counting characters of upper triangular groups abstract working over a finite field of order q let u n q be the group of upper triangular n n matrices with all diagonal entries equal to 1 it is known that all complex irreducible characters of u n q have degrees that are powers of q and that all powers q e occur for 0 e n where the upper bound n is defined by the formulas 2 m m m 1 and 2 m 1 m 2 it has been conjectured that the number of irreducible characters having degree q e is some polynomial in q with integer coefficients depending on n and e in this paper we construct explicit polynomials for the cases where e is one of n n 1 and 1 also a list of polynomials is presented that appear to give the correct character counts for n 9 several related results concerning pattern groups are also included

counting conjugacy classes for unitriangular matrices abstract let g n be the subgroup of gl n q consisting of upper unitriangular matrices a longstanding conjecture attributed to g higman states that the number of conjugacy classes of g n is given by a polynomial in q with integer coefficients recent results show that the resolution of this conjecture is linked to the classification of the canonical primitive connected matrices of g n in this paper we calculate the number of canonical primitive connected matrices a of g n satisfying that a the graph associated to a has two pivot lines and there are at most two edges connecting the pivot lines we prove that this number is a polynomial in q and it is connected with the combinatory of catalan numbers

multilevel algorithms for constrained compact fixed point problems in this paper the authors extend the multilevel algorithm of atkinson and brakhage for compact fixed point problems and the projected newton method of bertsekas to create a fast multilevel algorithm for parabolic boundary control problems having bound constraints on the control results are extended from finite dimension on constraint identification this approach permits both adaptive integration in time and inexact evaluation of the cost functional

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

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

applying gmres to the helmholtz equation with shifted laplacian preconditioning what is the largest shift for which wavenumber independent convergence is guaranteed there has been much recent research on preconditioning discretisations of the helmholtz operator delta k 2 k2 subject to suitable boundary conditions using a discrete version of the so called shifted laplacian delta k 2 mathrm i varepsilon k2 i for some varepsilon 0 0 this is motivated by the fact that as varepsilon increases the shifted problem becomes easier to solve iteratively despite many numerical investigations there has been no rigorous analysis of how to chose the shift in this paper we focus on the question of how large varepsilon can be so that the shifted problem provides a preconditioner that leads to k k independent convergence of gmres and our main result is a sufficient condition on varepsilon for this property to hold this result holds for finite element discretisations of both the interior impedance problem and the sound soft scattering problem with the radiation condition in the latter problem imposed as a far field impedance boundary condition note that we do not address the important question of how large varepsilon should be so that the preconditioner can easily be inverted by standard iterative methods

primal dual interior point methods for pde constrained optimization this paper provides a detailed analysis of a primal dual interior point method for pde constrained optimization considered are optimal control problems with control constraints in l p it is shown that the developed primal dual interior point method converges globally and locally superlinearly not only the easier l setting is analyzed but also a more involved l q analysis q is presented in l the set of feasible controls contains interior points and the frechet differentiability of the perturbed optimality system can be shown in the l q setting which is highly relevant for pde constrained optimization these nice properties are no longer available nevertheless a convergence analysis is developed using refined techniques in parti cular two norm techniques and a smoothing step are required the l q analysis with smoothing step yields global linear and local superlinear convergence whereas the l analysis without smoothing step yields only global linear convergence

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

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

relative springer isomorphisms let g be a simple algebraic group over the algebraically closed field k a slightly strengthened version of a theorem of t a springer says that under some mild restrictions on g and k there exists a g equivariant isomorphism of varieties u n where u denotes the unipotent variety of g and n denotes the nilpotent variety of g lieg such is called a springer isomorphism let b be a borel subgroup of g u the unipotent radical of b and u the lie algebra of u in this note we show that a springer isomorphism induces a b equivariant isomorphism u m u m where m is any unipotent normal subgroup of b and m liem we call such a map a relative springer isomorphism we also use relative springer isomorphisms to describe the geometry of u orbits in u

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

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

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

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

characters of the sylow p subgroups of the chevalley groups d4 pn abstract let u q be a sylow p subgroup of the chevalley groups d 4 q where q is a power of a prime p we describe a construction of all complex irreducible characters of u q and obtain a classification of these irreducible characters via the root subgroups which are contained in the center of these characters furthermore we show that the multiplicities of the degrees of these irreducible characters are given by polynomials in q 1 with nonnegative integer coefficients

conjugacy classes in unitriangular matrices let gn gn q be the group of the upper unitriangular matrices of size n n over fq the finite field with q pt elements higman has conjectured that for each n the number of conjugacy classes of elements of gn is a polynomial expression in q in this paper we develop the algorithm given in j algebra 177 1995 899 introducing new ideas and theoretical properties which lead us to get the conjugacy vector of gn for n 13 these vectors for n 5 resp for n 6 7 8 were given in j algebra 152 1 1992 1 resp j algebra 177 1995 899 in particular we conclude that for these values the number of conjugacy classes of gn is a polynomial in q with integral coefficients and degree n n 6 12 thus higman s conjecture holds for n 13 on the other hand for each positive integer n we find explicitly a set of q n n 6 12 different conjugacy classes

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

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

an algebraic multigrid method with guaranteed convergence rate we consider the iterative solution of large sparse symmetric positive definite linear systems we present an algebraic multigrid method which has a guaranteed convergence rate for the class of nonsingular symmetric m matrices with nonnegative row sum the coarsening is based on the aggregation of the unknowns a key ingredient is an algorithm that builds the aggregates while ensuring that the corresponding two grid convergence rate is bounded by a user defined parameter for a sensible choice of this parameter it is shown that the recursive use of the two grid procedure yields a convergence independent of the number of levels provided that one uses a proper amli cycle on the other hand the computational cost per iteration step is of optimal order if the mean aggregate size is large enough this cannot be guaranteed in all cases but is analytically shown to hold for the model poisson problem for more general problems a wide range of experiments suggests that there are no complexity issues and further demonstrates the robustness of the method the experiments are performed on systems obtained from low order finite difference or finite element discretizations of second order elliptic partial differential equations pdes the set includes two and three dimensional problems with both structured and unstructured grids some of them with local refinement and or reentering corner and possible jumps or anisotropies in the pde coefficients

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

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

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

solving elliptic control problems with interior point and sqp methods control and state constraints abstract we study optimal control problems for semilinear elliptic equations subject to control and state inequality constraints both boundary control and distributed control problems are considered with boundary conditions of dirichlet or neumann type by introducing suitable discretization schemes the control problem is transcribed into a nonlinear programming problem necessary conditions of optimality are discussed both for the continuous and the discretized control problem it is shown that the recently developed interior point method loqo of 35 is capable of solving these problems even for high discretizations four numerical examples with dirichlet and neumann boundary conditions are provided that illustrate the performance of the algorithm for different types of controls including bang bang controls

inexact central path following algorithms for optimal control problems a new approach to the numerical solution of optimal control problems including control and state constraints is presented like hybrid methods the approach aims at combining the advantages of direct and indirect methods unlike hybrid methods however our method is directly based on interior point concepts in function space realized via an adaptive multilevel scheme applied to the complementarity formulation and numerical continuation along the central path an adaptive stepsize control with respect to the duality gap parameter is worked out in the framework of affine invariant inexact newton methods finally the performance of our new type of algorithm is documented by a simple example within the range of our present theory and by the successful treatment of the well known intricate windshear problem outside this range

fast interior point solution of quadratic programming problems arising from pde constrained optimization interior point methods provide an attractive class of approaches for solving linear quadratic and nonlinear programming problems due to their excellent efficiency and wide applicability in this paper we consider pde constrained optimization problems with bound constraints on the state and control variables and their representation on the discrete level as quadratic programming problems to tackle complex problems and achieve high accuracy in the solution one is required to solve matrix systems of huge scale resulting from newton iteration and hence fast and robust methods for these systems are required we present preconditioned iterative techniques for solving a number of these problems using krylov subspace methods considering in what circumstances one may predict rapid convergence of the solvers in theory as well as the solutions observed from practical computations

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

rmcp relaxed mixed constraint preconditioners for saddle point linear systems arising in geomechanics abstract a major computational issue in the finite element fe integration of coupled consolidation equations is the repeated solution in time of the resulting discretized indefinite system because of ill conditioning the iterative solution which is recommended in large size 3d settings requires the computation of a suitable preconditioner to guarantee convergence in this paper the coupled system is solved by a krylov subspace method preconditioned by a relaxed mixed constraint preconditioner rmcp which is a generalization based on a parameter of the mixed constraint preconditioner mcp developed in 7 choice of optimal is driven by the spectral distribution of suitable symmetric positive definite spd matrices numerical tests performed on realistic 3d problems reveal that rmcp accelerates krylov subspace solvers by a factor up to three with respect to mcp

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

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

rational points on generalized flag varieties and unipotent conjugacy in finite groups of lie type let g be a connected reductive algebraic group defined over the finite field fq where q is a power of a good prime for g we write f for the frobenius morphism of g corresponding to the fq structure so that gf is a finite group of lie type let p be an f stable parabolic subgroup of g and u the unipotent radical of p in this paper we prove that the number of uf conjugacy classes in gf is given by a polynomial in q under the assumption that the centre of g is connected this answers a question of j alperin in 1 in order to prove the result mentioned above we consider for unipotent u gf the variety p0u of g conjugates of p whose unipotent radical contains u we prove that the number of fq rational points of p0u is given by a polynomial in q with integer coefficients moreover in case g is split over fq and u is split in the sense of 22 5 the coefficients of this polynomial are given by the betti numbers of p0u we also prove the analogous results for the variety pu consisting of conjugates of p that contain u

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

calculating conjugacy classes in sylow p subgroups of finite chevalley groups abstract in s m goodwin on the conjugacy classes in maximal unipotent subgroups of simple algebraic groups transform groups 11 1 2006 51 76 8 the first author outlined an algorithm for calculating a parametrization of the conjugacy classes in a sylow p subgroup u q of a finite chevalley group g q valid when q is a power of a good prime for g q in this paper we develop this algorithm and discuss an implementation in the computer algebra language gap using the resulting computer program we are able to calculate the parametrization of the conjugacy classes in u q when g q is of rank at most 6 in these cases we observe that the number of conjugacy classes of u q is given by a polynomial in q with integer coefficients

accuracy and convergence properties of the finite difference multigrid solution of an optimal control optimality system the finite difference multigrid solution of an optimal control problem associated with an elliptic equation is considered stability of the finite difference optimality system and optimal order error estimates in the discrete l2 norm and in the discrete h1 norm under minimum smoothness requirements on the exact solution are proved sharp convergence factor estimates of the two grid method for the optimality system are obtained by means of local fourier analysis a multigrid convergence theory is provided which guarantees convergence of the multigrid process towards weak solutions of the optimality system