mach reflection for the two dimensional burgers equation abstract we study shock reflection for the two 2d burgers equation this model equation is an asymptotic limit of the euler equations and retains many of the features of the full equations a von neumann type analysis shows that the 2d burgers equation has detachment sonic and crocco points in complete analogy with gas dynamics numerical solutions support the detachment sonic criterion for transition from regular to mach reflection there is also strong numerical evidence that the reflected shock in the 2d burgers mach reflection forms a smooth wave near the mach stem as proposed by colella and henderson in their study of the euler equations

diffraction of a weak shock by a wedge we match formal asymptotic expansions with differently scaled variables to obtain a uniform approximation to the similarity solution of the shock wedge diffraction problem

optimal preconditioners for systems defined by functions of toeplitz matrices abstract we propose several circulant preconditioners for systems defined by some functions g of toeplitz matrices a n in this paper we are interested in solving g a n x b by the preconditioned conjugate method or the preconditioned minimal residual method namely in the cases when g z are the functions e z sin z and cos z numerical results are given to show the effectiveness of the proposed preconditioners

clot norm minimization for continuous hands off control abstract in this paper we propose optimal control that is both sparse and continuous unlike previously proposed alternatives to maximum hands off control the maximum hands off control is the l 0 optimal or sparsest control among all feasible controls that are bounded by a specified value and transfer the state from a given initial state to the origin within a fixed time duration the proposed control is obtained via minimization of the clot combined l one and two norm of the control input along with the constraints on the state variable the constraints on the state variable ensures that the states are not blown up while achieving the optimal control by using the non smooth maximum principle we prove that the clot norm optimal control is unique and it is continuous in the time variable we show by numerical simulations that the clot control is continuous unlike l 0 optimal control or maximum hands off control and much sparser i e has longer time duration on which the control equals 0 than the conventional en elastic net control which is a convex combination of l 1 and squared l 2 norms

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

preconditioning of active set newton methods for pde constrained optimal control problems we address the problem of preconditioning a sequence of saddle point linear systems arising in the solution of pde constrained optimal control problems via active set newton methods with control and regularized state constraints we present two new preconditioners based on a full block matrix factorization of the schur complement of the jacobian matrices where the active set blocks are merged into the constraint blocks we discuss the robustness of the new preconditioners with respect to the parameters of the continuous and discrete problems numerical experiments on three dimensional problems are presented including comparisons with existing approaches based on preconditioned conjugate gradients in a nonstandard inner product

second order and stability analysis for state constrained elliptic optimal control problems with sparse controls an optimal control problem for a semilinear elliptic partial differential equation is discussed subject to pointwise control constraints on the control and the state the main novelty of the paper is the presence of the l 1 norm of the control as part of the objective functional that eventually leads to sparsity of the optimal control functions second order sufficient optimality conditions are analyzed they are applied to show the convergence of optimal solutions for vanishing l 2 regularization parameter for the control the associated convergence rate is estimated

the split levinson algorithm the classical levinson algorithm for computing the predictor polynomial relative to a real positive definite toeplitz matrix is shown to be redundant in complexity it can be broken down into two simpler algorithms either of which needs only to be processed this result can be interpreted in the framework of the theory of orthogonal polynomials on the real line as follows the symmetric and antisymmetric parts of the predictors relative to the sequence of toeplitz matrices constitute two families of polynomials orthogonal on the interval 1 1 with respect to some even weight functions it turns out that the recurrence relations for these orthogonal polynomials can be used efficiently to compute the desired predictor the resulting split levinson algorithm requires roughly one half the number of multiplications and the same number of additions as the classical levinson algorithm a simple derivation of cybenko s method for computing the pisarenko frequencies is obtained from the recurrence relations underlying the split levinson algorithm

parabolic control problems in measure spaces with sparse solutions optimal control problems in measure spaces lead to controls that have small support which is desirable e g in the context of optimal actuator placement for problems governed by parabolic partial differential equations well posedness is guaranteed in the space of square integrable measure valued functions which leads to controls with a spatial sparsity structure a conforming approximation framework allows one to derive numerically accessible optimality conditions as well as convergence rates in particular although the state is discretized the control problem can still be formulated and solved in the measure space numerical examples illustrate the structural features of the optimal controls

a note on the group inverse of some 2 2 block matrices over skew fields abstract let k be a skew field and km n be the set of all m n matrices over k in this paper we give the necessary and sufficient conditions to the existence and the representations of the group inverse for block matrix ax yb a b 0 under some conditions where a y km n x b kn m

an efficient and robust numerical algorithm for estimating parameters in turing systems we present a new algorithm for estimating parameters in reaction diffusion systems that display pattern formation via the mechanism of diffusion driven instability a modified discrete optimal control algorithm mdoca is illustrated with the schnakenberg and gierer meinhardt reaction diffusion systems using pde constrained optimization techniques the mdoca algorithm is a modification of a standard variable step gradient algorithm that yields a huge saving in computational cost the results of numerical experiments demonstrate that the algorithm accurately estimated key parameters associated with stationary target functions generated from the models themselves furthermore the robustness of the algorithm was verified by performing experiments with target functions perturbed with various levels of additive noise the mdoca algorithm could have important applications in the mathematical modeling of realistic turing systems when experimental data are available

approximation of sparse controls in semilinear equations by piecewise linear functions semilinear elliptic optimal control problems involving the l 1 norm of the control in the objective are considered a priori finite element error estimates for piecewise linear discretizations for the control and the state are proved these are obtained by a new technique based on an appropriate discretization of the objective function numerical experiments confirm the convergence rates

a spatial domain decomposition method for parabolic optimal control problems we present a non overlapping spatial domain decomposition method for the solution of linear quadratic parabolic optimal control problems the spatial domain is decomposed into non overlapping subdomains the original parabolic optimal control problem is decomposed into smaller problems posed on space time cylinder subdomains with auxiliary state and adjoint variables imposed as dirichlet boundary conditions on the space time interface boundary the subdomain problems are coupled through robin transmission conditions this leads to a schur complement equation in which the unknowns are the auxiliary state adjoint variables on the space time interface boundary the schur complement operator is the sum of space time subdomain schur complement operators the application of these subdomain schur complement operators is equivalent to the solution of an subdomain parabolic optimal control problem the subdomain schur complement operators are shown to be invertible and the application of their inverses is equivalent to the solution of a related subdomain parabolic optimal control problem we introduce a new family of neumann neumann type preconditioners for the schur complement system including several different coarse grid corrections we compare the numerical performance of our preconditioners with an alternative approach recently introduced by benamou

on practical conditions for the existence and uniqueness of solutions to the general equality quadratic programming problem we present practical conditions under which the existence and uniqueness of a finite solution to a given equality quadratic program may be examined different sets of conditions allow this examination to take place when null space range space or lagrangian methods are used to find stationary points for the quadratic program

potential theory for regular and mach reflection of a shock at a wedge if a plane shock hits a wedge a self similar pattern of reflected shocks travels outward as the shock moves forward in time the nature of the pattern is explored for weak incident shocks strength b and small wedge angles 2 w through potential theory a number of different scalings some study of mixed equations and matching asymptotics for the different scalings the self similar equations are of mixed type a linearization gives a linear mixed flow valid away from a sonic curve near the sonic curve a shock solution is constructed in another scaling except near the zone of interaction between the incident shock and the wall where a special scaling is used the parameter c1 2w 1 b ranges from 0 to here is the polytropic constant and c1 is the sound speed behind the incident shock for 2 regular reflection weak or strong can occur and the whole pattern is reconstructed to lowest order in shock strength for 1 2 mach reflection occurs and the flow behind the reflection is subsonic and can be constructed in principle with an open elliptic problem and matched the case 0 can be solved for 1 2 2 or even larger the flow behind a mach reflection may be transonic and further investigation must be made to determine what happens the basic pattern of reflection is an almost semi circular shock issuing for regular reflection from the reflection point on the wedge and for mach reflection matched with a local interaction flow assuming their nature choosing the least entropy generation the weak regular reflection will occur for sufficiently large von neumann paradox an accumulation point of vorticity occurs on the wedge above the leading point 1994 john wiley sons inc

eigenvalue estimates of an indefinite block triangular preconditioner for saddle point problems in this paper an indefinite block triangular preconditioner for symmetric saddle point problems is discussed the bounds for all the real eigenvalues of the preconditioned matrix are provided and the bounds for the real and imaginary parts of all the complex eigenvalues of the preconditioned matrix are also provided the corresponding theoretical results of paper by bergamaschi 2012 are improved numerical experiments are reported to confirm our theoretical results

high order essentially nonsocillatory schemes for hamilton jacobi equations hamilton jacobi h j equations are frequently encountered in applications e g in control theory and differential games h j equations are closely related to hyperbolic conservation laws in one space dimension the former is simply the integrated version of the latter similarity also exists for the multidimensional case and this is helpful in the design of difference approximations in this paper high order essentially nonoscillatory eno schemes for h j equations are investigated which yield uniform high order accuracy in smooth regions and sharply resolve discontinuities in the derivatives the eno scheme construction procedure is adapted from that for hyperbolic conservation laws the schemes are numerically tested on a variety of one dimensional and two dimensional problems including a problem related to control optimization and high order accuracy in smooth regions good resolution of discontinuities in the derivatives and convergence to viscosity solutions are observed

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

directional sparsity in optimal control of partial differential equations we study optimal control problems in which controls with certain sparsity patterns are preferred for time dependent problems the approach can be used to find locations for control devices that allow controlling the system in an optimal way over the entire time interval the approach uses a nondifferentiable cost functional to implement the sparsity requirements additionally bound constraints for the optimal controls can be included we study the resulting problem in appropriate function spaces and present two solution methods of newton type based on different formulations of the optimality system using elliptic and parabolic test problems we research the sparsity properties of the optimal controls and analyze the behavior of the proposed solution algorithms

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

domain decomposition methods for advection dominated linear quadratic elliptic optimal control problems abstract we present an optimization level domain decomposition dd preconditioner for the solution of advection dominated elliptic linear quadratic optimal control problems which arise in many science and engineering applications the dd preconditioner is based on a decomposition of the optimality conditions for the elliptic linear quadratic optimal control problem into smaller subdomain optimality conditions with dirichlet boundary conditions for the states and the adjoints on the subdomain interfaces these subdomain optimality conditions are coupled through robin transmission conditions for the states and the adjoints the parameters in the robin transmission condition depend on the advection this decomposition leads to a schur complement system in which the unknowns are the state and adjoint variables on the subdomain interfaces the schur complement operator is the sum of subdomain schur complement operators the application of which is shown to correspond to the solution of subdomain optimal control problems which are essentially smaller copies of the original optimal control problem we show that under suitable conditions the application of the inverse of the subdomain schur complement operators requires the solution of a subdomain elliptic linear quadratic optimal control problem with robin boundary conditions for the state numerical tests for problems with distributed and with boundary control show that the dependence of the preconditioners on mesh size and subdomain size is comparable to its counterpart applied to a single advection dominated equation these tests also show that the preconditioners are insensitive to the size of the control regularization parameter

a note on preconditioning nonsymmetric matrices the preconditioners for indefinite matrices of kkt form in m f murphy g h golub and a j wathen siam j sci comput 21 2000 pp 1969 1972 are extended to general nonsymmetric matrices

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

schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions domain decomposition methods provide powerful preconditioners for the iterative solution of the large systems of algebraic equations that arise in finite element or finite difference approximations

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

a schwarz domain decomposition method with gradient projection for optimal control governed by elliptic partial differential equations a b s t r a c t a domain decomposition method ddm is presented to solve the distributed optimal control problem the optimal control problem essentially couples an elliptic partial differential equation with respect to the state variable and a variational inequality with respect to the constrained control variable the proposed algorithm called sa gp algorithm consists of two iterative stages in the inner loops the schwarz alternating method sa is applied to solve the state and co state variables and in the outer loops the gradient projection algorithm gp is adopted to obtain the control variable convergence of iterations depends on both the outer and the inner loops which are coupled and affected by each other in the classical iteration algorithms a given tolerance would be reached after sufficiently many iteration steps but more iterations lead to huge computational cost for solving constrained optimal control problems most of the computational cost is used to solve pdes in this paper a proposed iterative number independent of the tolerance is used in the inner loops so as to save a lot of computational cost the convergence rate of l 2 error of control variable is derived also the analysis on how to choose the proposed iteration number in the inner loops is given some numerical experiments are performed to verify the theoretical results

preconditioning toeplitz plus diagonal linear systems using the sherman morrison woodbury formula in order to solve the toeplitz plus diagonal linear systems arising from image restorations efficiently we propose a sparse approximate inverse preconditioner based on the sherman morrison woodbury formula the preconditioner can be constructed through an incomplete factorization combined with some dropping strategies when the preconditioner is applied to the conjugate gradient method for solving the toeplitz plus diagonal linear systems numerical results show that our preconditioning method is more effective than other existing ones

on sparse optimal control for general linear systems in this paper we investigate an l 0 optimization problem with constraints in a form of volterra integral equation and the l infty norm in particular the equivalence theorem among the l p optimizations with p in 0 1 is derived which provides the following twofold extension of the existing results first these theoretical results enable us to solve sparse optimal control problems without imposing the finite dimensionality of the system to be controlled which was the crucial assumption for the derivation of the existing results second the relationship between the partial state constrained problem and output controllability is newly characterized

a class of quasilinear degenerate elliptic problems abstract we establish the existence of solutions for a class of quasilinear degenerate elliptic equations the equations in this class satisfy a structure condition which provides ellipticity in the interior of the domain and degeneracy only on the boundary equations of transonic gas dynamics for example satisfy this property in the region of subsonic flow and are degenerate across the sonic surface we prove that the solution is smooth in the interior of the domain but may exhibit singular behavior at the degenerate boundary the maximal rate of blow up at the degenerate boundary is bounded by the degree of degeneracy in the principal coefficients of the quasilinear elliptic operator our methods and results apply to the problems recently studied by several authors which include the unsteady transonic small disturbance equation the pressure gradient equations of the compressible euler equations and the singular quasilinear anisotropic elliptic problems and extend to the class of equations which satisfy the structure condition such as the shallow water equation compressible isentropic two dimensional euler equations and general two dimensional nonlinear wave equations our study provides a general framework to analyze degenerate elliptic problems arising in the self similar reduction of a broad class of two dimensional cauchy problems

representations of the drazin inverse for a class of block matrices abstract in this paper we give a formula for the drazin inverse of a block matrix f i i e 0 where e is square and i is the identity matrix further we get representations of the drazin inverse for matrices of the form m a b c 0 where a is square under some conditions expressed in terms of the individual blocks

optimal control of semilinear elliptic equations in measure spaces optimal control problems in measure spaces governed by semilinear elliptic equations are considered first order optimality conditions are derived and structural properties of their solutions in particular sparsity are discussed necessary and sufficient second order optimality conditions are obtained as well on the basis of the sufficient conditions stability of the solutions is analyzed highly nonlinear terms can be incorporated by utilizing an l infty omega regularity result for solutions of the first order necessary optimality conditions

efficient preconditioners for optimality systems arising in connection with inverse problems this paper is devoted to the numerical treatment of linear optimality systems os that arise in connection with inverse problems for partial differential equations if such inverse problems are regularized by tikhonov regularization then it follows from standard theory that the associated os is well posed provided that the regularization parameter alpha is positive and that the involved state equation satisfies suitable assumptions we explain and analyze how certain mapping properties of the operators appearing in the os can be employed to define efficient preconditioners for finite element fe approximations of such systems the key feature of the scheme is that the number of iterations needed to solve the preconditioned problem by the minimal residual method is bounded independently of the mesh parameter h used in the fe discretization and increases only moderately as alpha rightarrow0 more specifically if the stopping criterion for the iteration process is defined in terms of the associated energy norm then the number of iterations required in the severely ill posed case cannot grow faster than o ln alpha 2 our analysis is based on a careful study of the operators involved which yields the distribution of the eigenvalues of the preconditioned os finally the theoretical results are illuminated by a number of numerical experiments addressing both a model problem studied by borzi kunisch and kwak siam j control optim 41 2003 pp 1477 1497 and an inverse problem arising in connection with electrocardiography nielsen cai and lysaker math biosci 210 2007 pp 523 553

balancing neumann neumann methods for incompressible stokes equations this paper describes the unerlying mathematical model and the balancing neumann neumann methods are introduced and studied for incompressible stokes equations discretized with mixed finite or spectral elements with discontinuous pressures after decomposing the original domain of the problem into nonoverlapping subdomains the interior unknowns which are the interior velocity component and all except the constant pressure component of each subdomain problem are implicitly eliminated the resulting saddle point schur complement is solved with a krylov space method with a balancing neumann neumann preconditioner based on the solution of a coarse stokes problem with a few degrees of freedom per subdomain and on the solution of local stokes problems with natural neumann velocity and essential boundary conditions on the subdomains this preconditioner is of hybrid form in which the coarse problem is treated multiplicatively while the local problems are treated additively the condition number of the preconditioned operator is independent of the number of subdomains and is bounded from above by the product of the square of the logarithm of the local number of unknowns in each subdomain and the inverse of the inf sup constants of the discrete problem and of the coarse subproblem numerical results show that the method is quite fast they are also fully consistent with the theory

a domain decomposition method with coupled transmission conditions for the optimal control of systems governed by elliptic partial differential equations we prove the convergence of a nonoverlapping domain decomposition method with coupled transmission conditions applied to the optimal control of systems governed by elliptic partial differential equations pdes

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

approximate inverse circulant plus diagonal preconditioners for toeplitz plus diagonal matrices we consider the solutions of hermitian positive definite toeplitz plus diagonal systems t d x b where t is a toeplitz matrix and d is diagonal and positive however unlike the case of toeplitz systems no fast direct solvers have been developed for solving them in this paper we employ the preconditioned conjugate gradient method with approximate inverse circulant plus diagonal preconditioners to solving such systems the proposed preconditioner can be constructed and implemented efficiently using fast fourier transforms we show that if the entries of t decay away exponentially from the main diagonals the preconditioned conjugate gradient method applied to the preconditioned system converges very quickly numerical examples including spatial regularization for image deconvolution application are given to illustrate the effectiveness of the proposed preconditioner

spike controls for elliptic and parabolic pdes abstract we analyze the use of measures of the minimal norm to control elliptic and parabolic equations we prove the sparsity of the optimal control in the parabolic case we prove that the solution of the optimization problem is a borel measure supported in a set of lebesgue measure zero in both cases the approximate controllability can be achieved efficiently by means of controls that are activated in some finite number of pointwise locations we also analyze the corresponding dual problem

preconditioning pde constrained optimization with l1 sparsity and control constraints abstract pde constrained optimization aims at finding optimal setups for partial differential equations so that relevant quantities are minimized including nonsmooth l 1 sparsity promoting terms in the formulation of such problems results in more practically relevant computed controls but adds more challenges to the numerical solution of these problems the needed l 1 terms as well as additional inclusion of box control constraints require the use of semismooth newton methods we propose robust preconditioners for different formulations of the newton equation with the inclusion of a line search strategy and an inexact approach for the solution of the linear systems the resulting semismooth newton s method is reliable for practical problems our results are underpinned by a theoretical analysis of the preconditioned matrix numerical experiments illustrate the robustness of the proposed scheme

on partially inexact hss iteration methods for the complex symmetric linear systems in space fractional cnls equations the space fractional coupled nonlinear schrdinger cnls equations are discretized by an implicit conservative difference scheme with the fractional centered difference formula which is unconditionally stable the coefficient matrix of the discretized linear system is equal to the sum of a complex scaled identity matrix and a symmetric positive definite diagonal plus toeplitz matrix the hermitian and skew hermitian splitting hss method and the partially inexact hss pihss method are employed to solve the discretized linear system in the inner iteration processes of the hss method we only need to solve the linear sub systems associated with the hermitian part inexactly by the conjugate gradient cg method resulting in pihss iteration method theoretical analyses show that both hss and pihss methods are unconditionally convergent numerical examples are given to demonstrate the effectiveness of the hss iteration and the pihss iteration

a schur based algorithm for computing bounds to the smallest eigenvalue of a symmetric positive definite toeplitz matrix abstract recent progress in signal processing and estimation has generated c onsiderable interest in the problem of computing the smallest eigenvalue of symmetric positive definite toeplitz matrices several algorithms have been proposed in the literature they compute the smallest eigenvalue in an iterative fashion many of them relying on the levinson durbin solution of sequences of yule walker systems exploiting the properties of two algorithms recently developed for estimating a lower and an upper bound of the smallest singular value of upper triangular matrices respectively an algorithm for computing bounds to the smallest eigenvalue of a symmetric positive definite toeplitz matrix is derived the algorithm relies on the computation of the r factor of the qr factorization of the toeplitz matrix and the inverse of r the simultaneous computation of r and r 1 is efficiently accomplished by the generalized schur algorithm

computing the smallest eigenpair of a symmetric positive definite toeplitz matrix an algorithm for computing the smallest eigenvalue of a symmetric positive definite toeplitz matrix is presented the eigenvalue is approximated from below by newton s method applied to the characteristic polynomial of the matrix the newton s step is calculated by a levinson durbin type recursion simultaneously this recursion produces a realistic error bound of the actual approximation without additional computing effort as well as a simple and efficient way to compute the associated eigenvector

riemann problems for the two dimensional unsteady transonic small disturbance equation we study a two parameter family of riemann problems for the unsteady transonic small disturbance utsd equation also called the two dimensional burgers equation which is used to model the transition from regular to mach reflection for weak shock waves the related initial value problem consists of oblique shock data in the upper half plane with two parameters a and b corresponding to the slopes of the initial shock waves the study of quasi steady solutions leads to a problem that changes type when written in self similar coordinates the problem is hyperbolic in the region where the flow is supersonic and elliptic where the flow is subsonic in this paper we give a complete description of the flow in the hyperbolic region by resolving the hyperbolic wave interactions in the form of quasi one dimensional riemann problems in the region of physical space where the flow is subsonic we pose the related free boundary problems and discuss the behavior of the subsonic solution using results from our previo

inverse electrocardiographic source localization of ischemia an optimization framework and finite element solution with the goal of non invasively localizing cardiac ischemic disease using body surface potential recordings we attempted to reconstruct the transmembrane potential tmp throughout the myocardium with the bidomain heart model the task is an inverse source problem governed by partial differential equations pde our main contribution is solving the inverse problem within a pde constrained optimization framework that enables various physically based constraints in both equality and inequality forms we formulated the optimality conditions rigorously in the continuum before deriving finite element discretization thereby making the optimization independent of discretization choice such a formulation was derived for the l 2 norm tikhonov regularization and the total variation minimization the subsequent numerical optimization was fulfilled by a primal dual interior point method tailored to our problem s specific structure our simulations used realistic fiber included heart models consisting of up to 18 000 nodes much finer than any inverse models previously reported with synthetic ischemia data we localized ischemic regions with roughly a 10 false negative rate or a 20 false positive rate under conditions up to 5 input noise with ischemia data measured from animal experiments we reconstructed tmps with roughly 0 9 correlation with the ground truth while precisely estimating the tmp in general cases remains an open problem our study shows the feasibility of reconstructing tmp during the st interval as a means of ischemia localization

feti and neumann neumann iterative substructuring methods connections and new results the feti and neumann neumann families of algorithms are among the best know and most severely tested domain decomposition methods for elliptic partial differential equations they are iterative substructuring methods and have many algorithmic components in common but there are also differences the purpose of this paper is to further unify the theory for these two families of methods and to introduce a new family of feti algorithms bounds on the rate of convergence which are uniform with respect to the coefficients of a family of elliptic problems with heterogeneous coefficients are established for these new algorithms the theory for a variant of the neumann neumann algorithm is also redeveloped stressing similarities to that for the feti methods

computing the minimum eigenvalue of a symmetric positive definite toeplitz matrix by newton type methods a method for computing the smallest eigenvalue of a symmetric positive definite toeplitz matrix by application of newton s method to the characteristic polynomial has been recently introduced by mastronardi and boley siam j sci comput 20 1999 pp 1921 1927 though considerably slower than methods developed by the authors w mackens and h voss siam j matrix anal appl 18 1997 pp 521 534 w mackens and h voss linear algebra appl 275 276 1998 pp 401 415 h voss linear algebra appl 287 1999 pp 359 371 the new approach is conceptually much simpler in this paper we improve the performance of the new method substantially while keeping its simplicity

constraint interface preconditioning for the incompressible stokes equations we introduce a novel substructuring approach for solving the incompressible stokes equations for the case of enclosed flows we employ a simple distribution of the global pressure constraint to subdomains which allows for a natural decomposition into stokes subdomain problems with dirichlet data which are well posed and inf sup stable this approach yields a saddle point problem on the interface gamma involving an operator which is continuous and coercive on h 1 2 gamma and which is restricted to the interface trace space of functions satisfying the incompressibility constraint we derive the form of the constraints explicitly both for the continuous and for the discrete case this allows us to design directly a class of interface preconditioners of constraint type thus avoiding the need to formulate a coarse level problem our analysis indicates that the resulting solution method has performance independent of the mesh size while numerical results point to a mild dependence on the number of s

on solving block structured indefinite linear systems we consider 2 2 block indefinite linear systems whose 2 2 block is zero such systems arise in many applications we discuss two techniques that are based on modifying the 1 1 block in a way that makes the system easier to solve the main part of the paper focuses on an augmented lagrangian approach a technique that modifies the 1 1 block without changing the system size the choice of the parameter involved the spectrum of the linear system and its condition number are discussed and some analytical observations are provided a technique of deflating the 1 1 block is then introduced finally numerical experiments that validate the analysis are presented

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

a free boundary problem for a quasi linear degenerate elliptic equation regular reflection of weak shocks we prove the existence of a solution to the weak regular reflection problem for the unsteady transonic small disturbance utsd model for shock reflection by a wedge in weak regular reflection the state immediately behind the reflected shock is supersonic and constant the flow becomes subsonic further downstream the equation in self similar coordinates is degenerate at the sonic line the reflected shock becomes transonic and begins to curve there its position is the solution to a free boundary problem for the degenerate equation using the rankine hugoniot conditions along the reflected shock we derive an evolution equation for the transonic shock and an oblique derivative boundary condition at the unknown shock position by regularizing the degenerate problem we construct uniform bounds we apply local compactness arguments to extract a limit that solves the problem the solution is smooth in the interior and continuous up to the degenerate boundary this work completes a stage in our program to construct self similar solutions of two dimensional riemann problems in a series of papers we developed techniques for solving the degenerate elliptic equations that arise in self similar reductions of hyperbolic conservation laws in other papers especially in joint work with gary lieberman we developed techniques for solving free boundary problems of the type that arise from rankine hugoniot relations for the first time in this paper we combine these approaches and show that they are compatible although our construction is limited to a finite part of the unbounded subsonic region it suggests that this approach has the potential to solve a variety of problems in weak shock reflection including mach and von neumann reflection in the utsd equation and the analogous problems for the unsteady full potential equation 2002 john wiley sons inc

recurrence relations for the even and odd characteristic polynomials of a symmetric toeplitz matrix in a recent paper computation of the smallest even and odd eigenvalues of a symmetric positive definite toeplitz matrix siam j matrix anal appl 25 2004 949 963 melman proved a recurrence relation of the even and odd characteristic polynomials of a real symmetric toeplitz matrix t on which a symmetry exploiting method for computing the smallest eigenvalue of t can be based in this note we present a proof of the recurrence relation which is less technical and more transparent

preconditioning techniques for diagonal times toeplitz matrices in fractional diffusion equations the fractional diffusion equation is discretized by an implicit finite difference scheme with the shifted grunwald formula which is unconditionally stable the coefficient matrix of the discretized linear system is equal to the sum of a scaled identity matrix and two diagonal times toeplitz matrices standard circulant preconditioners may not work for such toeplitz like linear systems the main aim of this paper is to propose and develop approximate inverse preconditioners for such toeplitz like matrices an approximate inverse preconditioner is constructed to approximate the inverses of weighted toeplitz matrices by circulant matrices and then combine them together row by row because of toeplitz structure both the discretized coefficient matrix and the preconditioner can be implemented very efficiently by using fast fourier transforms theoretically we show that the spectra of the resulting preconditioned matrices are clustered around one thus krylov subspace methods with the proposed preconditione

preconditioned solution of state gradient constrained elliptic optimal control problems elliptic optimal control problems with pointwise state gradient constraints are considered a quadratic penalty approach is employed together with a semismooth newton iteration three different preconditioners are proposed and the ensuing spectral properties of the preconditioned linear newton saddle point systems are analyzed dependent on the penalty parameter a new bound for the smallest positive eigenvalue is proved since the analysis is carried out in function space it will ensure mesh independent convergence behavior of suitable krylov subspace methods such as minres also in discretized settings a path following strategy with a preconditioned inexact newton solver is implemented and numerical results are provided

newton picard based preconditioning for linear quadratic optimization problems with time periodic parabolic pde constraints we develop and investigate two preconditioners for a basic linear iterative splitting method for the numerical solution of linear quadratic optimization problems with time periodic parabolic pde constraints the resulting real valued linear system to be solved is symmetric indefinite we propose all at once symmetric indefinite preconditioners based on a newton picard approach which divides the variable space into slow and fast modes the division is performed either classically with eigenspace methods or with a novel two grid approach we prove mesh independent convergence for the classical newton picard preconditioner present a complexity analysis and show numerical results for the classical and the two grid preconditioners moreover the preconditioners compare favorably with existing symmetric positive definite schur complement preconditioners in a krylov method context

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

a two step even odd split levinson algorithm for toeplitz systems abstract we use a two step durbin method rather than the single step version in the even odd split levinson algorithm for strongly nonsingular real symmetric toeplitz systems with arbitrary right hand side thereby slightly reducing the complexity of this algorithm to 19 8 n 2 o n flops on a sequential machine we also present extensive numerical results comparing several levinson type methods

fast matrix splitting preconditioners for higher dimensional spatial fractional diffusion equations abstract the discretizations of two and three dimensional spatial fractional diffusion equations with the shifted finite difference formulas of the grunwald letnikov type can result in discrete linear systems whose coefficient matrices are of the form d t where d is a nonnegative diagonal matrix and t is a block toeplitz with toeplitz block matrix or a block toeplitz with each block being block toeplitz with toeplitz block matrix for these discrete spatial fractional diffusion matrices we construct diagonal and block circulant with circulant block splitting preconditioner for the two dimensional case and diagonal and block circulant with each block being block circulant with circulant block splitting preconditioner for the three dimensional case to further accelerate the convergence rates of krylov subspace iteration methods and we analyze the eigenvalue distributions for the corresponding preconditioned matrices theoretical results show that except for a small number of outliners the eigenvalues of the preconditioned matrices are located within a complex disk centered at 1 with the radius being exactly less than 1 and numerical experiments demonstrate that these structured preconditioners can significantly improve the convergence behavior of the krylov subspace iteration methods moreover this approach is superior to the geometric multigrid method and the preconditioned conjugate gradient methods incorporated with the approximate inverse circulant plus diagonal preconditioners in both iteration counts and computing times

measure valued directional sparsity for parabolic optimal control problems a directional sparsity framework allowing for measure valued controls in the spatial direction is proposed for parabolic optimal control problems it allows for controls which are localized in space where the spatial support is independent of time well posedness of the optimal control problems is established and the optimality system is derived it is used to establish structural properties of the minimizer an a priori error analysis for finite element discretization is obtained and numerical results illustrate the effects of the proposed cost functional and the convergence results

an optimization approach for well targeted transcranial direct current stimulation transcranial direct current stimulation tdcs is a noninvasive brain stimulation technique which modifies neural excitability by providing weak currents through scalp electrodes the aim of this study is to introduce and analyze a novel optimization method for safe and well targeted multiarray tdcs for optimization we consider an optimal control problem for a laplace equation with neumann boundary conditions with control and pointwise gradient state constraints we prove well posedness results for the proposed methods and provide computer simulation results in a highly realistic six compartment geometry adapted hexahedral head model for discretization of the proposed minimization problem the finite element method is employed and the existence of at least one minimizer to the discretized optimization problem is shown for numerical solution of the corresponding discretized problem we employ the alternating direction method of multipliers and comprehensively examine the cortical current flow field with regard to focality target intensity and orientation the numerical results reveal that the optimized current flow fields show significantly higher focality and in most cases higher directional agreement to the target vector in comparison to standard bipolar electrode montages

a sequential homotopy method for mathematical programming problems we propose a sequential homotopy method for the solution of mathematical programming problems formulated in abstract hilbert spaces under the guignard constraint qualification the method is equivalent to performing projected backward euler timestepping on a projected gradient antigradient flow of the augmented lagrangian the projected backward euler equations can be interpreted as the necessary optimality conditions of a primal dual proximal regularization of the original problem the regularized problems are always feasible satisfy a strong constraint qualification guaranteeing uniqueness of lagrange multipliers yield unique primal solutions provided that the stepsize is sufficiently small and can be solved by a continuation in the stepsize we show that equilibria of the projected gradient antigradient flow and critical points of the optimization problem are identical provide sufficient conditions for the existence of global flow solutions and show that critical points with emanating descent curves cannot be asymptotically stable equilibria of the projected gradient antigradient flow practically eradicating convergence to saddle points and maxima the sequential homotopy method can be used to globalize any locally convergent optimization method that can be used in a homotopy framework we demonstrate its efficiency for a class of highly nonlinear and badly conditioned control constrained elliptic optimal control problems with a semismooth newton approach for the regularized subproblems

improved schur complement preconditioners for block toeplitz systems with small size blocks in this paper we employ the preconditioned conjugate gradient method with the improved schur complement preconditioners for hermitian positive definite block toeplitz systems with small size blocks schur complement preconditioners have been proved to be an effective method for such block toeplitz systems ching et al 2007 the modification is based on taylor expansion approximation we prove that the matrices preconditioned by improved schur preconditioners have more clustered spectra compared to that of the schur complement preconditioners hence preconditioned conjugate gradient type methods will converge faster numerical examples are given to demonstrate the efficiency of the proposed method

on hss like iteration method for the space fractional coupled nonlinear schrodinger equations the implicit conservative difference scheme with the fractional centered difference formula which is unconditionally stable is employed to discretize the space fractional coupled nonlinear schrodinger equations the coefficient matrix of the discretized linear system is equal to the sum of a complex scaled identity matrix which can be written as the imaginary unit times the identity matrix and a symmetric diagonal plus toeplitz matrix in this paper the hss like iteration method is proposed to solve the discretized linear system theoretical analyses show that the hss like iteration method is unconditionally convergent numerical examples are presented to illustrate the effectiveness of the hss like iteration method

schwarz methods of neumann neumann type for three dimensional elliptic finite element problems several domain decomposition methods of neumann neumann type are considered for solving the large linear systems of algebraic equations that arise from discretizations of elliptic problems by finite elements we will only consider problems in three dimensions several new variants of the basic algorithm are introduced in a schwarz method framework that provides tools which have already proven very useful in the design and analysis of other domain decomposition and multi level methods the neumann neumann algorithms have several advantages over other domain decomposition methods the subregions which define the subproblems only share the boundary degrees of freedom with their neighbors the subregions can also be of quite arbitrary shape and many of the major components of the preconditioner can be constructed from subprograms available in standard finite element program libraries in its original form however the algorithm lacks a mechanism for global transportation of information and its performance therefore suffers when the number of subregions increases in the new variants of the algorithms considered in this paper the preconditioners include global components of low rank to overcome this difficulty bounds are established for the condition number of the iteration operator which are independent of the number of subregions and depend only polylogarithmically on the number of degrees of freedom of individual local subproblems results are also given for problems with arbitrarily large jumps in the coefficients across the interfaces separating the subregions 1995 john wiley sons inc

low rank solution to an optimization problem constrained by the navier stokes equations the numerical solution of pde constrained optimization problems subject to the nonstationary navier stokes equation is a challenging task while space time approaches often show favorable convergence properties they often suffer from storage problems here we propose to approximate the solution to the optimization problem in a low rank form which is similar to the model order reduction mor approach however in contrast to classical mor schemes we do not compress the full solution at the end of the algorithm but start our algorithm with low rank data and maintain this form throughout the iteration numerical experiments indicate that this approach reduces the computational costs by two orders of magnitude

a numerical study of riemann problems for the two dimensional unsteady transonic small disturbance equation we study a two parameter family of riemann problems for the unsteady transonic small disturbance utsd equation also called the two dimensional burgers equation the two parameters a and b which define oblique shock initial data correspond to the slopes of the initial shock waves in the upper half plane for each a and b the three constant states in the upper half plane satisfy the rankine hugoniot conditions across the shocks this leads to a two parameter family of oblique shock interaction problems in this paper we present a numerical study of global solution behavior for the values of a and b in a previously obtained bifurcation diagram our study supplements the related theoretical results and conjectures recently obtained by s canic and b l keyfitz we employ a high resolution numerical method which reveals fine solution structures our findings confirm theoretical results and conjectures about the solution patterns and deepen the understanding of the structure of several intricate wave in

a fast and stable preconditioned iterative method for optimal control problem of wave equations in this paper we develop a new central finite difference scheme in terms of both time and space for solving the first order necessary optimality systems that characterize the optimal control of wave equations the obtained new scheme is proved to be unconditionally convergent with a second order accuracy without the requirement of the courant friedrichs lewy condition on the corresponding grid ratio an efficient preconditioned iterative method is further developed for solving the discretized sparse linear system based on the relationship between the resultant matrix structure and the coupled pde optimality system numerical examples are presented to verify the theoretical analysis and to demonstrate the high efficiency of the proposed preconditioned iterative solver

symmetric schemes for computing the minimum eigenvalue of a symmetric toeplitz matrix abstract mackens and voss 8 9 presented two generalizations of a method of cybenko and van loan 4 for computing the smallest eigenvalue of a symmetric positive definite toeplitz matrix taking advantage of the symmetry or skew symmetry of the corresponding eigenvector both methods are improved considerably

fast iterative solution of elliptic control problems in wavelet discretization we investigate wavelet methods for the efficient numerical solution of a class of control problems constrained by a linear elliptic boundary value problem where the cost functional may contain fractional sobolev norms of the control and the state starting point is the formulation of the infinite dimensional control problem in terms of boundary adapted biorthogonal spline wavelets involving only l2 norms of wavelet expansion coefficients where different norms are realized by a diagonal scaling together with a riesz map and constraints in form of an l2 isomorphism the coupled system of equations resulting from optimization is solved by an inexact conjugate gradient cg method for the control which involves the approximate inversion of the primal and the adjoint operator using again cg iterations starting from a coarse discretization level we use nested iteration to solve the coupled system on successively finer uniform discretizations up to discretization error accuracy on each level the resulting inexact cg scheme is a fast solver it is of asymptotic optimal complexity in the sense that the overall computational effort to compute the solution up to discretization error on the finest grid is proportional to the number of unknowns on that grid a consequence of grid independent condition numbers of the linear operators in wavelet coordinates in the numerical examples we study the choice of different norms and the regularization parameter in the cost functional and their effect on the solution moreover for different situations the performance of the fully iterative inexact cg scheme is investigated confirming the theoretical results

improved block preconditioners for linear systems arising from half quadratic image restoration abstract in this paper the minimization problem with a half quadratic hq regularization for image restoration is studied for the structured linear system arising at each step of the newton method for solving the minimization problem we propose improved block preconditioners based on approximate inversion of the schur complement and matrix decomposition of the hessian matrix the approximate inverse of the schur complement is constructed by the taylor expansion we analyze the spectral properties of the preconditioned matrix and present eigenvalue bounds numerical results illustrate the efficiency of the proposed preconditioners

automatic preconditioning by limited memory quasi newton updating this paper proposes a preconditioner for the conjugate gradient method cg that is designed for solving systems of equations ax bi with different right hand side vectors or for solving a sequence of slowly varying systems ak x bk the preconditioner has the form of a limited memory quasi newton matrix and is generated using information from the cg iteration the automatic preconditioner does not require explicit knowledge of the coefficient matrix a and is therefore suitable for problems where only products of a times a vector can be computed numerical experiments indicate that the preconditioner has most to offer when these matrix vector products are expensive to compute and when low accuracy in the solution is required the effectiveness of the preconditioner is tested within a hessian free newton method for optimization and by solving certain linear systems arising in finite element models

circulant preconditioners for functions of hermitian toeplitz matrices abstract circulant preconditioners for function of matrices have been recently of interest in particular several authors proposed the use of the optimal circulant preconditioners as well as the superoptimal circulant preconditioners in this context and numerically illustrated that such preconditioners are effective for certain functions of toeplitz matrices motivated by their results we propose in this work the absolute value superoptimal circulant preconditioners and provide several theorems that analytically show the effectiveness of such circulant preconditioners for systems defined by functions of toeplitz matrices namely we show that the eigenvalues of the preconditioned matrices are clustered around 1 and rapid convergence of krylov subspace methods can therefore be expected moreover we show that our results can be extended to functions of block toeplitz matrices with toeplitz blocks provided that the optimal block circulant matrices with circulant blocks are used as preconditioners numerical examples are given to support our theoretical results

an improved empirical mode decomposition method based on the cubic trigonometric b spline interpolation algorithm abstract empirical mode decomposition emd is a new method presented recently for analyzing nonlinear and non stationary signals its basic idea is to decompose the signal into a series of complete orthogonal intrinsic mode functions imfs based on the local characteristics of the signal in time domain the key step of emd is to use the cubic spline interpolation to connect the maximum and minimum values of the signals into upper and lower envelopes respectively and then calculate the mean values of upper and lower envelopes based on the cubic trigonometric b spline interpolation algorithm a new improved method for emd is proposed named ctb emd in this paper in this method the interpolation curve is more flexible because of the adjustability of shape of the cubic trigonometric b splines curve thus the overshoot and undershoot problems in the cubic spline interpolation curve can be avoided and then the decomposition of the signal is more accurate and effect through numerical experiments we compare the effect of this method with other methods on decomposing simulation signals and real signals experimental results show that this method can decompose signals more effectively and accurately

preconditioned modified hermitian and skew hermitian splitting iteration methods for fractional nonlinear schrodinger equations abstract a variant of preconditioned modified hermitian and skew hermitian splitting pmhss iteration method is proposed for a class of toeplitz like complex linear equations arising from the space fractional coupled nonlinear schrodinger equations theoretical analysis show that the new method is convergent unconditionally moreover the choice of the theoretical optimal parameter is further studied in detail which minimize the upper bound of the spectral radius and relate to the extreme eigenvalues of the toeplitz matrix and the extreme absolute values of the diagonal matrix in addition the splitting can be used as a preconditioner for krylov subspaces method numerical experiments show that the proposed method is efficient and the optimal parameter is independent of the mesh size which is very useful in large sparse practical problem

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

proximal schemes for parabolic optimal control problems with sparsity promoting cost functionals abstractfast first order proximal methods that solve linear and bilinear parabolic optimal control problems with a sparsity cost functional are discussed weak convergence of these methods is proved and for benchmarking purposes the proposed inexact proximal schemes are compared to an inexact semi smooth newton method results of numerical experiments are presented to demonstrate the computational effectiveness of proximal schemes and to validate the theoretical estimates

an elliptic problem arising from the unsteady transonic small disturbance equation abstract we prove a theorem on existence of a weak solution of the dirichlet problem for a quasilinear elliptic equation with a degeneracy on one part of the boundary the degeneracy is of a type keldysh type associated with singular behavior blow up of a derivative at the boundary we define an associated operator which is continuous pseudo monotone and coercive and show that a weak solution displaying singular behavior at the boundary exists