two numerical methods for finding multiple solutions of a logistic equation abstract using two numerical methods we will show the existence of multiple solutions for the well known logistic equation u g x u 1 u for x with dirichlet boundary condition

weighted average importance sampling and defensive mixture distributions importance sampling uses observations from one distribution to estimate for another distribution by weighting the observations including the target distribution as one component of a mixture distribution bounds the weights and makes importance sampling more reliable the usual importance sampling estimate is a weighted average with weights that do not sum to 1 we discuss simple normalization and other more efficient normalization methods these innovations make importance sampling useful in a wider variety of problems we demonstrate with a case study of oil inventory reliability at a large utility

central limit theorem for sequential monte carlo methods and its application to bayesian inference the term sequential monte carlo methods or equivalently particle filters refers to a general class of iterative algorithms that performs monte carlo approximations of a given sequence of distributions of interest t we establish in this paper a central limit theorem for the monte carlo estimates produced by these computational methods this result holds under minimal assumptions on the distributions t and applies in a general framework which encompasses most of the sequential monte carlo methods that have been considered in the literature including the resample move algorithm of gilks and berzuini j r stat soc ser b stat methodol 63 2001 127 146 and the residual resampling scheme the corresponding asymptotic variances provide a convenient measurement of the precision of a given particle filter we study in particular in some typical examples of bayesian applications whether and at which rate these asymptotic variances diverge in time in order to assess the long term reliability of the considered algorithm

positive solutions for some indefinite nonlinear eigenvalue elliptic problems with robin boundary conditions abstract we consider a nonlinear eigenvalue problem with indefinite weight under robin boundary conditions we prove the existence and multiplicity of positive solutions to this end we carry out a detailed study of some linear eigenvalues problems and we use mainly bifurcation and sub supersolution methods

markov chains for exploring posterior distributions several markov chain methods are available for sampling from a posterior distribution two important examples are the gibbs sampler and the metropolis algorithm in addition several strategies are available for constructing hybrid algorithms this paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues on the theoretical side results from the theory of general state space markov chains can be used to obtain convergence rates laws of large numbers and central limit theorems for estimates obtained from markov chain methods these theoretical results can be used to guide the construction of more efficient algorithms for the practical use of markov chain methods standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation

stability radii of systems with stochastic uncertainty and their optimization by output feedback we consider linear plants controlled by dynamic output feedback which are subjected to blockdiagonal stochastic parameter perturbations the stability radii of these systems are characterized and it is shown that for real data the real and the complex stability radii coincide a corresponding result does not hold in the deterministic case even for perturbations of single output feedback type in a second part of the paper we study the problem of optimizing the stability radius by dynamic linear output feedback necessary and sufficient conditions are derived for the existence of a compensator which achieves a suboptimal stability radius these conditions consist of a parametrized riccati equation a parametrized liapunov inequality a coupling inequality and a number of linear matrix inequalities one for each disturbance term the corresponding problem in the deterministic case the optimal mu synthesis problem is still unsolved

convergence of adaptive mixtures of importance sampling schemes in the design of ecient simulation algorithms one is often beset with a poor choice of proposal distributions although the performances of a given kernel can clarify how adequate it is for the problem at hand a permanent on line modification of kernels causes concerns about the validity of the resulting algorithm while the issue is quite complex and most often intractable for mcmc algorithms the equivalent version for importance sampling algorithms can be validated quite precisely we derive sucient convergence conditions for a wide class of population monte carlo algorithms and show that rao blackwellized versions asymptotically achieve an optimum in terms of a kullback divergence criterion while more rudimentary versions simply do not benefit from repeated updating

stability radii of linear systems with respect to stochastic perturbations abstract real and complex stability radii for stable linear systems with respect to structured stochastic multiperturbations are introduced and completely characterized

combining multiple surrogate models to accelerate failure probability estimation with expensive high fidelity models in failure probability estimation importance sampling constructs a biasing distribution that targets the failure event such that a small number of model evaluations is sufficient to achieve a monte carlo estimate of the failure probability with an acceptable accuracy however the construction of the biasing distribution often requires a large number of model evaluations which can become computationally expensive we present a mixed multifidelity importance sampling mmfis approach that leverages computationally cheap but erroneous surrogate models for the construction of the biasing distribution and that uses the original high fidelity model to guarantee unbiased estimates of the failure probability the key property of our mmfis estimator is that it can leverage multiple surrogate models for the construction of the biasing distribution instead of a single surrogate model alone we show that our mmfis estimator has a mean squared error that is up to a constant lower than the mean squared errors of the corresponding estimators that uses any of the given surrogate models alone even in settings where no information about the approximation qualities of the surrogate models is available in particular our mmfis approach avoids the problem of selecting the surrogate model that leads to the estimator with the lowest mean squared error which is challenging if the approximation quality of the surrogate models is unknown we demonstrate our mmfis approach on numerical examples where we achieve orders of magnitude speedups compared to using the high fidelity model only

numerical simulation of singularly perturbed non linear elliptic boundary value problems using finite element method the present paper presents the finite element solution of two dimensional non linear singularly perturbed elliptic partial differential equation subject to appropriate dirichlet boundary conditions a new fifth order convergent newton type iterative method has been described and used to linearize the non linear problem the inclusion of this newton s method of fifth order convergence in finite element method for solving non linear system of equations reduces the number of iterations and hence the cost of computation to demonstrate the usefulness of the proposed scheme a non convex variational ginzburg landau equation is considered

control under stochastic multiplicative uncertainties part ii optimal design for performance this paper studies the optimal control design problem for linear discrete time systems with stochastic multiplicative uncertainties these uncertainties are assumed to be present in the control inputs and modeled as independent and identically distributed i i d random processes the optimal performance under study is defined in the mean square sense referred to as the mean square optimal h 2 performance it is shown that the mean square optimal h 2 control problem via state feedback can be solved using a mean square stabilizing solution to a modified algebraic riccati equation mare a necessary and sufficient condition for the existence of this solution is presented which constitutes a generalization of the solution to the classic optimal h 2 state feedback design problem whereas the latter can be obtained by solving an algebraic riccati equation are it is also proven that the optimal control design problem can be cast as an eigenvalue problem evp for the output feedback case with possible input delays we show that the mean square optimal h 2 control problem also amounts to solving an mare when the plant has no nonminimum phase zeros from the inputs to the measurement outputs that is the global optimal solution is obtained by solving an mare incorporating the delays the implication is that in this case a separation principle still holds

positive mountain pass solutions for a semilinear elliptic equation with a sign changing weight function abstract we use the mountain pass theorem to study the existence and multiplicity of positive solutions of the generalisation of the well known logistic equation u g x u x 1 u x with dirichlet boundary conditions to the case where g changes sign

limit theorems for weighted samples with applications to sequential monte carlo methods in the last decade sequential monte carlo methods smc emerged as a key tool in computational statistics see e g sequential monte carlo methods in practice 2001 springer new york monte carlo strategies in scientific computing 2001 springer new york complex stochastic systems 2001 109 173 these algorithms approximate a sequence of distributions by a sequence of weighted empirical measures associated to a weighted population of particles which are generated recursively despite many theoretical advances see e g j roy statist soc ser b 63 2001 127 146 ann statist 33 2005 1983 2021 feynman kac formulae genealogical and interacting particle systems with applications 2004 springer ann statist 32 2004 2385 2411 the large sample theory of these approximations remains a question of central interest in this paper we establish a law of large numbers and a central limit theorem as the number of particles gets large we introduce the concepts of weighted sample consistency and asymptotic normality and derive conditions under which the transformations of the weighted sample used in the smc algorithm preserve these properties to illustrate our findings we analyze smc algorithms to approximate the filtering distribution in state space models we show how our techniques allow to relax restrictive technical conditions used in previously reported works and provide grounds to analyze more sophisticated sequential sampling strategies including branching resampling at randomly selected times and so on

a finite element approach for finding positive solutions of a logistic equation with a sign changing weight function in this article we describe the mathematical modeling and use finite element approximation to study the existence and multiplicity of numerical positive solutions of well known generalized logistic equation with sign changing weight function we investigate the range of the real parameter involved in this problem to achieve the numerical solutions and discuss the behavior of the branch of the solutions using matlab 7 0

a new perspective on the robustness of markov jump linear systems this paper is concerned with the robust analysis and control of continuous time markov jump linear systems the centerpiece of the paper is an alternative to the small gain theorem which hinges on the robustification of a certain adjoint lyapunov operator by means of this technique it is proven that the small gain theorem of markov jump linear systems may sometimes be arbitrarily conservative even when nonlinear lipschitz disturbances are taken into account the adjoint approach on the other hand provides the maximal degree of robustness for this particular setup in addition we prove that the adjoint design of controllers is solved more efficiently than the design based on small gain analysis bearing these new facts in mind we derive an iterative algorithm for the design of robust controllers based on linear matrix inequalities by means of numerical examples regarding the robust control of an underactuated manipulator arm and of a simplified power systems model it is shown that the adjoint design methodology can be much more advantageous than its small gain counterpart

on linear and nonlinear fourth order eigenvalue problems with indefinite weight abstract we determine the principal eigenvalues of the linear indefinite weight problem u x r g x u x x 0 1 u 0 u 1 u 0 u 1 0 moreover we investigate the existence of positive solutions for the corresponding nonlinear indefinite weight problem where g 0 1 r is a continuous function which attains both positive and negative values f c r r and r is a parameter

a numerical method to find positive solution of semilinear elliptic dirichlet problems abstract in this work we present a numerical approach for finding positive solutions for the elliptic problems of the type u f x g u for x with dirichlet boundary condition we will show that in which range of this problem achieves a numerical solution and what is the behavior of the branch of this solutions

adaptive markov chain monte carlo through regeneration abstract markov chain monte carlo mcmc is used for evaluating expectations of functions of interest under a target distribution this is done by calculating averages over the sample path of a markov chain having as its stationary distribution for computational efficiency the markov chain should be rapidly mixing this sometimes can be achieved only by careful design of the transition kernel of the chain on the basis of a detailed preliminary exploratory analysis of an alternative approach might be to allow the transition kernel to adapt whenever new features of are encountered during the mcmc run however if such adaptation occurs infinitely often then the stationary distribution of the chain may be disturbed we describe a framework based on the concept of markov chain regeneration which allows adaptation to occur infinitely often but does not disturb the stationary distribution of the chain or the consistency of sample path averages

obituary to a j pritchard on the 12th of august 2007 tony pritchard professor of mathematics at the university of warwick died suddenly at the age of 70 from a heart attack while on holiday with his family on the island of lanzarote both of the authors have had the privilege of working with him for many years we have lost both a colleague and a friend the following notes are dedicated to his memory

sequential monte carlo samplers summary we propose a methodology to sample sequentially from a sequence of probability distributions that are defined on a common space each distribution being known up to a normalizing constant these probability distributions are approximated by a cloud of weighted random samples which are propagated over time by using sequential monte carlo methods this methodology allows us to derive simple algorithms to make parallel markov chain monte carlo algorithms interact to perform global optimization and sequential bayesian estimation and to compute ratios of normalizing constants we illustrate these algorithms for various integration tasks arising in the context of bayesian inference

rates of convergence of the hastings and metropolis algorithms we apply recent results in markov chain theory to hastings and metropolis algorithms with either independent or symmetric candidate distributions and provide necessary and sufficient conditions for the algorithms to converge at a geometric rate to a prescribed distribution pi in the independence case in mathbb r k these indicate that geometric convergence essentially occurs if and only if the candidate density is bounded below by a multiple of pi in the symmetric case in mathbb r only we show geometric convergence essentially occurs if and only if pi has geometric tails we also evaluate recently developed computable bounds on the rates of convergence in this context examples show that these theoretical bounds can be inherently extremely conservative although when the chain is stochastically monotone the bounds may well be effective

recursive monte carlo filters algorithms and theoretical analysis recursive monte carlo filters also called particle filters are a powerful tool to perform the computations in general state space models we discuss and compare the accept reject version with the more common sampling importance resampling version of the algorithm in particular we show how auxiliary variable methods and stratification can be used in the accept reject version and we compare dierent resampling techniques in a second part we show laws of large numbers and a central limit theorem for these monte carlo filters by simple induction arguments that need only weak conditions we also show that under stronger conditions the required sample size is independent of the length of the observed series ams 2000 subject classifications primary 62m09 secondary 60g35 60j22 65c05

adaptive approximate bayesian computation sequential techniques can be adapted to the abc algorithm to enhance its efficiency for instance when sisson et al 2007 introduced the abc prc algorithm the goal was to improve upon existing abc mcmc algorithms marjoram et al 2003 while the abc prc method is based upon the theoretical developments of del moral et al 2006 the application to the setting of approximate bayesian computation induces a bias in the approximation to the posterior distribution of interest as we demonstrate in this paper via both theoretical reasoning and experimental results it is however possible to devise an alternative version based on genuine importance sampling arguments that we call abc pmc in connection with the population monte carlo method introduced in cappe et al 2004 furthermore the abc pmc algorithm is simpler than the abc prc algorithm in that it does not require any backward transition kernel and proposes an automatic scaling of the forward kernel in this paper we demonstrate the applicability of abc pmc and compare its performances with abc prc

existence and multiplicity of positive solutions of a nonlinear eigenvalue problem with indefinite weight function this paper is concerned with the existence multiplicity and stability of positive solutions of an indefinite weight boundary value problemu email protected t f u 0 0

multifidelity importance sampling estimating statistics of model outputs with the monte carlo method often requires a large number of model evaluations this leads to long runtimes if the model is expensive to evaluate importance sampling is one approach that can lead to a reduction in the number of model evaluations importance sampling uses a biasing distribution to sample the model more efficiently but generating such a biasing distribution can be difficult and usually also requires model evaluations a different strategy to speed up monte carlo sampling is to replace the computationally expensive high fidelity model with a computationally cheap surrogate model however because the surrogate model outputs are only approximations of the high fidelity model outputs the estimate obtained using a surrogate model is in general biased with respect to the estimate obtained using the high fidelity model we introduce a multifidelity importance sampling mfis method which combines evaluations of both the high fidelity and a surrogate model it uses a surrogate model to facilitate the construction of the biasing distribution but relies on a small number of evaluations of the high fidelity model to derive an unbiased estimate of the statistics of interest we prove that the mfis estimate is unbiased even in the absence of accuracy guarantees on the surrogate model itself the mfis method can be used with any type of surrogate model such as projection based reduced order models and data fit models furthermore the mfis method is applicable to black box models i e where only inputs and the corresponding outputs of the high fidelity and the surrogate model are available but not the details of the models themselves we demonstrate on nonlinear and time dependent problems that our mfis method achieves speedups of up to several orders of magnitude compared to monte carlo with importance sampling that uses the high fidelity model only

the calculation of posterior distributions by data augmentation comment a noniterative sampling importance resampling alternative to the data augmentation algorithm for creating a few imputations when fractions of missing information are modest the sir algorithm

numerical solution of the three dimensional advection diffusion equation the study of advection diffusion equation continues to be an active field of research the subject has important applications to fluid dynamics as well as many other branches of science and engineering in this paper several different numerical techniques will be developed and compared for solving the three dimensional advection diffusion equation with constant coefficient these techniques are based on the two level fully explicit and fully implicit finite difference approximations the basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach developed from the 1974 work of warming and hyett this allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes another nice feature of the modified equivalent partial differential equation approach is that a high order of accuracy can be combined with excellent stability properties the new second order accurate methods are free of numerical diffusion the results of a numerical experiment are presented and the accuracy and central processor cpu time needed are discussed and compared

robust stabilization subject to structured uncertainties and mean power constraint abstract this paper deals with a robust stabilization problem for discrete time systems subject to multiple disturbances occurring in controller and actuating channel where both linear structured uncertainties and white gaussian noises are included the desired control law is aimed to robustly stabilize the system and to satisfy some pre specified mean power constraint simultaneously by the philosophy of the mixed h 2 h control a solvability condition is first derived for single input systems that reveals the intrinsic relation between the unstable poles of the plant and the disturbance parameters together with two cross coupled algebraic riccati equations the result is further generalized to multiple input systems with a sufficient condition given again by the unstable poles of the plant an example is included to illustrate the current results

moment stability of nonlinear discrete stochastic systems with time delays based on h representation technique in this paper mean square stability of nonlinear discrete stochastic systems with time delays is investigated we obtain the second order moment equations of discrete stochastic systems by virtue of the h representation technique by the comparison principle and the established moment equations some moment stability theorems for nonlinear discrete stochastic systems with time delays are established a numerical example is presented to illustrate the efficiency of our results at the end of the paper

global bifurcation of positive solutions from zero in nonlinearizable elliptic problems with indefinite weight abstract this paper is devoted to the study of global bifurcation of solutions from zero of some nonlinearizable eigenvalue problems for second order elliptic partial differential equations with indefinite weight the existence of global continua of solutions emanating from bifurcation intervals surrounding the trivial solutions corresponding to the principal eigenvalues of the linear problem obtained by setting the nonlinear term equal to zero and contained in the classes of positive and negative functions is shown

on discrete stochastic bilinear systems stability abstract mean square stability conditions for discrete time bilinear systems operating in a stochastic environment are given in this paper only independence and wide sense stationarity are required for the second order disturbance sequences involved thus dismissing ergodicity and zero mean assumptions stochastic stability conditions are derived by using a deterministic stability result for a class of separable nonlinear dynamical systems evolving in a banach space

brief stability radii of discrete time stochastic systems with respect to blockdiagonal perturbations we consider stochastic discrete time systems with multiplicative noise which are controlled by dynamic output feedback and subjected to blockdiagonal stochastic parameter perturbations stability radii for these systems are characterized via scaling techniques and it is shown that for real data the real and the complex stability radii coincide in a second part of the paper we investigate the problem of maximizing the stability radii by dynamic output feedback necessary and sufficient conditions are derived for the existence of a stabilizing compensator which ensures that the stability radius is above a prespecified level these conditions consist of parametrized matrix inequalities and a coupling condition