system on general grids including triangles quadrilaterals tetrahedra and hexahedra the time integration is performed by using a domain decomposition time splitting technique combined with multiterm fractional step diagonally implicit runge kutta methods the resulting scheme is unconditionally stable and computationally efficient as it reduces the global system to a collection of uncoupled subdomain problems that can be solved in parallel without the need for schwarz type iteration convergence analysis for both the semidiscrete and fully discrete schemes is presented

on the evaluation complexity of composite function minimization with applications to nonconvex nonlinear programming we estimate the worst case complexity of minimizing an unconstrained nonconvex composite objective with a structured nonsmooth term by means of some first order methods we find that it is unaffected by the nonsmoothness of the objective in that a first order trust region or quadratic regularization method applied to it takes at most mathcal o epsilon 2 function evaluations to reduce the size of a first order criticality measure below epsilon specializing this result to the case when the composite objective is an exact penalty function allows us to consider the objective and constraint evaluation worst case complexity of nonconvex equality constrained optimization when the solution is computed using a first order exact penalty method we obtain that in the reasonable case when the penalty parameters are bounded the complexity of reaching within epsilon of a kkt point is at most mathcal o epsilon 2 problem evaluations which is the same in order as the function evaluation complexity of steepest descent methods applied to unconstrained nonconvex smooth optimization

robust streamline tracing for the simulation of porous media flow on general triangular and quadrilateral grids streamline methods for subsurface flow simulation have received renewed attention as fast alternatives to traditional finite volume or finite element methods key aspects of streamline simulation are the accurate tracing of streamlines and the computation of travel time along individual streamlines in this paper we propose a new streamline tracing framework that enables the extension of streamline methods to unstructured grids composed of triangular or quadrilateral elements and populated with heterogeneous full tensor coefficients the proposed method is based on the mathematical framework of mixed finite element methods which provides approximations of the velocity field that are especially suitable for streamline tracing we identify and implement two classes of velocity spaces the lowest order raviart thomas space low order tracing and the brezzi douglas marini space of order one high order tracing both on triangles and quadrilaterals we discuss the implementation of the streamline tracing method in detail and we investigate the performance of the proposed tracing strategy by means of carefully designed test cases we conclude that for the same computational cost high order tracing is more accurate smaller error in the time of flight and robust less sensitive to grid distortion than low order tracing

asymptotic properties of bridge estimators in sparse high dimensional regression models we study the asymptotic properties of bridge estimators in sparse high dimensional linear regression models when the number of covariates may increase to infinity with the sample size we are particularly interested in the use of bridge estimators to distinguish between covariates whose coefficients are zero and covariates whose coefficients are nonzero we show that under appropriate conditions bridge estimators correctly select covariates with nonzero coefficients with probability converging to one and that the estimators of nonzero coefficients have the same asymptotic distribution that they would have if the zero coefficients were known in advance thus bridge estimators have an oracle property in the sense of fan and li j amer statist assoc 96 2001 1348 1360 and fan and peng ann statist 32 2004 928 961 in general the oracle property holds only if the number of covariates is smaller than the sample size however under a partial orthogonality condition in which the covariates of the zero coefficients are uncorrelated or weakly correlated with the covariates of nonzero coefficients we show that marginal bridge estimators can correctly distinguish between covariates with nonzero and zero coefficients with probability converging to one even when the number of covariates is greater than the sample size

on the finite volume reformulation of the mixed finite element method for elliptic and parabolic pde on triangles abstract a new resolution of parabolic and elliptic partial differential equations pdes based on the mixed finite element approximation on triangles has been recently developed 24 25 this new approach reduces the number of unknowns from fluxes or lagrange multiplier defined on edges to a single unknown per element in this paper we analyze this transformation mathematically and describe in details how to handle singular elements and singular edges for these singular elements the standard mixed method on triangles can always be made equivalent to a finite volume formulation where the finite volumes are obtained by aggregation of finite elements across singular edges the positive definiteness of the system matrix obtained with the new formulation is analyzed in details a criterion is given concerning the property of this matrix which show that its conditioning is related to the shape of the triangle and the contrast in parameters from one element to the adjacent ones numerical experiments are performed for elliptic and parabolic pdes the comparisons between an iterative solver pcg and a direct solver unifrontal multifrontal show that the direct solver is more efficient moreover its performance is not correlated with the system matrix conditioning it appears that the new formulation requires significantly less cpu time for elliptic pdes and is competitive for parabolic pdes the new formulation remains also accurate enough even in nearly singular situations

a stabilized mixed finite element method for darcy flow we develop new stabilized mixed finite element methods for darcy flow stability and an a priori error estimate in the stability norm are established a wide variety of convergent finite elements present themselves unlike the classical galerkin formulation which requires highly specialized elements an interesting feature of the formulation is that there are no mesh dependent parameters numerical tests confirm the theoretical results 2002 elsevier science b v all rights reserved

the numerical solution of diffusion problems in strongly heterogeneous non isotropic materials a new second order finite difference algorithm for the numerical solution of diffusion problems in strongly heterogeneous and non isotropic media is constructed on problems with rough coefficients or highly nonuniform grids the new algorithm is superior to all other algorithms we have compared it with for problems with smooth coefficients on smooth grids the method is comparable with other second order methods the new algorithm is formulated for logically rectangular grids and is derived using thesupport operators method a key idea in deriving the method was to replace the usual inner product of vector functions by an inner product weighted by the inverse of the material properties tensor and to use the flux operator defined as the material properties tensor times the gradient rather than the gradient as one of the basic first order operators in the support operators method the discrete analog of the flux operator must also be the negative adjoint of the discrete divergence in an inner product that is a discrete analog of the continuum inner product the resulting method is conservative and the discrete analog of the variable coefficient laplacian is symmetric and negative definite on nonuniform grids in addition on any grid the discrete divergence is zero on constant vectors the null space for the gradient is the constant functions and when the material properties are piecewise constant the discrete flux operator is exact for piecewise linear functions we compare the methods on some of the most difficult examples to be found in the literature

enhanced mass conservation in least squares methods for navier stokes equations there are many applications of the least squares finite element method for the numerical solution of partial differential equations because of a number of benefits that the least squares method has however one of the most well known drawbacks of the least squares finite element method is the lack of exact discrete mass conservation in some contexts due to the fact that the least squares method minimizes the continuity equation in l 2 norm in this paper we explore the reason for the mass loss and provide new approaches to retain the mass even in a severely underresolved grid

enriched multi point flux approximation for general grids it is well known that the two point flux approximation a numerical scheme used in most commercial reservoir simulators has o 1 error when grids are not k orthogonal in the last decade the multi point flux approximations have been developed as a remedy however non physical oscillations can appear when the anisotropy is really strong we found out the oscillations are closely related to the poor approximation of pressure gradient in the flux computation in this paper we propose the control volume enriched multi point flux approximation empfa for general diffusion problems on polygonal and polyhedral meshes non physical oscillations are not observed for realistic and strongly anisotropic heterogeneous material properties described by a full tensor exact linear solutions are recovered for grids with non planar interfaces and a first and second order convergence are achieved for the flux and scalar unknowns respectively

l_p norm regularization algorithms for optimization over permutation matrices optimization problems over permutation matrices appear widely in facility layout chip design scheduling pattern recognition computer vision graph matching etc since this problem is np hard due to the combinatorial nature of permutation matrices we relax the variable to be the more tractable doubly stochastic matrices and add an l p norm 0 p 1 regularization term to the objective function the optimal solutions of the l p regularized problem are the same as the original problem if the regularization parameter is sufficiently large a lower bound estimation of the nonzero entries of the stationary points and some connections between the local minimizers and the permutation matrices are further established then we propose an l p regularization algorithm with local refinements the algorithm approximately solves a sequence of l p regularization subproblems by the projected gradient method using a nonmonotone line search with the barzilai borwein step sizes its performance can be fu

a trust region strategy for minimization on arbitrary domains we present a trust region method for minimizing a general differentiable function restricted to an arbitrary closed set we prove a global convergence theorem the trust region method defines difficult subproblems that are solvable in some particular cases we analyze in detail the case where the domain is a euclidean ball for this case we present numerical experiments where we consider different hessian approximations

minimizing nonconvex functions for sparse vector reconstruction in this paper we develop a novel methodology for minimizing a class of nonconvex concave on the non negative orthant functions for solving an underdetermined linear system of equations as x when the solution vector s is known a priori to be sparse the proposed technique is based on locally replacing the original objective function by a quadratic convex function which is easily minimized the resulting algorithm is iterative and is absolutely converging to a fixed point of the original objective function for a certain selection of convex objective functions the class of algorithms called iterative reweighted least squares irls is shown to be a special case of the proposed methodology thus the proposed algorithms are a generalization and unification of the previous methods in addition we also propose a new class of algorithms with better convergence properties compared to the regular irls algorithms and hence can be considered as enhancements to these algorithms since the original objective functions are nonconvex the proposed algorithm is susceptible to convergence to a local minimum to alleviate this difficulty we propose a random perturbation technique that enhances the performance of the proposed algorithm the numerical results show that the proposed algorithms outperform some of the well known algorithms that are usually utilized for solving the same problem

some errors estimates for the box method we define and analyze several variants of the box method for discretizing elliptic boundary value problems in the plane our estimates show the error to be comparable to a standard galerkin finite element method using piecewise linear polynomials

comment on application of the mixed hybrid finite element approximation in a groundwater flow model luxury or necessity by r mose p siegel p ackerer and g chavent

newton ll method for the second order semi linear elliptic partial differential equations newton s method with first order system least squares fosls finite element method has been widely used to approximately solve a system of nonlinear partial differential equations adler et al 2010 9 codd et al 2003 10 manteuffel et al 2006 11 in this paper we propose to use the first order system ll method to find a correction in each newton s iteration which provides an l2 approximation of the second order semi linear elliptic partial differential equations while the typical newton fosls method providesh1 approximations the numerical tests have been conducted to validate the theory

superlinearly convergent approximate newton methods for lc 1 optimization problems in the literature the proof of superlinear convergence of approximate newton or sqp methods for solving nonlinear programming problems requires twice smoothness of the objective and constraint functions sometimes the second order derivatives of those functions are required to be lipschitzian in this paper we present approximate newton or sqp methods for solving nonlinear programming problems whose objective and constraint functions have locally lipschitzian derivatives and establishq superlinear convergence of these methods under the assumption that these derivatives are semismooth this assumption is weaker than the second order differentiability the extended linear quadratic programming problem in the fully quadratic case is an example of nonlinear programming problems whose objective functions have semismooth but not smooth derivatives

iterative reweighted minimization methods for l_p lp regularized unconstrained nonlinear programming in this paper we study general l p l p regularized unconstrained minimization problems in particular we derive lower bounds for nonzero entries of the first and second order stationary points and hence also of local minimizers of the l p l p minimization problems we extend some existing iterative reweighted l 1 l 1 mathrm irl 1 irl 1 and l 2 l 2 mathrm irl 2 irl 2 minimization methods to solve these problems and propose new variants for them in which each subproblem has a closed form solution also we provide a unified convergence analysis for these methods in addition we propose a novel lipschitz continuous epsilon approximation to vert x vert p p x p p using this result we develop new mathrm irl 1 irl 1 methods for the l p l p minimization problems and show that any accumulation point of the sequence generated by these methods is a first order stationary point provided that the approximation parameter epsilon is below a computable threshold value this is a remarkable result since all existing iterative reweighted minimization methods require that epsilon be dynamically updated and approach zero our computational results demonstrate that the new mathrm irl 1 irl 1 method and the new variants generally outperform the existing mathrm irl 1 irl 1 methods chen and zhou in 2012 foucart and lai in appl comput harmon anal 26 395 407 2009

an a posteriori error analysis of an augmented discontinuous galerkin formulation for darcy flow in this paper we develop an a posteriori error analysis for an augmented discontinuous garlerkin formulation applied to the darcy flow more precisely we derive a reliable and efficient a posteriori error estimator which consists of residual terms finally we present several numerical experiments showing the robustness of the method and the theoretical properties of the estimator thus confirming the capability of the corresponding adaptive algorithms to localize the inner layers the singularities and or the large stress regions of the exact solution

a smoothing sqp framework for a class of composite l_q lq minimization over polyhedron the composite l q 0 q 1 minimization problem over a general polyhedron has received various applications in machine learning wireless communications image restoration signal reconstruction etc this paper aims to provide a theoretical study on this problem first we derive the karush kuhn tucker kkt optimality conditions for local minimizers of the problem second we propose a smoothing sequential quadratic programming framework for solving this problem the framework requires a approximate solution of a convex quadratic program at each iteration finally we analyze the worst case iteration complexity of the framework for returning an epsilon kkt point i e a feasible point that satisfies a perturbed version of the derived kkt optimality conditions to the best of our knowledge the proposed framework is the first one with a worst case iteration complexity guarantee for solving composite l q minimization over a general polyhedron

a fully coupled multiphase flow and geomechanics solver for highly heterogeneous porous media abstract this paper introduces a fully coupled multiphase flow and geomechanics solver that can be applied to modeling highly heterogeneous porous media multiphase flow in deformable porous media is a multiphysics problem that considers the flow physics and rock physics simultaneously to model this problem the multiphase flow equations and geomechanical equilibrium equation must be tightly coupled conventional finite element modeling of coupled flow and geomechanics does not conserve mass locally since it uses continuous basis functions mixed finite element discretization that satisfies local mass conservation of the flow equation can be a good solution for this problem in addition the stabilized finite element method for discretizing the saturation equation minimizes numerical diffusion and provides better resolution of saturation solution in this work we developed a coupled multiphase flow and geomechanics solver that solves fully coupled governing equations namely pressure velocity saturation and geomechanical equilibrium equations the solver can deal with full tensor permeability and elastic stiffness for modeling a highly heterogeneous reservoir system the results of the numerical experiments are very encouraging we used the solver to simulate a reservoir system that has very heterogeneous permeability and elastic stiffness fields and found that the numerical solution captures the complex multiphysics of the system in addition we obtained higher resolution of saturation solution than with the conventional finite volume discretization this would help us make a better estimate of the numerical solution of complex multiphysics problems

goal oriented a posteriori error estimation for the travel time functional in porous media flows in this article we consider the a posteriori error estimation and adaptive mesh refinement for the numerical approximation of the travel time functional arising in porous media flows the key application of this work is in the safety assessment of radioactive waste facilities in this setting the travel time functional measures the time taken for a nonsorbing radioactive solute transported by groundwater to travel from a potential site deep underground to the biosphere to ensure the computability of the travel time functional we employ a mixed formulation of darcy s law and conservation of mass together with raviart thomas h mathrm div omega conforming finite elements the proposed a posteriori error bound is derived based on a variant of the standard dual weighted residual approximation which takes into account the lack of smoothness of the underlying functional of interest the proposed adaptive refinement strategy is tested on both a simple academic test case and a problem based on the