accelerating hessian free gauss newton full waveform inversion via l bfgs preconditioned conjugate gradient algorithm abstractfull waveform inversion fwi has emerged as a powerful strategy for estimating subsurface model parameters by iteratively minimizing the difference between synthetic data and observed data the hessian free hf optimization method represents an attractive alternative to newton type and gradient based optimization methods at each iteration the hf approach obtains the search direction by approximately solving the newton linear system using a matrix free conjugate gradient cg algorithm the main drawback with hf optimization is that the cg algorithm requires many iterations in our research we develop and compare different preconditioning schemes for the cg algorithm to accelerate the hf gauss newton gn method traditionally preconditioners are designed as diagonal hessian approximations we additionally use a new pseudo diagonal gn hessian as a preconditioner making use of the reciprocal property of green s function furthermore we have developed an l bfgs inverse hessian preconditioning

parallel preconditioners for monolithic solution of shear bands shear bands are one of the most fascinating instabilities in metals that occur under high strain rates they describe narrow regions on the order of micron scales where plastic deformations and significant heating are localized which eventually leads to fracture nucleation and failure of the component in this work shear bands are described by a set of four strongly coupled thermo mechanical equations discretized by a mixed finite element formulation a thermo viscoplastic flow rule is used to model the inelastic constitutive law and finite thermal conductivity is prescribed which gives rise to an inherent physical length scale governed by competition of shear heating and thermal diffusion the residual equations are solved monolithically by a newton type method at every time step and have been shown to yield mesh insensitive result the jacobian of the system is sparse and has a nonsymmetric block structure that varies with the different stages of shear bands formation the aim of the current work is to develop robust parallel preconditioners to gmres in order to solve the resulting jacobian systems efficiently the main idea is to design schur complements tailored to the specific block structure of the system and that account for the varying stages of shear bands we develop multipurpose preconditioners that apply to standard irreducible discretizations as well as our recent work on isogeometric discretizations of shear bands the proposed preconditioners are tested on benchmark examples and compared to standard state of practice solvers such as gmres ilu and lu direct solvers nonlinear and linear iterations counts as well as cpu times and computational speedups are reported and it is shown that the proposed preconditioners are robust efficient and outperform traditional state of the art solvers

self consistent solution of cosmological radiation hydrodynamics and chemical ionization we consider a pde system comprising compressible hydrodynamics flux limited diffusion radiation transport and chemical ionization kinetics in a cosmologically expanding universe under an operator split framework the cosmological hydrodynamics equations are solved through the piecewise parabolic method as implemented in the enzo community hydrodynamics code the remainder of the model including radiation transport chemical ionization kinetics and gas energy feedback form a stiff coupled pde system which we solve using a fully implicit inexact newton approach and which forms the crux of this paper the inner linear newton systems are solved using a schur complement formulation and employ a multigrid preconditioned conjugate gradient solver for the inner schur systems we describe this approach and provide results on a suite of test problems demonstrating its accuracy robustness and scalability to very large problems

a novel multigrid based preconditioner for heterogeneous helmholtz problems an iterative solution method in the form of a preconditioner for a krylov subspace method is presented for the helmholtz equation the preconditioner is based on a helmholtz type differential operator with a complex term a multigrid iteration is used for approximately inverting the preconditioner the choice of multigrid components for the corresponding preconditioning matrix with a complex diagonal is validated with fourier analysis multigrid analysis results are verified by numerical experiments high wavenumber helmholtz problems in heterogeneous media are solved indicating the performance of the preconditioner

optimality conditions for degenerate extremum problems with equality constraints in this paper we consider an optimization problem with equality constraints given in operator form as f x 0 where f x to y is an operator between banach spaces the paper addresses the case when the equality constraints are not regular in the sense that the frechet derivative f x is not onto in the first part of the paper we pursue an approach based on the construction of p regularity for p regular constrained optimization problems we formulate necessary conditions for optimality and derive sufficient conditions for optimality in the second part of the paper we consider a generalization of the concept of p regularity and derive generalized necessary conditions for optimality for an optimization problem that is neither regular nor p regular for this problem we show that the tangent cone to a level surface of f can consist of rays rather than lines this is in contrast to the regular and the p regular cases for which the tangent cone is always two sided we state that if the gradien

inexact newton type methods for the solution of steady incompressible viscoplastic flows with the supg pspg finite element formulation in this work we evaluate the performance of inexact newton type schemes to solve the nonlinear equations arising from the supg pspg finite element formulation of steady viscoplastic incompressible fluid flows the flow through an abrupt contraction and the rotational flow in eccentric annulus with power law and bingham rheologies are employed as benchmarks our results have shown that inexact schemes are more efficient than traditional newton type strategies

phase field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing this paper describes a novel numerical algorithm for simulating interfacial dynamics of non newtonian fluids the interface between two immiscible fluids is treated as a thin mixing layer across which physical properties vary steeply but continuously the property and evolution of the interfacial layer is governed by a phase field variable that obeys a cahn hilliard type of convection diffusion equation this circumvents the task of directly tracking the interface and produces the correct interfacial tension from the free energy stored in the mixing layer viscoelasticity and other types of constitutive equations can be incorporated easily into the variational phase field framework the greatest challenge of this approach is in resolving the sharp gradients at the interface this is achieved by using an efficient adaptive meshing scheme governed by the phase field variable the finite element scheme easily accommodates complex flow geometry and the adaptive meshing makes it possible to simulate large scale two phase systems of complex fluids in two dimensional and axisymmetric three dimensional implementations the numerical toolkit is applied here to drop deformation in shear and elongational flows rise of drops and retraction of drops and torii some of these solutions serve as validation of the method and illustrate its key features while others explore novel physics of viscoelasticity in the deformation and evolution of interfaces

generalized convexity and efficiency for non regular multiobjective programming problems with inequality type constraints abstract for multiobjective problems with inequality type constraints the necessary conditions for efficient solutions are presented these conditions are applied when the constraints do not necessarily satisfy any regularity assumptions and they are based on the concept of 2 regularity introduced by izmailov in general the necessary optimality conditions are not sufficient and the efficient solution set is not the same as the karush kuhn tucker points set so it is necessary to introduce generalized convexity notions in the multiobjective non regular case we give the notion of 2 kkt pseudoinvex ii problems this new concept of generalized convexity is both necessary and sufficient to guarantee the characterization of all efficient solutions based on the optimality conditions

generalized convexity in non regular programming problems with inequality type constraints abstract convexity plays a very important role in optimization for establishing optimality conditions different works have shown that the convexity property can be replaced by a weaker notion the invexity in particular for problems with inequality type constraints martin defined a weaker notion of invexity the karush kuhn tucker invexity hereafter kkt invexity that is both necessary and sufficient to obtain karush kuhn tucker type optimality conditions it is well known that for this result to hold the problem has to verify a constraint qualification i e it must be regular or non degenerate in non regular problems the classical optimality conditions are totally inapplicable meaningful results were obtained for problems with inequality type constraints by izmailov they are based on the 2 regularity condition of the constraints at a feasible point in this work we generalize martin s result to non regular problems by defining an analogous concept the 2 kkt invexity and using the characterization of the tangent cone in the 2 regular case and the necessary optimality condition given by izmailov

a non overlapping domain decomposition method with non matching grids for modeling large finite antenna arrays a non overlapping domain decomposition method ddm is proposed herein to solve maxwell equations in r 3 in this work the maxwell equations are discretized using a vector finite element method with hierarchical h curl vector basis functions there are two major ingredients in the proposed non overlapping ddm a a proper 1st order transmission condition to enforce field continuity across domain boundaries and b a cement technique to allow non matching grids for neighboring domains moreover a detail fourier analysis of the transmission condition for a canonical half space example is presented the analysis provides significant insights into the convergence behavior of the proposed non overlapping ddm for solving electromagnetic radiation problems such as the large finite antenna arrays particularly for the antenna arrays the proposed non overlapping ddm is extremely efficient since the formulation can easily incorporate geometrical repetitions exponentially tapered notch vivaldi antenna arrays with size up to 100x100 elements are solved on a common pc to validate the proposed non overlapping ddm

efficient preconditioner updates for shifted linear systems we present a technique for building effective and low cost preconditioners for sequences of shifted linear systems a alpha i x alpha b where a is symmetric positive definite and alpha 0 this technique updates a preconditioner for a available in the form of an ldl t factorization by modifying only the nonzero entries of the l factor in such a way that the resulting preconditioner mimics the diagonal of the shifted matrix and reproduces its overall behavior this approach is supported by a theoretical analysis as well as by numerical experiments showing that it works efficiently for a broad range of values of alpha

a review of algebraic multigrid abstract since the early 1990s there has been a strongly increasing demand for more efficient methods to solve large sparse unstructured linear systems of equations for practically relevant problem sizes classical one level methods had already reached their limits and new hierarchical algorithms had to be developed in order to allow an efficient solution of even larger problems this paper gives a review of the first hierarchical and purely matrix based approach algebraic multigrid amg amg can directly be applied for instance to efficiently solve various types of elliptic partial differential equations discretized on unstructured meshes both in 2d and 3d since amg does not make use of any geometric information it is a plug in solver which can even be applied to problems without any geometric background provided that the underlying matrix has certain properties

implementation of newton s method with an analytical jacobian to solve the 1d sea ice momentum equation new numerical solvers are being considered in response to the rising computational cost of properly solving the sea ice momentum equation at high resolution the jacobian free version of newton s method has allowed models to obtain the converged solution faster than other implicit solvers used previously to further improve on this recent development the analytical jacobian of the 1d sea ice momentum equation is derived and used inside newton s method the results are promising in terms of computational efficiency although robustness remains an issue for some test cases it is improved compared to the jacobian free approach in order to make use of the strong points of both the new and jacobian free methods a hybrid preconditioner using the picard and jacobian matrices to improve global and local convergence respectively is also introduced this preconditioner combines the robustness and computational efficiency of the previously used preconditioning matrices when solving the sea ice momentum equation

a taxonomy and comparison of parallel block multi level preconditioners for the incompressible navier stokes equations in recent years considerable effort has been placed on developing efficient and robust solution algorithms for the incompressible navier stokes equations based on preconditioned krylov methods these include physics based methods such as simple and purely algebraic preconditioners based on the approximation of the schur complement all these techniques can be represented as approximate block factorization abf type preconditioners the goal is to decompose the application of the preconditioner into simplified sub systems in which scalable multi level type solvers can be applied in this paper we develop a taxonomy of these ideas based on an adaptation of a generalized approximate factorization of the navier stokes system first presented in a quarteroni f saleri a veneziani factorization methods for the numerical approximation of navier stokes equations computational methods in applied mechanical engineering 188 2000 505 526 this taxonomy illuminates the similarities and differences among these preconditioners and the central role played by efficient approximation of certain schur complement operators we then present a parallel computational study that examines the performance of these methods and compares them to an additive schwarz domain decomposition dd algorithm results are presented for two and three dimensional steady state problems for enclosed domains and inflow outflow systems on both structured and unstructured meshes the numerical experiments are performed using mpsalsa a stabilized finite element code

on some krylov subspace based methods for large scale nonsymmetric algebraic riccati problems in the present paper we consider large scale nonsymmetric matrix riccati equations with low rank right hand sides these matrix equations appear in many applications such as transport theory wiener hopf factorization of markov chains applied probability and others we show how to apply directly krylov methods such as the extended block arnoldi algorithm to get low rank approximate solutions we also combine the newton method and block krylov subspace methods to get approximations of the desired minimal nonnegative solution we give some theoretical results and report some numerical experiments for the well known transport equation

continuation newton methods severely nonlinear problems can only be solved by some homotopy continuation method an example of a homotopy method is the continuous newton method which however must be discretized which leads to the damped step version of newton s method the standard newton iteration method for solving systems of nonlinear equations f u 0 must be modified in order to get global convergence i e convergence from any initial point the control of steplengths in the damped step newton method can lead to many small steps and slow convergence furthermore the applicability of the method is restricted in as much as it assumes a nonsingular and everywhere differentiable mapping f classical continuation methods are surveyed then a new method in the form of a coupled newton and load increment method is presented and shown to have a global convergence already from the start and second order of accuracy with respect to the load increment step and with less restrictive regularity assumptions than for the standard newton method the method is applied for an elastoplastic problem with hardening

scalable and efficient algorithms for the propagation of uncertainty from data through inference to prediction for large scale problems with application to flow of the antarctic ice sheet the majority of research on efficient and scalable algorithms in computational science and engineering has focused on the forward problem given parameter inputs solve the governing equations to determine output quantities of interest in contrast here we consider the broader question given a large scale model containing uncertain parameters possibly noisy observational data and a prediction quantity of interest how do we construct efficient and scalable algorithms to 1 infer the model parameters from the data the deterministic inverse problem 2 quantify the uncertainty in the inferred parameters the bayesian inference problem and 3 propagate the resulting uncertain parameters through the model to issue predictions with quantified uncertainties the forward uncertainty propagation problem we present efficient and scalable algorithms for this end to end data to prediction process under the gaussian approximation and in the context of modeling the flow of the antarctic ice sheet and its effect on loss of grounded ice to the ocean the ice is modeled as a viscous incompressible creeping shear thinning fluid the observational data come from satellite measurements of surface ice flow velocity and the uncertain parameter field to be inferred is the basal sliding parameter represented by a heterogeneous coefficient in a robin boundary condition at the base of the ice sheet the prediction quantity of interest is the present day ice mass flux from the antarctic continent to the ocean we show that the work required for executing this data to prediction process measured in number of forward and adjoint ice sheet model solves is independent of the state dimension parameter dimension data dimension and the number of processor cores the key to achieving this dimension independence is to exploit the fact that despite their large size the observational data typically provide only sparse information on model parameters this property can be exploited to construct a low rank approximation of the linearized parameter to observable map via randomized svd methods and adjoint based actions of hessians of the data misfit functional

a second order accurate in time implicit explicit imex integration scheme for sea ice dynamics current sea ice models use numerical schemes based on a splitting in time between the momentum and continuity equations because the ice strength is explicit when solving the momentum equation this can create unrealistic ice stress gradients when using a large time step as a consequence noise develops in the numerical solution and these models can even become numerically unstable at high resolution to resolve this issue we have implemented an iterated implicit explicit imex time integration method this imex method was developed in the framework of an already implemented jacobian free newton krylov solver the basic idea of this imex approach is to move the explicit calculation of the sea ice thickness and concentration inside the newton loop such that these tracers evolve during the implicit integration to obtain second order accuracy in time we have also modified the explicit time integration to a second order runge kutta approach and by introducing a second order backward difference method for the implicit integration of the momentum equation these modifications to the code are minor and straightforward by comparing results with a reference solution obtained with a very small time step it is shown that the approximate solution is second order accurate in time the new method permits to obtain the same accuracy as the splitting in time but by using a time step that is 10 times larger results show that the second order scheme is more than five times more computationally efficient than the splitting in time approach for an equivalent level of error

implementation of the jacobian free newton krylov method for solving the first order ice sheet momentum balance we have implemented the jacobian free newton krylov jfnk method for solving the first order ice sheet momentum equation in order to improve the numerical performance of the glimmer community ice sheet model glimmer cism the land ice component of the community earth system model cesm our jfnk implementation is based on significant re use of existing code for example our physics based preconditioner uses the original picard linear solver in glimmer cism for several test cases spanning a range of geometries and boundary conditions our jfnk implementation is 1 8 3 6 times more efficient than the standard picard solver in glimmer cism importantly this computational gain of jfnk over the picard solver increases when refining the grid global convergence of the jfnk solver has been significantly improved by rescaling the equation for the basal boundary condition and through the use of an inexact newton method while a diverse set of test cases show that our jfnk implementation is usually robust for some problems it may fail to converge with increasing resolution as does the picard solver globalization through parameter continuation did not remedy this problem and future work to improve robustness will explore a combination of picard and jfnk and the use of homotopy methods

a parallel multi domain solution methodology applied to nonlinear thermal transport problems in nuclear fuel pins this paper describes an efficient and nonlinearly consistent parallel solution methodology for solving coupled nonlinear thermal transport problems that occur in nuclear reactor applications over hundreds of individual 3d physical subdomains efficiency is obtained by leveraging knowledge of the physical domains the physics on individual domains and the couplings between them for preconditioning within a jacobian free newton krylov method details of the computational infrastructure that enabled this work namely the open source advanced multi physics amp package developed by the authors is described details of verification and validation experiments and parallel performance analysis in weak and strong scaling studies demonstrating the achieved efficiency of the algorithm are presented furthermore numerical experiments demonstrate that the preconditioner developed is independent of the number of fuel subdomains in a fuel rod which is particularly important when simulating different types of fuel rods finally we demonstrate the power of the coupling methodology by considering problems with couplings between surface and volume physics and coupling of nonlinear thermal transport in fuel rods to an external radiation transport code

the incorporation of the semi implicit linear equations into newton s method to solve radiation transfer equations abstract for large time steps the nonlinear equations of radiation transfer may not be solved adequately by the semi implicit linear approximation to yield physical solutions this deficiency is rectified in three steps the equations of the semi implicit linear method are modified the modified equations are incorporated into newton s method to solve nonlinear equations and the transfer equations are solved by the resulting method the new method also uses the photon free method to search for the solution in a lower dimensional space than the space of the underlying transfer equations two algorithms are developed from the new method they solve the modified semi implicit linear equations by different approaches the first is a physics approach it solves the linear equations approximately by the grey transport approximation the second is a mathematical approach it solves the linear equations exactly by the sherman morrison woodbury formula of linear algebra however both algorithms yield the solution to the nonlinear system derived by the photon free method therefore their solutions are equal to within the specified tolerance of the nonlinear solver moreover both methods can take advantage of the unconditional stability which comes with the implicit differencing of the time derivative the time step which both methods can take is much larger than the time step in which time discretization error is discernible we shall relate the mathematical approach to the photon free method numerical results for three test problems are presented

necessary optimality conditions for constrained optimization problems under relaxed constraint qualifications we derive first and second order necessary optimality conditions for set constrained optimization problems under the constraint qualification type conditions significantly weaker than robinson s constraint qualification our development relies on the so called 2 regularity concept and unifies and extends the previous studies based on this concept specifically in our setting constraints are given by an inclusion with an arbitrary closed convex set on the right hand side thus for the second order analysis some curvature characterizations of this set near the reference point must be taken into account

an inexact newton type method for inverse singular value problems in this paper an inexact newton type approach is proposed for solving inverse singular value problems we show that the method converges superlinearly this method can reduce significantly the oversolving problem of the newton type method and improve the efficiency numerical experiments is also presented to illustrate our results

nodal auxiliary space preconditioning in h curl and h div spaces in this paper we develop and analyze a general approach to preconditioning linear systems of equations arising from conforming finite element discretizations of h curl and h div elliptic variational problems the preconditioners exclusively rely on solvers for discrete poisson problems we prove mesh independent effectivity of the preconditioners by using the abstract theory of auxiliary space preconditioning the main tools are discrete analogues of so called regular decomposition results in the function spaces h curl and h div our preconditioner for h curl is similar to an algorithm proposed in r beck algebraic multigrid by component splitting for edge elements on simplicial triangulations tech rep sc 99 40 zib berlin germany 1999

investigation of multigrid preconditioner for integral equation fast analysis of electromagnetic scattering problems this paper proposes a multigrid preconditioner for integral equation fast analysis of electromagnetic scattering problems the multigrid model is constructed by recursively dividing every triangular element on the coarse level into 4 small ones the rao wilton glisson rwg basis functions defined on the coarse level are expressed with linear combinations of those on the fine level and the expansion coefficients of which are used to construct the restriction and interpolation operator to further accelerate the convergence of iterative solution a set of solenoidal basis functions on the coarse level are constructed additionally which are also expressed with linear combinations of rwg basis functions on the fine level and adopted to construct the second restriction and interpolation operator the two sets of coarsening techniques are then combined together to remove the low frequency components of the iterative error numerical results show that the proposed method can reduce both the steps of iteration and cpu time compared with the iterative solver without preconditioning

optimality conditions for irregular inequality constrained problems we consider feasible sets given by conic constraints where the cone defining the constraints is convex with nonempty interior we study the case where the feasible set is not assumed to be regular in the classical sense of robinson and obtain a constructive description of the tangent cone under a certain new second order regularity condition this condition contains classical regularity as a special case while being weaker when constraints are twice differentiable assuming that the cone defining the constraints is finitely generated we also derive a special form of primal dual optimality conditions for the corresponding constrained optimization problem our results subsume optimality conditions for both the classical regular and second order regular cases while still being meaningful in the more general setting in the sense that the multiplier associated with the objective function is nonzero

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

implicit finite incompressible elastodynamics with linear finite elements a stabilized method in rate form abstract we propose a stabilization method for linear tetrahedral finite elements suitable for the implicit time integration of the equations of nearly and fully incompressible nonlinear elastodynamics in particular we derive and discuss a generalized framework for stabilization and implicit time integration that can comprehensively be applied to the class of all isotropic hyperelastic models in this sense the presented development can be considered an important extension and complement to the stabilization approach proposed by the authors in previous work which was instead focused on explicit time integration and simple neo hookean models for nearly incompressible elasticity with the goal of computational efficiency we also present a two step block gauss seidel strategy for the time update of displacements velocities and pressures specifically a mixed system of equations for the velocity and pressure is updated implicitly in a first stage and the displacements are updated explicitly in a second stage the proposed mixed formulation is then embedded in newton type strategies for the nonlinear solution of the equations of motion various implicit time integration strategies are considered and particularly we focus on high frequency dissipation time integrators which are preferable in transient mechanics applications an extensive set of numerical computations with linear tetrahedral elements is presented to demonstrate the performance of the proposed approach

a survey on the high convergence orders and computational convergence orders of sequences abstract twenty years after the classical book of ortega and rheinboldt was published five definitions for the q convergence orders of sequences were independently and rigorously studied i e some orders characterized in terms of others by potra 1989 resp beyer ebanks and qualls 1990 the relationship between all the five definitions only partially analyzed in each of the two papers was not subsequently followed and moreover the second paper slept from the readers attention the main aim of this paper is to provide a rigorous selfcontained and as much as possible a comprehensive picture of the theoretical aspects of this topic as the current literature has taken away the credit from authors who obtained important results long ago moreover this paper provides rigorous support for the numerical examples recently presented in an increasing number of papers where the authors check the convergence orders of different iterative methods for solving nonlinear systems of equations tight connections between some asymptotic quantities defined by theoretical and computational elements are shown to hold

interrelation between laplace constants and the gradient distortion effect in laplace domain waveform inversion abstractlaplace domain waveform inversion wi is generally used to generate smooth initial velocity models for frequency or time domain full waveform inversion however in the inversion results of laplace domain wi anomalies such as salt domes are sometimes shifted we evaluate the gradient distortion effect that causes undesirable changes in parameter updates and found that this is caused by the relationship between the partial derivatives of laplace wavefields with respect to two different parameters by analyzing the gradient of the laplace domain misfit function we found that the gradient distortion effect increases as the laplace constants used in the laplace domain wi decrease the velocity model inverted in the laplace domain is generally blurred from shallower parameters to deeper parameters because the partial derivatives of the laplace wavefields with respect to shallower parameters tend to be larger than those of deeper parameters we found two solutions for suppressing the gradient dist

algebraic multigrid combined with domain decomposition for the finite element analysis of large scattering problems in the finite element analysis of electromagnetic scattering a recent algebraic multigrid method aghabarati and webb an algebraic multigrid method for the finite element analysis of large scattering problems ieee trans antennas propag vol 61 no 2 pp 809 817 feb 2013 has been shown to be an efficient way to solve the large sparse matrix equation however the effectiveness of the method decreases as the electrical size of the problem increases this limitation is overcome by combining it with a domain decomposition method ddm exact solution on each domain is not needed a single w cycle of the multigrid method is sufficient results are presented for scattering of plane waves by conducting and dielectric objects with dimensions ranging from 3 wavelengths to 20 wavelengths

multilevel methods for p adaptive finite element analysis of electromagnetic scattering in p adaptive finite element analysis the large sparse matrix that arises can be block structured according to the hierarchical level of the unknowns a multilevel preconditioner for the matrix is a v cycle that starts by applying gauss seidel to the highest level then the next level down and so on on the other side of the v gauss seidel is applied in the reverse order at the bottom of the v is the lowest order system which typically is solved exactly with a direct solver however for a complex geometry even the lowest order system may be too large for direct factorization here an alternative is proposed to continue the v cycle downwards first into a set of auxiliary node based spaces then through a series of progressively smaller matrices generated by an algebraic multigrid method the smallest matrix is solved by factorization the method is applied to p adaptive analysis of a five resonator iris filter a split ring resonator loaded waveguide a buckyball metallic frame surrounding a conducting sphere and a noncommensurate frequency selective surface tetrahedral elements up to fourth order are used the largest matrix has over 12 million rows and 0 6 billion nonzero entries

on projected newton krylov solvers for instationary laminar reacting gas flows numerical aspects of computational modeling of chemical vapor deposition are discussed large sparse strongly nonlinear algebraic systems are to be solved per time step for this inexact newton methods and preconditioned krylov subspace methods are suitable to ensure positivity of concentrations we propose a novel approach namely a projected inexact newton method unlike the commonly used method of clipping this conserves mass efficiency of several preconditionings is compared our numerical tests culminate in an unusually large computation namely a three dimensional case with 17 species and 26 reactions

a geometric newton method for oja s vector field newton s method for solving the matrix equation runs up against the fact that its zeros are not isolated this is due to a symmetry of f by the action of the orthogonal group we show how differential geometric techniques can be exploited to remove this symmetry and obtain a geometric newton algorithm that finds the zeros of f the geometric newton method does not suffer from the degeneracy issue that stands in the way of the original newton method

a parallel two level domain decomposition based one shot method for shape optimization problems summary a two level domain decomposition method is introduced for general shape optimization problems constrained by the incompressible navier stokes equations the optimization problem is first discretized with a finite element method on an unstructured moving mesh that is implicitly defined without assuming that the computational domain is known and then solved by some one shot lagrange newton krylov schwarz algorithms in this approach the shape of the domain its corresponding finite element mesh the flow fields and their corresponding lagrange multipliers are all obtained computationally in a single solve of a nonlinear system of equations highly scalable parallel algorithms are absolutely necessary to solve such an expensive system the one level domain decomposition method works reasonably well when the number of processors is not large aiming for machines with a large number of processors and robust nonlinear convergence we introduce a two level inexact newton method with a hybrid two level overlapping schwarz preconditioner as applications we consider the shape optimization of a cannula problem and an artery bypass problem in 2d numerical experiments show that our algorithm performs well on a supercomputer with over 1000 processors for problems with millions of unknowns copyright 2014 john wiley sons ltd

a meshless rbf method for computing a numerical solution of unsteady burgers type equations in this paper we discuss a meshless method for the computation of a numerical solution of unsteady coupled burgers type equations which are in the most general case nonlinear partial differential equations our approach is based on the interpolation of the solution by radial basis functions rbfs and is independent of the geometry of the domain the existence and unicity of such of an interpolant will be established by the mean of linear algebra arguments this spatial discretization transforms the initial problem into a system of ordinary differential equations odes we give a simple formulation of implicit runge kutta irk schemes that can be used to numerically solve this system of odes the main difficulty in the implementation of an irk method lies in the fact that it leads to a potentially large linear or nonlinear system that has to be solved at each time step a matrix gmres approach coupled with a newton based linearization technique in the nonlinear case will be used to numerically solve these systems some numerical examples will be provided to illustrate the performance of our approach

auxiliary space preconditioning in h 0 curl ω we adapt the principle of auxiliary space preconditioning as presented in j xu the auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids computing 56 1996 pp 215 235 to h curl elliptic variational problems discretized by means of edge elements the focus is on theoretical analysis within the abstract framework of subspace correction employing a helmholtz type splitting of edge element vector fields we can establish asymptotic h uniform optimality of the preconditioner defined by our auxiliary space method

parallel multilevel methods for implicit solution of shallow water equations with nonsmooth topography on the cubed sphere high resolution and scalable parallel algorithms for the shallow water equations on the sphere are very important for modeling the global climate in this paper we introduce and study some highly scalable multilevel domain decomposition methods for the fully implicit solution of the nonlinear shallow water equations discretized with a second order well balanced finite volume method on the cubed sphere with the fully implicit approach the time step size is no longer limited by the stability condition and with the multilevel preconditioners good scalabilities are obtained on computers with a large number of processors the investigation focuses on the use of semismooth inexact newton method for the case with nonsmooth topography and the use of two and three level overlapping schwarz methods with different order of discretizations for the preconditioning of the jacobian systems we test the proposed algorithm for several benchmark cases and show numerically that this approach converges well with smooth and nonsmooth bottom topography and scales perfectly in terms of the strong scalability and reasonably well in terms of the weak scalability on machines with thousands and tens of thousands of processors

nested multigrid vector and scalar potential finite element method for fast computation of two dimensional electromagnetic scattering nested multigrid techniques are combined with the ungauged vector and scalar potential formulation of the finite element method to accelerate the convergence of the numerical solution of two dimensional electromagnetic scattering problems the finite element modeling is performed on nested meshes of the same computational domain the conjugate gradient method is used to solve the resultant finite element matrix for the finest mesh while the nested multigrid vector and scalar potential algorithm acts as the preconditioner of the iterative solver numerical experiments are used to demonstrate the superior numerical convergence and efficient memory usage of the proposed algorithm

nonlinear krylov acceleration applied to a discrete ordinates formulation of the k eigenvalue problem we compare a variant of anderson mixing with the jacobian free newton krylov and broyden methods applied to an instance of the k eigenvalue formulation of the linear boltzmann transport equation we present evidence that one variant of anderson mixing finds solutions in the fewest number of iterations we examine and strengthen theoretical results of anderson mixing applied to linear problems

implementation of the jacobian free newton krylov method for solving the for solving the first order ice sheet momentum balance we have implemented the jacobian free newton krylov jfnk method for solving the rst order ice sheet momentum equation in order to improve the numerical performance of the community ice sheet model cism the land ice component of the community earth system model cesm our jfnk implementation is based on signicant re use of existing code for example our physics based preconditioner uses the original picard linear solver in cism for several test cases spanning a range of geometries and boundary conditions our jfnk implementation is 1 84 3 62 times more efficient than the standard picard solver in cism importantly this computational gain of jfnk over the picard solver increases when rening the grid global convergence of the jfnk solver has been signicantly improved by rescaling the equation for the basal boundary condition and through the use of an inexact newton method while a diverse set of test cases show that our jfnk implementation is usually robust for some problems it may fail to converge with increasing resolution as does the picard solver globalization through parameter continuation did not remedy this problem and future work to improve robustness will explore a combination of picard and jfnk and the use of homotopy methods

high order approximations and generalized necessary conditions for optimality in this paper we derive generalized necessary conditions for optimality for an optimization problem with equality and inequality constraints in a banach space the equality constraints are given in operator form as q x in x f x 0 where f x rightarrow y is an operator between banach spaces the inequality constraints are given by smooth functionals or by closed convex sets models of this type are common in the optimal control problem the paper addresses the case when the frechet derivative f x is not onto and hence the classical lyusternik theorem does not apply to describe the tangent space to q in this case the classical euler lagrange type necessary conditions are trivially satisfied generating abnormal cases a high order generalization of the lyusternik theorem derived earlier u ledzewicz and h schattler nonlinear anal 34 1998 pp 793 815 is used to calculate high order tangent cones to the equality constraint at points x in q where f x is not onto combining these with high order approximating cones related to the other constraints of the problem feasible cones respectively cones of decrease a high order generalization of the dubovitskii milyutin theorem is given and then applied to derive generalized necessary conditions for optimality these conditions reduce to classical conditions for normal cases but they give new and nontrivial conditions for abnormal cases

an algebraic multigrid method for the finite element analysis of large scattering problems an efficient iterative solver is proposed for the linear matrix equation that arises in the analysis of large scattering problems by the frequency domain finite element method a krylov method is preconditioned by a technique that approximately solves equivalent problems in two auxiliary spaces a space of scalar functions and a space of piecewise linear nodal vector functions on each space the traditional algebraic multigrid method is employed further the shifted laplace idea is used to improve the performance of the solver as the frequency increases results are reported for a waveguide cavity filter and three free space scatterers a conducting sphere a metallic frequency selective surface and a metamaterial lens made of split ring resonators containing dielectric and metal

inexact block newton methods for solving nonlinear equations in the paper two parallelizable inexact block newton methods are presented for solving large and sparse nonlinear equations the basic idea is simple and direct combining the simplified newton method with the component averaging cav method parallel comput 27 2001 777 results in an inexact newton method called simplified newton cav method parallel tests of the algorithm are implemented on a message passing distributed memory multiprocessor architecture such as a cluster of workstations the results show that the new algorithm can achieve good performance moreover as a development of the simplified newton cav method the overlapped block newton cav method is further proposed and discussed

comparison of multifrequency selection strategies for simultaneous source full waveform inversion abstractthe computational cost of full waveform inversion fwi is a major obstacle to estimating the velocity model for large scale problems one way to reduce the overall cost of waveform inversion is by adopting a simultaneous source strategy in other words multiple sources are simultaneously fired to simulate supershot gathers and thereby reduce the number of seismic modeling simulations that are performed during the inversion however the use of simultaneous sources introduces crosstalk artifacts that arise from the interference among the sources that constitute a supershot we analyzed the influence of different simultaneous multifrequency selection strategies on crosstalk artifacts our analysis focused on a frequency domain fwi algorithm that is implemented with simultaneous sources that are randomly encoded with random time shifts in the multiscale conventional fwi strategies a finite set of discrete frequencies was selected and the inversion was carried out sequentially from low to high fr

on sufficiency of the kuhn tucker conditions if all the functions fo x f x f x are convex on c then these conditions are also sufficient various classes of functions have been defined for the purpose of weakening this limitation of convexity in mathematical programming mangasarian 11 has speculated that pseudo convexity of x and quasiconvexity off x are the weakest conditions that can be imposed so that the above conditions are sufficient for optimality it will be shown that there are other wide classes of functions for which the conditions are sufficient 545

goal oriented error estimation and h adaptivity for maxwell s equations goal oriented error estimates based on dual solutions are derived for the s parameters of a waveguide cavity resonator and a computationally cheap method is proposed to compute these numerical results show that the error estimators are relatively accurate usually well within a factor 2 from the exact errors adaptive mesh refinement based on these estimates recovers optimal convergence rates for complete and incomplete first and second order curl conforming finite elements of nedelec type despite singularities at reentrant corners it is also shown numerically that the convergence rate is independent of the element order when singularities are present and the mesh is refined uniformly

nonlinear feti dp and bddc methods a unified framework and parallel results parallel newton krylov feti dp finite element tearing and interconnecting dual primal domain decomposition methods are fast and robust solvers e g for nonlinear implicit problems in structural mechanics in these methods the nonlinear problem is first linearized and then decomposed into loosely coupled linear problems which can be solved in parallel by changing the order of the operations new parallel communication can be constructed where the loosely coupled local problems are nonlinear we discuss different nonlinear feti dp methods which are equivalent when applied to linear problems but which show a different performance for nonlinear problems moreover a new unified framework is introduced which casts all nonlinear feti dp domain decomposition approaches discussed in the literature into a single algorithm furthermore the equivalence of nonlinear feti dp methods to specific nonlinearly right preconditioned newton krylov methods is shown for the methods using nested newton iteration

lagrange multipliers and optimality lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write first order optimality conditions formally as a system of equations modern applications with their emphasis on numerical methods and more complicated side conditions than equations have demanded deeper understanding of the concept and how it fits into a larger theoretical picture a major line of research has been the nonsmooth geometry of one sided tangent and normal vectors to the set of points satisfying the given constraints another has been the game theoretic role of multiplier vectors as solutions to a dual problem interpretations as generalized derivatives of the optimal value with respect to problem parameters have also been explored lagrange multipliers are now being seen as arising from a general rule for the subdifferentiation of a nonsmooth objective function which allows black and white constraints to be replaced by penalty expressions this paper traces su

development of a continuity preserving segregated method for incompressible navier stokes equations the present study aims to develop a new method for obtaining the non oscillatory incompressible navier stokes solutions on the non staggered grids within the segregated grid framework the divergence free equation is chosen to replace one of the momentum equations so as to preserve the fluid incompressibility for the sake of numerical accuracy the five point stencil convection diffusion reaction scheme is developed to obtain the nodally exact solution for this chosen momentum equation the validity of the proposed mass preserving navier stokes method is justified by solving the three problems which are amenable to analytical solutions the simulated solution quality is shown to outperform that of the conventional segregated approach besides gaining a very high spatial rate of convergence

goal oriented model constrained optimization for reduction of large scale systems optimization oriented reduced order models should target a particular output functional span an applicable range of dynamic and parametric inputs and respect the underlying governing equations of the system to achieve this goal we present an approach for determining a projection basis that uses a goal oriented model constrained optimization framework the mathematical framework permits consideration of general dynamical systems with general parametric variations and is applicable to both linear and nonlinear systems results for a simple linear model problem of the two dimensional heat equation demonstrate the ability of the goal oriented approach to target a particular output functional of interest application of the methodology to a more challenging example of a subsonic blade row governed by the unsteady euler flow equations shows a significant advantage of the new method over the proper orthogonal decomposition

adaptive eigenspace for multi parameter inverse scattering problems abstract a nonlinear optimization method is proposed for inverse scattering problems in the frequency domain when the unknown medium is characterized by one or several spatially varying parameters the time harmonic inverse medium problem is formulated as a pde constrained optimization problem and solved by an inexact truncated newton type method combined with frequency stepping instead of a grid based discrete representation each parameter is projected to a separate finite dimensional subspace which is iteratively adapted during the optimization each subspace is spanned by the first few eigenfunctions of a linearized regularization penalty functional chosen a priori the small and slowly increasing finite number of eigenfunctions effectively introduces regularization into the inversion and thus avoids the need for standard tikhonov type regularization numerical results illustrate the accuracy and efficiency of the resulting adaptive eigenspace regularization for single and multi parameter problems including the well known marmousi model from geosciences

implicit boundary equations for conservative navier stokes equations abstract co existence of the physical and numerical boundary conditions makes implicit boundary treatment a particularly difficult problem in modern cfd simulations previous studies adopted space time extrapolation or specially designed partial differential equations on the boundaries that are different from those of interior points they are often formulated in terms of primitive variables and are very challenging for complicated boundary types to be converted to and implemented in the conservative variables that are preferred in numerical simulations more importantly different boundary equations or different extrapolation techniques may compromise the stability accuracy or convergence rate of the a stable schemes that are developed for interior points a new methodology for implicit boundary treatment is proposed in this study by introducing a simple correction matrix t a set of generalized equations q t i t r are developed in terms of conservative variables it is applicable for both the interior domain t 0 and the boundaries t 0 it is in a partial differential equation form satisfies the boundary conditions accurately independent of the time and spatial discretizations any one sided schemes can be used on the boundaries but still maintain the upwind property implicit solution techniques are made significantly easy to implement using for example the data parallel lower upper relation method and the newton method combined with the gmres method for subsidiary iterations numerical experiments show that the proposed methodology produces stable simulations for very large cfl numbers and preserve the imposed boundary values accurately

incomplete jacobian newton method for nonlinear equations this paper presents a new modified newton method for nonlinear equations this method uses a part of elements of the jacobian matrix to obtain the next iteration point and is refereed to as the incomplete jacobian newton ijn method the ijn method may be fit for solving large scale nonlinear equations with dense jacobian the conditions of linear superlinear and quadratic convergence of the ijn method are given and the local convergence results are analyzed and proved some special ijn algorithms are designed and numerical experiments are given the results show that the ijn method is promising

an overlapping domain decomposition preconditioning method for monolithic solution of shear bands metals subjected to high strain rate impact often exhibit a sudden and profound drop in the material s load bearing capability a ductile failure phenomenon known as shear banding shear bands characterized as material instabilities are driven by shear heating and described as narrow regions that have sustained intense plastic deformation and high temperature rise this coupled thermo mechanical localization problem can be formulated as a nonlinear system with two balance equations momentum and energy and two constitutive laws for elasticity and plasticity in our formulation mixed finite elements are used to discretize the equations in space and an implicit finite difference scheme is used to advance the system in time where at every time step a newton type method is used to solve the system monolithically to that end a block jacobian matrix is formed analytically using g teaux derivatives the resulting jacobian is sparse nonsymmetric and its sparsity pattern and eigenvalue content vary with the different stages of shear bands formation consequently posing a significant challenge to iterative solvers to address this issue an overlapping domain decomposition preconditioner that takes into account the physics of shear bands and the domain in which it forms is proposed the key idea is to resolve the shear band domain accurately while only solving the remaining domain approximately with selective updates when necessary the preconditioner is formed on the basis of an additive schwartz method that is applied to a schur complement partition of the system the proposed preconditioner is implemented in parallel and tested on three different benchmark problems as compared against off the shelf solvers we study the effect of hh and kk refinement that are obtained from isogeometric discretizations of the system the overlapping strategy and its parallel performance excellent performance is demonstrated in serial and parallel on all benchmark examples

inexact overlapped block broyden methods for solving nonlinear equations in this paper a parallelizable overlapped block broyden method is presented for solving large systems of nonlinear equations the basic idea is to perform the block broyden iteration described in siam j sci comput 18 1997 1367 for the overlapped blocks and then assemble the overlapping solutions in a weighted average manner at each iteration a family of nonlinear overlapped solvers could be generated combining it with some iterative or direct linear solvers the conditions under which the algorithm is locally convergent are studied and many useful techniques regarding the implementation are also considered

inexact newton methods for inverse eigenvalue problems in this paper we survey some of the latest development in using inexact newton methods for solving inverse eigenvalue problems these methods require the solutions of nonsymmetric and large linear systems i e the large jacobian equations one can solve these systems by iterative methods inner iterations however iterative methods usually oversolve the problem in the sense that they require far more inner iterations than is required for the convergence of the newton outer iterations the inexact methods can avoid the oversolving problem and hence improve the efficiency the convergence rate of the inexact methods are superlinear and a good tradeoff between the required inner and outer iterations can be obtained

an improved algebraic multigrid method for solving maxwell s equations we propose two improvements to the reitzinger and schoberl algebraic multigrid amg method for solving the eddy current approximations to maxwell s equations the main focus in the reitzinger schoberl method is to maintain null space properties of the weak nabla times nabla times operator on coarse grids while these null space properties are critical they are not enough to guarantee h independent convergence of the overall multigrid method we illustrate how the reitzinger schoberl amg method loses h independence due to the somewhat limited approximation property of the grid transfer operators we present two improvements to these operators that not only maintain the important null space properties on coarse grids but also yield significantly improved multigrid convergence rates the first improvement is based on smoothing the reitzinger schoberl grid transfer operators the second improvement is obtained by using higher order nodal interpolation to derive the corresponding amg interpolation operators while not completely h independent the resulting amg cg method demonstrates improved convergence behavior while maintaining low operator complexity

analysis of a multigrid algorithm for time harmonic maxwell equations this paper considers a multigrid algorithm suitable for efficient solution of indefinite linear systems arising from finite element discretization of time harmonic maxwell equations in particular a backslash multigrid cycle is proven to converge at rates independent of refinement level if certain indefinite block smoothers are used the method of analysis involves comparing the multigrid error reduction operator with that of a related positive definite multigrid operator this idea has previously been used in multigrid analysis of indefinite second order elliptic problems however the maxwell application involves a nonelliptic indefinite operator with the help of a few new estimates the earlier ideas can still be applied some numerical experiments with lowest order nedelec elements are also reported

scalable parallel approach for high fidelity steady state aeroelastic analysis and adjoint derivative computations aeroelastic systems achieve the best performance when the aerodynamic shape and structural sizing are optimized concurrently but such an optimization is challenging when high fidelity aerodynamic and structural models are required this paper addresses this challenge through several significant improvements fully coupled newton krylov methods are presented for the solution of aerostructural systems and for the corresponding adjoint systems the coupled adjoint method presented can compute gradients with respect to thousands of multidisciplinary design variables accurately and efficiently this is enabled by several improvements in the computation of the multidisciplinary terms in the coupled adjoint the parallel scalability of the methods is demonstrated for a full aircraft configuration using an euler computational fluid dynamics model with more than 8 106 state variables and a detailed structural finite element model of the wing with more than 1 106 degrees of freedom the coupled newton krylov metho

an integral equation approach to the incompressible navier stokes equations in two dimensions we present a collection of methods for solving the incompressible navier stokes equations in the plane that are based on a pure stream function formulation the advantages of this approach are twofold first the velocity is automatically divergence free and second complicated nonlocal boundary conditions for the vorticity are avoided the disadvantage is that the solution of a nonlinear fourth order partial differential equation is required by recasting this partial differential equation as an integral equation we avoid the ill conditioning which hampers finite difference and finite element methods in this environment by using fast algorithms for the evaluation of volume integrals we are able to solve the equations using o m or o m log m operations where m is the number of points in the discretization of the domain

a domain decomposition method with nonconformal meshes for finite periodic and semi periodic structures we present a domain decomposition method as a preconditioner for krylov type solvers to model complex electromagnetic problems containing periodicities the method reduces memory requirements by decomposing the original problem into several nonoverlapping sub domains the 1st order transmission condition is employed on interfaces between adjacent sub domains to enforce continuity of electromagnetic fields and to ensure the sub domain problems are well posed by following the spirit of duality paring a symmetric system is obtained to reduce the computational burdens of the present method the finite element tearing and interconnecting like algorithm is adopted this algorithm results in the computation of the so called numerical green s function which can be compressed efficiently via a rank revealing matrix factorization algorithm the final system matrix is solved by krylov solvers instead of classical stationary solvers to improve the convergence of iterative solvers several robust implementation details are discussed and the choice of some popular krylov subspace solvers is studied through numerical examples

choosing the forcing terms in an inexact newton method an inexact newton method is a generalization of newton s method for solving f x 0 f mathbb r n to mathbb r n in which at the kth iteration the step s k from the current approximate solution x k is required to satisfy a condition f x k f x k s k leqslant eta k f x k for a forcing term eta k in 0 1 in typical applications the choice of the forcing terms is critical to the efficiency of the method and can affect robustness as well promising choices of the forcing terms are given their local convergence properties are analyzed and their practical performance is shown on a representative set of test problems

enhanced nonlinear iterative techniques applied to a nonequilibrium plasma flow we study the application of enhanced nonlinear iterative methods to the steady state solution of a system of two dimensional convection diffusion reaction partial differential equations that describe the partially ionized plasma flow in the boundary layer of a tokamak fusion reactor this system of equations is characterized by multiple time and spatial scales and contains highly anisotropic transport coefficients due to a strong imposed magnetic field we use newton s method to linearize the nonlinear system of equations resulting from an implicit finite volume discretization of the governing partial differential equations on a staggered cartesian mesh the resulting linear systems are neither symmetric nor positive definite and are poorly conditioned preconditioned krylov iterative techniques are employed to solve these linear systems we investigate both a modified and a matrix free newton krylov implementation with the goal of reducing cpu cost associated with the numerical formation of the jacobian a combination of a damped iteration mesh sequencing and a pseudotransient continuation technique is used to enhance global nonlinear convergence and cpu efficiency gmres is employed as the krylov method with incomplete lower upper ilu factorization preconditioning the goal is to construct a combination of nonlinear and linear iterative techniques for this complex physical problem that optimizes trade offs between robustness cpu time memory requirements and code complexity it is shown that a mesh sequencing implementation provides significant cpu savings for fine grid calculations performance comparisons of modified newton krylov and matrix free newton krylov algorithms will be presented

on a one dimensional optimization problem derived from the efficiency analysis of newton pcg like algorithms the newton pcg preconditioned conjugate gradient like algorithms are usually very efficient however their efficiency is mainly supported by the numerical experiments recently a new kind of newton pcg like algorithms is derived in j optim theory appl 105 2000 97 superiority analysis on truncated newton method with preconditioned conjugate gradient technique for optimization in preparation by the efficiency analysis it is proved from the theoretical point of view that their efficiency is superior to that of newton s method for the special cases where newton s method converges with precise q order 2 and 2 respectively in the process of extending such kind of algorithms to the more general case where newton s method has no fixed convergence order the first is to get the solutions to the one dimensional optimization problems with many different parameter values of if these problems were solved by numerical method one by one the computation cost would reduce the efficiency of the newton pcg algorithm and therefore is unacceptable in this paper we overcome the difficulty by deriving an analytic expression of the solution to the one dimensional optimization problem with respect to the parameter