canonical forms for linear differential algebraic equations with variable coefficients abstract we give a new set of local characterizing quantities for the treatment of linear differential algebraic equations with variable coefficients this leads to new global invariances under which we can find a generalization of the global or differentiation index and get a new existence and uniqueness theorem

multigrid methods for propagators in lattice gauge theories multigrid methods were invented for the solution of discretized partial differential equations in ordered systems the slowness of traditional algorithms is overcome by updates on various length scales in this overview article we discuss generalizations of multigrid methods for disordered systems in particular for propagators in lattice gauge theories a discretized nonabelian gauge theory can be formulated as a system of statistical mechanics where the gauge field degrees of freedom are su n matrices on the links of the lattice these su n matrices appear as random coefficients in dirac equations we aim at finding an efficient method by which one can solve dirac equations without critical slowing down if this could be achieved monte carlo simulations of quantum chromodynamics the theory of the strong interaction would be accelerated considerably in principle however the methods discussed can be used in arbitrary space time dimension and for arbitrary gauge group moreover there are applications in multigrid monte carlo simulations and for the definition of block spins and blocked gauge fields in monte carlo renormalization group studies as a central result it was found that the geometric multigrid method works in principle in arbitrarily disordered gauge fields finally an overview is given of other approaches to the propagator problem in lattice gauge theories

two grid analysis of minimal residual smoothing as a multigrid acceleration technique we analyze the two level method accelerated by a minimal residual smoothing mrs technique the two grid analysis is sufficient for our purpose because our mrs acceleration scheme is only applied on the finest level of the multigrid method we prove that the mrs acceleration scheme is a semi iterative method with respect to the underlying two level iteration and that the mrs accelerated two level method is a polynomial acceleration of first order we explain why mrs may not effectively accelerate standard multigrid method for solving poisson like problems the iteration matrices for the mrs accelerated coarse grid correction operator and the mrs accelerated two level operator are obtained we give bounds for the residual reduction rates of the accelerated two level method numerical experiments are employed to support the analytical results

acceleration methods for total variation based image denoising in this paper we apply a fixed point method to solve the total variation based image denoising problem an algebraic multigrid method is used to solve the corresponding linear equations krylov subspace acceleration is adopted to improve convergence in the fixed point iteration a good initial guess for this outer iteration at finest grid is obtained by combining fixed point iteration and geometric multigrid interpolation successively from the coarsest grid to the finest grid numerical experiments demonstrate that this method is efficient and robust even for images with large noise to signal ratios

building blocks for odd even multigrid with applications reduced to systems abstract building blocks yielding an efficient implementation of the odd even multigrid method for the poisson problem in the reference domain 0 1 d d 2 3 are described modifications needed to transform these techniques to solve reduced linear systems representing boundary value problems in arbitrary domains are given a new way to define enriched coarser subspaces in the multilevel realization is proposed numerical examples demonstrating the efficiency of developed multigrid methods are included

derivative and proportional state feedback for linear descriptor systems with variable coefficients abstract we study linear descriptor systems with rectangular variable coefficient matrices using local and global equivalence transformations we introduce normal and condensed forms and get sets of characteristic quantities these quantities allow us to decide whether a linear descriptor system with variable coefficients is regularizable by derivative and or proportional state feedback or not regularizable by feedback means for us that there exists a feedback which makes the closed loop system uniquely solvable for every consistent initial vector

on the computation of multi dimensional solution manifolds of parametrized equations a new algorithm is presented for computing vertices of a simplicial triangulation of thep dimensional solution manifold of a parametrized equationf x 0 wheref is a nonlinear mapping fromr n tor m p n m 1 an essential part of the method is a constructive algorithm for computing moving frames on the manifold that is of orthonormal bases of the tangent spaces that vary smoothly with their points of contact the triangulation algorithm uses these bases together with a chord form of the gauss newton process as corrector to compute the desired vertices the jacobian matrix of the mapping is not required at all the vertices but only at the centers of certain local triangulation patches several numerical examples show that the method is very efficient in computing triangulations even around singularities such as limit points and bifurcation points this opens up new possibilities for determining the form and special features of such solution manifolds

solvability conditions consistency and weak consistency for linear differential algebraic equations and time invariant singular systems the general case we present several solvability concepts for linear differential algebraic equations daes with constant coefficients on the positive time axis as well as for the associated singular systems and investigate under which conditions these concepts are met next we derive necessary and sufficient conditions for global consistency of initial conditions for the dae as well as for the system and generalize these conditions with respect to our concept of weak consistency our distributional approach enables us to generalize results in an earlier paper where singular systems are assumed to have a regular pencil in the sense of gantmacher in particular we establish that global weak consistency in the system sense is equivalent to impulse controllability

regularization of descriptor systems by output feedback conditions are given under which a descriptor or generalized state space system can be regularized by output feedback it is shown that under these conditions proportional and derivative output feedback controls can be constructed such that the closed loop system is regular and has index at most one this property ensures the solvability of the resulting system of dynamic algebraic equations a reduced form is given that allows the system properties as well as the feedback to be determined the construction procedures used to establish the theory are based only on orthogonal matrix decompositions and can therefore be implemented in a numerically stable way

acceleration and stabilization properties of minimal residual smoothing technique in multigrid we analyze the standard multigrid method accelerated by a minimal residual smoothing mrs technique we show that mrs can accelerate the convergence of the slow residual components thus accelerates the overall multigrid convergence we prove that under certain hypotheses mrs stabilizes the divergence of certain slow residual components and thus stabilizes the divergent multigrid iteration the analysis is customarily conducted on the two level method

minimal residual smoothing in multi level iterative method a minimal residual smoothing mrs technique is employed to accelerate the convergence of the multi level iterative method by smoothing the residuals of the original iterative sequence the sequence with smoothed residuals is re introduced into the multi level iterative process the new sequence generated by this acceleration procedure converges much faster than both the sequence generated by the original multi level method and the sequence generated by mrs technique the cost of this acceleration scheme is independent of the original operator and in many cases is negligible the emphasis of this paper is on the practical implementation of mrs acceleration techniques in the multi level method the discussions are focused on the two level method because the acceleration scheme is only applied on the finest level of the multi level method numerical experiments using the proposed mrs acceleration scheme to accelerate both the two level and multi level methods are conducted to show the efficiency and the cost effectiveness of this acceleration scheme

descriptor systems without controllability at infinity this paper concerns the structure that can be achieved by feedback in descriptor systems that lack controllability at infinity staircase and double staircase condensed forms obtained through a sequence of orthogonal state transformations display when and how feedback can be used to achieve minimal index furthermore they reveal that the modes that are uncontrollable at infinity have a fixed minimal index that cannot be reduced by feedback however this fixed higher index part of the control system is constrained to be zero in an appropriate coordinate system provided the initial conditions are consistent the remainder is a reduced order system that is controllable at infinity that can be made to have index one by feedback

numerical solution of hybrid systems of differential algebraic equations abstract we present a mathematical framework for general over and under determined hybrid switched systems of differential algebraic equations hdaes we give a systematic formulation of hdaes and discuss existence and uniqueness of solutions the treatment of the switch points and how to perform consistent initialization at switch points we show how numerical solution methods for daes can be adapted for hdaes and present a numerical results for these methods for the real world example of simulating an automatic gearbox

krylov subspace acceleration of nonlinear multigrid with application to recirculating flows this paper deals with the combination of two solution methods multigrid and gmres siam j sci comput 14 1993 pp 856 869 the generality and parallelizabili nonlinear pdes as the preconditioner for a nonlinear krylov subspace method we use the full approximation storage fas scheme math comp 31 1977 pp 333 390 a nonlinear multigrid method the nonlinear krylov acceleration is applied also on coarse grids so that recirculating incompressible flow problems discretized with a higher order upwind scheme can be solved efficiently

fourier analysis of gmres m preconditioned by multigrid this paper deals with convergence estimates of gmres m saad and schultz siam j sci statist comput 7 1986 pp 856 869 preconditioned by multigrid brandt math comp 31 1977 pp 333 390 hackbusch multi grid methods and applications springer berlin 1985 fourier analysis is a well known and useful tool in the multigrid community for the prediction of two grid convergence rates brandt math comp 31 1977 pp 333 390 stuben and trottenberg in multigrid methods lecture notes in math 960 k stuben and u trottenberg eds springer berlin pp 1 176 this analysis is generalized here to the situation in which multigrid is a preconditioner since it is possible to obtain the whole spectrum of the two grid iteration matrix a preconditioned krylov subspace acceleration method like gmres m implicitly builds up a minimal residual polynomial the determination of the polynomial coefficients is easily possible and can be done explicitly since from fourier analysis a simple block diagonal two grid iteration matrix results based on the gmres m polynomial sharp theoretical convergence estimates can be obtained which are compared with estimates based on the spectrum of the iteration matrix several numerical scalar test problems are computed in order to validate the theoretical predictions

multi level minimal residual smoothing a family of general purpose mutigrid acceleration techniques abstract we employ multi level minimal residual smoothing mrs as a pre optimization technique to accelerate standard multigrid convergence the mrs method is used to improve the current multigrid iterate by smoothing its corresponding residual before the latter is projected to the coarse grid we develop different schemes for implementing mrs technique on the finest grid and on the coarse grids and several versions of the inexact mrs technique numerical experiments are conducted to show the efficiency of the multi level and inexact mrs techniques

on recombining iterants in multigrid algorithms and problems with small islands this article deals with nearly singular problems and with problems having small islands or other small features not visible on a coarse grid in which cases usual multigrid solvers converge slowly or even diverge in these and many other problems the main difficulty is related to a few badly approximated components the numerical experiments reported below show that by recombining multigrid iterants such components are eliminated effectively and the usual multigrid asymptotic convergence rate is restored

a geometric multigrid solver for the free surface equation in environmental models featuring irregular coastlines a recently developed multigrid method botto 2013 based on the concept of volume fraction has been tested for the inversion of the helmholz type equation for the free surface in the environmental public domain code coherens the volume fraction concept is particularly interesting for coarse grid cells that are agglomerated from both dry and wet fine grid cells at irregular coastlines at these locations modifying the prolongation operator and the coarse grid discretization using the volume fraction results in better convergence however as convergence deteriorates in the case of small elongated islands that tend to disappear by the multigrid coarsening procedure a correction is proposed yielding good convergence rates irrespective of the presence of small or large islands the method is tested extensively for the inversion of the academic poisson equation larger test cases solving the helmholz type equation prove the applicability for real life applications of environmental flows

a nonlinear truck model and its treatment as a multibody system abstract a planar vertical truck model with nonlinear suspension and its multibody system formulation are presented the equations of motion of the model form a system of differential algebraic equations daes all equations are given explicitly including a complete set of parameter values consistent initial values and a sample road excitation thus the truck model allows various investigations of the specific dae effects and represents a test problem for algorithms in control theory mechanics of multibody systems and numerical analysis several numerical tests show the properties of the model

classical and generalized solutions of time dependent linear differential algebraic equations abstract a coordinate free reduction procedure is developed for linear time dependent differential algebraic equations that transforms their solutions into solutions of smaller systems of ordinary differential equations the procedure applies to classical as well as distribution solutions in the case of analytic coefficients the hypotheses required for the reduction not only are necessary for the validity of the existence and uniqueness results but even allow for the presence of singularities straightforward extensions including undetermined systems and systems with nonanalytic coefficients are also discussed

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

controllability and observability of linear time varying singular systems controllability and observability of analytically solvable linear time varying singular systems are considered our analysis includes the possible impulsive zero input response based on the definitions of controllability and observability naturally extended from the ones adopted in the time invariant setting sufficient and necessary conditions for these two properties are derived furthermore according to the specific definitions of these concepts used here the authors show that the dual relationship between controllability and observability does not hold

acceleration of five point red black gauss seidel in multigrid for poisson equation abstract a new relaxation analysis and two acceleration schemes are proposed for the five point red black gauss seidel smoothing in multigrid for solving a two dimensional poisson equation for a multigrid v cycle we discovered that underrelaxation is applicable to restriction half cycle and overrelaxation is applicable to interpolation half cycle numerical experiments using modified multigrid v cycle algorithms show that our simple acceleration schemes accelerate the convergence rate by as much as 34 with negligible cost this result is contrary to the existing belief that sor is not suitable for using as a smoother in multigrid for poisson equation because the gain in computational savings would not pay for the cost of implementing it more important is the idea of employing different parameters to accelerate the reduction of low and high frequency errors separately our discovery offers a new way for sor smoothing in multigrid

smooth factorizations of matrix valued functions and their derivatives in this paper we study the numerical factorization of matrix valued functions in order to apply them in the numerical solution of differential algebraic equations with time varying coefficients the main difficulty is to obtain smoothness of the factors and a numerically accessible form of their derivatives we show how this can be achieved without numerical differentiation if the derivative of the given matrix valued function is known these results are then applied in the numerical solution of differential algebraic riccati equations for this a numerical algorithm is given and its properties are demonstrated by a numerical example

technical communique impulse observability and impulse controllability of linear time varying singular systems impulse controllability and impulse observability of linear time varying singular systems in a standard canonical form are examined we focus on the fast subsystems with constant rank n matrices sufficient and necessary conditions are derived and presented directly in terms of system parameters conditions obtained for these two properties are algebraically dual to each other

a parallel multilevel technique for solving the bidomain equation on a human heart with purkinje fibers and a torso model in this paper we present a multigrid method and its implementation on parallel computers to solve the bidomain equation that appears in excitation propagation analysis of the human heart with the torso the bidomain equation is discretized with the finite element method on a composite mesh composed of a fine voxel mesh around the heart and a coarse voxel mesh covering the torso the extracellular potential problem on the torso is formulated as a variational problem with a constraint at the interface of the fine and coarse meshes we show that this formulation naturally satisfies the conservation property of the electric currents and fits into the multilevel adaptive solution technique framework we also present our special treatment of the purkinje fiber network in the multigrid algorithm where it is modeled as multiway branching lines connected to the nodes in the voxel mesh of the heart a parallel implementation of the proposed multigrid algorithm on distributed memory computers is presented and the performance is evaluated using real life applications

a new look at pencils of matrix valued functions abstract matrix pencils depending on a parameter and their canonical forms under equivalence are discussed the study of matrix pencils or generalized eigenvalue problems is often motivated by applications from linear differential algebraic equations daes based on the weierstrass kronecker canonical form of the underlying matrix pencil one gets existence and uniqueness results for linear constant coefficients daes in order to study the solution behavior of linear daes with variable coefficients one has to look at new types of equivalence transformations this then leads to new canonical forms and new invariances for pencils of matrix valued functions we give a survey of recent results for square pencils and extend these results to nonsquare pencils furthermore we partially extend the results for canonical forms of hermitian pencils and give new canonicalforms there too based on these results we obtain new existence and uniqueness theorems for differential algebraic systems which generalize the classical results of weierstrass and kronecker

regularization of linear descriptor systems with variable coefficients we study linear descriptor control systems with rectangular variable coefficient matrices we introduce condensed forms for such systems under equivalence transformations and use these forms to detect whether the system can be transformed to a uniquely solvable closed loop system via state or derivative feedback we show that under some mild assumptions every such system consists of an underlying square subsystem that behaves essentially like a standard state space system plus some solution components that are constrained to be zero

exact singular behavior of positive solutions to nonlinear elliptic equations with a hardy potential abstract in this paper we study the singular behavior at x 0 of positive solutions to the equation u x 2 u x u p x 0 where r n n 3 is a bounded domain with 0 and p 1 2 are given constants for the case n 2 2 4 the singular behavior of all the positive solutions is completely classified in the recent paper 5 here we determine the exact singular behavior of all the positive solutions for the remaining case n 2 2 4 in sharp contrast to the case n 2 2 4 where several converging blow up rates of u x are possible as x 0 we show that when n 2 2 4 every positive solution u x blows up in the same fashion lim x 0 x 2 p 1 u x 2 p 1 2 p 1 2 n 1 p 1

disturbance impulse controllability in descriptor system the descriptor system ex ax gu hv with external disturbance v is considered to characterize the possibility to cancel the impulses produced by external disturbance v through using control u the concept of disturbance impulse controllability is introduced

a simple multigrid scheme for solving the poisson equation with arbitrary domain boundaries we present a new multigrid scheme for solving the poisson equation with dirichlet boundary conditions on a cartesian grid with irregular domain boundaries this scheme was developed in the context of the adaptive mesh refinement amr schemes based on a graded octree data structure the poisson equation is solved on a level by level basis using a one way interface scheme in which boundary conditions are interpolated from the previous coarser level solution such a scheme is particularly well suited for self gravitating astrophysical flows requiring an adaptive time stepping strategy by constructing a multigrid hierarchy covering the active cells of each amr level we have designed a memory efficient algorithm that can benefit fully from the multigrid acceleration we present a simple method for capturing the boundary conditions across the multigrid hierarchy based on a second order accurate reconstruction of the boundaries of the multigrid levels in case of very complex boundaries small scale features become smaller than the discretization cell size of coarse multigrid levels and convergence problems arise we propose a simple solution to address these issues using our scheme the convergence rate usually depends on the grid size for complex grids but good linear convergence is maintained the proposed method was successfully implemented on distributed memory architectures in the ramses code for which we present and discuss convergence and accuracy properties as well as timing performances

the index of general nonlinear daes in the last few years there has been considerable research on differential algebraic equations daes f t y y 0 where f y is identically singular much of the mathematical effort has focused on computing a solution that is assumed to exist more recently there has been some discussion of solvability of daes there has historically been some imprecision in the use of the two key concepts of solvability and index for daes the index is also important in control and systems theory but with different terminology the consideration of increasingly complex nonlinear daes makes a clear and correct development necessary this paper will try to clarify several points concerning the index after establishing some new and more precise terminology that we need some inaccuracies in the literature will be corrected the two types of indices most frequently used the differentiation index and the perturbation index are defined with respect to solutions of unperturbed problems examples are given to show that these indices can be very different for the same problem we define new maximum indices which are the maxima of earlier indices in a neighborhood of the solution over a set of perturbations and show that these indices are simply related to each other these indices are also related to an index defined in terms of jacobians