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

a fast numerical method for solving the three dimensional stokes equations in the presence of suspended particles abstract a new fast numerical method for solving the three dimensional stokes equations in the presence of suspended particles is presented the fluid dynamics equations are solved on a lattice a particle is represented by a set of points each of which moves at the local fluid velocity and is not constrained to lie on the lattice these points are coupled by forces which resist deformation of the particle these forces contribute to the force density in the stokes equations as a result a single set of fluid dynamics equations holds at all points of the domain and there are no internal boundaries particle size shape and deformability may be prescribed computational work increases only linearly with the number of particles so large numbers 500 1000 of particles may be studied efficiently the numerical method involves implicit calculation of the particle forces by minimizing an energy function and solution of a finite difference approximation to the stokes equations using the fourier toeplitz method the numerical method has been implemented to run on all cray computers the implementation exploits the cray s vectorized arithmetic and on machines with insufficient central memory it performs efficient disk 1 o while storing most of the data on disk applications of the method to sedimentation of one two and many particle systems are described trajectories and settling speeds for two particle sedimentation and settling speed for multiparticle sedimentation from initial distributions on a cubic lattice or at random give good quantitative agreement with existing theories

stability analysis for the immersed fiber problem a linear stability analysis is performed on a two dimensional version of the immersed fiber problem formulated by c peskin to model the flow of fluid in the presence of a mesh of moving elastic fibers the purpose of the analysis is to isolate the modes in the solution which are associated with the fiber and thereby determine the effect of the presence of a fiber on the fluid the results are used not only to make conclusions about the stability of the problem but also to suggest guidelines for developing numerical methods for flows with immersed fibers

spectral analysis of the discrete helmholtz operator preconditioned with a shifted laplacian shifted laplace preconditioners have attracted considerable attention as a technique to speed up convergence of iterative solution methods for the helmholtz equation in this paper we present a comprehensive spectral analysis of the helmholtz operator preconditioned with a shifted laplacian our analysis is valid under general conditions the propagating medium can be heterogeneous and the analysis also holds for different types of damping including a radiation condition for the boundary of the computational domain by combining the results of the spectral analysis of the preconditioned helmholtz operator with an upper bound on the gmres residual norm we are able to provide an optimal value for the shift and to explain the mesh dependency of the convergence of gmres preconditioned with a shifted laplacian we illustrate our results with a seismic test problem

an immersed boundary method with formal second order accuracy and reduced numerical viscosity a formally second order accurate immersed boundary method is presented and tested in this paper we apply this new scheme to simulate the flow past a circular cylinder and study the effect of numerical viscosity on the accuracy of the computation by comparing the numerical results with those of a first order method the numerical evidence shows that the new scheme has less numerical viscosity and is therefore a better choice for the simulation of high reynolds number flows with immersed boundaries

a second order virtual node algorithm for nearly incompressible linear elasticity in irregular domains we present a cut cell method in r 2 for enforcing dirichlet and neumann boundary conditions with nearly incompressible linear elastic materials in irregular domains virtual nodes on cut uniform grid cells are used to provide geometric flexibility in the domain boundary shape without sacrificing accuracy we use a mixed formulation utilizing a mac type staggered grid with piecewise bilinear displacements centered at cell faces and piecewise constant pressures at cell centers these discretization choices provide the necessary stability in the incompressible limit and the necessary accuracy in cut cells numerical experiments suggest second order accuracy in l we target high resolution problems and present a class of geometric multigrid methods for solving the discrete equations for displacements and pressures that achieves nearly optimal convergence rates independent of grid resolution

matrix dependent prolongations and restrictions in a blackbox multigrid solver multigrid methods are studied for the solution of linear systems resulting from the 9 point discretization of a general linear second order elliptic partial differential equation in two dimensions the rate of convergence of standard multigrid methods often deteriorates when the coefficients in the differential equation are discontinuous or when dominating first order terms are present these difficulties may be overcome by choosing the prolongation and restriction operators in a special way a novel way to do this is proposed as a result a blackbox solver written in standard fortran 77 has been developed numerical experiments for several hard test problems are described and comparison is made with other algorithms the standard mg method and a method introduced by kettler a significant improvement of robustness and efficiency is found

immersed interface methods for stokes flow with elastic boundaries or surface tension a second order accurate interface tracking method for the solution of incompressible stokes flow problems with moving interfaces on a uniform cartesian grid is presented the interface may consist of an elastic boundary immersed in the fluid or an interface between two different fluids the interface is represented by a cubic spline along which the singularly supported elastic or surface tension force can be computed the stokes equations are then discretized using the second order accurate finite difference methods for elliptic equations with singular sources developed in our previous paper siam j numer anal 31 1994 pp 1019 1044 the resulting velocities are interpolated to the interface to determine the motion of the interface an implicit quasi newton method is developed that allows reasonable time steps to be used

a second order virtual node algorithm for stokes flow problems with interfacial forces discontinuous material properties and irregular domains we present numerical methods for the solution of the stokes equations that handle interfacial discontinuities discontinuous material properties and irregular domains the discretization couples a lagrangian representation of the material interface with eulerian representations of the fluid velocity and pressure the methods are efficient easy to implement and yield discretely divergence free velocities that are second order accurate in l for the special case of continuous fluid viscosity we present a method that decouples the stokes equations into three poisson interface problems which we use the techniques in bedrossian 2010 1 to solve we also solve a fourth poisson equation to enforce a discrete divergence free condition in this case we discretize all equations using an embedded approach on a uniform mac grid employing virtual nodes and duplicated cells at the interfaces these additional degrees of freedom allow for accurate resolution of discontinuities in the fluid stress at the material interface in the case of discontinuous viscosity we require a lagrange multiplier term to enforce continuity of the fluid velocity we provide a novel discretization of this term that accurately resolves constant pressure null modes we show that the accurate resolution of these modes significantly improves performance the discrete coupled equations for the velocity pressure and lagrange multipliers are in the form of a symmetric kkt system however if both fluids have the same viscosity then all four linear systems involved are symmetric positive definite with three of the four having the standard 5 point laplace stencil everywhere numerical results indicate second order accuracy for the velocities and first order accuracy for the pressure in the general case for the continuous viscosity case numerical results indicate second order accuracy for both velocities and pressure

constrained maximum likelihood estimation of two level covariance structure model via em type algorithms in this paper the constrained maximum likelihood estimation of a two level covariance structure model with unbalanced designs is considered the two level model is reformulated as a single level model by treating the group level latent random vectors as hypothetical missing data then the popular em algorithm is extended to obtain the constrained maximum likelihood estimates for general nonlinear constraints the multiplier method is used at them step to find the constrained minimum of the conditional expectation an accelerated em gradient procedure is derived to handle linear constraints the empirical performance of the proposed em type algorithms is illustrated by some artifical and real examples

numerical simulation of a cylinder in uniform flow in this study a virtual boundary technique is applied to the numerical simulation of stationary and moving cylinders in uniform flow this approach readily allows the imposition of a no slip boundary within the flow field by a feedback forcing term added to the momentum equations in the present work this technique is used with a high order finite difference method effectively eliminating spurious oscillations caused by the feedback forcing when used with spectrally discretized flow solvers very good agreement is found between the present calculations and previous computational and experimental results for steady and time dependent flow at low reynolds numbers

immersed boundary smooth extension the immersed boundary method is a simple efficient and robust numerical scheme for solving pde in general domains yet it only achieves first order spatial accuracy near embedded boundaries in this paper we introduce a new high order numerical method which we call the immersed boundary smooth extension ibse method the ibse method achieves high order accuracy by smoothly extending the unknown solution of the pde from a given smooth domain to a larger computational domain enabling the use of simple cartesian grid discretizations e g fourier spectral methods the method preserves much of the flexibility and robustness of the original ib method in particular it requires minimal geometric information to describe the boundary and relies only on convolution with regularized delta functions to communicate information between the computational grid and the boundary we present a fast algorithm for solving elliptic equations which forms the basis for simple high order implicit time methods for parabolic pde and implicit explicit methods for related nonlinear pde we apply the ibse method to solve the poisson heat burgers and fitzhugh nagumo equations and demonstrate fourth order pointwise convergence for dirichlet problems and third order pointwise convergence for neumann problems

tether force constraints in stokes flow by the immersed boundary method on a periodic domain the immersed boundary method is an algorithm for simulating the interaction of immersed elastic bodies or boundaries with a viscous incompressible fluid the immersed elastic material is represented in the fluid equations by a system or field of applied forces the particular case of stokes flow with applied forces on a periodic domain involves two related mathematical complications one of these is that an arbitrary constant vector may be added to the fluid velocity and the other is the constraint that the integral of the applied force must be zero typically forces defined on a freely floating elastic immersed boundary or body satisfy this constraint but there are many important classes of forces that do not for example the so called tether forces that are used to prescribe the simulated configuration of an immersed boundary possibly in a time dependent manner typically do not sum to zero another type of force that does not have zero integral is a uniform force density that may be used to simulate an overall pressure gradient driving flow through a system we present a method for periodic stokes flow that when used with tether points admits the use of all forces irrespective of their integral over the domain a byproduct of this method is that the additive constant velocity associated with periodic stokes flow is uniquely determined indeed the additive constant is chosen at each time step so that the sum of the tether forces balances the sum of any other forces that may be applied

immersed boundary smooth extension ibse a high order method for solving incompressible flows in arbitrary smooth domains abstract the immersed boundary method is a simple efficient and robust numerical scheme for solving pde in general domains yet for fluid problems it only achieves first order spatial accuracy near embedded boundaries for the velocity field and fails to converge pointwise for elements of the stress tensor in a previous work we introduced the immersed boundary smooth extension ibse method a variation of the ib method that achieves high order accuracy for elliptic pde by smoothly extending the unknown solution of the pde from a given smooth domain to a larger computational domain enabling the use of simple cartesian grid discretizations in this work we extend the ibse method to allow for the imposition of a divergence constraint and demonstrate high order convergence for the stokes and incompressible navier stokes equations up to third order pointwise convergence for the velocity field and second order pointwise convergence for all elements of the stress tensor the method is flexible to the underlying discretization we demonstrate solutions produced using both a fourier spectral discretization and a standard second order finite difference discretization

idr s a family of simple and fast algorithms for solving large nonsymmetric systems of linear equations we present idr s a new family of efficient short recurrence methods for large nonsymmetric systems of linear equations the new methods are based on the induced dimension reduction idr method proposed by sonneveld in 1980 idr s generates residuals that are forced to be in a sequence of nested subspaces although idr s behaves like an iterative method in exact arithmetic it computes the true solution using at most n n s matrix vector products with n the problem size and s the codimension of a fixed subspace we describe the algorithm and the underlying theory and present numerical experiments to illustrate the theoretical properties of the method and its performance for systems arising from different applications our experiments show that idr s is competitive with or superior to most bi cg based methods and outperforms bi cgstab when s 1

unconditionally stable discretizations of the immersed boundary equations the immersed boundary ib method is known to require small timesteps to maintain stability when solved with an explicit or approximately implicit method many implicit methods have been proposed to try to mitigate this timestep restriction but none are known to be unconditionally stable and the observed instability of even some of the fully implicit methods is not well understood in this paper we prove that particular backward euler and crank nicolson like discretizations of the nonlinear immersed boundary terms of the ib equations in conjunction with unsteady stokes flow can yield unconditionally stable methods we also show that the position at which the spreading and interpolation operators are evaluated is not relevant to stability so as long as both operators are evaluated at the same location in time and space we further demonstrate through computational tests that approximate projection methods which do not provide a discretely divergence free velocity field appear to have a stabilizing influence for these problems and that the implicit methods of this paper when used with the full navier stokes equations are no longer subject to such a strict timestep restriction and can be run up to the cfl constraint of the advection terms

gpu implementation of a helmholtz krylov solver preconditioned by a shifted laplace multigrid method a helmholtz equation in two dimensions discretized by a second order finite difference scheme is considered krylov methods such as bi cgstab and idr s have been chosen as solvers since the convergence of the krylov solvers deteriorates with increasing wave number a shifted laplace multigrid preconditioner is used to improve the convergence the implementation of the preconditioned solver on cpu central processing unit is compared to an implementation on gpu graphics processing units or graphics card using cuda compute unified device architecture the results show that preconditioned bi cgstab on gpu as well as preconditioned idr s on gpu is about 30 times faster than on cpu for the same stopping criterion

a multigrid method enhanced by krylov subspace iteration for discrete helmholtz equations standard multigrid algorithms have proven ineffective for the solution of discretizations of helmholtz equations in this work we modify the standard algorithm by adding gmres iterations at coarse levels and as an outer iteration we demonstrate the algorithm s effectiveness through theoretical analysis of a model problem and experimental results in particular we show that the combined use of gmres as a smoother and outer iteration produces an algorithm whose performance depends relatively mildly on wave number and is robust for normalized wave numbers as large as 200 for fixed wave numbers it displays grid independent convergence rates and has costs proportional to the number of unknowns

a computational model of the cochlea using the immersed boundary method this thesis describes a two dimensional computational model of the cochlea inner ear solved using a modified form of peskin s immersed boundary method this method is an extension of peskin s method that was originally applied to solving a model of the heart 1972 this method solves the time dependent incompressible navier stokes equations in the presence of immersed boundaries the fluid equations are specified on a fixed eulerian grid while the immersed boundaries are specified on a moving lagrangian grid the immersed boundaries are used to represent the structures of the cochlea the rigid bone the actively moving stapes and the flexible membranes the immersed boundaries exert forces locally on the fluid these local forces are seen by the fluid as external forces that are added to the other forces pressure and viscous acting on the fluid the computational method divides into three separate parts first the forces exerted by the points of the immersed boundary are calculated and extended to the fluid grid second the fluid equations are solved and the velocities are restricted to the immersed boundary points finally the immersed boundary points are moved to their new locations at the new local velocities the immersed boundary forces are either specified such that the boundary moves at a given velocity zero or non zero or they are specified such that the forces obey some given force law for example hooke s law these forces can be calculated either explicitly or implicitly or a combination of both for the fluid the navier stokes equations are solved on a doubly periodic rectangular grid in a second order accurate manner using a projection method developed by bell colella and glaz 1988 the extension and restriction of the boundary forces and fluid velocities to and from the fluid grid are also done in a second order accurate manner the cochlea is modelled as two fluid chambers separated by a flexible partition whose stiffness varies exponentially along its length the stapes is represented by a moving piston and the bony outer walls are allowed to be either straight or tapered a travelling wave propagates along the flexible partition under the influence of the moving piston and the dependence of this wave on cochlea geometry and partition stiffness is studied also included are studies of bone conduction and transient signal analysis finally comparisons of the model results are made to experimental data and asymptotic results

how large a shift is needed in the shifted helmholtz preconditioner for its effective inversion by multigrid the shifted helmholtz operator has received a lot of attention over the past decade as a preconditioner for the iterative solution of the helmholtz equation the idea is that if one uses a small complex shift the shifted helmholtz operator is still close to the original helmholtz operator and could thus be an effective preconditioner it was shown in m j gander i g graham and e a spence numer math 53 2015 pp 573 579 that the shift can be at most o k to prove rigorously wave number independent convergence of the preconditioned system solved with gmres provided the preconditioner is inverted exactly in practice however the preconditioner is inverted only approximately and if one shifts enough this can be done effectively by standard multigrid methods we show in this paper that for a finite element discretization the shift has to be at least o k 2 to be able to invert the shifted helmholtz preconditioner using multigrid there is therefore a gap between being a good precondi

simulating the dynamics and interactions of flexible fibers in stokes flows the dynamics of slender filaments or fibers suspended in stokesian fluids are fundamental to understanding many flows arising in physics biology and engineering such filaments can have aspect ratios of length to radius ranging from a few tens to several thousands full discretizations of such 3d flows are very costly instead we employ a nonlocal slender body theory that yields an integral equation along the filament centerline relating the force exerted on the body to the filament velocity this hydrodynamical description takes into account the effect of the filament on the fluid and is extended to capture the interaction of multiple filaments as mediated by the intervening fluid we consider filaments that are inextensible and elastic replacing the force in the slender body integral equation by an explicit expression that uses euler bernoulli theory to model bending and tensile forces yields an integral expression for the velocity of the filament centerlines coupled to auxiliary integro differential equations for the filament tensions based on a regularized version of these slender body equations that is asymptotically equivalent to the original formulation we construct a numerical method which uses a combination of finite differences implicit time stepping to avoid severe stability constraints and special quadrature methods for nearly singular integrals we present simulations of single flexible filaments as well as multiple interacting filaments evolving in a background shear flow these simulations show shear induced buckling and relaxation of the filaments leading to the storage and release of elastic energy these dynamics are responsible for the development of positive first normal stress differences commonly associated with visco elastic fluids that are suspensions of microscopic elastic fibers