on nonautonomous linear systems of differential and difference equations with r symmetric coefficient matrices abstract let c n n i denote the set of continuous n n matrices on an interval i we say that r c n n i is a nontrivial k involution if r p l 0 k 1 l i d l p 1 where e 2 i k d 0 d 1 d k 1 n and p p l 0 k 1 u l with u l c d l d l i we say that a c n n i is r symmetric if r t a t r 1 t a t t i and we show that if a is r symmetric then solving x a t x or x a t x f t reduces to solving k independent d l d l systems 0 l k 1 we consider the asymptotic behavior of the solutions in the case where i t 0 finally we sketch analogous results for linear systems of difference equations

fast algorithms for bayesian uncertainty quantification in large scale linear inverse problems based on low rank partial hessian approximations we consider the problem of estimating the uncertainty in large scale linear statistical inverse problems with high dimensional parameter spaces within the framework of bayesian inference when the noise and prior probability densities are gaussian the solution to the inverse problem is also gaussian and is thus characterized by the mean and covariance matrix of the posterior probability density unfortunately explicitly computing the posterior covariance matrix requires as many forward solutions as there are parameters and is thus prohibitive when the forward problem is expensive and the parameter dimension is large however for many ill posed inverse problems the hessian matrix of the data misfit term has a spectrum that collapses rapidly to zero we present a fast method for computation of an approximation to the posterior covariance that exploits the low rank structure of the preconditioned by the prior covariance hessian of the data misfit analysis of an infinite dimensional model convection diffusion problem and numerical experiments on large scale three dimensional convection diffusion inverse problems with up to 1 5 million parameters demonstrate that the number of forward pde solves required for an accurate low rank approximation is independent of the problem dimension this permits scalable estimation of the uncertainty in large scale ill posed linear inverse problems at a small multiple independent of the problem dimension of the cost of solving the forward problem

the constrained inverse eigenvalue problem and its approximation for normal matrices abstract in this paper the constrained inverse eigenvalue problem and associated approximation problem for normal matrices are considered the solvability conditions and general solutions of the constrained inverse eigenvalue problem are presented and the expression of the solution for the optimal approximation problem is obtained

matrices a such that as 1r ra with rk i abstract we study matrices a c n n such that a s 1 r r a where r k i n and s k are nonnegative integers with k 2 such matrices are called r s 1 k potent matrices the s 0 case corresponds to matrices such that a r a r 1 with r k i n and is studied using spectral properties of the matrix r for s 1 various characterizations of the class of r s 1 k potent matrices and relationships between these matrices and other classes of matrices are presented

inverse eigenproblem for centrosymmetric and centroskew matrices and their approximation in this paper we first give the solvability condition for the following inverse eigenproblem iep given a set of vectors xi i 1m in cn and a set of complex numbers i i 1m find a centrosymmetric or centroskew matrix c in rn n such that xi i 1m and i i 1m are the eigenvectors and eigenvalues of c respectively we then consider the best approximation problem for the ieps that are solvable more precisely given an arbitrary matrix b in rn n we find the matrix c which is the solution to the iep and is closest to b in the frobenius norm we show that the best approximation is unique and derive an expression for it

two dimensional transonic aerodynamic design method this paper demonstrates the capabilities of a new transonic two dimensional design method based on the simultaneous solution of multiple streamtubes coupled through the position of and pressure at the streamline interfaces this allows the specification of either the airfoil shape direct analysis mode or the airfoil surface pressure distribution inverse design mode the nonlinear system of equations is formulated in a conservative manner which guarantees the correct treatment of shocks and is solved by a rapid newton solution method viscous effects can also be included through a coupled integral boundary layer analysis the first set of results shows the effect of different far field treatments demonstrating the improvement in accuracy obtained by including the second order doublet terms in addition to the usual first order vortex term the results are also compared to those obtained by specifying straight far field streamlines corresponding to solid wall wind tunnel experiments or constant far field pressure corresponding to freejet experiments to show the sensitivity to the farfield distance in the second set of results the design method is used to design a transonic airfoil with c 1 000 at a oo 0 70 it is shown that the off design performance is improved by specifying a surface pressure distribution with a very weak shock

procrustes problems and associated approximation problems for matrices with k involutory symmetries abstract we say that a matrix r c n n is k involutary if its minimal polynomial is x k 1 for some k 2 so r k 1 r 1 and the eigenvalues of r are 1 2 k 1 where e 2 i k let 0 1 k 1 if r c m m a c m n s c n n and r and s are k involutory we say that a is r s symmetric if ras 1 a and a is r s symmetric if ras a let l be the class of m n r s symmetric matrices or the class of m n r s symmetric matrices given x c n t and b c m t we characterize the matrices a in l that minimize ax b frobenius norm and given an arbitrary w c m n we find the unique matrix a l that minimizes both ax b and a w we also obtain necessary and sufficient conditions for existence of a l such that ax b and assuming that the conditions are satisfied characterize the set of all such a

a hessian based method for uncertainty quantification in global ocean state estimation national science foundation u s collaboration in mathematical geosciences grant 0934404

scalable and efficient algorithms for the propagation of uncertainty from data through inference to prediction for large scale problems with application to flow of the antarctic ice sheet the majority of research on efficient and scalable algorithms in computational science and engineering has focused on the forward problem given parameter inputs solve the governing equations to determine output quantities of interest in contrast here we consider the broader question given a large scale model containing uncertain parameters possibly noisy observational data and a prediction quantity of interest how do we construct efficient and scalable algorithms to 1 infer the model parameters from the data the deterministic inverse problem 2 quantify the uncertainty in the inferred parameters the bayesian inference problem and 3 propagate the resulting uncertain parameters through the model to issue predictions with quantified uncertainties the forward uncertainty propagation problem we present efficient and scalable algorithms for this end to end data to prediction process under the gaussian approximation and in the context of modeling the flow of the antarctic ice sheet and its effect on loss of grounded ice to the ocean the ice is modeled as a viscous incompressible creeping shear thinning fluid the observational data come from satellite measurements of surface ice flow velocity and the uncertain parameter field to be inferred is the basal sliding parameter represented by a heterogeneous coefficient in a robin boundary condition at the base of the ice sheet the prediction quantity of interest is the present day ice mass flux from the antarctic continent to the ocean we show that the work required for executing this data to prediction process measured in number of forward and adjoint ice sheet model solves is independent of the state dimension parameter dimension data dimension and the number of processor cores the key to achieving this dimension independence is to exploit the fact that despite their large size the observational data typically provide only sparse information on model parameters this property can be exploited to construct a low rank approximation of the linearized parameter to observable map via randomized svd methods and adjoint based actions of hessians of the data misfit functional

inverse eigenproblems and associated approximation problems for matrices with generalized symmetry or skew symmetry let r cn n be a nontrivial involution i e r r 1 i we say that a cn n is r symmetric r skew symmetric if rar a rar a let s be one of the following subsets of cn n i r symmetric matrices ii hermitian r symmetric matrices iii r skew symmetric matrices iv hermitian r skew symmetric matrices let z cn m with rank z m and diag 1 m the inverse eigenproblem consists of finding z such that the set s z a s az z is nonempty and to find the general form of a s z in all cases we use the special spectral properties of s to essentially characterize the set of admissible pairs z and the special structure of the members of s to obtain the general solution of the inverse eigenproblem given an arbitrary b s the approximation problem consists of finding the unique matrix ab s z that best approximates b in the frobenius norm it is not necessary to assume that r r in connection with the inverse eigenproblem for r symmetric or r skew symmetric matrices however we impose this additional assumption in connection with the inverse eigenproblem for hermitian r symmetric or r skew symmetric matrices and in connection with the approximation problem for i iv

on a matrix group constructed from an r s 1 k potent matrix abstract let r c n n be a k involutory matrix that is r k i n for some integer k 2 and let s be a nonnegative integer a matrix a c n n is called an r s 1 k potent matrix if a satisfies r a a s 1 r in this paper a matrix group corresponding to a fixed r s 1 k potent matrix is explicitly constructed and properties of this group are derived and investigated this group is then reconciled with the classical matrix group g a that is associated with a generalized group invertible matrix a

efficient methods for large scale linear inversion using a geostatistical approach 1 in geophysical inverse problems such as estimating the unknown parameter field from noisy observations of dependent quantities e g hydraulic conductivity from head observations stochastic bayesian and geostatistical approaches are frequently used to obtain best estimates and conditional realizations it is required to perform several matrix matrix computations involving the covariance matrix of the discretized field of the parameters in realistic three dimensional fields that are finely discretized these operations as performed in conventional algorithms become extremely expensive and even prohibitive in terms of memory and computational requirements using hierarchical matrices we show how to reduce the complexity of forming approximate matrix vector products involving the covariance matrices in log linear complexity for an arbitrary distribution of points and a wide variety of generalized covariance functions the resulting system of equations is solved iteratively using a matrix free krylov subspace approach furthermore we show how to generate unconditional realizations using an approximation to the square root of the covariance matrix using chebyshev matrix polynomials and use the above to generate conditional realizations we demonstrate the efficiency of our method on a few standard test problems such as interpolation from noisy observations and contaminant source identification

matrices with multiple symmetry properties applications of centrohermitian and perhermitian matrices abstract twelve known symmetry patterns of matrices are combined with three modest patterns to form a steiner triple system we investigate matrices satisfying more than one symmetry pattern we show how a group of operators on gl n c gives rise to distinct types of matrices which satisfy sets of patterns and which give unique decompositions of matrices into components of each type these give a new characterization of normal and unitary matrices we extend symmetry patterns to vectors to study spectral properties of these matrices when a skew symmetric basis of eigenvectors exist we can infer symmetry properties of these matrices

scalable posterior approximations for large scale bayesian inverse problems via likelihood informed parameter and state reduction two major bottlenecks to the solution of large scale bayesian inverse problems are the scaling of posterior sampling algorithms to high dimensional parameter spaces and the computational cost of forward model evaluations yet incomplete or noisy data the state variation and parameter dependence of the forward model and correlations in the prior collectively provide useful structure that can be exploited for dimension reduction in this setting both in the parameter space of the inverse problem and in the state space of the forward model to this end we show how to jointly construct low dimensional subspaces of the parameter space and the state space in order to accelerate the bayesian solution of the inverse problem as a byproduct of state dimension reduction we also show how to identify low dimensional subspaces of the data in problems with high dimensional observations these subspaces enable approximation of the posterior as a product of two factors i a projection of the posterior onto a low dimensional parameter subspace wherein the original likelihood is replaced by an approximation involving a reduced model and ii the marginal prior distribution on the high dimensional complement of the parameter subspace we present and compare several strategies for constructing these subspaces using only a limited number of forward and adjoint model simulations the resulting posterior approximations can rapidly be characterized using standard sampling techniques e g markov chain monte carlo two numerical examples demonstrate the accuracy and efficiency of our approach inversion of an integral equation in atmospheric remote sensing where the data dimension is very high and the inference of a heterogeneous transmissivity field in a groundwater system which involves a partial differential equation forward model with high dimensional state and parameters

characterization and properties of r s commutative matrices let r pdiag 0im0 1im1 k 1imk 1 p 1 cm m and s qdiag 0 in0 1 in1 k 1 ink 1 q 1 cn n where m0 m1 mk 1 m n0 n1 nk 1 n 0 1 k 1 are distinct complex numbers and zktozk 0 1 k 1 we say that a cm n is r s commutative if ra as we characterize the class of r s commutative matrices and extend results obtained previously for the case where l e2 il k and l l modk 0 l k 1 with zk our results are independent of 0 1 k 1 so long as they are distinct i e if ra as for some choice of 0 1 k 1 all distinct then ra as for arbitrary of 0 1 k 1

dimension independent likelihood informed mcmc many bayesian inference problems require exploring the posterior distribution of high dimensional parameters that represent the discretization of an underlying function this work introduces a family of markov chain monte carlo mcmc samplers that can adapt to the particular structure of a posterior distribution over functions two distinct lines of research intersect in the methods developed here first we introduce a general class of operator weighted proposal distributions that are well defined on function space such that the performance of the resulting mcmc samplers is independent of the discretization of the function second by exploiting local hessian information and any associated low dimensional structure in the change from prior to posterior distributions we develop an inhomogeneous discretization scheme for the langevin stochastic differential equation that yields operator weighted proposals adapted to the non gaussian structure of the posterior the resulting dimension independent and likelihood informed dili mcmc samplers may be useful for a large class of high dimensional problems where the target probability measure has a density with respect to a gaussian reference measure two nonlinear inverse problems are used to demonstrate the efficiency of these dili samplers an elliptic pde coefficient inverse problem and path reconstruction in a conditioned diffusion

algorithms for solving the inverse problem associated with k a k a s 1 in previous papers the authors introduced and characterized a class of matrices called k s 1 potent also they established a method to construct these matrices the purpose of this paper is to solve the associated inverse problem several algorithms are developed in order to find all involutory matrices k satisfying k a s 1 k a for a given matrix a c n n and a given natural number s the cases s 0 and s 1 are separately studied since they produce different situations in addition some examples are presented showing the numerical performance of the methods

the solutions to linear matrix equations ax b ya d with k involutory symmetries let rcmm and scnn be nontrivial k involutions if their minimal polynomials are both xk1 for some k2 i e rk1 r1i and sk1 s1i we say that acmn is r s symmetric if ras1 a and a is r s symmetric if ras a with 0 1 k1 and 0 let s be one of the subsets of all r s symmetric and r s symmetric matrices given xcnr ycsm bcmr and dcsn we characterize the matrices a in s that minimize axb2 yad2 frobenius norm under the assumption that r and s are unitary moreover among the set s x y b d s of the minimizers of axb2 yad2 min we find the optimal approximate matrix as x y b d that minimizes ag to a given unstructural matrix gcmn we also present the necessary and sufficient conditions such that ax b ya d is consistent in s if the conditions are satisfied we characterize the consistent solution set of all such a finally a numerical algorithm and some numerical examples are given to illustrate the proposed results

bayesian calibration of a large scale geothermal reservoir model by a new adaptive delayed acceptance metropolis hastings algorithm 1 the aim of this research is to estimate the parameters of a large scale numerical model of a geothermal reservoir using markov chain monte carlo mcmc sampling within the framework of bayesian inference all feasible parameters that are consistent with the measured data are summarized by the posterior distribution and hence parameter estimation and uncertainty quantification are both given by calculating expected values of statistics of interest over the posterior distribution it appears to be computationally infeasible to use the standard metropolis hastings algorithm mh to sample the high dimensional computationally expensive posterior distribution to improve the sampling efficiency a new adaptive delayed acceptance mh algorithm adamh is implemented to adaptively build a stochastic model of the error introduced by the use of a reduced order model this use of adaptivity differs from existing adaptive mcmc algorithms that tune proposal distributions of the metropolis hastings algorithm mh though adamh also implements that technique for the 3 d geothermal reservoir model we present here adamh shows a great improvement in the computational efficiency of the mcmc sampling and promising results for parameter estimation and uncertainty quantification are obtained this algorithm could offer significant improvement in computational efficiency when implementing sample based inference in other large scale inverse problems

characterization and properties of matrices with generalized symmetry or skew symmetry let r cn n be a nontrivial involution i e r r 1 i we say that a cn n is r symmetric r skew symmetric if rar a rar a there are positive integers r and s with r s n and matrices p cn r and q cn s such that p p ir q q is rp p and rq q we give an explicit representation of an arbitrary r symmetric matrix a in terms of p and q and show that solving az w and the eigenvalue problem for a reduce to the corresponding problems for matrices app cr r and aqq cs s we also express a 1 in terms of app 1 and aqq 1 under the additional assumption that r r we show that moore penrose inversion and singular value decomposition of a reduce to the corresponding problems for app and aqq we give similar results for r skew symmetric matrices these results are known for the case where r j i n j 1 i j 1n however our proofs are simpler even in this case we say that a cn n is r conjugate if rar a where r rn n and r r 1 i in this case r a is r symmetric and i a is r skew symmetric so our results provide explicit representations for r conjugate matrices in terms of p and q which are now in rn r and rn s respectively we show that solving az w inverting a and the eigenvalue problem for a reduce to the corresponding problems for a related matrix s rn n if rt r this is also true for moore penrose inversion and singular value decomposition of a

characterization and properties of matrices with k involutory symmetries we say that a matrix r cn n is k involutory if its minimal polynomial is xk 1 for some k 2 so rk 1 r 1 and the eigenvalues of r are 1 2 k 1 where e2 i k let 0 1 k 1 if r cm m a cm n s cn n and r and s are k involutory we say that a is r s symmetric if ras a we show that an r s symmetric matrix a can be usefully represented in terms of matrices fl cc l dl 0 l k 1 where cl and dl are respectively the dimensions of the l eigenspaces of r and s this continues a theme initiated in an earlier paper with the same title in which we assumed that 1 we say that a k involution is equidimensional with width d if all of its eigenspaces have dimension d we show that if r and s are equidimensional k involutions with widths d1 and d2 respectively then r s symmetric matrices are closely related to generalized circulants ras r r s 0k 1 where a0 a1 ak 1 cd1 d2 for this case our results are new even if 1 we also give an explicit formula for the moore penrose inverse of a unilevel block circulant as r r s 0k 1 for any 0 1 k 1 generalizing a result previously obtained for the case where gcd k 1

the solvability conditions for the inverse eigenvalue problems of centro symmetric matrices this paper involves related inverse eigenvalue problems of centro symmetric matrices and their optimal approximation the sufficient and necessary conditions under which the solvable problems of inverse eigenvalue and the general provided form of the solution furthermore the algorithm to compute the optimal approximate solution and some numerical experiments are given

faims a fast algorithm for the inverse medium problem with multiple frequencies and multiple sources for the scalar helmholtz equation we propose an algorithm to compute an approximate singular value decomposition svd of least squares operators related to linearized inverse medium problems with multiple events such factorizations can be used to accelerate matrix vector multiplications and to precondition iterative solvers we describe the algorithm in the context of an inverse scattering problem for the low frequency time harmonic wave equation with broadband and multi point illumination this model finds many applications in science and engineering e g seismic imaging subsurface imaging impedance tomography non destructive evaluation and diffuse optical tomography we consider small perturbations of the background medium and by invoking the born approximation we obtain a linear least squares problem the scheme we describe in this paper constructs an approximate svd of the born operator the operator in the linearized least squares problem the main feature of the method is that it can accelerate the application of the born operator to a vector if n w is the number of illumination frequencies n s the number of illumination locations n d the number of detectors and n the discretization size of the medium perturbation a dense singular value decomposition of the born operator requires o min n sn wn d n 2xmax n sn wn d n operations the application of the born operator to a vector requires o n wn s m n work where m n is the cost of solving a forward scattering problem we propose an approximate svd method that under certain conditions reduces these work estimates significantly for example the asymptotic cost of factorizing and applying the born operator becomes o m n n w we provide numerical results that demonstrate the scalability of the method

properties of unilevel block circulants let a a0 a1 ak 1 cd1 d2 e 2 i k fl m 0k 1 lmam 0 l k 1 and fa l 0k 1fl all operations in indices are modulo k it is well known that if d1 d2 1 then as r r s 0k 1 fa where 1k lm l m 0k 1 however to our knowledge it has not been emphasized that fa plays a fundamental role in connection with all the matrices as r r s 0k 1 0 k 1 with d1 d2 arbitrary we begin by adapting a theorem of ablow and jenner with d1 d2 1 to the case where d1 and d2 are arbitrary we show that a as r r s 0k 1 if and only if a u fap where u and p are related to p is unitary and u is invertible in fact unitary if and only if gcd k 1 in which case we say that a is a proper circulant we prove the following for proper circulants a as r r s 0k 1 i a br s r s 0k 1 with bm 1k l 0k 1 lmfl 0 m k 1 ii solving az w reduces to solving flul v l 0 l k 1 where v0 v1 vk 1 depend only on w iii a singular value decomposition of a can be obtained from singular value decompositions off0 f1 fk 1 iv the least squares problem for a reduces to independent least squares problems for f0 f1 fk 1 v if d1 d2 d the eigenvalues of as r r s 0k 1 are the eigenvalues of f0 f1 fk 1 and the corresponding eigenvectors of a are easily obtainable from those of f0 f1 fk 1 vi if d1 d2 d and 1 then the eigenvalue problem for as r r s 0k 1 reduces to eigenvalue problems for d d matrices related to f0 f1 fk 1 in a manner depending upon

taylor approximation and variance reduction for pde constrained optimal control under uncertainty abstract in this work we develop a scalable computational framework for the solution of pde constrained optimal control problems under high dimensional uncertainty specifically we consider a mean variance formulation of the control objective and employ a taylor expansion with respect to the uncertain parameter field either to directly approximate the control objective or as a control variate for variance reduction the expressions for the mean and variance of the taylor approximation are known analytically although their evaluation requires efficient computation of the trace of the preconditioned hessian of the control objective we propose to estimate this trace by solving a generalized eigenvalue problem using a randomized algorithm that only requires the action of the hessian on a small number of random directions then the computational work does not depend on the nominal dimension of the uncertain parameter but depends only on the effective dimension i e the rank of the preconditioned hessian thus ensuring scalability to high dimensional problems moreover to increase the estimation accuracy of the mean and variance of the control objective by the taylor approximation we use it as a control variate for variance reduction which results in considerable computational savings several orders of magnitude compared to a plain monte carlo method in summary our approach amounts to solving an optimal control problem constrained by the original pde the generalized eigenvalue equations at a small number of eigenfunctions and a set of linearized pdes that arise from the computation of the gradient and hessian of the control objective with respect to the uncertain parameter we use the lagrangian formalism to derive expressions for the gradient with respect to the control and apply a gradient based optimization method to solve the problem we demonstrate the accuracy efficiency and scalability of the proposed computational method for two examples with high dimensional uncertain parameters subsurface flow in a porous medium modeled as an elliptic pde and turbulent jet flow modeled by the reynolds averaged navier stokes equations coupled with a nonlinear advection diffusion equation characterizing model uncertainty in particular for the latter more challenging example we show scalability of our algorithm up to one million parameters resulting from discretization of the uncertain parameter field

randomized matrix free trace and log determinant estimators we present randomized algorithms for estimating the trace and determinant of hermitian positive semi definite matrices the algorithms are based on subspace iteration and access the matrix only through matrix vector products we analyse the error due to randomization for starting guesses whose elements are gaussian or rademacher random variables the analysis is cleanly separated into a structural deterministic part followed by a probabilistic part our absolute bounds for the expectation and concentration of the estimators are non asymptotic and informative even for matrices of low dimension for the trace estimators we also present asymptotic bounds on the number of samples columns of the starting guess required to achieve a user specified relative error numerical experiments illustrate the performance of the estimators and the tightness of the bounds on low dimensional matrices and on a challenging application in uncertainty quantification arising from bayesian optimal experimental design

resolving seismic anisotropy sparse matrix methods for geophysical inverse problems two techniques for the singular value decomposition svd of large sparse matrices are directly applicable to geophysical inverse problems subspace iteration and the lanczos method both methods require very little in core storage and efficiently compute a set of singular values and singular vectors a comparison of the singular value and vector estimates of these iterative approaches with the results of a conventional in core svd algorithm demonstrates their accuracy hence it is possible to conduct an assessment of inversion results for larger sparse inverse problems such as those arising in seismic tomography as an example we examine the resolution matrix associated with a crosswell seismic inversion of first arrival times for lateral variations in anisotropy the application to a set of first arrival times from a crosswell survey at the grimsel laboratory emphasizes the utility of including anisotropy in a traveltime inversion the isotropic component of the estimated velocity structure appears to be well constrained even when anisotropy is included in the inversion in the case of the grimsel experiment we are able to resolve a fracture zone in agreement with borehole fracture intersections elements of the resolution matrix reveal moderate averaging among anisotropy coefficients as well as between anisotropy coefficients and source receiver static terms the information on anisotropy such as the directions of maximum velocity appears sensitive to lateral variations in velocity and must be interpreted with some caution

on the eigenvectors of symmetric toeplitz matrices this paper presents a number of results concerning the eigenvectors of a symmetric toeplitz matrix and the location of the zeros of the filters eigenfilters whose coefficients are the elements of the eigenvectors one of the results is that the eigenfilters corresponding to the maximum and minimum eigenvalues if distinct have their zeros on the unit circle while the zeros of the other eigenfilters may or may not have their zeros on the unit circle even if the zeros of the eigenfilters of a matrix are all on the unit circle the matrix need not be toeplitz examples are given to illustrate the different properties

adaptive hessian based nonstationary gaussian process response surface method for probability density approximation with application to bayesian solution of large scale inverse problems we develop an adaptive hessian based non stationary gaussian process gp response surface method for approximating a probability density function pdf that exploits its structure particularly the hessian of its negative logarithm of particular interest to us are pdfs that arise from the bayesian solution of large scale inverse problems which imply very expensive to evaluate pdfs the method can be considered as a piecewise adaptive gaussian approximation in which a gaussian tailored to the local hessian of the negative log probability density is constructed for each subregion in high dimensional parameter space the task of efficiently partitioning the parameter space into subregions is done implicitly through hessian informed membership probability functions the gp machinery is then employed to glue all local gaussian approximations into a global analytical response surface that is far cheaper to evaluate than the original expensive probability density the resulting response surface is also equipp

large scale hydraulic tomography and joint inversion of head and tracer data using the principal component geostatistical approach pcga the stochastic geostatistical inversion approach is widely used in subsurface inverse problems to estimate unknown parameter fields and corresponding uncertainty from noisy observations however the approach requires a large number of forward model runs to determine the jacobian or sensitivity matrix thus the computational and storage costs become prohibitive when the number of unknowns m and the number of observations n increase to overcome this challenge in large scale geostatistical inversion the principal component geostatistical approach pcga has recently been developed as a matrix free geostatistical inversion strategy that avoids the direct evaluation of the jacobian matrix through the principal components low rank approximation of the prior covariance and the drift matrix with a finite difference approximation as a result the proposed method requires about k runs of the forward problem in each iteration independently of m and n where k is the number of principal components and can be much less than m and n for large scale inverse problems furthermore the pcga is easily adaptable to different forward simulation models and various data types for which the adjoint state method may not be implemented suitably in this paper we apply the pcga to representative subsurface inverse problems to illustrate its efficiency and scalability the low rank approximation of the large dimensional dense prior covariance matrix is computed through a randomized eigen decomposition a hydraulic tomography problem in which the number of observations is typically large is investigated first to validate the accuracy of the pcga compared with the conventional geostatistical approach then the method is applied to a large scale hydraulic tomography with 3 million unknowns and it is shown that underlying subsurface structures are characterized successfully through an inversion that involves an affordable number of forward simulation runs lastly we present a joint inversion of head and tracer test data using modflow and mt3dms as coupled black box forward simulation solvers these applications demonstrate the advantages of the pcga i e the scalability to high dimensional inverse problems and the ability to utilize multiple forward models as black boxes

x tmcmc adaptive kriging for bayesian inverse modeling abstract the bayesian inference of models associated with large scale simulations is prohibitively expensive even for massively parallel architectures we demonstrate that we can drastically reduce this cost by combining adaptive kriging with the population based transitional markov chain monte carlo tmcmc techniques for uni modal posterior probability distribution functions pdf the proposed hybrid method can reduce the computational cost by an order of magnitude with the same computational resources for complex posterior pdf landscapes we show that it is necessary to further extend the tmcmc by langevin adjusted proposals the proposed hybrid method exhibits high parallel efficiency we demonstrate the capabilities of our method on test bed problems and on high fidelity simulations in structural dynamics

euler based inverse method for turbomachine blades part 2 three dimensional flows the aerodynamic design method for turbomachine blades reported in part 1 is extended to three dimensions in this inverse method the blade pressure loading i e pressure difference between blade upper and lower surfaces and the blade thickness distributions are prescribed and the corresponding blade camber surface is sought the inverse problem is formulated using a pressure loading boundary condition across the blade surfaces and modification of the blade geometry is achieved using the flow tangency conditions along the blade surfaces the method is demonstrated for the design of axial flow machines ranging from the subsonic to the supersonic flow regimes

matrices a such that ra as 1r when rk i abstract this paper examines matrices a c n n such that ra a s 1 r where r k i the identity matrix and where s and k are nonnegative integers with k 2 spectral theory is used to characterize these matrices the cases s 0 and s 1 are considered separately since they are analyzed by different techniques

improved design of subcritical and supercritical cascades using complex characteristics and boundary layer correction the method of complex characteristics and hodograph transformation for the design of shockless airfoils was extended to design supercritical cascades with high solidities and large inlet angles this capability was achieved by introducing a conformal mapping of the hodograph domain onto an ellipse and expanding the solution in terms of tchebycheff polynomials a computer code was developed based on this idea a number of airfoils designed with the code are presented various supercritical and subcritical compressor turbine and propeller sections are shown the lag entrainment method for the calculation of a turbulent boundary layer was incorporated to the inviscid design code the results of this calculation are shown for the airfoils described the elliptic conformal transformation developed to map the hodograph domain onto an ellipse can be used to generate a conformal grid in the physical domain of a cascade of airfoils with open trailing edges with a single transformation a grid generated with this transformation is shown for the korn airfoil previously announced in star as n83 24474

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

on the topography of the cost functional in linear and nonlinear inverse problems we analyze through linear algebra the topography of the cost functional in linear and nonlinear inverse problems with the aim of illuminating general characteristics to a firstorder approximation the local data misfit function in any inverse problem is valley shaped and elongated in the directions of the null space of the jacobian and or in the directions of the smallest singular values in nonlinear inverse problems valleys persist however local minima might also coexist in the misfit space and might be related to nonlinear effects ignored by the gauss newton approximation to the hessian the regularization term designed to provide convexity to the misfit function or to noise in the data furthermore noise perturbs the size of the equivalence region making location of solutions easier but finding a global minimum harder in the case of existence understanding the behavior of the cost functional is an important step in the developing techniques to appraise inverse solutions and estimate uncertainties caused by noise incomplete sampling regularization and more fundamentally simplified physical models

euler based inverse method for turbomachine blades i two dimensional cascades a three dimensio nal inverse method for the aerodynamic design of turbomachine blades using robust timemarching algorithms for the numerical solutions of the euler equations is proposed in this inverse method the circumferential mass averaged tangential velocity or the blade loading is the primary specified flow quantity and the corresponding blade geometry is sought after the presence of the blades is represented by a periodic array of discrete body forces which is included in the equations of motion a four stage runge kutta time stepping scheme is used to march a finite volume formulation of the unsteady euler equations to a steady state solution modification of the blade geometry during this time marching process is achieved using the flow tangency conditions along the blade surfaces in this paper the method is demonstrated for the design of two dimensional infinitely thin cascaded blades ranging from the subsonic to the supersonic flow regimes including cases with rotational flows and complex shock structures in the flow passage

minimization problems for r s symmetric and r s skew symmetric matrices let r cm m and s cn n be nontrivial involutions thus r r 1 i and s s 1 i we say that a cm n is r s symmetric r s skew symmetric if ras a ras a let s be the class of m n r s symmetric matrices or the class of m n r s skew symmetric matrices let z cn q and w cm q we study the following problems i give necessary and sufficient conditions for the existence of an a s such that az w and find all such matrices if the conditions are met ii find z w mina s az w and characterize the class s z w a s az w z w iii if b cm n is arbitrary find z w b mina s z w a b and find a s z w such that a b z w b we obtain explicit formulas for z w b z w and all the matrices in question

further comments on on the eigenvectors of symmetric toeplitz matrices attention is drawn to some previous work on the matrices discussed in the comments of datta and morgera on the above paper

properties of multilevel block α circulants abstract in a previous paper we characterized unilevel block circulants a a s r r s 0 n 1 a m c d 1 d 2 0 m n 1 in terms of the discrete fourier transform f a f 0 f 1 f n 1 of a a 0 a 1 a n 1 defined by f l 1 n m 0 n 1 e 2 i l m n a m we showed that most theoretical and computational problems concerning a can be conveniently studied in terms of corresponding problems concerning the fourier coefficients f 0 f 1 f n 1 individually in this paper we show that analogous results hold for k 1 level matrices where the first k levels have block circulant structure and the entries at the k 1 st level are unstructured rectangular matrices

hermitian hermitian r symmetric and hermitian r skew symmetric procrustes problems let r cn n be a nontrivial unitary involution i e r r r 1 i we say that a cn n is r symmetric r skew symmetric if rar a rar a let s be one of the following subsets of cn n i hermitian matrices ii hermitian r symmetric matrices iii hermitian r skew symmetric matrices given z w cn m we characterize the matrices a in s that minimize az w frobenius norm and given an arbitrary e cn n we find the unique matrix among the minimizers of az w in s that minimizes a e we also obtain necessary and sufficient conditions for existence of a s such that az w and assuming that the conditions are satisfied characterize the set of all such a