the primal dual active set strategy as a semismooth newton method this paper addresses complementarity problems motivated by constrained optimal control problems it is shown that the primal dual active set strategy which is known to be extremely efficient for this class of problems and a specific semismooth newton method lead to identical algorithms the notion of slant differentiability is recalled and it is argued that the max function is slantly differentiable in lp spaces when appropriately combined with a two norm concept this leads to new local convergence results of the primal dual active set strategy global unconditional convergence results are obtained by means of appropriate merit functions

on schwarz type smoothers for saddle point problems in this paper we consider additive schwarz type iteration methods for saddle point problems as smoothers in a multigrid method each iteration step of the additive schwarz method requires the solutions of several small local saddle point problems this method can be viewed as an additive version of a multiplicative vanka type iteration well known as a smoother for multigrid methods in computational fluid dynamics it is shown that under suitable conditions the iteration can be interpreted as a symmetric inexact uzawa method in the case of symmetric saddle point problems the smoothing property an important part in a multigrid convergence proof is analyzed for symmetric inexact uzawa methods including the special case of the additive schwarz type iterations as an example the theory is applied to the crouzeix raviart mixed finite element for the stokes equations and some numerical experiments are presented

finite element approximation of a phase field model for void electromigration we consider a fully practical finite element approximation of the nonlinear degenerate parabolic system begin align gamma textstyle frac partial u partial t nabla b u nabla w alpha phi 0 quad w gamma delta u gamma 1 psi u nonumber quad nabla c u nabla phi 0 nonumber end align subject to an initial condition u 0 cdot in 1 1 on u and flux boundary conditions on all three equations here gamma in mathbb r 0 alpha in mathbb r geq 0 psi is a nonsmooth double well potential and c u 1 u b u 1 u 2 are degenerate coefficients diffusional mobility the degeneracy in b restricts u cdot cdot in 1 1 the above in the limit gamma rightarrow 0 models the evolution of voids by surface diffusion in an electrically conducting solid in addition to showing stability bounds for our approximation we prove convergence and hence existence of a solution to this nonlinear degenera

numerical analysis of the cahn hilliard equation with a logarithmic free energy a fully discrete finite element method for the cahn hilliard equation with a logarithmic free energy based on the backward euler method is analysed existence and uniqueness of the numerical solution and its convergence to the solution of the continuous problem are proved two iterative schemes to solve the resulting algebraic problem are proposed and some numerical results in one space dimension are presented

qmr a quasi minimal residual method for non hermitian linear systems the biconjugate gradient bcg method is the natural generalization of the classical conjugate gradient algorithm for hermitian positive definite matrices to general non hermitian linear systems unfortunately the original bcg algorithm is susceptible to possible breakdowns and numerical instabilities in this paper we present a novel bcg like approach the quasi minimal residual qmr method which overcomes the problems of bcg an implementation of qmr based on a look ahead version of the nonsymmetric lanczos algorithm is proposed it is shown how bcg iterates can be recovered stably from the qmr process some further properties of the qmr approach are given and an error bound is presented finally numerical experiments are reported

finite element approximation of the cahn hilliard equation with degenerate mobility we consider a fully practical finite element approximation of the cahn hilliard equation with degenerate mobility textstyle frac partial u partial t del b u del gamma lap u psi u where b cdot geq 0 is a diffusional mobility and psi cdot is a homogeneous free energy in addition to showing well posedness and stability bounds for our approximation we prove convergence in one space dimension furthermore an iterative scheme for solving the resulting nonlinear discrete system is analyzed we also discuss how our approximation has to be modified in order to be applicable to a logarithmic homogeneous free energy finally some numerical experiments are presented

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 multigrid method for the cahn hilliard equation with obstacle potential we present a multigrid finite element method for the deep quench obstacle cahn hilliard equation the non smooth nature of this highly nonlinear fourth order partial differential equation make this problem particularly challenging the method exhibits mesh independent convergence properties in practice for arbitrary time step sizes in addition numerical evidence shows that this behaviour extends to small values of the interfacial parameter c several numerical examples are given including comparisons with existing alternative solution methods for the cahn hilliard equation

convergence of the cahn hilliard equation to the hele shaw model we prove that level surfaces of solutions to the cahn hilliard equation tend to solutions of the hele shaw problem under the assumption that classical solutions of the latter exist the method is based on a new matched asymptotic expansion for solutions a spectral analysis for linearizd operators and an estimate for the difference between the true solutions and certain approximate ones

the kosterlitz thouless phase in a hierarchical model the author uses renormalisation group methods to study the long distance behaviour of hierarchical cos phi 2 lattice field theories in the kosterlitz thouless phase

computationally efficient approach for the minimization of volume constrained vector valued ginzburg landau energy functional the minimization of volume constrained vector valued ginzburg landau energy functional is considered in the present study it has many applications in computational science and engineering like the conservative phase separation in multiphase systems such as the spinodal decomposition phase coarsening in multiphase systems color image segmentation and optimal space partitioning a computationally efficient algorithm is presented to solve the space discretized form of the original optimization problem the algorithm is based on the constrained nonmonotone l 2 gradient flow of ginzburg landau functional followed by a regularization step which is resulted from the tikhonov regularization term added to the objective functional that lifts the solution from the l 2 function space into h 1 space the regularization step not only improves the convergence rate of the presented algorithm but also increases its stability bound the step size selection based on the barzilai borwein approach is adapted to improve the convergence rate of the introduced algorithm the success and performance of the presented approach is demonstrated throughout several numerical experiments to make it possible to reproduce the results presented in this work the matlab implementation of the presented algorithm is provided as the supplementary material developing an efficient algorithm to solve conservative gradient flow of multiphase ginzburg landau functional performing the constrained minimization of energy functional on the h 1 space using non monotone optimization path by applying the barzilai borwein step size selection approach the matlab implementation of the presented algorithms is given as supplementary material

upper bounds for coarsening for the deep quench obstacle problem the deep quench obstacle problem models phase separation at low temperatures during phase separation domains of high and low concentration are formed then coarsen or grow in average size of interest is the time dependence of the dominant length scales of the system relying on recent results by novick cohen and shishkov discrete contin dyn syst b 25 251 272 2009 we demonstrate upper bounds for coarsening for the deep quench obstacle problem with either constant or degenerate mobility for the case of constant mobility we obtain upper bounds of the form t1 3 at early times as well as at times t for which e t le frac 1 overline u 2 4 where e t denotes the free energy for the case of degenerate mobility we get upper bounds of the form t1 3 or t1 4 at early times depending on the value of e 0 as well as bounds of the form t1 4 whenever e t le frac 1 overline u 2 4

computation of matrix chain products part ii this paper considers the computation of matrix chain products of the form m 1 times m 2 times cdots times m n 1 if the matrices are of different dimensions the order in which the product is computed affects the number of operations an optimum order is an order which minimizes the total number of operations we present some theorems about an optimum order of computing the matrices based on these theorems an o n log n algorithm for finding an optimum order will be presented in part ii

numerical approximation of phase field based shape and topology optimization for fluids we consider the problem of finding optimal shapes of fluid domains the fluid obeys the navier stokes equations inside a holdall container we use a phase field approach using diffuse interfaces to describe the domain of free flow we formulate a corresponding optimization problem where flow outside the fluid domain is penalized the resulting formulation of the shape optimization problem is shown to be well posed hence there exists a minimizer and first order optimality conditions are derived for the numerical realization we introduce a mass conserving gradient flow and obtain a cahn hilliard type system which is integrated numerically using the finite element method an adaptive concept using reliable residual based error estimation is exploited for the resolution of the spatial mesh the overall concept is numerically investigated and comparison values are provided

on multi grid methods for variational inequalities we consider here a general class of algorithms for the numerical solution of variational inequalities a convergence proof is given and in particular a multi grid method is described numerical results are presented for the finite difference discretization of an obstacle problem for minimal surfaces

rigidity of the three dimensional hierarchical coulomb gas a random set of points in euclidean space is called rigid or hyperuniform if the number of points falling inside any given region has significantly smaller fluctuations than the corresponding number for a set of i i d random points this phenomenon has received considerable attention in recent years due to its appearance in random matrix theory the theory of coulomb gases and zeros of random analytic functions however most of the published results are in dimensions one and two this paper gives the first proof of hyperuniformity in a coulomb type system in dimension three known as the hierarchical coulomb gas this is a simplified version of the actual 3d coulomb gas the interaction potential in this model inspired by dyson s hierarchical model of the ising ferromagnet has a hierarchical structure and is locally an approximation of the coulomb potential hyperuniformity is proved at both macroscopic and microscopic scales with upper and lower bounds for the order of fluctuations that match up to logarithmic factors the fluctuations have cube root behavior in agreement with a well known prediction for the 3d coulomb gas for completeness analogous results are also proved for the 2d hierarchical coulomb gas and the 1d hierarchical log gas

dynamic programming and graph optimization problems abstract several classes of graph optimization problems which can be solved using dynamic programming are known to have more efficient tailor made algorithms this paper discusses four such classes and the underlying constraints on their subproblem interrelationships that yield these efficient algorithms these classes are also extended to handle more general cost functions

monotone multigrid methods for elliptic variational inequalities i we derive fast solvers for discrete elliptic variational inequalities of the first kind obstacle problems as resulting from the approximation of related continuous problems by piecewise linear finite elements using basic ideas of successive subspace correction we modify well known relaxation methods by extending the set of search directions extended underrelaxations are called monotone multigrid methods if they are quasioptimal in a certain sense by construction all monotone multigrid methods are globally convergent we take a closer look at two natural variants the standard monotone multigrid method and a truncated version for the considered model problems the asymptotic convergence rates resulting from the standard approach suffer from insufficient coarse grid transport while the truncated monotone multigrid method provides the same efficiency as in the unconstrained case

multilevel projection algorithm for solving obstacle problems abstract obstacle problems are nonlinear free boundary problems and the computation of approximate solutions can be difficult and expensive little work has been done on effective numerical methods of such problems this paper addresses some aspects of this issue discretizing the problem in a continuous piecewise linear finite element space gives a quadratic programming problem with inequality constraints a new method called the multilevel projection mp method is established in this paper the mp algorithm extends the multigrid method for linear equations to nonlinear obstacle problems the convergence theorems of this method are also proved a numerical example presented shows our error estimate is sharp and the mp algorithm is robust

triangulations tilings and certain block triangular matrices for a convex polygonp withn sides a partitioning ofp inton 2 nonoverlapping triangles each of whose vertices is a vertex ofp is called a triangulation or tiling and each triangle is a tile each tile has a given cost associated with it which may differ one from another this paper considers the problem of finding a tiling ofp such that the sum of the costs of the tiles used is a minimum and explores the curiosity that an abstract formulation of it can be cast as a linear program further the special structure of the linear program permits a recursive o n 3 algorithm

multigrid algorithms for variational inequalities we present an iterative scheme for obstacle problems described by variational inequalities at each step of the iteration the method leads to a reduced linear resp nonlinear algebraic system whi

preconditioning for allen cahn variational inequalities with non local constraints the solution of allen cahn variational inequalities with mass constraints is of interest in many applications this problem can be solved both in its scalar and vector valued form as a pde constrained optimization problem by means of a primal dual active set method at the heart of this method lies the solution of linear systems in saddle point form in this paper we propose the use of krylov subspace solvers and suitable preconditioners for the saddle point systems numerical results illustrate the competitiveness of this approach

an efficient parallel dynamic programming algorithm abstract antonio tsai and huang proposed a scheme in 1991 to parallelize the standard dynamic programming approach to solve combinatorial multistage problems however their dynamic programming approach is restricted to those multistage problems where the decision made at each stage depends only on decisions made in the stage immediately preceding it for many interesting problems the decision at each stage depends on the decisions made at all the previous stages and therefore their approach doesn t apply the matrix chain multiplication problem longest common subsequence problem and optimal polygon triangulation problem are some examples of such problems we also present techniques for parallelizing the dynamic programming solution to such problems the parallel algorithm we develop for a pram has complexity n employing n 2 processors since the traditional sequential algorithm for such problems is n 3 our parallel algorithm is an optimal parallel algorithm based on this traditional algorithm we also describe the results of our experiments that are in conformity with our theoretical complexity results we also compare and contrast our result with results obtained by earlier researchers and show that our parallel algorithm has optimal efficiency of 100 with respect to the traditional dynamic programming algorithm

front migration in the nonlinear cahn hilliard equation the method of matched asymptotic expansions is used to describe solutions of the nonlinear cahn hilliard equation for phase separation in n 1 space dimensions the expansion is formally valid when the thickness of internal transition layers is small compared with the distance separating layers and with their radii of curvature on the dominant slowest timescale the interface velocity is determined by the mean curvature of the interface by a non local relation which is identical to that in a well known quasi static model of solidification which exhibits a shape instability discovered by mullins sekerka j appl phys 34 323 329 1963 on a faster timescale the cahn hilliard equation regularizes a classic two phase stefan problem similarity solutions of the two phase stefan problem should describe boundary layers existence and uniqueness of such similarity solutions which admit metastable states is proved rigorously in an appendix

an octree multigrid method for quasi static maxwell s equations with highly discontinuous coefficients in this paper we develop an octree discretization for maxwell s equations in the quasi static regime we then use this discretization in order to develop a multigrid method for maxwell s equations with highly discontinuous coefficients we test our algorithms and compare it to other multilevel algorithms

inpainting of binary images using the cahn hilliard equation image inpainting is the filling in of missing or damaged regions of images using information from surrounding areas we outline here the use of a model for binary inpainting based on the cahn hilliard equation which allows for fast efficient inpainting of degraded text as well as super resolution of high contrast images

unconditionally energy stable time stepping scheme for cahn morral equation an unconditionally energy stable time stepping scheme is introduced to solve cahn morral like equations in the present study it is constructed based on the combination of david eyre s time stepping scheme and schur complement approach although the presented method is general and independent of the choice of homogeneous free energy density function term logarithmic and polynomial energy functions are specifically considered in this paper the method is applied to study the spinodal decomposition in multi component systems and optimal space tiling problems a penalization strategy is developed in the case of later problem to avoid trivial solutions extensive numerical experiments demonstrate the success and performance of the presented method according to the numerical results the method is convergent and energy stable independent of the choice of time stepsize its matlab implementation is included in the appendix for the numerical evaluation of algorithm and reproduction of the presented results extension of eyre s convex concave splitting scheme to multiphase systems efficient solution of spinodal decomposition in multi component systems efficient solution of least perimeter periodic space partitioning problem developing a penalization strategy to avoid trivial solutions presentation of matlab implementation of the introduced algorithm

preconditioning for vector valued cahn hilliard equations the solution of vector valued cahn hilliard systems is of interest in many applications we discuss strategies for the handling of smooth and nonsmooth potentials as well as for different types of constant mobilities concerning the nonsmooth systems the necessary bound constraints are incorporated via the moreau yosida regularization technique we develop effective preconditioners for the efficient solution of the linear systems in saddle point form numerical results illustrate the efficiency of our approach in particular we numerically show mesh and phase independence of the developed preconditioner in the smooth case the results in the nonsmooth case are also satisfying and the preconditioned version always outperforms the unpreconditioned one

a mesh independence result for semismooth newton methods for a class of semismooth operator equations a mesh independence result for generalized newton methods is established the main result states that the continuous and the discrete newton process when initialized properly converge q linearly with the same rate the problem class considered in the paper includes mcp function based reformulations of first order conditions of a class of control constrained optimal control problems for partial differential equations for which a numerical validation of the theoretical results is given

renormalization group flow of a hierarchical sine gordon model by partial differential equations we use a renormalization group differential equation to rigorously control the renormalization group flow in a hierarchical lattice sine gordon field theory in the kosterlitz thouless phase

a multigrid finite element solver for the cahn hilliard equation a multigrid finite element solver for the cahn hilliard equation is presented that has mesh independent convergence rates for any time step size including in the important limit e 0 which is examined via numerical examples numerics are performed for a number of test problems which show that the features of the cahn hilliard equation minimising interface measure lyapunov energy functional etc are preserved we also explore the use of this solver in conjunction with adaptive time stepping and adaptive mesh strategies

ground states and hyperuniformity of the hierarchical coulomb gas in all dimensions stochastic point processes with coulomb interactions arise in various natural examples of statistical mechanics random matrices and optimization problems often such systems due to their natural repulsion exhibit remarkable hyperuniformity properties that is the number of points landing in any given region fluctuates at a much smaller scale compared to that of a set of i i d random points a well known conjecture from physics appearing in the works of jancovici lebowitz manificat martin and yalcin see jancovici et al in j stat phys 72 3 4 773 787 1993 lebowitz in phys rev a 27 1491 1494 1983 martin and yalcin in j stat phys 22 4 435 463 1980 states that the variance of the number of points landing in a set should grow like the surface area instead of the volume unlike i i d random points in a recent beautiful work chatterjee in rigidity of the three dimensional hierarchical coulomb gas 2017 arxiv preprint arxiv 1708 01965 to appear in probability theory and related fields chatterjee gave the first proof of such a result in dimension three for a coulomb type system known as the hierarchical coulomb gas inspired by dyson s hierarchical model of the ising ferromagnet dyson in phys rev 92 1331 1338 1953 dyson in commun math phys 12 2 91 107 1969 however the case of dimensions greater than three had remained open in this paper we establish the correct fluctuation behavior up to logarithmic factors in all dimensions greater than three for the hierarchical model using similar methods we also prove sharp variance bounds for smooth linear statistics which were unknown in any dimension bigger than two a key intermediate step is to obtain precise results about the ground states of such models whose behavior can be interpreted as hierarchical analogues of various crystalline conjectures predicted for energy minimizing systems and could be of independent interest

numerical analysis of vanka type solvers for steady stokes and navier stokes flows we consider vanka type smoothers for solving stokes and navier stokes problems in each iteration step this smoother requires the solution of several small local subproblems over finite element blocks it is shown that for particular choices for the blocks the algorithm always converges to the solution of the stokes problem and under suitable conditions to the solution of the navier stokes problem the convergence properties are analyzed and numerical examples are presented

fast solution of cahn hilliard variational inequalities using implicit time discretization and finite elements we consider the efficient solution of the cahn hilliard variational inequality using an implicit time discretization which is formulated as an optimal control problem with pointwise constraints on the control by applying a semi smooth newton method combined with a moreau yosida regularization technique for handling the control constraints we show superlinear convergence in function space at the heart of this method lies the solution of large and sparse linear systems for which we propose the use of preconditioned krylov subspace solvers using an effective schur complement approximation numerical results illustrate the competitiveness of this approach