nonlinear inversion of geoelectric data acquired across 3d objects using a finite element approach we introduce a new finite element based scheme for the fast nonlinear inversion of large 3d geoelectric data sets acquired around isolated objects or across the earth s surface the principal novelty of this scheme is the combination of a versatile finite element approach with 1 a method involving minimization of an objective function using a conjugate gradient algorithm that includes an adjoint field technique for efficiently establishing the objective function gradient and 2 parabolic interpolation for estimating suitable inversion step lengths this scheme is capable of handling large volumes of data acquired using diverse electrode configurations located around or across 3d structures only three solutions to the forward problem are required for each iteration computation of the jacobian matrix which might require computers with a large amount of memory is not necessary to minimize artificial irregularities in the inverted models particularly near the electrodes we smooth the model parameter

a finite volume solution to the geophysical electromagnetic forward problem using unstructured grids abstractthe application of unstructured grids can improve the solution of total field electromagnetic problems because these grids allow efficient local refinement of the mesh at the locations of high field gradients unstructured grids also provide the flexibility required for representing arbitrary topography and subsurface interfaces we investigated the generalization of the standard yee s staggered scheme to unstructured tetrahedral voronoi grids using a finite volume approach we discretized the helmholtz equation for the electric field in the frequency domain and solved the problem to find the projection of the total electric field along the edges of the tetrahedral elements to compute the electric and magnetic fields at the observation points an interpolation technique was employed which uses the edge vector interpolation functions of the tetrahedral elements to verify the presented scheme two examples were evaluated which revealed the computation of the total and secondary fields due to ele

an lp analog to aak theory for p 2 we develop an lp analog to aak theory on the unit circle that interpolates continuously between the case p which classically solves for best uniform meromorphic approximation and the case p 2 which is equivalent to h2 best rational approximation we apply the results to the uniqueness problem in rational approximation and to the asymptotic behaviour of poles of best meromorphic approximants to functions with two branch points as pointed out by a referee part of the theory extends to every p 1 when the definition of the hankel operator is suitably generalized this we discuss in connection with the recent manuscript by v a prokhorov submitted for publication

3d finite element forward modeling of electromagnetic data using vector and scalar potentials and unstructured grids abstractwe present a finite element solution to the 3d electromagnetic forward modeling problem in the frequency domain the method is based on decomposing the electric field into vector and scalar potentials in the helmholtz equation and in the equation of conservation of charge edge element and nodal element basis functions were used respectively for the vector and scalar potentials this decomposition was performed with the intention of satisfying the continuity of the tangential component of the electric field and the normal component of the current density across the interelement boundaries therefore finding an efficient solution to the problem the computational domain was subdivided into unstructured tetrahedral elements the system of equations was discretized using the galerkin variant of the weighted residuals method with the approximated vector and scalar potentials as the unknowns of a sparse linear system a generalized minimum residual solver with an incomplete lu preconditioner was use

robust h disturbance minimization by duality abstract the nonstandard optimization problem inf sup w i pq e i vpq e i 0 2 q h n n plays a basic role in robust feedback analysis here its solutions are characterized using banach space duality theory and shown to satisfy a flatness or allpass condition

regularity for solutions of nonlinear evolution equations with nonlinear perturbations abstract the approximate controllability for the nonlinear control system with nonlinear monotone hemicontinuous and the nonlinear perturbations is studied the existence uniqueness and a variation of solutions of the system are also given

duality theory of robust disturbance attenuation abstract the nonstandard minimization problem inf q h w 1 pq vpq underlies many questions of robust control here its solutions are characterized using banach space duality theory and shown to be allpass the results are applied to the optimization of weighted sensitivity to disturbances in feedback systems subject to large plant uncertainty

nonlinear evolution equations with nonmonotonic perturbations soit v un espace de banach reflexif densement et continument plonge dans un espace de hilbert reel h on considere l existence des solutions du probleme aux valeurs initiales du dt au g u f 0 t t u 0 u 0 ou a est un operateur monotone de v dans v g v h est une application continue et f 0 t v est une fonction mesurable

resinvm3d a 3d resistivity inversion package we have developed an open source 3d matlab based resistivity inversion package the forward solution to the governing partial differential equation is efficiently computed using a second order finite volume discretization coupled with a preconditioned biconjugate stabilized gradient algorithm using the analytical solution to a potential field in a homogeneous half space we evaluate the accuracy of our numerical forward solution and subsequently develop a source correction factor that reduces forward modeling errors associated with boundary effects and source electrode singularities for the inversion algorithm we have implemented an inexact gauss newton solver with the model update being calculated using a preconditioned conjugate gradient algorithm the inversion uses a combination of zero and first order tikhonov regularization two synthetic examples demonstrate the usefulness of this code the first example considers a surface resistivity survey with 3813 measurements the discretized model s

efficient microcomputer based finite difference resistivity modeling via polozhii decomposition finite difference modeling for direct current resistivity problems can be cast in a form allowing rapid and efficient solution on a microcomputer almost a million nodes can be accommodated on a microcomputer with only 256 kilobytes of memory the polozhii decomposition procedure polozhii 1965 allows a three dimensional 3 d problem with no more than a one dimensional 1 d property variation to be transformed into a series of decoupled and independent one dimensional problems the decomposition utilizes matrix transforms analogous to analytic spatial fourier and hankel transforms these matrix transforms along with recursive formulas for the solution of the 1 d problems yield a process which can be thought of as a continuation operator when compared with current popular methods of potential field continuation using the fourier transform the result is that only a single plane of node potentials needs to be determined fully by setting up more complicated models as separate regions with differing 1

on the schmidt pairs of multivariable hankel operators and robust control abstract we study here a particular application of the theory of hankel operators to robust control the techniques of superoptimal nehari extension developed by young are employed to derive some properties of a particular controller of a rational function g called the superoptimally robust stabilizing controller using this controller we generalize to the multivariable case some results due to fuhrmann and ober we also give a characterization of all plants g which are stabilized by the same superoptimally robust stabilizing controller

factorization and approximation problems for matrix functions the purpose of this paper is to develop a new approach to certain approximation and factorization problems for matrix valued functions the approach is based on studying properties of maximizing vectors of hankel operators with matrix valued symbols and on the solution of the so called recovery problem for unitary valued matrix functions in the case of scalar functions such problems were studied in detail in pk it turns out however that the case of matrix functions is considerably more complicated than the scalar case in this paper we solve the recovery problem with the help of hankel operators this solution also allows us to give a new approach to wiener hopf factorization problems and obtain some new results another application we consider here is the problem of finding a superoptimal analytic approximation to a matrix function on the unit circle we improve earlier results on hereditary properties of such approximations see py1 and on continuity properties of the operator of superoptimal approximation see py3 recall that for a bounded m x n matrix function b on the unit circle tf the hankel operator hp is defined on the hardy class h2 cn of cn valued functions by hdf padf f e h cn

on the sum of superoptimal singular values in this paper we study the following extremal problem and its relevance to the sum of the so called superoptimal singular values of a matrix function given an m n matrix function when is there a matrix function in the set akn m such that ttrace dm sup akn m ttrace dm the set akn m is defined by akn m def h01 mn m l1 mn m 1 rank k a e t to address this extremal problem we introduce hankel type operators on spaces of matrix functions and prove that this problem has a solution if and only if the corresponding hankel type operator has a maximizing vector the main result of this paper is a characterization of the smallest number k for which ttrace dm equals the sum of all the superoptimal singular values of an admissible matrix function e g a continuous matrix function for some function akn m moreover we provide a representation of any such function when is an admissible very badly approximable unitary valued n n matrix function

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

a note on essential spectra and norms of mixed hankel toeplitz operators in this brief hankel toeplitz operators which occur in feedback theory e g in the minimization of mixed h sensitivity and complementary sensitivity will be considered a method of computing their spectra eigenvectors and norms will be presented for infinite dimensional systems subject to continuous weightings

weighted extremal domains and best rational approximation abstract let f be holomorphically continuable over the complex plane except for finitely many poles and branch points contained in the unit disk we prove that best rational approximants to f of degree n in the l 2 sense on the unit circle have poles that asymptotically distribute according to the equilibrium measure on the compact set outside of which f is single valued and which has minimal green capacity in the disk among all such sets this provides us with n th root asymptotics of the approximation error by conformal mapping we deduce further estimates in approximation by rational or meromorphic functions to f in the l 2 sense on more general jordan curves encompassing the poles and branch points the key to these approximation theoretic results is a characterization of extremal domains of holomorphy for f in the sense of a weighted logarithmic potential which is the technical core of the paper

singular values and invariant factors of matrix sums and products abstract some remarks are made concerning the range of individual singular values of sums and products of two complex square matrices when these are allowed to vary in their orbits under unitary equivalence an analogous question is considered for invariant factors of products of matrices over a principal ideal domain

well posedness of h optimal control problems this paper considers the effect of perturbations of a nominal single input single output linear plant or a band of uncertainty on the solution to certain frequency domain optimal control problems weighted h infty sensitivity minimization mixed and robust sensitivity minimization will be considered along with problems of more general type a brief discussion of h p optimal control problems will also be given typical examples will be presented where optimal sensitivity does not depend continuously on the plant ill posedness and conditions for well posedness will be given it is demonstrated that similar discontinuities can occur for more general problems also suggested are ways of defining an optimization problem so that continuous dependence of the infimum is ensured

an investigation of finite element modeling for electrical and electromagnetic data in three dimensions the finite element method can be used to solve the differential equations which describe electrical and electromagnetic em field behavior the equations are respectively poisson s equation and the vector damped wave equation the finite element equations are derived in both cases using the minimum theorem while both tetrahedral and hexahedral elements may be used for the modeling of the resistivity problem only hexahedral elements give satisfactory results for the em problem a disadvantage of the relatively simple mesh design used in the approach described here is the presence of long thin elements such elements have very poor interpolating properties and they adversely affect the rate of convergence of the overrelaxation technique used in solving the resulting system of linear equations for the modeling of resistivity data over an earth with one plane of symmetry the system of equations typically has about 9000 unknowns about 50 000 unknowns are needed to give a satisfactory solution to an

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

linear control theory with an h 0e optimality criterion this expository paper sets out the principal results in bf h infty control theory in the context of continuous time linear systems the focus is on the mathematical theory rather than computational methods

superoptimal analytic approximations of matrix functions we study the approximation of a bounded matrix valued function g on the unit circle by functions q bounded and analytic in the unit disc we show that if g is continuous then there is a unique q for which the error g q has a strong minimality property involving not only the l norm of g q but also the suprema of its subsequent singular values we obtain structural properties of the error g q and show that certain smoothness properties of g are inherited by q e g membership of besov spaces

schur products and matrix completions we prove that a necessary and sufficient condition for a given partially positive matrix to have a positive completion is that a certain schur product map defined on a certain subspace of matrices is a positive map by analyzing the positive elements of this subspace we obtain new proofs of the results of h dym and i gohberg and grone johnson sa and wolkowitz linear algebra appl 58 1984 109 124 linear algebra appl 36 1981 1 24 we also obtain a new proof of a result of u haagerup decomposition of completely bounded maps on operation algebras preprint characterizing the norm of schur product maps and a new hahn banach type extension theorem for these maps finally we obtain generalizations of many of these results to matrices of operators which we apply to the study of representations of certain subalgebras of the n n matrices

fuzzy constrained lp norm inversion of direct current resistivity data abstractwe evaluate the use of a fuzzy c means clustering procedure to improve an inverted 2d resistivity model within the iterative error minimization procedure the algorithm is coded in matlab language for the lp norm inversion of 2d direct current resistivity data and is referred to as fuzzy constrained inversion fci two additional input parameters are required to be provided by the interpreter 1 the number of geologic units in the model i e the number of clusters and 2 the mean resistivity values of each geologic unit i e cluster center values of the geologic units the efficacy of our approach is evaluated by tests carried on the synthetic and field electrical resistivity tomography ert data inversion results from the fci algorithm are presented for conventional l1 and l2 norm minimization techniques fci indicates improvement over conventional inversion approaches in differentiating the geologic units if a proper number of the geologic units is provided to the algorithm inappro

a spectral characterization of h optimal feedback performance and its efficient computation abstract we study the spectral properties of a toeplitz hankel operator which arises in the context of the mixed sensitivity h optimization problem and whose largest eigenvalue characterizes the optimal achievable performance 0 the existence of such an operator was first shown by verma and jonckheere 26 who also noted the potential numerical advantage of computing eo through its eigenvalue characterization rather than through the iteration here we investigate this operator in detail with the objective of efficiency computing its spectrum we define an adjoint linear quadratic problem that involves the same toeplitz hankel operator as shown by jonckheere and silverman 13 16 consequently a finite polynomial algorithm allows 0 to be characterized as simply as the largest root of a polynomial finally a computationally more attractive state space algorithm emerges from the h t 8 lq relationship this algorithm yields a very good accuracy evaluation of the performance 0 by solving just one algebraic riccati equation thorough exploitation of this algorithm results in a drastic computation reduction with respect to the standard e iteration

kkm fan s lemma for solving nonlinear vector evolution equations with nonmonotonic perturbations abstract in this paper we are interested in the existence of solutions of the following initial value problem du dt a u g u f on 0 t with u 0 u 0 where a v v is a monotone operator g v v is a nonlinear nonmonotone operator and f 0 t v is a measurable function by means of a recent generalization of the famous kkm fan s lemma

the use of summary representation for electromagnetic modeling the method of summary representation developed by g n polozhii is a quasi analytical method for solving self adjoint finite difference boundary value problems expressed on regular meshes in principle the method should allow considerable savings in computing time as well as improved accuracy when compared to commonly used finite difference schemes we have used summary representation as the basis for a new hybrid scheme to solve the two dimensional helmholtz equation for electromagnetic modeling the theory behind this hybrid scheme is presented preliminary results for the two dimensional problem show that substantial computing time and storage savings can be made

robust controllers for uncertain linear multivariable systems abstract this paper is addressed to three distinct yet related topics in the design of controllers for imprecisely known linear multivariable systems in the first part it is supposed that the plant to be stabilized is subject to additive or multiplicative uncertainties and necessary and sufficient conditions are derived for the existence of a controller that stabilizes all plants within this band of uncertainty in the second part in contrast with the first part it is supposed that the number of unstable poles of the plant to be stabilized is not precisely known the type of plant uncertainty is the so called stable factor uncertainty and necessary and sufficient conditions are given for robust stabilization in the third part the model of uncertainty is a ball in the space of rational matrices metrized by the so called graph metric and sufficient conditions for robust stabilization are derived

hybrid calculations of the three dimensional electromagnetic response of buried conductors the hybrid method for computing the electromagnetic response of a three dimensional conductor in a layered conducting half space consists of solving a finite element problem in a localized region containing the conductor and using integral equation methods to obtain the fields outside that region the original scheme obtains the boundary values by iterating between the integral equation solution and the finite element solution after making an initial guess based on primary values from the field a two dimensional interpolation scheme is then used to speed the evaluation of the 105 to 106 green s function convolution integrals required by most problems the two algorithms presented are modifications of the original scheme both contain a search routine to identify a set of unique points where the convolution integral evaluations are required by replacing the two dimensional interpolation with a one dimensional interpolation and reading the convolution integrals from a reference table computation time

a functional approach to lqg balancing the paper has as its theme a circle of problems related to lqg balancing with a special emphasis on the related problems of model reduction and robust stabilization the aim is to present a unified point of view to both previously described problems the unification is achieved by focusing on the study of three functions and the relationships between them and the various operators that are associated with these functions with an arbitrary transfer function g one can associate canonically two objects the conjugate inner function which is based on the nrcf and the r which is associated with the lqg controller of the function g the approach that is taken is functional emphasizing operators a balanced realization of a stable g arises as a matrix representation of the shift realization with respect to a basis made out of suitably normalized hankel singular vectors a similar result holds for lqg balanced realizations here the underlying hankel operator we study is h r where r u m v n and u v s

langevin equation involving three fractional orders in this paper the existence and uniqueness of initial value problems for nonlinear langevin equation involving three fractional orders are discussed we use a new norm that is convenient for the fractional and singular differential equations this norm and the contraction mapping principles are the main tools for investigating the existence and uniqueness of the desired issue the fractional derivatives are described in caputo sense

singularity removal a refinement of resistivity modeling techniques modeling techniques commonly exhibit errors of 3 to 10 percent or more in the calculation of apparent resistivities over earth models for which analytic solutions are easily available a singularity occurs in the solution of any elliptic partial differential equation for which the forcing function is not smooth the inability to adequately represent in discrete space a discontinuous function in this case the delta function describing the introduction of current at a point commonly results in numerical error near the source of a modeled singularity inspection of an integrated finite difference method for modeling the dc resistivity geophysical technique indicates much of the error encountered is of singular origin a procedure is herein detailed by which the singularity is mathematically removed from the modeling process and reintroduced as a last step thus preventing it from contributing to the numerical error using this procedure the average error in apparent resistivity values for a model of a polar dipole traverse over a nonconducting sphere is reduced by 40 percent for a dipole dipole traverse of a two layer model the error decreases by 75 percent and in the case of a wenner profile of a model of a vertically faulted earth the average error is diminished by 90 percent

solutions of nonlinear evolution inclusions abstract in this paper the solution of the nonlinear evolution inclusion problem of the form u t b t u t f t is studied in this problem the operators are of type m or type s which are different from those of pseudo monotone operators that had been studied by many authors at the same time we study the perturbation problem in fact many kinds of evolution equations can be generalized by this problem the former results are improved and generalized by our conclusions and we will give more applications

a state space approach to super optimal h sup infinity control problems a numerically stable algorithm is presented for solving the strengthened or super optimal model matching problem the steps of the algorithm follows closely those given by n j young 1986 except that for each step a reliable implementation using state space models is provided

hermitian solutions of the equation x q nx 1n we consider the matrix equation x q nx 1n its hermitian solutions are parametrized in terms of the generalized lagrangian eigenspaces of a certain matrix pencil we show that the equation admits both a largest and a smallest solution the largest solution corresponds to the unique positive definite solution the smallest solution is the unique negative definite solution if and only if n is nonsingular if n is singular no negative definite solution exists an interesting relation between the given equation and a standard algebraic riccati equation of kalman filtering theory is also obtained finally we present an algorithm which converges to the positive definite solution for a wide range of initial conditions

3d time domain simulation of electromagnetic diffusion phenomena a finite element electric field approach we present a finite element time domain fetd approach for the simulation of 3d electromagnetic em diffusion phenomena the finite element algorithm efficiently simulates transient electric fields and the time derivatives of magneticfields in general anisotropic earth media excited by multiple arbitrarily configured electric dipoles with various signal waveforms to compute transient electromagnetic fields theelectricfielddiffusionequationistransformedinto asystemofdifferentialequationsviagalerkin smethodwith homogeneous dirichlet boundary conditions to ensure numerical stability and an efficient time step the system of the differential equations is discretized in time using an implicit backward euler scheme the resultant fetd matrix vector equation is solved using a sparse direct solver along with a fill inreducedorderingtechnique whenadvancingthesolution in time the fetd algorithm adjusts the time step by examining whether or not the current step size can be doubled without unacceptably affecting the accuracy of the solution to simulate a step off source waveform the 3d fetd algorithm also incorporates a 3d finite element direct current fedcalgorithmthatsolvespoisson sequationusingasecondarypotentialmethodforageneralanisotropicearthmodel examples of controlled source fetd simulations are compared with analytic and or 3d finite difference time domain solutions and are used to confirm the accuracy and efficiencyofthe3dfetdalgorithm

a nonlinear matrix equation connected to interpolation theory in this paper we study the matrix equation x q a x c 1a where q is an n n positive definite matrix c is an mn mn positive semidefinite matrix a is an arbitrary mn n matrix and x is the m m block diagonal matrix with on each diagonal entry the n n matrix x we are interested in the existence and uniqueness of solutions which are contained in a certain subset of the set of the positive definite matrices under the condition that c q these solutions play a role in an optimal interpolation theory problem interpolation theory and its applications mathematics and its applications vol 428 kluwer academic dordrecht 1997 chapter 7

the finite layer method for groundwater flow models the finite layer method flm is an extension of the finite strip method familiar in structural engineering the idea behind the method is to discretize two space dimensions using truncated fourier series approximating variations in the third via finite elements the eigenfunctions used in the fourier expansions are orthogonal and consequently the galerkin integrations decouple the weighted residual equations associated with different fourier modes the method therefore reduces three dimensional problems to sets of independent matrix equations that one can solve either sequentially on a microcomputer or concurrently on a parallel processor the latter capability makes the method suitable for such computationally intensive applications as optimization and inverse problems four groundwater flow applications are presented to demonstrate the effectiveness of flm as a forward solver

invariance properties of thematic factorizations of matrix functions abstract we study the problem of invariance of indices of thematic factorizations such factorizations were introduced by v v peller and n j young 1994 j funct anal 120 300 343 for studying superoptimal approximation by bounded analytic matrix functions as shown by peller and young the indices may depend on the choice of a thematic factorization we introduce the notion of a monotone thematic factorization the main result shows that under natural assumptions a matrix function that admits a thematic factorization also admits a monotone thematic factorization and the indices of a monotone thematic factorization are uniquely determined by the matrix function itself we obtain similar results for so called partial thematic factorizations

state space algorithm for the computation of superoptimal matrix interpolating functions a state space algorithm is studied which generates the unique superoptimal nehari extension of a general rational matrix the procedure is to use a set of all pass transformations to sequentially minimize each frequency dependent singular value of the interpolating function in a dimension peeling algorithm these all pass transformations are determined by the maximal schmidt pairs of a sequence of hankel operators the process terminates when the original problem is reduced to one of rank one at this stage all the available degrees of freedom have been exhausted the work is an extension of that by young 1986 and gives a concrete state space implementation of his operator theoretic arguments in addition bounds are given on the minimum achievable values for s1 e sup ersi e j i 1 2 rank g0 and also the mcmillan degree of the final superoptimal extension here si denotes the ith singular value of a frequency dependent matrix and the numbering is taken to be in decrea

approximation by analytic matrix functions the four block problem abstract the four block problem is a generalization of nehari s problem for matrix functions it plays an important role in h optimal control theory it is well known that nehari s problem for a continuous scalar function has a unique solution however in the matrix case the situation is quite different v v peller and n j young 1994 j funct anal 120 300 343 studied superoptimal solutions of nehari s problem they minimize not only the l norm of the corresponding matrix function but also the essential suprema of all further singular values it was shown that for h c matrix functions nehari s problem has a unique superoptimal solution in this paper we study superoptimal solutions of the four block problem and we find a natural condition under which such a superoptimal solution is unique our result is new even in the case of nehari s problem we study some related problems such as thematic factorizations invariance of indices and inequalities between the singular values of the four block operator and the superoptimal singular values

duality theory for mimo robust disturbance rejection banach space duality theory is used to characterize the solutions of a nonstandard h sup infinity optimization problem which is shown to be allpass in general and unique in the single input single output siso ease the theory leads to a numerical solution of duality and convex optimization which is applied to an example for a limiting case of sharp cutoff filters an explicit solution of the optimal robust disturbance attenuation problem ordap resembling the two arc theorem of complex analysis is derived

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

extremal classical interpolation problems matrix case in this paper we try to find amongst the solutions w z of the corresponding interpolation problem the solution which satisfies an additional extremal condition we show that degenerate interpolation problems play an important role in the theory of extremal interpolation problems at the end of the paper we accomplish a comparison of our approach with former known approaches

mihaly matrix completions moments and sums of hermitian squares by mihaly bakonyi and hugo j woerdeman princeton series in applied mathematics princeton university press 2011 xii 518 pp hardback isbn 978 0 691 12889 4

existence and uniqueness of solutions to nonlinear evolution equations with locally monotone operators abstract in this paper we establish the existence and uniqueness of solutions for nonlinear evolution equations on a banach space with locally monotone operators which is a generalization of the classical result for monotone operators in particular we show that local monotonicity implies pseudo monotonicity the main results are applied to pde of various types such as porous medium equations reaction diffusion equations the generalized burgers equation the navier stokes equation the 3d leray model and the p laplace equation with non monotone perturbations

hardy spaces of the conjugate beltrami equation we study hardy spaces of solutions to the conjugate beltrami equation with lipschitz coefficient on dini smooth simply connected planar domains in the range of exponents 1 infty we analyse their boundary behaviour and certain density properties of their traces we derive on the way an analog of the fatou theorem for the dirichlet and neumann problems associated with the equation mbox div sigma nabla u 0 with l p boundary data

identical pseudospectra of any geometric multiplicity abstract if a b are n n complex matrices such that the singular values of zi n a are the same as those of zi n b for each z c then a and b are similar

fractional diffusion analysis of the electromagnetic field in fractured media part i 2d approach we address a 2d finite difference fd frequency domain modeling algorithm based on the theory of fractional diffusion of electromagnetic em fields which is generated by an infinite line source lying above a fractured geological medium the presence of fractures in the subsurface usually containing highly conductive pore fluids gives rise to spatially hierarchical flow paths of induced em eddy currents the diffusion of em eddy currents in such formations is anomalous generalizing the classical gaussian process described by the conventional maxwell equations based on the continuous time random walk ctrw theory the diffusion of em eddy currents in a rough medium is governed by the fractional maxwell equations here we model the em response of a 2d subsurface containing fractured zones based on the fractional maxwell equations the governing equation in the frequency domain is discretized using the fd approach the resulting equation system is solved by the multifrontal massively parallel solver mumps we find excellent agreement between the fd and analytic solutions for a rough half space model then fd solutions are calculated for a 2d fault zone model with variable conductivity and roughness we illustrate a case in which a rough fault zone would not be resolved by classical diffusion modeling even if its conductivity contrasts with the background

on backward shift algorithm for estimating poles of systems abstract in this paper we present an algorithm for estimating poles of linear time invariant systems by using the backward shift operator we prove that poles of rational functions including zeros and multiplicities are solutions to an algebraic equation which can be obtained by taking backward shift operator to the shifted cauchy kernels in the unit disc case the algorithm is accordingly developed for frequency domain identification we also prove the robustness of this algorithm some illustrative examples are presented to show the efficiency in systems with distinguished and multiple poles

positive semidefinite matrices with a given sparsity pattern abstract this paper concerns the cone of positive semidefinite matrices which have zeros in prescribed entries one of the main purposes is to obtain information and if possible describe the ranks of external matrices in such cones in terms of the pattern of prescribed zeros in particular we study those patterns for which the ranks of extremal matrices are less than a given number

treatment of singularity in the method of boundary integral equations for 2 5d electromagnetic modeling abstractfast and accurate simulation of responses of electromagnetic logging while drilling tools in 2d and 3d formations is important for further development of the proactive geosteering service which is currently based on the 1d layered model inversion in search of an optimal method of forward simulation we have developed boundary integral equations in a 2d formation model with planar boundaries a peculiarity of the method is that the number of grid nodes required for reaching a certain accuracy and hence the computation time increases as the transmitter approaches the boundary to retain efficiency in the case of transmitter boundary proximity we use the following modification of the method the main term of the calculated electromagnetic field namely the solution for the two layered model corresponding to the closest boundary is calculated by the well known explicit formulas whereas the anomalous field is found from the system of integral equations the modified method is implemented and tes

extensions of band matrices with band inverses abstract let r ij be a given set of i j matrices for i j 1 n and i j m where 0 m n 1 necessary and sufficient conditions are established for the existence and uniqueness of an invertible block matrix f ij i j 1 n such that f ij r ij for i j m f admits either a left or right block triangular factorization and f 1 ij 0 for i j m the well known conditions for an invertible block matrix to admit a block triangular factorization emerge for the particular choice m n 1 the special case in which the given r ij are positive definite in an appropriate sense is explored in detail and an inequality which corresponds to burg s maximal entropy inequality in the theory of covariance extension is deduced the block toeplitz case is also studied

factorization of operator valued polynomials in several non commuting variables abstract a version of fejer riesz factorization and factorization of positive operator valued polynomials in several non commuting variables holds the proofs use arveson s extension theorem and matrix completions

unitary interpolants factorization indices and infinite hankel block matrices abstract the existence uniqueness and construction of unitary n n matrix valued functions j j j in wiener like algebras on the circle with prescribed matrix fourier coefficients j j for j 0 are studied in particular if j then such an exists with j if and only if 0 1 where v denotes the infinite block hankel matrix j k v j k 0 1 acting in the sequence space l n 2 one of the main results is that the nonnegative factorization indices of every such are uniquely determined by the given data in terms of the dimensions of the kernels of i v v whereas the negative factorization indices are arbitrary it is also shown that there is a unique such if and only if the data forces all the factorization indices to be nonnegative and simple conditions for that and a formula for in terms of certain schmidt pairs of 0 are given the results depend upon a fine analysis of the structure of the kernels of i v v and of the one step extension problem of adamjan arov and krein funct anal appl 2 1968 1 18 isometric interpolants for the nonsquare case are also considered

3d vector gravity potential and line integrals for the gravity anomaly of a rectangular prism with 3d variable density contrast three dimensional rectangular prisms are building blocks for calculating gravity anomalies from irregular 3d mass bodies with spatially variable density contrasts a 3d vector gravity potential is defined for a 3d rectangular prism with density contrast varying in depth and horizontally the vertical component of the gravity anomaly equals the flux of the 3d vector gravity potential through the enclosed surface of the prism thus the 3d integral for the gravity anomaly is reduced to a 2d surface integral in turn a 2d vector gravity potential is defined the vertical component of the gravity anomaly equals the net circulation of the 2d vector gravity potential along the enclosed contour bounding the surfaces of the prism the 3d integral for the gravity anomaly is reduced to 1d line integrals further analytical or numerical solutions can then be obtained from the line integrals depending on the forms of the density contrast functions if an analytical solution cannot be obtained the line integral method is semianalytical requiring numerical quadratures to be carried out at the final stages singularity and discontinuity exist in the algorithm and the method of exclusive infinitesimal sphere or circle is effective to remove them then the vector potential line integral method can calculate the gravity anomaly resulting from a rectangular prism with density contrast varying simply in one direction and sophisticatedly in three directions the advantage of the method is that the constraint to the form of the density contrast is greatly reduced and the numerical calculation for the gravity anomaly is fast

efficient 2 5d electromagnetic modeling using boundary integral equations abstractfast and accurate simulation of responses of logging while drilling lwd electromagnetic em tools in complex 2d and 3d formations is very important for reconstruction of the resistivity distribution in proactive geosteering currently real time interpretation is based on the 1d parametric inversion advances in fast simulation beyond the 1d model would open a way for real time 2d inversion we developed implemented numerically and tested an efficient method for simulation of lwd em tools in complex 2d formations with an arbitrary 3d position of the tool the method is based on the boundary integral equations for the tangential components of the field and the fourier transform which reduces the problem to a series of 1d integral equations computations are carried out simultaneously for the whole set of measurement points with the same matrix which provides a short computation time per point we verified the method by comparing our results with those obtained by the well known explicit 1d m

2 d and 3 d interpretation of electrical tomography measurements part 2 the inverse problem the interpretation of borehole to borehole electrical measurements requires solving an inverse problem for a given class of model geometries the usual approach to an inverse problem includes a model dependent task i e forward modeling and a generic task i e an optimization scheme we have developed an optimization algorithm using a nonlinear inversion technique this algorithm allows recovery of a possible resistivity distribution in an investigated zone between two boreholes or in the vicinity of them this resistivity distribution is defined as a set of 2 d or 3 d volumes of constant resistivity the inversion procedure minimizes a least squares term plus a damping term this latter term seeks to minimize the roughness of the solution an improved form of this smoothness term may enhance the spatial resolution of the resistivity image assuming that the resistivity contrast is known a priori this reconstruction algorithm has been tested for both 2 d and 3 d geometries these inversion tests we

mimo disturbance and plant uncertainty attenuation by feedback this paper investigates the ability of feedback to reduce plant and disturbance uncertainties in the multiple input multiple output mimo case by solving two fundamental problems posed by zames in the late seventies the first problem is termed the mimo extension of the optimal robust disturbance attenuation problem the second is a filtering of plant uncertainty two degree of freedom feedback problem the optimal solutions of these problems are characterized using banach space duality theory and shown to satisfy an allpass condition and an extremal identity moreover the duality description leads to a dual pair of optimizations and the introduction of two nonstandard matrix norms in particular the primal dual optimization reduces naturally to approximate finite dimensional convex optimizations within desired tolerance whereas the computation of the nonstandard matrix norms are shown to be equivalent to specific semi definite programming problems and a numerical solution based on convex programming is provided it is also shown using douglas range inclusion theorem that performance is a monotonic function of uncertainty and some qualitative implications for feedback are derived

operator theoretic approach to the optimal two disk problem the nonstandard two disk problem plays a fundamental role in robust feedback optimization here it is shown via banach space duality theory that its solutions satisfy an extremal identity and may be viewed as a dual extremal kernel of a particular l sup 1 optimization problem a novel operator theoretic framework to characterize explicitly its solutions is developed in particular the two disk optimization is shown to be equal to the induced norm of a specific operator defined on a projective tensor product space involving a non hilbert version of a vector valued h sup 2 space moreover this operator is shown to be a combination of multiplication and toeplitz operators under certain conditions existence of maximal vectors is established leading to an explicit formula for the optimal controller an infinite matrix representation with respect to a canonical basis is derived together with an algorithm to compute it the norm of the relevant operator is approximated by special finite dimensional optimizations whose solutions lead to solving semi definite programming problems involving the computation of a matrix projective tensor norm

a one dimensional heat equation with mixed boundary conditions abstract we study a nonlinear one dimensional heat equation with nonmonotone perturbation and with mixed boundary conditions that can even be discontinuous we show that we can balance these two main difficulties in order to obtain existence of globally defined strong solutions for this class of problems the main tools used are the theory of monotone operators and the galerkin method

on superoptimal approximation by analytic and meromorphic matrix valued functions abstract it is a well known fact that for any continuous scalar valued function on the unit circle there is a unique best approximation in the hardy class h of bounded analytic functions however in the matrix valued case a best approximation by bounded analytic functions is almost never unique to make it unique one has to impose additional assumptions in 1986 n j young introduced the so called superoptimal solution of the nehari problem the word superoptimal means that we are seeking f h matrix valued to minimize not only sup f 1 sup s0 f 1 but also the suprema of further singular values sj f j 1 it was proved by v v peller and n j young that for matrix valued function in h c such superoptiminal solution is unique in this paper an alternative geometric approach to the problem is presented it allows us to obtain another proof of the uniqueness and some new results uniqueness of the superoptimal approximation by meromorphic functions a generalization of the adamyan arov krein result on approximation of scalar valued functions inequalities between s numbers of a hankel operator and superoptimal singular values our approach works well for operator valued functions as for matrix valued ones

on the optimal control of strongly nonlinear evolution equations abstract in this paper we study optimal control problems governed by nonlinear evolution equations with nonmonotone nonlinearities in the state first we prove an existence theorem then using the penalty method we obtain necessary conditions for optimality from them we derive a bang bang characterization of optimal controls finally an example of a nonlinear parabolic optimal control system is worked out in detail

a finite element solution for the transient electromagnetic response of an arbitrary two dimensional resistivity distribution we present a numerical method for solving maxwell s equations in the case of an arbitrary two dimensional resistivity distribution excited by an infinite current line the electric field is computed directly in the time domain the computations are carried out in the lower half space only because exact boundary conditions are used on the free surface the algorithm follows the finite element approach which leads after space discretization to an equation system with a sparse matrix time stepping is done with an implicit time scheme at each time step the solution of the equation system is provided by the fast system iccg 0 the resulting algorithm produces good results even when large resistivity contrasts are involved we present a test of the algorithm s performance in the case of a homogeneous earth with a reasonable grid the relative error with respect to the analytical solution does not exceed 1 percent even 2 s after the source is turned off

a polynomial approach to hankel norm and balanced approximations abstract a new unified approach to problems of hankel norm and balanced approximations is presented which is based on a combination of polynomial algebra and the geometry of invariant subspaces contrary to state space methods where contact with external properties of systems is indirect the approach presented yields new insights into basic properties of hankel norm approximation and balanced realizations several approximation results are interpreted geometrically in terms of projections also duality results in this area are printed out

positive noncommutative polynomials are sums of squares hilbert s 17th problem concerns expression of polynomials on r n as a sum of squares it is well known that many positive polynomials are not sums of squares see re d a for excellent surveys in this paper we consider symmetric noncommutative polynomials and call one matrix positive if whenever matrices of any size are substituted for the variables in the polynomial the matrix value which the polynomial takes is positive semidefinite the result in this paper is a polynomial is matrix positive if and only if it is a sum of squares

2 d resistivity modeling an approach to arrays parallel to the strike direction an algorithm for two dimensional electrical resistivity modeling using the finite element method with mixed boundary conditions can calculate the electrical potential along any arbitrary direction in the particular case of the direction parallel to the strike of the structure a numerical singularity occurs we resolved it by calculating the potential near the singularity and improving the involved interpolation of the transformed potential investigating the shape of potential in the fourier transformed space we conclude that this interpolation may combine logarithmic and exponential functions

time varying optimal disturbance minimization in presence of plant uncertainty the optimal robust disturbance rejection problem plays an important role in feedback control theory here its time varying version is solved explicitly in terms of duality and operator theory in particular the optimum is shown to satisfy a time varying allpass property moreover optimal performance is given in terms of the norm of a bilinear form the latter depends on a lower triangular projection and a multiplication operator defined on special versions of spaces of compact operators

positive extensions fejer riesz factorization and autoregressive filters in two variables in this paper we treat the two variable positive extension problem for trigonometric polynomials where the extension is required to be the reciprocal of the absolute value squared of a stable polynomial this problem may also be interpreted as an autoregressive filter design problem for bivariate stochastic processes we show that the existence of a solution is equivalent to solving a finite positive definite matrix completion problem where the completion is required to satisfy an additional low rank condition as a corollary of the main result a necessary and sufficient condition for the existence of a spectral fejerriesz factorization of a strictly positive two variable trigonometric polynomial is given in terms of the fourier coefficients of its reciprocal tools in the proofs include a specific two variable kronecker theorem based on certain elements from algebraic geometry as well as a two variable christoffel darboux like formula the key ingredient is a matrix valued polynomial that appears in a parametrized version of the schur cohn test for stability the results also have consequences in the theory of two variable orthogonal polynomials where a spectral matching result is obtained as well as in the study of inverse formulas for doubly indexed toeplitz matrices finally numerical results are presented for both the autoregressive filter problem and the factorization problem

robust stabilization over communication channels in the presence of unstructured uncertainty this technical note is concerned with the problem of controlling plants over communication channels where the plant is subject to two types of unstructured uncertainty additive uncertainty and stable coprime factor uncertainty necessary lower bounds on the rate of transmission or channel capacity c for robust stabilization are computed explicitly in particular it is shown that the lower bound in the additive uncertainty case corresponds to a fixed point of a particular function in the stable coprime factor uncertainty case the derivation relies on linear fractional transformation concepts the results are important in determining the minimum channel capacity needed in order to stabilize plants subject to unstructured uncertainty over communication channels for instance the bounds obtained can be used to analyze the effect of uncertainty on the channel capacity an illustrative example is provided

unitary interpolants and factorization indices of matrix functions abstract for an n n bounded matrix function we study unitary interpolants u i e unitary valued functions u such that u j j j u for which the toeplitz operator t u is fredholm we give a new approach based on superoptimal singular values and thematic factorizations we describe wiener hopf factorization indices of u in terms of superoptimal singular values of and thematic indices of f where f is a superoptimal approximation of by bounded analytic matrix functions the approach essentially relies on the notion of a monotone thematic factorization introduced in ap in the last section we discuss hereditary properties of unitary interpolants in particular for matrix functions of class h c we study unitary interpolants u of class qc