stochastic homogenization of certain nonconvex hamilton jacobi equations abstract in this paper we prove the stochastic homogenization of certain nonconvex hamilton jacobi equations the nonconvex hamiltonians which are generally uneven and inseparable are generated by a sequence of quasiconvex hamiltonians and a sequence of quasiconcave hamiltonians through the min max formula we provide a monotonicity assumption on the contact values between those stably paired hamiltonians so as to guarantee the stochastic homogenization

rank revealing qr factorizations abstract an algorithm is presented for computing a column permutation pi and a qr factorization a pi qr of an m by n m or n matrix a such that a possible rank deficiency of a will be revealed in the triangular factor r having a small lower right block for low rank deficient matrices the algorithm is guaranteed to reveal the rank of a and the cost is only slightly more than the cost of one regular qr factorization a posteriori upper and lower bounds on the singular values of a are derived and can be used to infer the numerical rank of a keywords qr factorization rank deficient matrices least squares computation subset selection rank singular values

nematic nematic phase separation in binary mixtures of thick and thin hard rods results from onsager like theories the fundamental nature of the nematic nematic n n phase separation in binary mixtures of rigid hard rods is analyzed within the onsager second virial theory and the extension of parsons and lee which includes a treatment of the higher body contributions the particles of each component are modeled as hard spherocylinders of different diameter d1d2 but equal length l1 l2 l in the case of a system which is restricted to be fully aligned parallel rods we provide an analytical solution for the spinodal boundary for the limit of stability of n n demixing only a single region of n n coexistence bounded at lower pressures densities by a n n critical point is possible for such a system the full numerical solution with the parsons lee extension also indicates that depending on the length of the particles there is a range of values of the diameter ratio d d2 d1 where the n n phase coexistence is closed off by a critical point at lower pressure a second region of n n coexistence can be found at even lower pressures for certain values of the parameters this region is bounded by an upper n n critical point the two n n coexistence regions can also merge to give a single region of n n coexistence extending to very high pressure without a critical point by including the higher order contributions to the excluded volume end effects in the onsager theory we prove analytically that the existence of the n n lower critical point is a direct consequence of the finite size of the particles a new analytical equation of state is derived for the nematic phase using the gaussian approximation in the case of onsager limit infinite aspect ratio we show that the n n phase behavior obtained using the parsons lee approach substantially deviates from that with the onsager theory for the n n transition due to the nonvanishing third and higher order virial coefficients we also provide a detailed discussion of the n n phase behavior of recent experimental results for mixtures of thin and thick rods of the same length for which the onsager and parsons lee theories can provide a qualitative description

a divide and conquer method for unitary and orthogonal eigenproblems leth n xn be a unitary upper hessenberg matrix whose eigenvalues and possibly also eigenvectors are to be determined we describe how this eigenproblem can be solved by a divide and conquer method in which the matrixh is split into two smaller unitary upper hessenberg matricesh 1 andh 2 by a rank one modification ofh the eigenproblems forh 1 andh 2 can be solved independently and the solutions of these smaller eigenproblems define a rational function whose zeros on the unit circle are the eigenvalues ofh the eigenvector ofh can be determined from the eigenvalues ofh and the eigenvectors ofh 1 andh 2 the outlined splitting of unitary upper hessenberg matrices into smaller such matrices is carried out recursively this gives rise to a divide and conquer method that is suitable for implementation on a parallel computer whenh n xn is orthogonal the divide and conquer scheme simplifies and is described separately our interest in the orthogonal eigenproblem stems from applications in signal processing numerical examples for the orthogonal eigenproblem conclude the paper

homogenization of hamilton jacobi bellman equations with respect to time space shifts in a stationary ergodic medium we consider a family fu t x g 0 of solutions to the equation u t c u 2ch t x ru d0 with the terminal datau t x du x assuming that the dependence of the hamiltonian h t x p on time and space is realized through shifts in a stationary ergodic random medium and that h is convex inp and satisfies certain growth and regularity conditions we show the almost sure locally uniform convergence in time and space of u t x as 0 to the solution u t x of a deterministic averaged equation u t c h ru d 0 u t x d u x the effective hamiltonian h is given by a variational formula

stochastic homogenization of a front propagation problem with unbounded velocity we study the homogenization of hamilton jacobi equations which arise in front propagation problems in stationary ergodic media our results are obtained for fronts moving with possible unbounded velocity we show by an example that the homogenized hamiltonian which always exists may be unbounded in this context we show convergence results if we start with a compact initial front on the other hand if the media satisfies a finite range of dependence condition we prove that the effective hamiltonian is bounded and obtain classical homogenization in this context

stochastic homogenization of hamilton jacobi bellman equations we study the homogenization of some hamilton jacobi bellman equations with a vanishing second order term in a stationary ergodic random medium under the hyperbolic scaling of time and space imposing certain convexity growth and regularity assumptions on the hamiltonian we show the locally uniform convergence of solutions of such equations to the solution of a deterministic effective first order hamilton jacobi equation the effective hamiltonian is obtained from the original stochastic hamiltonian by a minimax formula our homogenization results have a large deviations interpretation for a diffusion in a random environment c 2006 wiley periodicals inc

error estimates and convergence rates for the stochastic homogenization of hamilton jacobi equations we present exponential error estimates and demonstrate an algebraic convergence rate for the homogenization of level set convex hamilton jacobi equations in i i d random environments the first quantitative homogenization results for these equations in the stochastic setting by taking advantage of a connection between the metric approach to homogenization and the theory of first passage percolation we obtain estimates on the fluctuations of the solutions to the approximate cell problem in the ballistic regime away from flat spot of the effective hamiltonian in the sub ballistic regime on the flat spot we show that the fluctuations are governed by an entirely different mechanism and the homogenization may proceed without further assumptions at an arbitrarily slow rate we identify a necessary and sufficient condition on the law of the hamiltonian for an algebraic rate of convergence to hold in the sub ballistic regime and show under this hypothesis that the two rates may be merged to yield comprehensive error estimates and an algebraic rate of convergence for homogenization our methods are novel and quite different from the techniques employed in the periodic setting although we benefit from previous works in both first passage percolation and homogenization the link between the rate of homogenization and the flat spot of the effective hamiltonian which is related to the nonexistence of correctors is a purely random phenomenon observed here for the first time

a schur decomposition for hamiltonian matrices abstract a schur type decomposition for hamiltonian matrices is given that relies on unitary symplectic similarity transformations these transformations preserve the hamiltonian structure and are numerically stable making them ideal for analysis and computation using this decomposition and a special singular value decomposition for unitary symplectic matrices a canonical reduction of the algebraic riccati equation is obtained which sheds light on the sensitivity of the nonnegative definite solution after presenting some real decompositions for real hamiltonian matrices we look into the possibility of an orthogonal symplectic version of the qr algorithm suitable for hamiltonian matrices a finite step initial reduction to a hessenberg type canonical form is presented however no extension of the francis implicit shift technique was found and reasons for the difficulty are given

stochastic homogenization of nonconvex hamilton jacobi equations in one space dimension abstract we prove stochastic homogenization for a general class of coercive nonconvex hamilton jacobi equations in one space dimension some properties of the effective hamiltonian arising in the nonconvex case are also discussed

properties of a quadratic matrix equation and the solution of the continuous time algebraic riccati equation abstract we discuss some properties of a quadratic matrix equation with some restrictions then use these results on the algebraic riccati equation to get a new algorithm the algorithm sufficiently takes account of the structure of the associated matrix hence it is very effective

homogenization for stochastic hamilton jacobi equations homogenization asks whether average behavior can be discerned from partial differential equations that are subject to high frequency fluctuations when those fluctuations result from a dependence on two widely separated spatial scales we prove homogenization for certain stochastic hamilton jacobi partial differential equations the idea is to use the subadditive ergodic theorem to establish the existence of an average in the infinite scale separation limit in some cases we also establish a central limit theorem

matrix factorizations for symplectic qr like methods abstract symplectic qr like methods use symplectic or unitary symplectic similarity transformations instead of the usual unitary similarity transformations in the qr process a fundamental problem for the development of such methods is the choice of a suitable type of decomposition a sr corresponding to the qr decomposition where s is symplectic or unitary symplectic decompositions of this type are studied with regard to their application in a qr like process

on hamiltonian and symplectic hessenberg forms abstract characterizations are given for the hamiltonian matrices that can be reduced to hamiltonian hessenberg form and the symplectic matrices that can be reduced to symplectic hessenberg form by orthogonal symplectic similarity transformations the reduction to these special hessenberg forms is the missing link in the solution of the open problem of constructing a stable structure preserving qr like method of complexity o n 3 for the computation of invariant subspaces of hamiltonian and symplectic matrices our considerations lead us to propose an approach to the computation of lagrangian invariant subspaces of a hamiltonian or symplectic matrix

the qr algorithm for unitary hessenberg matrices abstract let h be an n n unitary right hessenberg matrix with positive subdiagonal elements using what we call the schur parameterization of h we show how one step of the shifted qr algorithm for h can be carried out in o n arithmetic operations coupled with the shift strategy of eberlein and huang 3 this will permit computation of the spectrum of h to machine precision in o n 2 operations one potential application is the computation of gauss szego quadrature formulas 12 given the associated schur parameters 7 the weights can also be computed by direct analogy with 6

stochastic homogenization of nonconvex hamilton jacobi equations a counterexample this paper provides a counterexample to hamilton jacobi homogenization in the nonconvex case for general stationary ergodic environments

min max formulas and other properties of certain classes of nonconvex effective hamiltonians this paper is the first attempt to systematically study properties of the effective hamiltonian overline h arising in the periodic homogenization of some coercive but nonconvex hamilton jacobi equations firstly we introduce a new and robust decomposition method to obtain min max formulas for a class of nonconvex overline h secondly we analytically and numerically investigate other related interesting phenomena such as quasi convexification and breakdown of symmetry of overline h from other typical nonconvex hamiltonians finally in the appendix we show that our new method and those a priori formulas from the periodic setting can be used to obtain stochastic homogenization for the same class of nonconvex hamilton jacobi equations some conjectures and problems are also proposed

a numerically stable structure preserving method for computing the eigenvalues of real hamiltonian or symplectic pencils a new method is presented for the numerical computation of the generalized eigenvalues of real hamiltonian or symplectic pencils and matrices the method is numerically backward stable and preserves the structure i e hamiltonian or symplectic in the case of a hamiltonian matrix the method is closely related to the square reduced method of van loan but in contrast to that method which may suffer from a loss of accuracy of order sqrt varepsilon where varepsilon is the machine precision the new method computes the eigenvalues to full possible accuracy

a new method for computing the stable invariant subspace of a real hamiltonian matrix abstract a new backward stable structure preserving method of complexity o n3 is presented for computing the stable invariant subspace of a real hamiltonian matrix and the stabilizing solution of the continuous time algebraic riccati equation the new method is based on the relationship between the invariant subspaces of the hamiltonian matrix h and the extended matrix 0 h h 0 and makes use of the symplectic urv like decomposition that was recently introduced by the authors

a rigidity result for effective hamiltonians with 3 mode periodic potentials abstract we continue studying an inverse problem in the theory of periodic homogenization of hamilton jacobi equations proposed in 14 let v 1 v 2 c r n be two given potentials which are z n periodic and h 1 h 2 be the effective hamiltonians associated with the hamiltonians 1 2 p 2 v 1 1 2 p 2 v 2 respectively a main result in this paper is that if the dimension n 2 and each of v 1 v 2 contains exactly 3 mutually non parallel fourier modes then h 1 h 2 v 1 x v 2 x c x 0 for all x t 2 r 2 z 2 for some c q 0 and x 0 t 2 when n 3 the scenario is slightly more subtle and a complete description is provided for any dimension these resolve partially a conjecture stated in 14 some other related results and open problems are also discussed

exact and approximate correctors for stochastic hamiltonians the 1 dimensional case we perform a qualitative investigation of critical hamilton jacobi equations with stationary ergodic hamiltonian in dimension 1 we show the existence of approximate correctors give characterizing conditions for the existence of correctors provide lax type representation formulae and establish comparison principles the results are applied to look into the corresponding effective hamiltonian and to study a homogenization problem in the analysis a crucial role is played by tools from stochastic geometry such as for instance closed random stationary sets

a sublinear variance bound for solutions of a random hamilton jacobi equation we estimate the variance of the value function for a random optimal control problem the value function is the solution w of a hamilton jacobi equation with random hamiltonian h p x k p v x in dimension d 2 it is known that homogenization occurs as 0 but little is known about the statistical fluctuations of w our main result shows that the variance of the solution w is bounded by o log the proof relies on a modified poincare inequality of talagrand