multigrid methods for saddle point problems stokes and lame systems we develop new multigrid methods for a class of saddle point problems that include the stokes system in fluid flow and the lame system in linear elasticity as special cases the new smoothers in the multigrid methods involve optimal preconditioners for the discrete laplace operator we prove uniform convergence of the w w cycle algorithm in the energy norm and present numerical results for w w cycle and v v cycle algorithms

learning conditional preference networks conditional preference networks cp nets have recently emerged as a popular language capable of representing ordinal preference relations in a compact and structured manner in this paper we investigate the problem of learning cp nets in the well known model of exact identification with equivalence and membership queries the goal is to identify a target preference ordering with a binary valued cp net by interacting with the user through a small number of queries each example supplied by the user or the learner is a preference statement on a pair of outcomes in this model we show that acyclic cp nets are not learnable with equivalence queries alone even if the examples are restricted to swaps for which dominance testing takes linear time by contrast acyclic cp nets are what is called attribute efficiently learnable when both equivalence queries and membership queries are available we indeed provide a learning algorithm whose query complexity is linear in the description size of the target concept but only logarithmic in the total number of attributes interestingly similar properties are derived for tree structured cp nets in the presence of arbitrary examples our learning algorithms are shown to be quasi optimal by deriving lower bounds on the vc dimension of cp nets in a nutshell our results reveal that active queries are required for efficiently learning cp nets in large multi attribute domains

local error estimates for supg solutions of advection dominated elliptic linear quadratic optimal control problems we derive local error estimates for the discretization of optimal control problems governed by linear advection diffusion partial differential equations pdes using the streamline upwind petrov galerkin supg stabilized finite element method we show that if the supg method is used to solve optimization problems governed by an advection dominated pde the convergence properties of the supg method is substantially different from the convergence properties of the supg method applied for the solution of an advection dominated pde the reason is that the solution of the optimal control problem involves another advection dominated pde the so called adjoint equation whose advection field is just the negative of the advection of the governing pdes for the solution of the optimal control problem a coupled system involving both the original governing pde as well as the adjoint pde must be solved we show that in the presence of a boundary layer the local error between the solution of the supg discretized optimal control problem and the solution of the infinite dimensional problem is only of first order even if the error is computed locally in a region away from the boundary layer in the presence of interior layers we prove optimal convergence rates for the local error in a region away from the layer between the solution of the supg discretized optimal control problems and the solution of the infinite dimensional problem numerical examples are presented to illustrate some of the theoretical results

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

multigrid methods for saddle point problems darcy systems we design and analyze multigrid methods for the saddle point problems resulting from raviart thomas nedelec mixed finite element methods for the darcy system in porous media flow uniform convergence of the w cycle algorithm in a nonstandard energy norm is established extensions to general second order elliptic problems are also addressed

optimal control of the convection diffusion equation using stabilized finite element methods in this paper we analyze the discretization of optimal control problems governed by convection diffusion equations which are subject to pointwise control constraints we present a stabilization scheme which leads to improved approximate solutions even on corse meshes in the convection dominated case moreover the in general different approaches optimize then discretize and discretize then optimize coincide for the proposed discretization scheme this allows for a symmetric optimality system at the discrete level and optimal order of convergence

convergence analysis of all at once multigrid methods for elliptic control problems under partial elliptic regularity in this paper we consider the convergence theory for an all at once multigrid method for a distributed optimal control problem such an analysis has been recently done j schoberl r simon and w zulehner siam j numer anal 49 2011 pp 1482 1503 here we give a new proof which is based on a more straightforward approach the main benefit of this new approach is the possibility to extend the analysis to domains where full elliptic regularity i e h 2 regularity for the poisson problem cannot be guaranteed

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

a robust numerical method for a control problem involving singularly perturbed equations we consider an unconstrained linear quadratic optimal control problem governed by a singularly perturbed convection reaction diffusion equation we discretize the optimality system by using standard piecewise bilinear finite elements on the graded meshes introduced by duran and lombardi in du an and lombardi 2005 2006 we prove convergence of this scheme in addition when the state equation is a singularly perturbed reaction diffusion equation we derive quasi optimal a priori error estimates for the approximation error of the optimal variables on anisotropic meshes we present several numerical experiments when the state equation is both a reaction diffusion and a convection reaction diffusion equation these numerical experiments reveal a competitive performance of the proposed solution technique

the mortar finite element method with lagrange multipliers the present paper deals with a variant of a non conforming domain decomposition technique the mortar finite element method in the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices and the edges in 3d of subdomains it is shown that written under primal hybrid formulation the approximation problem issued from a discretization of a second order elliptic equation in 2d is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms finally the parallelization advantages consequence of this variant are also addressed

an hdg method for distributed control of convection diffusion pdes abstract we propose a hybridizable discontinuous galerkin hdg method to approximate the solution of a distributed optimal control problem governed by an elliptic linear convection diffusion pde we use degree k polynomials to approximate the state adjoint state their fluxes and the optimal control and we show the approximations converge with order k 1 in the l 2 norm finally we use a simple element by element postprocessing scheme to obtain new superconvergent approximations of the state dual state and the control we show the postprocessed variables converge with order k 2 in the l 2 norm we present 2d and 3d numerical experiments to illustrate our theoretical results

multigrid methods for saddle point problems oseen system abstract we develop and analyze multigrid methods for the oseen system in fluid flow we show that the w cycle algorithm is a uniform contraction if the number of smoothing steps is sufficiently large numerical results that illustrate the performance of the methods are also presented

exact learning boolean functions via the monotone theory we study the learnability of boolean functions from membership and equivalence queries we develop the monotone theory that proves 1 any boolean function is learnable in polynomial time in its minimal disjunctive normal form size its minimal conjunctive normal form size and the number of variables n in particular 2 decision trees are learnable our algorithms are in the model of exact learning with membership queries and unrestricted equivalence queries the hypotheses to the equivalence queries and the output hypotheses are depth 3 formulas

local error analysis of discontinuous galerkin methods for advection dominated elliptic linear quadratic optimal control problems this paper analyzes the local properties of the symmetric interior penalty upwind discontinuous galerkin sipg method for the numerical solution of optimal control problems governed by linear reaction advection diffusion equations with distributed controls the theoretical and numerical results presented in this paper show that for advection dominated problems the convergence properties of the sipg discretization can be superior to the convergence properties of stabilized finite element discretizations such as the streamline upwind petrov galerkin supg method for example we show that for a small diffusion parameter the sipg method is optimal in the interior of the domain this is in sharp contrast to supg discretizations for which it is known that the existence of boundary layers can pollute the numerical solution of optimal control problems everywhere even into domains where the solution is smooth and as a consequence in general reduces the convergence rates to only first order in order to prove

felicity a matlab c toolbox for developing finite element methods and simulation modeling this paper describes a matlab c finite element toolbox called felicity for simulating various types of systems of partial differential equations e g coupled elliptic parabolic problems usin

multigrid methods for saddle point problems optimality systems abstract we develop multigrid methods for an elliptic distributed optimal control problem on convex domains that are robust with respect to a regularization parameter we prove the uniform convergence of the w cycle algorithm and demonstrate the performance of v cycle and w cycle algorithms through numerical experiments

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

a priori and a posteriori error analysis of edge stabilization galerkin method for the optimal control problem governed by convection dominated diffusion equation in this paper we study an edge stabilization galerkin approximation scheme for the constrained optimal control problem governed by convection dominated diffusion equation the method uses least square stabilization of the gradient jumps across element edges a priori and a posteriori error estimates are derived for both the state co state and the control the theoretical results are illustrated by two numerical experiments

fuzzy soft set theory applied to bck bci algebras abstract in the paper p k maji r biswas a r roy fuzzy soft sets j fuzzy math 9 3 2001 589 602 maji et al introduced the concept of fuzzy soft sets as a generalization of the standard soft sets and presented an application of fuzzy soft sets in a decision making problem the aim of this manuscript is to apply fuzzy soft set for dealing with several kinds of theories in bck bci algebras the notions of fuzzy soft bck bci algebras closed fuzzy soft ideals and fuzzy soft p ideals are introduced and related properties are investigated

some observations on babu vs ka and brezzi theories summary some observations are made on abstract error estimates for galerkin approximations based on babuska brezzi conditions a basic error estimate due to babuska is sharpened by means of an identity that p i p for any nontrivial idempotent operator p some remarks are also made on the brezzi s theory for mixed variational problems and their galerkin approximations