finite element approximations of a ladyzhenskaya model for stationary incompressible viscous flow some finite element approximation procedures are presented for a model proposed by ladyzhenskaya for stationary incompressible viscous flow the approximate problems are proved to be well posed and stable under standard assumptions on the finite element families the solutions of the approximate problems converge to the solution of the original problem under minimum regularity assumptions some error estimates are derived the optimal order of accuracy is assured with or even without using exact integration rules in the approximation procedure iterative methods for solving the discrete nonlinear problems and comments on some computational experiments are provided special attention is also paid to the common properties as well as differences between the approximation procedure presented here and the approximation for the stationary navier stokes equations

high lewis number combustion wavefronts a perturbative melnikov analysis the wavefronts associated with a one dimensional combustion model with arrhenius kinetics and no heat loss are analyzed within the high lewis number perturbative limit this situation in which fuel diffusivity is small in comparison to that of heat is appropriate for highly dense fluids a formula for the wavespeed is established by a nonstandard application of melnikov s method and slow manifold theory from dynamical systems and compared to numerical results a simple characterization of the wavespeed correction is obtained it is proportional to the ratio between the exothermicity parameter and the lewis number the perturbation method developed herein is also applicable to more general coupled reaction diffusion equations with strongly differing diffusivities the stability of the wavefronts is also tested using a numerical evans function method

detection of signals by information theoretic criteria a new approach is presented to the problem of detecting the number of signals in a multichannel time series based on the application of the information theoretic criteria for model selection introduced by akaike aic and by schwartz and rissanen mdl unlike the conventional hypothesis testing based approach the new approach does not requite any subjective threshold settings the number of signals is obtained merely by minimizing the aic or the mdl criteria simulation results that illustrate the performance of the new method for the detection of the number of signals received by a sensor array are presented

a nonlinear programming algorithm for solving semidefinite programs via low rank factorization in this paper we present a nonlinear programming algorithm for solving semidefinite programs sdps in standard form the algorithm s distinguishing feature is a change of variables that replaces the symmetric positive semidefinite variable x of the sdp with a rectangular variable r according to the factorization x rr t the rank of the factorization i e the number of columns of r is chosen minimally so as to enhance computational speed while maintaining equivalence with the sdp fundamental results concerning the convergence of the algorithm are derived and encouraging computational results on some large scale test problems are also presented

on model order estimation for partially observed markov chains abstract model order estimation for the partially observed markov chain pomc also called the hidden markov model is addressed model order is defined as the number of independent model parameters simulation is used to investigate the small sample performance of a number of model order criteria as applied to pomc models with low state dimension and binary observations consistency of these criteria in the context of pomc models with discrete state and observation space is also investigated a criterion due to akaike is shown not to be consistent simulation results indicate that akaike s criterion can provide more accurate estimation in the small sample case

variational bound on energy dissipation in plane couette flow we present numerical solutions to the extended doering constantin variational principle for upper bounds on the energy dissipation rate in turbulent plane couette flow using the compound matrix technique in order to reformulate this principle s spectral constraint we derive a system of equations that is amenable to numerical treatment in the entire range from low to asymptotically high reynolds numbers our variational bound exhibits a minimum at intermediate reynolds numbers and reproduces the busse bound in the asymptotic regime as a consequence of a bifurcation of the minimizing wavenumbers there exist two length scales that determine the optimal upper bound the effective width of the variational profile s boundary segments and the extension of their flat interior part

boundary layer techniques for solving the helmholtz equation in the presence of small inhomogeneities we consider solutions to the helmholtz equation in two and three dimensions based on layer potential techniques we provide for such solutions a rigorous systematic derivation of complete asymptotic expansions of perturbations resulting from the presence of diametrically small inhomogeneities with constitutive parameters different from those of the background medium it is expected that our results will find important applications for developing effective algorithms for reconstructing small dielectric inhomogeneities from boundary measurements

on the determination of the grad div criterion abstract grad div stabilization adding a term grad div u to penalize violation of incompressibility has proven to be a useful tool in the simulation of incompressible flows such a term requires a choice of the coefficient and studies have begun appearing with various suggestions for its value we give an analysis herein that provides a restricted range of possible values for the coefficient in 3d turbulent flows away from walls if u l denote the large scale velocity and length respectively and is the signal to noise ratio of the body force estimates suggest that should be restricted to the range 2 24 l u 2 4 r e l u mesh independent case 2 24 l u 2 4 h l 4 3 l u mesh dependent case

an automatic orthonormalization method for solving stiff boundary value problems abstract a new initial value method is described based on a remark by drury for solving stiff linear differential two point cigenvalue and boundary value problems the method is extremely reliable it is especially suitable for high order differential systems and it is capable of accommodating realms of stiffness which other methods cannot reach the key idea behind the method is to decompose the stiff differential operator into two non stiff operators one of which is nonlinear the nonlinear one is specially chosen so that it advances an orthonormal frame indeed the method is essentially a kind of automatic orthonormalization the second is auxiliary but it is needed to determine the required function the usefulness of the method is demonstrated by calculating some eigenfunctions for an orr sommerfeld problem when the reynolds number is as large as 10

an algebraic formula for the accelerated computation of the low frequency scattering coefficients an explicit expression for the rapid computation of the low frequency scattering coefficients is presented although the demonstration of the methodology is restricted to the simple case of an acoustically soft spherical scatterer the introduced main concept can be applied to a wide class of scattering problems the procedure based on algebraic calculations only provides an analytic expression of the corresponding scattering coefficients in a straightforward fashion the proposed algorithm is readily implemented furnishing an efficient way for obtaining any order approximation of the total field in the exterior of the scatterer

decomposition in conic optimization with partially separable structure decomposition techniques for linear programming are difficult to extend to conic op timization problems with general nonpolyhedral convex cones because the conic inequalities introduce an additional nonlinear coupling between the variables however in many applications the convex cones have a partially separable structure that allows them to be characterized in terms of simpler lower dimensional cones the most important example is sparse semidefinite programming with a chordal sparsity pattern here partial separability derives from the clique decomposition theorems that characterize positive semidefinite and positive semidefinite completable matrices with chordal sparsity patterns the paper describes a decomposition method that exploits partial separability in conic linear optimization the method is based on spingarn s method for equality constrained convex optimization combined with a fast interior point method for evaluating proximal operators

the compound matrix method for ordinary differential systems abstract a generalization of the compound matrix method is presented to deal with eigenvalue and boundary value problems involving unstable systems of ordinary differential equations details are given for fourth and sixth order problems next is shown that a simple equivalence relation exists between the compounds of the solution matrices of the eigenvalue problems and their adjoints and how this relation can be exploited to simplify the calculation of the adjoint eigenfunctions is discussed using the orr sommerfeld problem as an example it is also shown how the techniques used in the derivation of certain auxiliary systems which play a crucial role in the generalization of the compound matrix method can also provide an alternate method for the direct computation of certain quantities defined in terms of the eigenfunctions finally there is a brief discussion of the relationship between the compound matrix method anf the riccati method

stability and instability of solitary waves of the fifth order kdv equation a numerical framework abstract the spectral problem associated with the linearization about solitary waves of the generalized fifth order kdv equation is formulated in terms of the evans function a complex analytic function whose zeros correspond to eigenvalues a numerical framework based on a fast robust shooting algorithm on exterior algebra spaces is introduced the complete algorithm has several new features including a rigorous numerical algorithm for choosing starting values a new method for numerical analytic continuation of starting vectors the role of the grassmannian g 2 c 5 in choosing the numerical integrator and the role of the hodge star operator for relating 2 c 5 and 3 c 5 and deducing a range of numerically computable forms for the evans function the algorithm is illustrated by computing the stability and instability of solitary waves of the fifth order kdv equation with polynomial nonlinearity

optimization with affine homogeneous quadratic integral inequality constraints we introduce a new technique to optimize a linear cost function subject to an affine homogeneous quadratic integral inequality i e the requirement that a homogeneous quadratic integral functional affine in the optimization variables is nonnegative over a space of functions defined by homogeneous boundary conditions such problems arise in control and stability or input to state output analysis of systems governed by partial differential equations pdes particularly fluid dynamical systems we derive outer approximations for the feasible set of a homogeneous quadratic integral inequality in terms of linear matrix inequalities lmis and show that a convergent sequence of lower bounds for the optimal cost can be computed with a sequence of semidefinite programs sdps we also obtain inner approximations in terms of lmis and sum of squares constraints so upper bounds for the optimal cost and strictly feasible points for the integral inequality can be computed with sdps we present quinopt an open source add on to yalmip to aid the formulation and solution of our sdps and demonstrate our techniques on problems arising from the stability analysis of pdes

grassmannian spectral shooting we present a new numerical method for computing the pure point spectrum associated with the linear stability of coherent structures in the context of the evans function shooting and matching approach all the relevant information is carried by the flow projected onto the underlying grassmann manifold we show how to numerically construct this projected flow in a stable and robust manner in particular the method avoids representation singularities by in practice choosing the best coordinate patch representation for the flow as it evolves the method is analytic in the spectral parameter and of complexity bounded by the order of the spectral problem cubed for large systems it represents a competitive method to those recently developed that are based on continuous orthogonalization we demonstrate this by comparing the two methods in three applications boussinesq solitary waves autocatalytic travelling waves and the ekman boundary layer

fast computation of the maslov index for hyperbolic linear systems with periodic coefficients the maslov index is a topological property of periodic orbits of finite dimensional hamiltonian systems that is widely used in semiclassical quantization quantum chaology stability of waves and classical mechanics the maslov index is determined from the analysis of a linear hamiltonian system with periodic coefficients in this paper a numerical scheme is devised to compute the maslov index for hyperbolic linear systems when the phase space has a low dimension the idea is to compute on the exterior algebra of the ambient vector space where the lagrangian subspace representing the unstable subspace is reduced to a line when the exterior algebra is projectified the lagrangian subspace always forms a closed loop the idea is illustrated by application to hamiltonian systems on a phase space of dimension 4 the theory is used to compute the maslov index for the spectral problem associated with periodic solutions of the fifth order korteweg de vries equation

variational bounds on shear driven turbulence and turbulent boussinesq convection abstract based on earlier studies by hopf 1941 doering and constantin 1992 1994 1995 have recently formulated a new background technique for obtaining upper bounds on turbulent fluid flow quantities this method produces upper bounds on the limit supremum of long time averages making no statistical assumptions about the flow in contrast to the well known howard busse approach the full optimisation problems posed by this method for the momentum transport in turbulent couette flow and the heat transport with zero background flow in turbulent boussinesq convection are solved here for the first time at asymptotically large reynolds number and rayleigh number within busse s multiple boundary layer approximation to extract the best lowest possible upper bounds available intriguingly the original bounds isolated by busse 1969 1970 within the confines of statistical stationarity are recovered exactly using this new formalism the optimal background velocity profile for turbulent couette flow is found to be shearless in the interior thus differing from busse s 1 4 mean shear result in the convective case an interesting degeneracy in the formulation of the background variational problem leads to an indeterminacy in the optimal background temperature profile only for one special choice is the isothermal core feature of busse s mean profile recovered

numerical exterior algebra and the compound matrix method the compound matrix method which was first proposed for numerically integrating systems of differential equations in hydrodynamic stability on k 2 3 dimensional subspaces of mathbb c n by using compound matrices as coordinates is reformulated in a coordinate free way using exterior algebra spaces bigwedge k mathbb c n

instability of the hocking stewartson pulse and its implications for three dimensional poiseuille flow the linear stability problem for the hocking stewartson pulse obtained by linearizing the complex ginzburg landau cgl equation is formulated in terms of the evans function a complex analytic function whose zeros correspond to stability exponents a numerical algorithm based on the compound matrix method is developed for computing the evans function using values in the cgl equation associated with spanwise modulation of plane poiseuille flow we show that the hocking stewartson pulse associated with points along the neutral curve is always linearly unstable due to a real positive eigenvalue implications for the spanwise structure of nonlinear poiseuille problem between parallel plates are also discussed

computing the maslov index of solitary waves part 2 phase space with dimension greater than four this paper extends the theory of the maslov index of solitary waves in part 1 to the case where the phase space is of dimension greater than four the starting point is hamiltonian pdes in one space dimension and time whose steady part is a hamiltonian ode with a phase space of dimension six or greater this steady hamiltonian ode is the main focus of the paper homoclinic orbits of the steady ode represent solitary waves of the pde and one of the properties of the homoclinic orbits is the maslov index we develop formulae for the maslov index the maslov angle and its subangles in an exterior algebra framework and develop numerical algorithms to compute them in addition a new numerical approach based on a discrete qr algorithm is proposed the maslov index is of interest for classifying solitary waves and as an indicator of stability or instability of the solitary wave in the time dependent problem the theory is applied to a class of reaction diffusion equations the longwave shortwave resonance equations and the seventh order kdv equation

state of the art a survey of partially observable markov decision processes theory models and algorithms this paper surveys models and algorithms dealing with partially observable markov decision processes a partially observable markov decision process pomdp is a generalization of a markov decision process which permits uncertainty regarding the state of a markov process and allows for state information acquisition a general framework for finite state and action pomdp s is presented next there is a brief discussion of the development of pomdp s and their relationship with other decision processes a wide range of models in such areas as quality control machine maintenance internal auditing learning and optimal stopping are discussed within the pomdp framework lastly algorithms for computing optimal solutions to pomdp s are presented

local minima and convergence in low rank semidefinite programming the low rank semidefinite programming problem lrsdpr is a restriction of the semidefinite programming problem sdp in which a bound r is imposed on the rank of x and it is well known that lrsdpr is equivalent to sdp if r is not too small in this paper we classify the local minima of lrsdpr and prove the optimal convergence of a slight variant of the successful yet experimental algorithm of burer and monteiro 5 which handles lrsdpr via the nonconvex change of variables x rrt in addition for particular problem classes we describe a practical technique for obtaining lower bounds on the optimal solution value during the execution of the algorithm computational results are presented on a set of combinatorial optimization relaxations including some of the largest quadratic assignment sdps solved to date

exploiting sparsity in semidefinite programming via matrix completion i general framework a critical disadvantage of primal dual interior point methods compared to dual interior point methods for large scale semidefinite programs sdps has been that the primal positive semidefinite matrix variable becomes fully dense in general even when all data matrices are sparse based on some fundamental results about positive semidefinite matrix completion this article proposes a general method of exploiting the aggregate sparsity pattern over all data matrices to overcome this disadvantage our method is used in two ways one is a conversion of a sparse sdp having a large scale positive semidefinite matrix variable into an sdp having multiple but smaller positive semidefinite matrix variables to which we can effectively apply any interior point method for sdps employing a standard block diagonal matrix data structure the other way is an incorporation of our method into primal dual interior point methods which we can apply directly to a given sdp in part ii of this article we will investigate an implementation of such a primal dual interior point method based on positive definite matrix completion and report some numerical results

energy transfer and dissipation in forced isotropic turbulence a model for the reynolds number dependence of the dimensionless dissipation rate ce was derived from the dimensionless karman howarth equation resulting in ce ce c rl o 1 rl where rl is the integral scale reynolds number the coefficients c and ce arise from asymptotic expansions of the dimensionless second and third order structure functions this theoretical work was supplemented by direct numerical simulations dns of forced isotropic turbulence for integral scale reynolds numbers up to rl 5875 r 435 which were used to establish that the decay of dimensionless dissipation with increasing reynolds number took the form of a power law rln with exponent value n 1 000 0 009 and that this decay of ce was actually due to the increase in the taylor surrogate u3 l the model equation was fitted to data from the dns which resulted in the value c 18 9 1 3 and in an asymptotic value for ce in the infinite reynolds number limit of ce 0 468 0 006

stability of viscous detonations for majda s model abstract using analytical and numerical evans function techniques we examine the spectral stability of strong detonation wave solutions of a version of majda s scalar model for a reacting gas mixture with an arrhenius type ignition function we introduce an energy estimate to limit possible unstable eigenvalues to a compact region in the unstable complex half plane and we use a numerical approximation of the evans function to search for possible unstable eigenvalues in this region our results show for the parameter values tested that these waves are spectrally stable combining these numerical results with the pointwise green function analysis of lyng raoofi texier zumbrun g lyng m raoofi b texier k zumbrun pointwise green function bounds and stability of combustion waves j differential equations 233 2 2007 654 698 we conclude that these waves are nonlinearly stable this represents the first demonstration of nonlinear stability for detonation wave solutions of a majda type model without a smallness assumption notably our results indicate that for this simplified scalar model there does not occur either in a normal parameter range or in the limit of high activation energy hopf bifurcation to galloping or pulsating solutions as is observed in the full reactive navier stokes equations this answers in the negative for this model a question posed by majda as to whether such scalar detonation models capture this aspect of detonation behavior

enhancement of near cloaking part ii the helmholtz equation the aim of this paper is to extend the method of ammari et al commun math phys 2012 to scattering problems we construct very effective near cloaking structures for the scattering problem at a fixed frequency these new structures are before using the transformation optics layered structures and are designed so that their first scattering coefficients vanish inside the cloaking region any target has near zero scattering cross section for a band of frequencies we analytically show that our new construction significantly enhances the cloaking effect for the helmholtz equation

approximation of stationary statistical properties of dissipative dynamical systems time discretization we consider temporal approximation of stationary statistical properties of dissipative infinite dimensional dynamical systems we demonstrate that stationary statistical properties of the time discrete approximations i e numerical scheme converge to those of the underlying continuous dissipative infinite dimensional dynamical system under three very natural assumptions as the time step approaches zero the three conditions that are sufficient for the convergence of the stationary statistical properties are 1 uniform dissipativity of the scheme in the sense that the union of the global attractors for the numerical approximations is pre compact in the phase space 2 convergence of the solutions of the numerical scheme to the solution of the continuous system on the unit time interval 0 1 uniformly with respect to initial data from the union of the global attractors 3 uniform continuity of the solutions to the continuous dynamical system on the unit time interval 0 1 uniformly for initial data from the union of the global attractors the convergence of the global attractors is established under weaker assumptions an application to the infinite prandtl number model for convection is discussed

kolmogorov theory via finite time averages abstract several relations from the kolmogorov theory of fully developed three dimensional turbulence are rigorously established for finite time averages over leray hopf weak solutions of the navier stokes equations the navier stokes equations are considered with periodic boundary conditions and an external forcing term the main parameter is the grashof number associated with the forcing term the relations rigorously proved in this article include estimates for the energy dissipation rate the kolmogorov wavenumber the taylor wavenumber the reynolds number and the energy cascade process for some estimates the averaging time depends on the macroscale wavenumber and the kinematic viscosity alone while for others such as the kolmogorov energy dissipation law and the energy cascade the estimates depend also on the grashof number as compared with earlier works by some of the authors the more physical concept of finite time average is replacing the concept of infinite time average used before

on the maslov index of multi pulse homoclinic orbits multi pulse homoclinic orbits of hamiltonian systems on can be classified by a sequence of integers in this paper we find the surprising result that this string of integers encodes the value of the maslov index of the homoclinic orbit our results include a computable formulation of the maslov index for homoclinic orbits and a robust numerical method for the evaluation of the maslov index

the orr sommerfeld equation on infinite intervals numerical and asymptotic methods have been developed the last several years to solve the orr sommerfeld os stability equation for boundary layers and other unbounded domain flows these are surveyed and some of the abstract mathematical questions they raise are mentioned there has been a conventional wisdom among fluid dynamicists that it has not been proved that the eigenvalues of the os equations are complete on an unbounded domain there is something to this since without the continuous spectrum which occurs for boundary layer flows the eigenvalues are not complete however much has been proved concerning the nature of the spectrum and the expansion properties of the generalized eigenfunctions the results of miklavcic and williams and of bakenko and herron form the core of the survey

convergence of time averaged statistics of finite element approximations of the navier stokes equations when discussing numerical solutions of the navier stokes equations especially when turbulent flows are concerned there are at least two questions that can be raised what is meaningful to compute how to determine the fidelity of the computed solution with respect to the true solution this paper takes a step towards the answer of these questions for turbulent flows we consider long time averages of weak solutions of the navier stokes equations rather than strong solutions we present error estimates for the time averaged energy dissipation rate drag and lift most of them under the assumption of small reynolds generalized grashof number for shear flows we address the question of fidelity of the computed solution with respect to the true solution in view of kolmogorov s energy cascade theory

variational bounds in dissipative systems abstract we present a general method for estimating the space time average dissipation in a large class of dissipative pdes the method leads to variational problems with spectral side conditions

an algebraic calculation method for the acoustic low frequency expansion abstract in the present work we propose an analytical method for the algebraic calculation of the low frequency expansion of the solution of a scalar scattering problem upon a smooth starshaped scatterer the method is based on the far field expansion theorem introduced by atkinson and wilcox in the middle of the last century applied in the low frequency realm in this view the need for solving elliptical boundary value problems to obtain the low frequency approximations is replaced by an analytical procedure easily encoded in an algorithm including only algebraic operators as a demonstration example the method is applied to the acoustic scattering problem with plane wave excitation upon three different non penetrable spherical scatterers the first low frequency approximations are deduced yielding the validity of the proposed method by both recovering already known results and accurately deriving higher orders of approximation

analysis of mesh effects on turbulent flow statistics abstract turbulence models such as the smagorinsky model herein are used to represent the energy lost from resolved to under resolved scales due to the energy cascade i e non linearity analytic estimates of the energy dissipation rates of a few turbulence models have recently appeared but none yet study the energy dissipation restricted to resolved scales i e after spatial discretization with h micro scale we do so herein for the smagorinsky model upper bounds are derived on the computed time averaged energy dissipation rate e u h for an under resolved mesh h for turbulent shear flow for coarse mesh size o r e 1 h l it is proven e u h c s h 2 l 5 c s 4 h l 5 2 c s 4 h 3 2 u 3 l where u and l are global velocity and length scale and c s and are model parameters this upper bound being independent of the viscosity at high reynolds number is in accord with the equilibrium dissipation law kolmogrov s conventional turbulence theory this estimate suggests over dissipation for any of c s 0 and 0 consistent with numerical evidence on the effects of model viscosity without wall damping function moreover the analysis indicates that the turbulent boundary layer is a more important length scale for shear flow than the kolmogorov microscale

two very accurate and efficient methods for computing eigenvalues and eigenfunctions in porous convection problems we develop the compound matrix method and the chebyshev tau method to be applicable to linear and nonlinear stability problems for convection in porous media in a natural way it is shown how to obtain highly accurate answers to problems which may be stiff and spurious eigenvalues are avoided a detailed analysis is provided for a porous convection problem of much current interest namely convection with a horizontally varying temperature gradient

on the shock wave spectrum for isentropic gas dynamics with capillarity abstract we consider the stability problem for shock layers in slemrod s model of an isentropic gas with capillarity we show that these traveling waves are monotone in the weak capillarity case and become highly oscillatory as the capillarity strength increases using a spectral energy estimate we prove that small amplitude monotone shocks are spectrally stable we also show that monotone shocks have no unstable real spectrum regardless of amplitude this implies that any instabilities of these monotone traveling waves if they exist must occur through a hopf like bifurcation where one or more conjugate pairs of eigenvalues cross the imaginary axis we then conduct a systematic numerical evans function study which shows that monotone and mildly oscillatory profiles in an adiabatic gas are spectrally stable for moderate values of shock and capillarity strengths in particular we show that the transition from monotone to nonmonotone profiles does not appear to trigger any instabilities

the orr sommerfeld equation on a manifold the most effective and widely used methods for integrating the orr sommerfeld equation by shooting are the continuous orthogonalization method and the compound matrix method in this paper we consider this problem from a differential geometric point of view a new definition of orthogonalization is presented restriction of the orr sommerfeld to a complex stiefel manifold and this definition leads to a new formulation of continuous orthogonalization which differs in a precise and interesting geometric way from existing orthogonalization routines present orthogonalization methods based on davey9s algorithm are shown to have a different differential geometric interpretation restriction of the orr sommerfeld equation to a complex grassmanian manifold this leads us to introduce the concept of a grassmanian integrator which preserves linear independence and not necessarily orthogonality using properties of grassmanian manifolds and their tangent spaces a new grassmanian integrator is introduced that generalizes davey9s algorithm furthermore it is shown that the compound matrix method is a dual grassmanian integrator it uses plucker coordinates for integrating on a grassmanian manifold and this characterization suggests a new algorithm for constructing the compound matrices extension of the differential geometric framework to general systems of linear ordinary differential equations is also discussed

an efficient shooting algorithm for evans function calculations in large systems abstract in evans function computations of the spectra of asymptotically constant coefficient linear operators a basic issue is the efficient and numerically stable computation of subspaces evolving according to the associated eigenvalue ode for small systems a fast shooting algorithm may be obtained by representing subspaces as single exterior products j c alexander r sachs linear instability of solitary waves of a boussinesq type equation a computer assisted computation nonlinear world 2 4 1995 471 507 l q brin numerical testing of the stability of viscous shock waves ph d thesis indiana university bloomington 1998 l q brin numerical testing of the stability of viscous shock waves math comp 70 235 2001 1071 1088 l q brin k zumbrun analytically varying eigenvectors and the stability of viscous shock waves in seventh workshop on partial differential equations part i 2001 rio de janeiro mat contemp 22 2002 19 32 t j bridges g derks g gottwald stability and instability of solitary waves of the fifth order kdv equation a numerical framework physica d 172 1 4 2002 190 216 for large systems however the dimension of the exterior product space quickly becomes prohibitive growing as n k where n is the dimension of the system written as a first order ode and k typically n 2 is the dimension of the subspace we resolve this difficulty by the introduction of a simple polar coordinate algorithm representing pure monomial products as scalar multiples of orthonormal bases for which the angular equation is a numerically optimized version of the continuous orthogonalization method of drury davey a davey an automatic orthonormalization method for solving stiff boundary value problems j comput phys 51 2 1983 343 356 l o drury numerical solution of orr sommerfeld type equations j comput phys 37 1 1980 133 139 and the radial equation is evaluable by quadrature notably the polar coordinate method preserves the important property of analyticity with respect to parameters

efficient computation of analytic bases in evans function analysis of large systems in evans function computations of the spectra of asymptotically constant coefficient linearized operators of large systems a problem that becomes important is the efficient computation of global analytically varying bases for invariant subspaces of the limiting coefficient matrices in the case that the invariant subspace is spectrally separated from its complementary invariant subspace we propose an efficient numerical implementation of a standard projection based algorithm of kato for which the key step is the solution of an associated sylvester problem this may be recognized as the analytic cousin of a c k algorithm developed by dieci and collaborators based on orthogonal projection rather than eigenprojection as in our case for a one dimensional subspace it reduces essentially to an algorithm of bridges derks and gottwald based on path finding and continuation methods

variational bounds on energy dissipation in incompressible flows iii convection building on a method of analysis for the navier stokes equations introduced by hopf math ann 117 764 1941 a variational principle for upper bounds on the largest possible time averaged convective heat flux is derived from the boussinesq equations of motion when supplied with appropriate test background fields satisfying a spectral constraint reminiscent of an energy stability condition the variational formulation produces rigorous upper bounds on the nusselt number nu as a function of the rayleigh number ra for the case of vertical heat convection between parallel plates in the absence of sidewalls a simplified but rigorous formulation of the optimization problem yields the large rayleigh number bound nu ensuremath le 0 167 mathrm ra 1 2 1 nonlinear euler lagrange equations for the optimal background fields are also derived which allow us to make contact with the upper bound theory of howard j fluid mech 17 405 1963 for statistically stationary flows the structure of solutions of the euler lagrange equations are elucidated from the geometry of the variational constraints which sheds light on busse s j fluid mech 37 457 1969 asymptotic analysis of general solutions to howard s euler lagrange equations the results of our analysis are discussed in the context of theory recent experiments and direct numerical simulations textcopyright 1996 the american physical society

error analysis of a hybrid method for computing lyapunov exponents in a previous paper beyn and lust in numer math 113 357 375 2009 we suggested a numerical method for computing all lyapunov exponents of a dynamical system by spatial integration with respect to an ergodic measure the method extended an earlier approach of aston and dellnitz comput methods appl mech eng 170 223 237 1999 for the largest lyapunov exponent by integrating the diagonal entries from the qr decomposition of the jacobian for an iterated map in this paper we provide an asymptotic error analysis of the method for the case in which all lyapunov exponents are simple we employ oseledec multiplicative ergodic theorem and impose certain hyperbolicity conditions on the invariant subspaces that belong to neighboring exponents the resulting error expansion shows that one step of extrapolation is enough to obtain exponential decay of errors

optimal bounds with semidefinite programming an application to stress driven shear flows we introduce an innovative numerical technique based on convex optimization to solve a range of infinite dimensional variational problems arising from the application of the background method to fluid flows in contrast to most existing schemes we do not consider the euler lagrange equations for the minimizer instead we use series expansions to formulate a finite dimensional semidefinite program sdp whose solution converges to that of the original variational problem our formulation accounts for the influence of all modes in the expansion and the feasible set of the sdp corresponds to a subset of the feasible set of the original problem moreover sdps can be easily formulated when the fluid is subject to imposed boundary fluxes which pose a challenge for the traditional methods we apply this technique to compute rigorous and near optimal upper bounds on the dissipation coefficient for flows driven by a surface stress we improve previous analytical bounds by more than 10 times and show that the bounds become independent of the domain aspect ratio in the limit of vanishing viscosity we also confirm that the dissipation properties of stress driven flows are similar to those of flows subject to a body force localized in a narrow layer near the surface finally we show that sdp relaxations are an efficient method to investigate the energy stability of laminar flows driven by a surface stress

geometric phase in the hopf bundle and the stability of non linear waves we develop a stability index for the travelling waves of non linear reaction diffusion equations using the geometric phase induced on the hopf bundle s2n 1 cns2n 1 cn this can be viewed as an alternative formulation of the winding number calculation of the evans function whose zeroes correspond to the eigenvalues of the linearization of reaction diffusion operators about the wave the stability of a travelling wave can be determined by the existence of eigenvalues of positive real part for the linear operator our method of geometric phase for locating and counting eigenvalues is inspired by the numerical results in way s dynamics in the hopf bundle the geometric phase and implications for dynamical systems way 2009 we provide a detailed proof of the relationship between the phase and eigenvalues for dynamical systems defined on c2c2 and sketch the proof of the method of geometric phase for cncn and its generalization to boundary value problems implementing the numerical method modified from way 2009 we conclude with open questions inspired from the results

exploiting sparsity in semidefinite programming via matrix completion ii implementation and numerical results in part i of this series of articles we introduced a general framework of exploiting the aggregate sparsity pattern over all data matrices of large scale and sparse semidefinite programs sdps when solving them by primal dual interior point methods this framework is based on some results about positive semidefinite matrix completion and it can be embodied in two different ways one is by a conversion of a given sparse sdp having a large scale positive semidefinite matrix variable into an sdp having multiple but smaller positive semidefinite matrix variables the other is by incorporating a positive definite matrix completion itself in a primal dual interior point method the current article presents the details of their implementations we introduce new techniques to deal with the sparsity through a clique tree in the former method and through new computational formulae in the latter one numerical results over different classes of sdps show that these methods can be very efficient for some problems

combustion waves in hydraulically resistant porous media in a special parameter regime abstract in this paper we study the stability of fronts in a reduction of a well known pde system that is used to model the combustion in hydraulically resistant porous media more precisely we consider the original pde system under the assumption that one of the parameters of the model the lewis number is chosen in a specific way and with initial conditions of a specific form for a class of initial conditions then the number of unknown functions is reduced from three to two for the reduced system the existence of combustion fronts follows from the existence results for the original system the stability of these fronts is studied here by a combination of energy estimates and numerical evans function computations and nonlinear analysis when applicable we then lift the restriction on the initial conditions and show that the stability results obtained for the reduced system extend to the fronts in the full system considered for that specific value of the lewis number the fronts that we investigate are proved to be either absolutely unstable or convectively unstable on the nonlinear level

computing the maslov index of solitary waves part 1 hamiltonian systems on a four dimensional phase space abstract when solitary waves are characterized as homoclinic orbits of a finite dimensional hamiltonian system they have an integer valued topological invariant the maslov index we develop a new robust numerical algorithm to compute the maslov index to understand its properties and to study the implications for the stability of solitary waves the algorithm reported here is developed in the exterior algebra representation which leads to a fast algorithm with some novel properties new results on the maslov index for solitary wave solutions of reaction diffusion equations the fifth order korteweg de vries equation and the long wave short wave resonance equations are presented part 1 considers the case of a four dimensional phase space and part 2 considers the case of a 2 n dimensional phase space with n 2

dynamic scaling laws in two types of extended hamiltonian systems at dissipation onset the transition to dissipation in one dimensional extended hamiltonian systems with saddle node bifurcations of stationary solutions is characterized three different systems are studied i nonlinear schrodinger flow past a localized obstacle ii sine gordon pendulum chains forced by a local torque iii electrically charged nonlinear schrodinger flows in case i no frequency gap is present in the dispersion relation in contrast in cases ii and iii a minimum frequency for propagating waves exists in the gapless case the growth rates of the unstable modes and the frequency of supercritical soliton emission are found to scale as the square root of the bifurcation parameter no subcriticality is observed in contrast when a frequency gap is present subcritical soliton emission takes place logarithmic and one fourth power scaling laws are found respectively at the bottom and top of the subcriticality window