sums of squares in real analytic rings let a be an analytic ring we show 1 a has finite pythagoras number if and only if its real dimension is 2 and 2 if every positive semidefinite element of a is a sum of squares then a is real and has real dimension 2

the optimal projection equations for reduced order state estimation the singular measurement noise case the optimal projection equations for reduced order state estimation are generalized to allow for singular i e colored measurement noise the noisy and noise free measurements serve as inputs to dynamic and static estimators respectively the optimal solution is characterized by necessary conditions which involve a pair of oblique projections corresponding to reduced estimator order and singular measurement noise intensity

stochastic inverse heat conduction using a spectral approach an adjoint based functional optimization technique in conjunction with the spectral stochastic finite element method is proposed for the solution of an inverse heat conduction problem in the presence of uncertainties in material data process conditions and measurement noise the ill posed stochastic inverse problem is restated as a conditionally well posed l 2 optimization problem the gradient of the objective function is obtained in a distributional sense by defining an appropriate stochastic adjoint field the l 2 optimization problem is solved using a conjugate gradient approach accuracy and effectiveness of the proposed approach is appraised with the solution of several stochastic inverse heat conduction problems

hierarchical bayesian inference for ill posed problems via variational method this paper investigates a novel approximate bayesian inference procedure for numerically solving inverse problems a hierarchical formulation which determines automatically the regularization parameter and the noise level together with the inverse solution is adopted the framework is of variational type and it can deliver the inverse solution and regularization parameter together with their uncertainties calibrated it approximates the posteriori probability distribution by separable distributions based on kullback leibler divergence two approximations are derived within the framework and some theoretical properties e g variance estimate and consistency are also provided algorithms for their efficient numerical realization are described and their convergence properties are also discussed extensions to nonquadratic regularization nonlinear forward models are also briefly studied numerical results for linear and nonlinear cauchy type problems arising in heat conduction with both smooth and nonsmooth solutions are presented for the proposed method and compared with that by markov chain monte carlo the results illustrate that the variational method can faithfully capture the posteriori distribution in a computationally efficient way

optimal robust filtering with time varying parameter uncertainty stochastic linear systems subject to time varying parameter uncertainties affecting both system dynamics and noise statistics are considered a linear filter is used to estimate a linear combination of the states of the system when the filter is given the stabilizing solution of a suitable riccati equation is shown to yield an upper bound for the covariance of the estimator error the main problem addressed in the paper is the design of an optimal robust filter that minimizes such a covariance bound necessary conditions are given for the existence of an optimal reduced order robust filter as well as necessary and sufficient conditions for the full order case the computation of the optimal filter calls for the solution of a riccati equation that generalizes the standard riccati equation for the kalman filtering problem a numerical example is provided in which the new filter is compared with both the kalman and h filters

some remarks on pythagorean real curve germs let k be a real closed field a real ap curve over k is a 1 dimensional excellent henselian local real domain with residue field k a 1 dimensional noetherian local ring is arf if emb dim b mult b for every local ring b infinitely near to a j lipman amer j math 93 1971 649 685 for n 1 the 2nth pythagoras number p2n of a commutative ring a is the least p 1 p such that any sum of 2nth powers in a is a sum of no more than p2nth powers in a a main purpose of this paper is to affirm the following conjectures proposed by ruiz j algebra 94 1985 no 1 126 144 let a be a real ap curve and let a be pythagorean i e p2 1 then i a is arf ii every local ring infinitely near to a is pythagorean actually the authors obtain a finer result for a real ap curve a the following assertions are equivalent 1 a is arf 2 a is pythagorean 3 p2n 1 for some n 4 p2n 1 for all n here 2 1 is exactly conjecture i and 1 2 reduces conjecture ii to the obvious fact that if a is arf every local ring infinitely near to a is arf too of course the result contains some additional insight into the study of pythagoras s numbers even of higher order of real curve germs

reduction of modes for the solution of inverse natural convection problems abstract the inverse natural convection problem of determining heat flux at the bottom wall of a two dimensional cavity from temperature measurement in the domain is investigated by means of the karhunen loeve galerkin procedure the karhunen loeve galerkin procedure which is a type of galerkin method that employs the empirical eigenfunctions of the karhunen loeve decomposition as basis functions can reduce nonlinear partial differential equations to sets of minimal number of ordinary differential equations by limiting the solution space to the smallest linear subspace that is sufficient to describe the observed phenomena previously it had been demonstrated that the problems of optimal control of burgers equation h m park m w lee y d jang comput methods appl mech engrg 166 1998 289 308 and the navier stokes equation h m park m w lee comput methods appl mech engrg 1999 in press can be solved very efficiently through the reduction of modes based on the karhunen loeve galerkin procedure in the present investigation this technique is applied to the solution of inverse natural convection problem of estimating unknown wall heat flux the performance of the present technique of inverse analysis using the karhunen loeve galerkin procedure is assessed in comparison with a traditional technique employing the boussinesq equation and is found to be very accurate as well as efficient

boundary knot method for some inverse problems associated with the helmholtz equation the boundary knot method is an inherently meshless integration free boundary type radial basis function collocation technique for the solution of partial differential equations in this paper the method is applied to the solution of some inverse problems for the helmholtz equation including the highly ill posed cauchy problem since the resulting matrix equation is badly ill conditioned a regularized solution is obtained by employing truncated singular value decomposition while the regularization parameter for the regularization method is provided by the l curve method numerical results are presented for both smooth and piecewise smooth geometry the stability of the method with respect to the noise in the data is investigated by using simulated noisy data the results show that the method is highly accurate computationally efficient and stable and can be a competitive alternative to existing methods for the numerical solution of the problems copyright 2005 john wiley sons ltd

a variational bayesian method to inverse problems with impulsive noise we propose a novel numerical method for solving inverse problems subject to impulsive noises which possibly contain a large number of outliers the approach is of bayesian type and it exploits a heavy tailed t distribution for data noise to achieve robustness with respect to outliers a hierarchical model with all hyper parameters automatically determined from the given data is described an algorithm of variational type by minimizing the kullback leibler divergence between the true posteriori distribution and a separable approximation is developed the numerical method is illustrated on several one and two dimensional linear and nonlinear inverse problems arising from heat conduction including estimating boundary temperature heat flux and heat transfer coefficient the results show its robustness to outliers and the fast and steady convergence of the algorithm

robust state estimation for uncertain systems with averaged integral quadratic constraints this paper considers a robust state estimation problem for a new class of uncertain systems the new uncertainty class introduced in the paper involves structured uncertainties which are required to satisfy a certain averaged integral quadratic constraint this uncertainty class combines aspects of norm bounded and stochastic uncertainty descriptions the solution to the robust state estimation problem is obtained by solving a parametrized riccati differential equation of the game type

expanded uncertainty quantification in inverse problems hierarchical bayes and empirical bayes a common way to account for uncertainty in inverse problems is to apply bayes rule and obtain a posterior distribution of the quantities of interest given a set of measurements a conventional bayesian treatment however requires assuming specific values for parameters of the prior distribution and of the distribution of the measurement errors e g the standard deviation of the errors in practice these parameters are often poorly known a priori and choosing a particular value is often problematic moreover the posterior uncertainty is computed assuming that these parameters are fixed if they are not well known a priori the posterior uncertainties have dubious value this paper describes extensions to the conventional bayesian treatment that assign uncertainty to the parameters defining the prior distribution and the distribution of the measurement errors these extensions are known in the statistical literature as empirical bayes and hierarchical bayes we demonstrate the practical applicati

recursive state estimation for uncertain systems with an integral quadratic constraint this paper considers a robust state estimation problem for a class of uncertain systems where the noise and uncertainty are modeled deterministically via an integral quadratic constraint the robust state estimation problem involves constructing the set of all possible states at the current time consistent with given output measurements and the integral quadratic constraint

conjugate gradient method for the robin inverse problem associated with the laplace equation this paper studies a non linear inverse problem associated with the laplace equation of identifying the robin coefficient from boundary measurements a variational formulation of the problem is suggested thereby transforming it into an optimization problem mathematical properties relevant to its numerical computation are established the optimization problem is solved using the conjugate gradient method in conjunction with the discrepancy principle and the algorithm is implemented using the boundary element method numerical results are presented for several benchmark problems with both exact and noisy data and the convergence of the algorithm with respect to mesh refinement and decreasing the amount of noise in the data is investigated copyright 2006 john wiley sons ltd

a stochastic inverse solution for transient groundwater flow parameter identification and reliability analysis in this paper a stochastic approach is developed for solving the inverse problem of parameter identification for transient groundwater flow adjoint state equations are derived for the stochastic partial differential equation spde relating transient head and log hydraulic conductivity perturbations the derived equations can be used to calculate the covariance matrix of head observations and the cross covariance matrix between head observations and log hydraulic conductivity measurements for each observation time as well as the covariance and cross covariance matrices between different observation times the number of simulation runs required for obtaining these matrices is equal to the number of head observations the reliability of model prediction is evaluated through the variance estimate method using adjoint sensitivity analysis and the cokriging estimate the reliability evaluation of model prediction can be used as an aid in the optimization of experimental design for sampling strategies a numerical example is given to illustrate the stochastic inverse procedure for a general transient flow problem the example shows that using head observations of all observation times simultaneously produces much better results than using them sequentially commonly known as the quasi steady approach the example also shows that it is possible to obtain a reliable stochastic inverse solution for meeting a given prediction objective based on a limited number of observations

high order collocation methods for differential equations with random inputs recently there has been a growing interest in designing efficient methods for the solution of ordinary partial differential equations with random inputs to this end stochastic galerkin methods appear to be superior to other nonsampling methods and in many cases to several sampling methods however when the governing equations take complicated forms numerical implementations of stochastic galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations on the other hand the traditional sampling methods e g monte carlo methods are straightforward to implement but they do not offer convergence as fast as stochastic galerkin methods in this paper a high order stochastic collocation approach is proposed similar to stochastic galerkin methods the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence however the numerical implementation of stochastic collocation is trivial as it requires only repetitive runs of an existing deterministic solver similar to monte carlo methods the computational cost of the collocation methods depends on the choice of the collocation points and we present several feasible constructions one particular choice based on sparse grids depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods

a probabilistic approach to multivariable robust filtering and open loop control a new approach to robust filtering prediction and smoothing of discrete time signal vectors is presented linear time invariant filters are designed to be insensitive to spectral uncertainty in signal models the goal is to obtain a simple design method leading to filters which are not overly conservative modeling errors are described by sets of models parameterized by random variables with known covariances these covariances could either be estimated from data or be used as robustness tuning knobs a robust design is obtained by minimizing the spl hscrsub 2 norm or equivalently the mean square estimation error averaged with respect to the assumed model errors a polynomial solution based on an averaged spectral factorization and a unilateral diophantine equation is derived the robust estimator is referred to as a cautious wiener filter it turns out to be only slightly more complicated to design than an ordinary wiener filter the methodology can be applied to any open loop filtering or control problem in particular we illustrate this for the design of robust multivariable feedforward regulators decoupling and model matching filters

boundary velocity control of incompressible flow with an application to viscous drag reduction an optimal boundary control problem for the navier stokes equations is presented the control is the velocity on the boundary which is constrained to lie in a closed convex subset of h 1 2 of the boundary a necessary condition for optimality is derived computations are done when the control set is actually finite dimensional resulting in an application to viscous drag reduction

fast bayesian approach for parameter estimation this paper presents two techniques i e the proper orthogonal decomposition pod and the stochastic collocation method scm for constructing surrogate models to accelerate the bayesian inference approach for parameter estimation problems associated with partial differential equations pod is a model reduction technique that derives reduced order models using an optimal problem adapted basis to effect significant reduction of the problem size and hence computational cost scm is an uncertainty propagation technique that approximates the parameterized solution and reduces further forward solves to function evaluations the utility of the techniques is assessed on the non linear inverse problem of probabilistically calibrating scalar robin coefficients from boundary measurements arising in the quenching process and non destructive evaluation a hierarchical bayesian model that handles flexibly the regularization parameter and the noise level is employed and the posterior state space is explored by the markov chain monte carlo the numerical results indicate that significant computational gains can be realized without sacrificing the accuracy

pythagoras numbers of real irreducible algebroid curves and gram matrices abstract using elementary techniques of linear algebra in relation with gram matrices we bound the pythagoras number of a real irreducible algebroid curve by its multiplicity

geometric convergence and central limit theorems for multidimensional hastings and metropolis algorithms we develop results on geometric ergodicity of markov chains and apply these and other recent results in markov chain theory to multidimensional hastings and metropolis algorithms for those based on random walk candidate distributions we find sufficient conditions for moments and moment generating functions to converge at a geometric rate to a prescribed distribution by phrasing the conditions in terms of the curvature of the densities we show that the results apply to all distributions with positive densities in a large class which encompasses many commonly used statistical forms from these results we develop central limit theorems for the metropolis algorithm converse results showing non geometric convergence rates for chains where the rejection rate is not bounded away from unity are also given these show that the negative definiteness property is not redundant

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

robust filtering and feedforward control based on probabilistic descriptions of model errors abstract a new approach to robust estimation of signals prediction of time series and robust feedforward control is considered modelling errors are parametrized by random variables with known covariances a robust design is obtained by minimizing the squared estimation error averaged both with respect to model errors and the noise a polynomial equations approach based on averaged spectral factorizations and averaged diophantine equations is derived mild solvability conditions guarantee the existence of stable optimal filters and feedforward regulators the robust design turns out to be no more complicated than the design of an ordinary wiener filter or lqg regulator the proposed approach avoids two drawbacks of minimax design first probabilistic descriptions of model uncertainties may have soft bounds these are more readily obtainable in a noisy environment than the hard bounds required for minimax design furthermore not only the range of uncertainties but also their likelihood is taken into account common model deviations will have a greater impact on an estimator design than do very rare worst cases the conservativeness is thus reduced

on the pythagoras numbers of real analytic rings abstract we show that the pythagoras number of a real analytic ring of dimension 2 is finite bounded by a function of the multiplicity and the codimension

inverse problems for metal oxide semiconductor field effect transistor contact resistivity inverse problems arising from the identification of transistor contact resistivity and contact window location are studied a one point boundary measurement of the potential is shown to be sufficient to identify each from a one parameter monotone family and such identification is both stable and continuously dependent on the parameter for a nonmonotone family of locations with resistivity given similar results are obtained for measurement made on almost all boundary points

reducing the sensitivity to parameter variations of a minimum order reduced order observer a quantitative upper hound is established which defines the extent to which a minimum order reduced order non adaptive observer may be rendered insensitive to parameter variations that upper bound depends on the number of measurements and the location of questionable parameters within the system matrices an integral squared error objective function is explicitly solved that objective function may be used to assess the sensitivity of the system to parameter errors a numerical example is presented for a continuous observer the results are extended to the discrete ease

empirical reduction of modes for the shape identification problems of heat conduction systems abstract the karhunen loeve galerkin procedure is a type of galerkin method that employs as basis functions the empirical eigenfunctions of the karhunen loeve decomposition and proved to be a powerful tool for reducing the degree of freedom of partial differential equations int j numer methods engrg 41 1998 1131 comput methods appl mech engrg 188 2000 165 in the present investigation we extend the applicability of the karhunen loeve galerkin procedure to systems of variable domains and apply it to the shape identification problems of heat conduction systems where the unknown boundary shape is estimated from temperature measurements on the other boundary the efficiency and accuracy of the present technique are assessed as compared to the traditional technique employing the original partial differential equation

robust reduced order nonstrictly proper state estimation via the optimal projection equations with guaranteed cost bounds a state estimation design problem involving parametric plant uncertainties is considered an estimation error bound suggested by multiplicative white noise modeling is utilized for guaranteeing robust estimation over a specified range of parameter uncertainties necessary conditions that generalize the optimal projection equations for reduced order state estimation are used to characterize the estimator that minimizes the error bound the design equations thus effectively serve as sufficient conditions for synthesizing robust estimators additional features include the presence of a static estimation gain in conjunction with the dynamic kalman estimator to obtain a nonstrictly proper estimator

feedback control of natural convection abstract an applicable method is developed for the identification and feedback control of natural convection the boussinesq equation is reduced to a small set of ordinary differential equations by means of the karhunen loeve galerkin procedure int j heat mass transfer 39 1996 3311 based on this low dimensional dynamic model a feedback control synthesis is constructed by first performing an extended kalman filter estimate of the velocity and temperature fields to treat the measurement errors and then developing the optimal feedback law by means of the linear quadratic regulator theory the present method allows for the practical implementation of modern control concepts to many flow systems including natural convection

control input separation by actuation mode expansion for flow control problems a control input separation method is proposed for reduced order modelling in boundary control problems the dynamics of flow systems are typically described by partial differential equations where the input affects the system through boundary conditions from a control design perspective it is most desirable and natural to employ finite dimensional representations in which the input enters the dynamics directly the method proposed here to resolve the input from the boundary conditions is based on obtaining a proper orthogonal decomposition of the unforced flow of the system and then augmenting this decomposition by optimally computed actuation modes built using snapshots of the actuated flow a reduced order galerkin model is then derived for this expansion in which the input appears as an explicit term in the system dynamics the model reduces exactly to the original baseline case under zero input conditions the proposed method is then compared to an existing input separation technique namely the s

velocity log upscaling based on reversible jump markov chain monte carlo simulated annealing abstractupscaling log measurements acquired at high frequencies and correlating them with corresponding low frequency values from surface seismic and vertical seismic profile data is a challenging task we have applied a sampling technique called the reversible jump markov chain monte carlo rjmcmc method to this problem a key property of our approach is that it treats the number of unknowns itself as a parameter to be determined specifically we have considered upscaling as an inverse problem in which we considered the number of coarse layers layer boundary depths and material properties as the unknowns the method applies bayesian inversion with rjmcmc sampling and uses simulated annealing to guide the optimization at each iteration the algorithm will randomly move a boundary in the current model add a new boundary or delete an existing boundary in each case a random perturbation is applied to backus average values we have developed examples showing that the mismatch between seismograms com

sums of squares in real rings let a be an excellent ring we show that if the real dimension of a is at least three then a has in finite pythagoras number and there exists a positive semidefinite element in a which is not a sum of squares in a

the optimal projection equations for reduced order state estimation first order necessary conditions for optimal steady state reduced order state estimation for a linear time invariant plant in the presence of correlated disturbance and nonsingular measurement noise are derived in a new and highly simplified form in contrast to the lone matrix riccati equation arising in the full order kalman filter case the optimal steady state reduced order estimator is characterized by three matrix equations one modified riccati equation and two modified lyapunov equations coupled by a projection whose rank is precisely equal to the order of the estimator and which determines the optimal estimator gains this coupling is a graphic reminder of the suboptimality of proposed approaches involving either model reduction followed by full order estimator design or full order estimator design followed by estimator reduction techniques the results given here complement recently obtained results which characterize the optimal reduced order model by means of a pair of coupled modified lyapunov equations 7 and the optimal fixed order dynamic compensator by means of a coupled system of two modified riceati equations and two modified lyapunov equations 6

the optimal projection equations for reduced order discrete time state estimation for linear systems with multiplicative white noise the optimal projection equations obtained in 2 3 for reduced order discrete time state estimation are generalized to include the effects of state and measurement dependent noise to provide a model of parameter uncertainty in contrast to the single matrix riccati equation arising in the full order kalman filter case the optimal steady state reduced order discrete time estimator is characterized by three matrix equations one modified riccati equation and two modified lyapunov equations coupled by both an oblique projection and stochastic effects

comparing robustness of the kalman h_ infty and ufir filters this paper provides a comparative analysis for robustness of the kalman filter kf h infty filter derived using the game theory and unbiased finite impulse response ufir filter which ignores the noise statistics and initial values a comparison is provided for gaussian models by studying the effects of errors and disturbing factors on the bias correction gain it is shown that the rule of thumb of optimal filtering in terms of accuracy text ufir typically does not hold in the real world implying errors in the noise statistics mismodeling temporary uncertainties and difficulties in filter tuning to optimal mode under such conditions the filters are related to each other as text kf lessgtr h infty a justification of this statement is provided analytically and confirmed by simulations and experimentally based on two state polynomial and harmonic models

a new numerical method for the inverse source problem from a bayesian perspective in this work a new numerical method for the inverse problem of determining a spacewise dependent heat source for a parabolic heat equation is developed we reconstruct the unknown heat source by an augmented tikhonov regularization a tr method derived from a bayesian perspective the a tr method could determine the regularization parameter and detect the noise level automatically numerical results for several benchmark test problems indicate that the a tr method is an accurate and flexible method to determine the unknown spacewise dependent heat source copyright 2010 john wiley sons ltd

a bayesian inference approach to the ill posed cauchy problem of steady state heat conduction this paper studies a bayesian inference approach to the cauchy problem in steady state heat conduction of probabilistically calibrating the boundary temperature the prior modeling is achieved via the markov random field and its regularizing property is investigated a hierarchical bayesian model is adopted for selecting the regularization parameter and detecting the noise level automatically the posterior state space is explored using the markov chain monte carlo for obtaining relevant statistics two augmented tikhonov regularization methods that could determine the regularization parameter and the noise level are proposed and analyzed numerical results indicate that the bayesian inference approach could yield an accurate estimate of the solution with its uncertainties quantified and the augmented tikhonov regularization methods are accurate and flexible copyright 2008 john wiley sons ltd

a meshless method for some inverse problems associated with the helmholtz equation in this paper a new numerical scheme based on the method of fundamental solutions is proposed for the numerical solution of some inverse boundary value problems associated with the helmholtz equation including the cauchy problem since the resulting matrix equation is badly ill conditioned a regularized solution is obtained by employing truncated singular value decomposition while the regularization parameter for the regularization method is provided by the l curve method numerical results are presented for problems on smooth and piecewise smooth domains with both exact and noisy data and the convergence and stability of the scheme are investigated these results show that the proposed scheme is highly accurate computationally efficient stable with respect to the noise in the data and convergent with respect to decreasing the amount of data noise and increasing the distance between the physical and fictitious boundaries and could be considered as a competitive alternative to existing methods for these problems

stochastic spectral methods for efficient bayesian solution of inverse problems we present a reformulation of the bayesian approach to inverse problems that seeks to accelerate bayesian inference by using polynomial chaos pc expansions to represent random variables evaluation of integrals over the unknown parameter space is recast more efficiently as monte carlo sampling of the random variables underlying the pc expansion we evaluate the utility of this technique on a transient diffusion problem arising in contaminant source inversion the accuracy of posterior estimates is examined with respect to the order of the pc representation the choice of pc basis and the decomposition of the support of the prior the computational cost of the new scheme shows significant gains over direct sampling

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

a priori space time separated representation for the reduced order modeling of low reynolds number flows abstract reduced order models roms in fluid dynamics are nowadays mostly developed by performing a projection of the navier stokes equations onto a low dimensional space basis this basis is usually obtained through proper orthogonal decomposition pod which remains one the most efficient techniques to compress precomputed data the main drawback of a posteriori pod based roms is however their lacks of reliability as their parameters are varied preventing their direct uses within optimization algorithms the goal of the present article is to obtain an a priori low dimensional space time separated representation of the fluid fields without precomputed data the approach is based on the use of space time proper generalized decomposition pgd definitions which are successfully applied in several fields but whose uses in fluid dynamics remain scarce their applications to the navier stokes equations are indeed not straightforward due to the pressure velocity coupling the divergence free constraint and the non linear convective term the roms are built here from a space time weak formulation of the chorin temam prediction correction scheme more particularly a priori space time separated representations are obtained by applying the galerkin based progressive pgd definition the minimax pgd definition is moreover experimented on a linear stokes simplified case the related algorithms are explicitly given and illustrated on a transient lid driven cavity flow it is shown that the a priori space time separated representations converge to the full model solution as the decomposition order increases the ability of the resulting rom to learn iteratively from its own error is highlighted the progressive pgd algorithm can be used to effectively enrich incomplete pod and pgd precomputed space bases such as those obtained with different parameter values this may allow to ensure the accuracy of a rom as the parameter is varied which is of crucial interest for optimization problems

boundary control of the navier stokes equation by empirical reduction of modes abstract the karhunen loeve galerkin procedure is a type of galerkin methods that employs the empirical eigenfunctions of the karhunen loeve decomposition as basis functions this technique can reduce nonlinear partial differential equations to sets of minimal number of ordinary differential equations by limiting the solution space to the smallest linear subspace that is sufficient to describe the observed phenomena previously 1 it has been shown that one dimensional burgers equation is reduced to a low dimensional model by this method which is employed to solve boundary optimal control problems very efficiently the present paper demonstrates that the karhunen loeve galerkin procedure can be extended to solve problems of the boundary optimal control of multidimensional navier stokes equations since the reduction of modes in the multidimensional case is much larger than that in the one dimensional case the present technique is found to be more powerful when applied to the control problems of the navier stokes equation than those of the burgers equation

the selection of prior distributions by formal rules abstract subjectivism has become the dominant philosophical foundation for bayesian inference yet in practice most bayesian analyses are performed with so called noninformative priors that is priors constructed by some formal rule we review the plethora of techniques for constructing such priors and discuss some of the practical and philosophical issues that arise when they are used we give special emphasis to jeffreys s rules and discuss the evolution of his viewpoint about the interpretation of priors away from unique representation of ignorance toward the notion that they should be chosen by convention we conclude that the problems raised by the research on priors chosen by formal rules are serious and may not be dismissed lightly when sample sizes are small relative to the number of parameters being estimated it is dangerous to put faith in any default solution but when asymptotics take over jeffreys s rules and their variants remain reasonable choices we also provide an annotated b

a karhunen loeve least squares technique for optimization of geometry of a blunt body in supersonic flow a novel karhunen loeve kl least squares model for the supersonic flow of an inviscid calorically perfect ideal gas about an axisymmetric blunt body employing shock fitting is developed the kl least squares model is used to accurately select an optimal configuration which minimizes drag accuracy and efficiency of the kl method is compared to a pseudospectral method employing global lagrange interpolating polynomials kl modes are derived from pseudospectral solutions at mach 3 5 from a uniform sampling of the design space and subsequently employed as the trial functions for a least squares method of weighted residuals results are presented showing the high accuracy of the method with less than 10 kl modes close agreement is found between the optimal geometry found using the kl model to that found from the pseudospectral solver not including the cost of sampling the design space and building the kl model the kl least squares method requires less than half the central processing unit time as the pseudospectral method to achieve the same level of accuracy a decrease in computational cost of several orders of magnitude as reported in the literature when comparing the kl method against discrete solvers is shown not to hold for the current problem the efficiency is lost because the nature of the nonlinearity renders a priori evaluation of certain necessary integrals impossible requiring as a consequence many costly reevaluations of the integrals

a computational quasi reversiblility method for cauchy problems for laplace s equation this work concerns the use of the method of quasi reversibility for solving cauchy problems for laplace s equation the paper begins with a derivation of an error estimate for this method using car

on the pythagoras numbers of real analytic curves we show that the pythagoras number of a real analytic curve is the supremum of the pythagoras numbers of its singularities or that supremum plus 1 this includes cases when the pythagoras number is infinite

bayesian linearized avo inversion a new linearized avo inversion technique is developed in a bayesian framework the objective is to obtain posterior distributions for p wave velocity s wave velocity and density distributions for other elastic parameters can also be assessed for example acoustic impedance shear impedance and p wave to s wave velocity ratio the inversion algorithm is based on the convolutional model and a linearized weak contrast approximation of the zoeppritz equation the solution is represented by a gaussian posterior distribution with explicit expressions for the posterior expectation and covariance hence exact prediction intervals for the inverted parameters can be computed under the specified model the explicit analytical form of the posterior distribution provides a computationally fast inversion method tests on synthetic data show that all inverted parameters were almost perfectly retrieved when the noise approached zero with realistic noise levels acoustic impedance was the best determined parameter

technical communique an lmi approach to guaranteed cost control of linear uncertain time delay systems the guaranteed cost control problem via memoryless state feedback controllers is studied in this paper for a class of linear time delay systems with norm bounded time varying parametric uncertainty and a given quadratic cost function sufficient conditions for the existence of guaranteed cost controllers are given in terms of linear matrix inequalities lmis furthermore a convex optimization problem with lmi constraints is formulated to design the optimal guaranteed cost controller which minimizes the guaranteed cost of the closed loop uncertain system

robust observers for systems with parameters abstract in many cases linear multivariable models which describe physical systems contain structured parametric type uncertainties specifically it may be that the state space model is expressed in a form which involves parameters we first consider circumstances in which a parameter independent observer can be constructed for the system and suggest conditions that guarantee its existence we also give conditions under which such a robust observer can be subsequently used with additional dynamics in a parameter free configuration for regulation

robust nonfragile kalman filtering for uncertain linear systems with estimator gain uncertainty the note is concerned with the problem of a robust nonfragile kalman filter design for a class of uncertain linear systems with norm bounded uncertainties the designed state estimator can tolerate multiplicative uncertainties in the state estimator gain matrix the robust nonfragile state estimator designs are given in terms of solutions to algebraic riccati equations the designs guarantee known upper bounds on the steady state error covariance a numerical example is given to illustrate the results

an efficient computational method of boundary optimal control problems for the burgers equation abstract the burgers equation is a simple one dimensional model of the navier stokes equation in the present paper we suggest an efficient method of solving optimal boundary control problems of the burgers equation which is practical as well as mathematically rigorous our eventual purpose is to extend this technique to the control problems of viscous fluid flows the present method is based on the karhunen loeve decomposition which is a technique of obtaining empirical eigenfunctions from the experimental or numerical data of a system employing these empirical eigenfunctions as basis functions of a galerkin procedure one can a priori limit the function space considered to the smallest linear subspace that is sufficient to describe the observed phenomena and consequently reduce the burgers equation to a set of ordinary differential equations with a minimum degree of freedom the resulting low dimensional model of burgers equation is shown to simulate the original system almost exactly the present algorithm is well suited for the problems of control or optimization where one has to solve the governing equation repeatedly but one can also estimate the approximate solution space based on the range of control variables the present method of solving boundary control problems of burgers equation employing the lowdimensional model obtained by the karhunen loeve galerkin procedure is found to yield accurate results in a very efficient way when the minimization of the objective function is performed using a conjugate gradient method

an alternating iterative algorithm for the cauchy problem associated to the helmholtz equation in this paper the iterative algorithm proposed by kozlov et al comput maths math phys 31 1991 45 for obtaining approximate solutions to the ill posed cauchy problem for the helmholtz equation is analysed the technique is then numerically implemented using the boundary element method bem the numerical results confirm that the iterative bem produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data an efficient stopping regularising criterion is also proposed

a bayesian inference approach to identify a robin coefficient in one dimensional parabolic problems this paper investigates a nonlinear inverse problem associated with the heat conduction problem of identifying a robin coefficient from boundary temperature measurement a bayesian inference approach is presented for the solution of this problem the prior modeling is achieved via the markov random field mrf the use of a hierarchical bayesian method for automatic selection of the regularization parameter in the function estimation inverse problem is discussed the markov chain monte carlo mcmc algorithm is used to explore the posterior state space numerical results indicate that mrf provides an effective prior regularization and the bayesian inference approach can provide accurate estimates as well as uncertainty quantification to the solution of the inverse problem

expectation propagation for nonlinear inverse problems with an application to electrical impedance tomography in this paper we study a fast approximate inference method based on expectation propagation for exploring the posterior probability distribution arising from the bayesian formulation of nonlinear inverse problems it is capable of efficiently delivering reliable estimates of the posterior mean and covariance thereby providing an inverse solution together with quantified uncertainties some theoretical properties of the iterative algorithm are discussed and the efficient implementation for an important class of problems of projection type is described the method is illustrated with one typical nonlinear inverse problem electrical impedance tomography with complete electrode model under sparsity constraints numerical results for real experimental data are presented and compared with that by markov chain monte carlo the results indicate that the method is accurate and computationally very efficient

implicit sampling for hierarchical bayesian inversion and applications in fractional multiscale diffusion models abstract this paper presents an implicit sampling method for hierarchical bayesian inverse problems a widely used approach for sampling posterior distribution is markov chain monte carlo mcmc however the samples generated by mcmc are usually strongly correlated which may lead to a small size of effective samples the implicit sampling method proposed in chorin and tu 2009 can generate independent samples and capture inherent non gaussian features of the posterior in implicit sampling method the posterior samples are generated by a map and distribution around the maximum a posterior map point for practical applications some parameters in prior density are often unknown and the hierarchical bayesian formulation is used to estimate the map point effectively it is applied to the bayesian inverse problems of multi term time fractional diffusion models in heterogeneous media to effectively capture the heterogeneity effect we present a mixed generalized multiscale finite element method mixed gmsfem to substantially speed up the bayesian inversion finally we carry out a few numerical examples to effectively identify various unknown inputs

prior information and uncertainty in inverse problems solving any inverse problem requires understanding the uncertainties in the data to know what it means to fit the data we also need methods to incorporate data independent prior information to eliminate unreasonable models that fit the data both of these issues involve subtle choices that may significantly influence the results of inverse calculations the specification of prior information is especially controversial how does one quantify information what does it mean to know something about a parameter a priori in this tutorial we discuss bayesian and frequentist methodologies that can be used to incorporate information into inverse calculations in particular we show that apparently conservative bayesian choices such as representing interval constraints by uniform probabilities as is commonly done when using genetic algorithms for example may lead to artificially small uncertainties we also describe tools from statistical decision theory that can be used to characterize the performance of inv

iterative estimation of reflectivity and image texture least squares migration with an empirical bayes approach in many geophysical inverse problems smoothness assumptions on the underlying geology are used to mitigate the effects of nonuniqueness poor data coverage and noise in the data and to improve the quality of the inferred model parameters within a bayesian inference framework a priori assumptions about the probabilistic structure of the model parameters can impose such a smoothness constraint analogous to regularization in a deterministic inverse problem we have considered anempirical bayes generalization of the kirchhoff based least squares migration lsm problem we have developed a novel methodology for estimation ofthe reflectivitymodel andregularization parameters using a bayesian statistical framework that treats both of these as random variables to be inferred from the data hence rather than fixing the regularization parameters prior to inverting for the image we allow the data to dictate where to regularize estimating these regularization parameters givesusinformation about the degree of conditional correlation or lack thereof between neighboring image parameters and subsequently incorporating this information in the final model produces more clearly visible discontinuities in the estimated image the inference framework is verified on 2d synthetic data sets in which the empirical bayes imaging results significantly outperform standard lsm images we note that although we evaluated this method within the context of seismic imaging it is in fact a general methodology that can be applied to any linear inverse problem in which there are spatially varying correlations in the model parameter space

robust static and dynamic output feedback stabilization deterministic and stochastic perspectives three parallel gaps in robust feedback control theory are examined sufficiency versus necessity deterministic versus stochastic uncertainty modeling and stability versus performance deterministic and stochastic output feedback control problems are considered with both static and dynamic controllers the static and dynamic robust stabilization problems involve deterministically modeled bounded but unknown measurable time varying parameter variations while the static and dynamic stochastic optimal control problems feature state control and measurement dependent white noise general sufficiency conditions for the deterministic problems are obtained using lyapunov s direct method while necessary conditions for the stochastic problems are derived as a consequence of minimizing a quadratic performance criterion the sufficiency tests are then applied to the necessary conditions to determine when solutions of the stochastic optimization problems also solve the deterministic robust stability problems as an additional application of the deterministic result the modified riccati equation approach of petersen and hollot is generalized in the static case and extended to dynamic compensation

a bayesian approach to characterizing uncertainty in inverse problems using coarse and fine scale information the bayesian approach allows one to easily quantify uncertainty at least in theory in practice however the markov chain monte carlo mcmc method can be computationally expensive particularly in complicated inverse problems we present a methodology for improving the speed and efficiency of an mcmc analysis by combining runs on different scales by using a coarser scale the chain can run faster particularly when there is an external forward simulator involved in the likelihood evaluation and better explore the posterior being less likely to become stuck in local maxima we discuss methods for linking the coarse chain back to the original fine scale chain of interest the resulting coupled chain can thus be run more efficiently without sacrificing the accuracy achieved at the finer scale