brief paper a robust override scheme enforcing strict output constraints for a class of strictly proper systems this paper presents an override controller which ensures that constrained output variables retain certain prescribed strict bounds the class of nominal closed loop systems considered is strictly proper and minimum phase assuming the first markov parameter to be full rank and for each output measurement constraint there is one available actuator the advantage of the considered class of nominal systems is that an output constraint translates directly into a state constraint for which it is possible to use a particular non smooth lyapunov function the non smooth lyapunov function is defined by the level of the output constraint creating an invariant set for which the strict output constraints are satisfied the override strategy is designed to have only a minimal effect on the nominal control loop when no constraint is violated

a multipoint flux mixed finite element method we develop a mixed finite element method for single phase flow in porous media that reduces to cell centered finite differences on quadrilateral and simplicial grids and performs well for discontinuous full tensor coefficients motivated by the multipoint flux approximation method where subedge fluxes are introduced we consider the lowest order brezzi douglas marini bdm mixed finite element method a special quadrature rule is employed that allows for local velocity elimination and leads to a symmetric and positive definite cell centered system for the pressures theoretical and numerical results indicate second order convergence for pressures at the cell centers and first order convergence for subedge fluxes second order convergence for edge fluxes is also observed computationally if the grids are sufficiently regular

on convergence of block centered finite differences for elliptic problems we consider linear selfadjoint elliptic problems with neumann boundary conditions in rectangular domains we demonstrate that with sufficiently smooth data the discrete l 2 norm errors for tensor product block centered finite differences in both the approximate solution and its first derivatives are second order for all nonuniform grids extensions to nonselfadjoint and parabolic problems are discussed

quasi m matrix multifamily continuous darcy flux approximations with full pressure support on structured and unstructured grids in three dimensions new multifamilies of flux continuous vertex centered finite volume methods are presented for the full tensor pressure equation with general discontinuous coefficients for any cell type in three dimensions the new schemes are flux continuous with full pressure support over each subcell with continuous pressure imposed across each control volume subface in contrast to earlier formulations full pressure continuity across subfaces leads to a quasi positive formulation that minimizes spurious oscillations in discrete pressure solutions for strongly anisotropic full tensor fields the multifamily formulation permits maximum flexibility in quadrature yielding improved solution resolution the earlier methods are pointwise continuous in pressure and flux with tetrahedral pressure support which leads to a more limited quadrature range which is shown to cause spurious oscillations in the solution for strong full tensor fields an m matrix analysis of the three dimensional schemes identifies bounding limits for the schemes to possess a local discrete maximum principle conditions for the schemes to be positive definite are also derived

mixed finite elements for second order elliptic problems in three variables two families of mixed finite elements one based on simplices and the other on cubes are introduced as alternatives to the usual raviart thomas nedelec spaces these spaces are analogues of those introduced by brezzi douglas and marini in two space variables error estimates inl2 andh s are derived

constrained stabilization problems for linear plants linear systems with magnitude and rate constraints on both the state and control variables are considered for such systems semi global and global constrained stabilization problems are formulated when state feedback controllers are used necessary and sufficient conditions for the solvability of the formulated problems are developed moreover design methodologies for such constrained stabilization problems are presented an important aspect of our development here is a taxonomy of constraints to show clearly for what type of constraints what can or cannot be achieved

computational bases for rtk and bdmk on triangles in this article we derive computational bases for raviart thomas rt and brezzi douglas marini bdm vector approximation spaces on a triangulation of a domain in r 2 the basis functions defined on the reference triangle have a lagrangian property the continuity of the normal component of the approximation across the edges in the triangulation is satisfied by the use of the piola transformation and the lagrangian property of the basis functions a numerical example is given demonstrating the approximation property of the bases

linear systems with state and control constraints the theory and application of maximal output admissible sets the initial state of an unforced linear system is output admissible with respect to a constraint set y if the resulting output function satisfies the pointwise in time condition y t in y t or 0 the set of all possible such initial conditions is the maximal output admissible set o sub infinity the properties of o sub infinity and its characterization are investigated in the discrete time case it is generally possible to represent o sub infinity or a close approximation of it by a finite number of functional inequalities practical algorithms for generating the functions are described in the continuous time case simple representations of the maximal output admissible set are not available however it is shown that the discrete time results may be used to obtain approximate representations

a control volume mixed finite element formulation for tetrahedral elements summary the control volume mixed finite element method is formulated for and applied to a computational domain consisting of a tetrahedral partitioning to solve the steady groundwater flow equations test functions consistent with piecewise constant and piecewise linear pressure distributions are used in the formulation comparisons are made with a standard mixed finite element formulation using lowest order raviart thomas basis functions results suggest that the control volume based formulation is a viable alternative to the standard formulation copyright 2015 john wiley sons ltd

mixed finite elements for elliptic problems with tensor coefficients as cell centered finite differences we present an expanded mixed finite element approximation of second order elliptic problems containing a tensor coefficient the mixed method is expanded in the sense that three variables are explicitly approximated namely the scalar unknown the negative of its gradient and its flux the tensor coefficient times the negative gradient the resulting linear system is a saddle point problem in the case of the lowest order raviart thomas elements on rectangular parallelepipeds we approximate this expanded mixed method by incorporating certain quadrature rules this enables us to write the system as a simple cell centered finite difference method requiring the solution of a sparse positive semidefinite linear system for the scalar unknown for a general tensor coefficient the sparsity pattern for the scalar unknown is a 9 point stencil in two dimensions and 19 points in three dimensions existing theory shows that the expanded mixed method gives optimal order approximations in the l 2 and h s norms and superconvergence is obtained between the l 2 projection of the scalar variable and its approximation we show that these rates of convergence are retained for the finite difference method if h denotes the maximal mesh spacing then the optimal rate is o h the superconvergence rate o h 2 is obtained for the scalar unknown and rate o h 3 2 for its gradient and flux in certain discrete norms moreover the full o h 2 is obtained in the strict interior of the domain computational results illustrate these theoretical results

pegasos primal estimated sub gradient solver for svm we describe and analyze a simple and effective stochastic sub gradient descent algorithm for solving the optimization problem cast by support vector machines svm we prove that the number of iterations required to obtain a solution of accuracy epsilon is tilde o 1 epsilon where each iteration operates on a single training example in contrast previous analyses of stochastic gradient descent methods for svms require omega 1 epsilon 2 iterations as in previously devised svm solvers the number of iterations also scales linearly with 1 where is the regularization parameter of svm for a linear kernel the total run time of our method is tilde o d lambda epsilon where d is a bound on the number of non zero features in each example since the run time does not depend directly on the size of the training set the resulting algorithm is especially suited for learning from large datasets our approach also extends to non linear kernels while working solely on the primal objective function though in this case the runtime does depend linearly on the training set size our algorithm is particularly well suited for large text classification problems where we demonstrate an order of magnitude speedup over previous svm learning methods

a multipoint flux mixed finite element method on distorted quadrilaterals and hexahedra in this paper we develop a new mixed finite element method for elliptic problems on general quadrilateral and hexahedral grids that reduces to a cell centered finite difference scheme a special non symmetric quadrature rule is employed that yields a positive definite cell centered system for the pressure by eliminating local velocities the method is shown to be accurate on highly distorted rough quadrilateral and hexahedral grids including hexahedra with non planar faces theoretical and numerical results indicate first order convergence for the pressure and face fluxes

a composite mixed finite element for hexahedral grids a new mixed finite element method on three dimensional hexahedral meshes for second order elliptic problems is proposed this finite element is a composite element it is shown to have optimal convergence properties and it is applied to a hydrogeology problem

a simple a posteriori estimate on general polytopal meshes with applications to complex porous media flows this paper develops an posteriori error estimate for lowest order locally conservative methods on meshes consisting of very general polytopal elements we focus on the ease of implementation of the methodology based on h1 conforming potential reconstruction and h div conforming flux reconstruction in particular the evaluation of our estimates merely consists in some local matrix vector multiplications where on each mesh element the matrices are either directly inherited from the given numerical method or trivially constructed from the element geometry while the vectors are the degrees of freedom on the given element we then apply this methodology to unsteady nonlinear coupled degenerate problems describing complex multiphase flows in porous media here on each step of the time marching scheme linearization procedure and linear algebraic solver we distinguish the corresponding error components this leads to an easy to implement and fast to run adaptive algorithm with simultaneously guaranteed overall precision and optimal efficiency ensured through the use of adaptive stopping criteria together with adaptive space and time mesh refinements numerous numerical experiments on practical problems in two and three space dimensions illustrate the performance of our methodology

nonlinear tracking control in the presence of state and control constraints a generalized reference governor this paper proposes a new approach to reference governor design as in prior literature the governor accepts input commands and modifies their evolution so that specified pointwise in time constraints on state and control variables are satisfied the new approach applies to general discrete time and continuous time nonlinear systems with uncertainties it relies on safety properties provided by sublevel sets of equilibria parameterized functions these functions need not be lyapunov functions and the corresponding sublevel sets need not be positively invariant technical conditions that capture the bare essentials of what is needed are identified and the usual desirable properties of reference governors are established the new approach significantly broadens the class of methods available for constructing the nonlinear function that is required in the implementation of the reference governors this advantage is illustrated in a nonlinear control problem where off line computer based simulation is the basis for constructing the nonlinear function

bundle methods for regularized risk minimization a wide variety of machine learning problems can be described as minimizing a regularized risk functional with different algorithms using different notions of risk and different regularizers examples include linear support vector machines svms gaussian processes logistic regression conditional random fields crfs and lasso amongst others this paper describes the theory and implementation of a scalable and modular convex solver which solves all these estimation problems it can be parallelized on a cluster of workstations allows for data locality and can deal with regularizers such as l1 and l2 penalties in addition to the unified framework we present tight convergence bounds which show that our algorithm converges in o 1 e steps to e precision for general convex problems and in o log 1 e steps for continuously differentiable problems we demonstrate the performance of our general purpose solver on a variety of publicly available data sets

a multipoint flux mixed finite element method on hexahedra we develop a mixed finite element method for elliptic problems on hexahedral grids that reduces to cell centered finite differences the paper is an extension of our earlier paper for quadrilateral and simplicial grids m f wheeler and i yotov siam j numer anal 44 2006 pp 2082 2106 the construction is motivated by the multipoint flux approximation method and it is based on an enhancement of the lowest order brezzi douglas duran fortin bddf mixed finite element spaces on hexahedra in particular there are four fluxes per face one associated with each vertex a special quadrature rule is employed that allows for local velocity elimination and leads to a symmetric and positive definite cell centered system for the pressures theoretical and numerical results indicate first order convergence for pressures and subface fluxes on sufficiently regular grids as well as second order convergence for pressures at the cell centers second order convergence for face fluxes is also observed computationally

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

from mixed finite elements to finite volumes for elliptic pdes in two and three dimensions the link between mixed finite element mfe and finite volume fv methods applied to elliptic partial differential equations has been investigated by many authors recently a fv formulation of the mixed approach has been developed this approach was restricted to 2d problems with a scalar for the parameter used to calculate fluxes from the state variable gradient this new approach is extended to 2d problems with a full parameter tensor and to 3d problems the objective of this new formulation is to reduce the total number of unknowns while keeping the same accuracy this is achieved by defining one new variable per element for the 2d case with full parameter tensor this new formulation exists for any kind of triangulation it allows the reduction of the number of unknowns to the number of elements instead of the number of edges no additional assumptions are required concerning the averaging of the parameter in hetero geneous domains for 3d problems we demonstrate that the new formulation cannot exist for a general 3d tetrahedral discretization unlike in the 2d problem however it does exist when the tetrahedrons are regular or deduced from rectangular parallelepipeds and allows reduction of the number of unknowns numerical experiments and comparisons between both formulations in 2d show the efficiency of the new formulation copyright 2003 john wiley sons ltd

discretization on unstructured grids for inhomogeneous anisotropic media part i derivation of the methods discretization methods are proposed for control volume formulations on polygonal and triangular grid cells in two space dimensions the methods are applicable for any system of conservation laws where the flow density is defined by a gradient law like darcy s law for porous media flow a strong feature of the methods is the ability to handle media inhomogeneities in combination with full tensor anisotropy this paper gives a derivation of the methods and the relation to previously published methods is also discussed a further discussion of the methods including numerical examples is given in the companion paper part ii siam j sci comput pp 1717 1736

residual flux based a posteriori error estimates for finite volume and related locally conservative methods we derive in this paper a posteriori error estimates for discretizations of convection diffusion reaction equations in two or three space dimensions our estimates are valid for any cell centered finite volume scheme and in a larger sense for any locally conservative method such as the mimetic finite difference covolume and other we consider meshes consisting of simplices or rectangular parallelepipeds and also provide extensions to nonconvex cells and nonmatching interfaces we allow for the cases of inhomogeneous and anisotropic diffusion dispersion tensors and of convection dominance the estimates are established in the energy semi norm for a locally postprocessed approximate solution preserving the conservative fluxes and are of residual type they are fully computable all occurring constants are evaluated explicitly and locally efficient give a local lower bound on the error times an efficiency constant so that they can serve both as indicators for adaptive refinement and for the actual control of the error they are semi robust in the sense that the local efficiency constant only depends on local variations in the coefficients and becomes optimal as the local peclet number gets sufficiently small numerical experiments confirm their accuracy

on the algorithmic implementation of multiclass kernel based vector machines in this paper we describe the algorithmic implementation of multiclass kernel based vector machines our starting point is a generalized notion of the margin to multiclass problems using this notion we cast multiclass categorization problems as a constrained optimization problem with a quadratic objective function unlike most of previous approaches which typically decompose a multiclass problem into multiple independent binary classification tasks our notion of margin yields a direct method for training multiclass predictors by using the dual of the optimization problem we are able to incorporate kernels with a compact set of constraints and decompose the dual problem into multiple optimization problems of reduced size we describe an efficient fixed point algorithm for solving the reduced optimization problems and prove its convergence we then discuss technical details that yield significant running time improvements for large datasets finally we describe various experiments with our approach comparing it to previously studied kernel based methods our experiments indicate that for multiclass problems we attain state of the art accuracy

set membership identification of nonlinear systems in the paper the problem of identifying nonlinear dynamic systems described in nonlinear regression form is considered using finite and noise corrupted measurements most methods in the literature are based on the estimation of a model within a finitely parametrized model class describing the functional form of involved nonlinearities a key problem in these methods is the proper choice of the model class typically realized by a search from the simplest to more complex ones linear bilinear polynomial neural networks etc in this paper an alternative approach based on a set membership framework is presented not requiring assumptions on the functional form of the regression function describing the relations between measured input and output but assuming only some information on its regularity given by bounds on its gradient in this way the problem of considering approximate functional forms is circumvented moreover noise is assumed to be bounded in contrast with statistical methods which rely on assumptions such as stationarity ergodicity uncorrelation type of distribution etc whose validity may be difficult to test reliably and is lost in presence of approximate modeling in this paper necessary and sufficient conditions are given for the validation of the considered assumptions an optimal interval estimate of the regression function is obtained providing its uncertainty range for any assigned regressor values the set estimate allows to derive an optimal identification algorithm giving estimates with minimal guaranteed l p error on the assigned domain of the regressors the properties of the optimal estimate are investigated and its worst case l p identification error is evaluated the presented approach is tested and compared with other nonlinear methods on the identification of a water heater a mechanical system with input saturation and a vehicle with controlled suspensions

superconvergence of mixed finite element approximations over quadrilaterals a superconvergence result is established in this article for approximate solutions of second order elliptic equations by mixed finite element methods over quadrilaterals the superconvergence indicates an accuracy of cal o h k 2 for the mixed finite element approximation if the raviart thomas or brezzi douglas fortin marini elements of order k are employed with optimal error estimate of cal o h k 1 numerical experiments are presented to illustrate the theoretical result

regularized bundle methods for convex and non convex risks machine learning is most often cast as an optimization problem ideally one expects a convex objective function to rely on efficient convex optimizers with nice guarantees such as no local optima yet non convexity is very frequent in practice and it may sometimes be inappropriate to look for convexity at any price alternatively one can decide not to limit a priori the modeling expressivity to models whose learning may be solved by convex optimization and rely on non convex optimization algorithms the main motivation of this work is to provide efficient and scalable algorithms for non convex optimization we focus on regularized unconstrained optimization problems which cover a large number of modern machine learning problems such as logistic regression conditional random fields large margin estimation etc we propose a novel algorithm for minimizing a regularized objective that is able to handle convex and non convex smooth and non smooth risks the algorithm is based on the cutting plane technique and on the idea of exploiting the regularization term in the objective function it may be thought as a limited memory extension of convex regularized bundle methods for dealing with convex and non convex risks in case the risk is convex the algorithm is proved to converge to a stationary solution with accuracy e with a rate o 1 e where is the regularization parameter of the objective function under the assumption of a lipschitz empirical risk in case the risk is not convex getting such a proof is more difficult and requires a stronger and more disputable assumption yet we provide experimental results on artificial test problems and on five standard and difficult machine learning problems that are cast as convex and non convex optimization problems that show how our algorithm compares well in practice with state of the art optimization algorithms

approximation by quadrilateral finite elements we consider the approximation properties of finite element spaces on quadrilateral meshes the finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element it is known that for affine isomorphisms a necessary and sufficient condition for approximation of order r 1 in l p and order r in w 1 p is that the given space of functions on the reference element contain all polynomial functions of total degree at most r in the case of bilinear isomorphisms it is known that the same estimates hold if the function space contains all polynomial functions of separate degree r we show by means of a counterexample that this latter condition is also necessary as applications we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements

error analysis of piecewise constant pressure approximations of darcy s law abstract in this paper we present a short discussion on some finite element formulations for linear elliptic problems for the sake of simplicity we consider the poisson equation p f taking the notation from darcy s law among the zillions of such methods we shall concentrate our attention on fem leading to a final system of linear algebraic equations mp f where each unknown pi represents the constant value of the approximated pressure ph in a single element it is indeed well known that for some applications there is a certain demand for these types of schemes

invariance control for a class of cascade nonlinear systems we consider the control of partially linear cascade systems using switching control of the states of the linear subsystem we give sufficient conditions under which feedback of the linear states with switching gains guarantees both exponential stability of the linear subsystem and positive invariance of a prespecified region in state space we refer to a control scheme incorporating these two objectives as invariance control semiglobal asymptotic stabilization follows under some additional conditions the key idea of our design is to keep a given state space region positively invariant by switching on the boundary of the region thus the transient response of the system can be kept within prescribed bounds which is important in many practical applications our approach can also be viewed as an alternative to high gain designs the results in this paper can be extended to nonlinear cascades and even to noncascaded systems

a faster cutting plane algorithm with accelerated line search for linear svm faster cutting plane algorithms with accelerated line search are proposed to solve linear svm it proposes a novel linear time line search solver while the existing strategy spends o mlogm time an optimized explicit piecewise linear function finding algorithm for multiclass linear svm is derived it can be proved to reduce the total svm training time experiments demonstrate the effectiveness of the proposed algorithm cutting plane algorithm cpa is a generalization of iterative first order gradient method in which the objective function is approximated successively by supporting hyperplanes cpa has been tailored to solve regularized loss minimization in machine learning by exploiting the regularization structure in particular for linear support vector machine svm embedding a line search procedure effectively remedies the fluctuations of function value and speeds up the convergence in practical issue however the existing line search strategy based on sorting algorithm takes o mlogm time in this paper we propose a more effective line search solver which spends only linear time it can be extended to multiclass svm in which an optimized explicit piecewise linear function finding algorithm is prearranged the total svm training time is proved to reduce theoretically and experiments consistently confirm the effectiveness of the proposed algorithms

min max control of constrained uncertain discrete time linear systems for discrete time uncertain linear systems with constraints on inputs and states we develop an approach to determine state feedback controllers based on a min max control formulation robustness is achieved against additive norm bounded input disturbances and or polyhedral parametric uncertainties in the state space matrices we show that the finite horizon robust optimal control law is a continuous piecewise affine function of the state vector and can be calculated by solving a sequence of multiparametric linear programs when the optimal control law is implemented in a receding horizon scheme only a piecewise affine function needs to be evaluated on line at each time step the technique computes the robust optimal feedback controller for a rather general class of systems with modest computational effort without needing to resort to gridding of the state space

double families of quasi positive darcy flux approximations with highly anisotropic tensors on structured and unstructured grids this paper focuses on flux continuous pressure equation approximation for strongly anisotropic media previous work on families of flux continuous schemes for solving the general geometry permeability tensor pressure equation has focused on single parameter families these schemes have been shown to remove the o 1 errors introduced by standard two point flux reservoir simulation schemes when applied to full tensor flow approximation improved convergence of the schemes has also been established for specific quadrature points however these schemes have conditional m matrices depending on the strength of the off diagonal tensor coefficients when applied to cases involving full tensors arising from strongly anisotropic media the point wise continuous schemes can fail to satisfy the maximum principle and induce severe spurious oscillations in the numerical pressure solution new double family flux continuous locally conservative schemes are presented for the general geometry permeability tensor pressure equation the new double family formulation is shown to expand on the current single parameter range of existing conditional m matrix schemes the conditional m matrix bounds on a double family formulation are identified for both quadrilateral and triangle cell grids a quasi positive qm matrix analysis is presented that classifies the behaviour of the new schemes with respect to double family quadrature in regions beyond the m matrix bounds the extension to double family quadrature is shown to be beneficial resulting in novel optimal anisotropic quadrature schemes the new methods are applied to strongly anisotropic full tensor field problems and yield results with sharp resolution with only minor or practically zero spurious oscillations

nonquadratic lyapunov functions for robust control abstract this paper deals with the problem of controlling a linear continuous time system with structured time varying parameter uncertainties and input disturbances with a lyapunov function approach in contrast with most of the previous results in the literature we do not confine our attention to the class of quadratic lyapunov functions conversely the basic motivation of this paper is to determine whether there exist other functions that can be conveniently used as candidate lyapunov functions this question has a positive answer the proposed class is that of polyhedral norms or more generally of polyhedral minkowski functionals we show that the class of these functions is universal in the sense that if the problem of ultimately bounding the state in an assigned convex set via state feedback control can be solved via a lyapunov function and a continuous state feedback compensator then it can be solved via a polyhedral lyapunov function and a possibly different continuous control moreover we show that the control can be piecewise linear a numerical technique for constructing the controller is presented for the case in which the uncertainty constraint sets are polyhedral

adaptive subgradient methods for online learning and stochastic optimization we present a new family of subgradient methods that dynamically incorporate knowledge of the geometry of the data observed in earlier iterations to perform more informative gradient based learning metaphorically the adaptation allows us to find needles in haystacks in the form of very predictive but rarely seen features our paradigm stems from recent advances in stochastic optimization and online learning which employ proximal functions to control the gradient steps of the algorithm we describe and analyze an apparatus for adaptively modifying the proximal function which significantly simplifies setting a learning rate and results in regret guarantees that are provably as good as the best proximal function that can be chosen in hindsight we give several efficient algorithms for empirical risk minimization problems with common and important regularization functions and domain constraints we experimentally study our theoretical analysis and show that adaptive subgradient methods outperform state of the art yet non adaptive subgradient algorithms

output regulation of linear plants subject to constraints output regulation problems for continuous time linear systems with state and or input constraints are studied the problems are formulated in global and semi global setting by using state or full information feedback the goal of this paper is to develop solvability conditions for the posed problems moreover appropriate regulators are constructed under the solvability conditions to state the solvability conditions clearly a taxonomy of constraints is introduced which delineates the constraints into several categories such a taxonomy of constraints provides a classification of linear plants with constraints and identifies what types of output regulation problems are solvable results developed here include as a special case the results obtained in the literature for systems with only input constraints the constraint taxonomy also identifies some intrinsically hard constraints non right invertible constraints for which the solvability conditions of global semi global output regulation problems are not clear yet as a special case of output regulation we also consider tracking problems with constraints it is shown that if there exists a state feedback controller with a stabilizing domain of attraction then one can find a regulator with a tracking domain of attraction arbitrarily close to the stabilizing domain