convergence rates for tikhonov regularization of a two coefficient identification problem in an elliptic boundary value problem we investigate the convergence rates for tikhonov regularization of the problem of simultaneously estimating the coefficients q and a in the neumann problem for the elliptic equation rm div q nabla u au f rm in omega q partial u partial n g on the boundary partial omega omega subset mathbb r d d geq 1 when u is imprecisely given by z delta in h 1 omega u z delta h 1 omega le delta delta 0 we regularize this problem by minimizing the strictly convex functional of q a begin array lll int limits omega left q nabla u q a z delta 2 a u q a z delta 2 right dx quad rho left q q 2 l 2 omega a a 2 l 2 omega right end array over the admissible set k where 0 is the regularization parameter and q a is an a priori estimate of the true pair q a which is identified and consider the unique solution of these minimization problem as the regularized one to that of the inverse problem we obtain the convergence rate mathcal o sqrt delta as 0 and for the regularized solutions under the simple and weak source condition rm there is a function w in v rm such that u prime q dagger a dagger w q dagger q a dagger a with q dagger a dagger being the q a minimum norm solution of the coefficient identification problem u the frechet derivative of u v the sobolev space on which the boundary value problem is considered our source condition is without the smallness requirement on the source function which is popularized in the theory of regularization of nonlinear ill posed problems furthermore some concrete cases of our source condition are proved to be simply the requirement that the sought coefficients belong to certain smooth function spaces

identification of transmissivity coefficients by mollification techniques part ii n dimensional elliptic problems abstract we analyze the numerical identification of the transmissivity coefficient in the n dimensional model for linear elliptic equations this problem arises in a number of important practical problems in science the identification of this coefficient is an inverse problem which is highly ill posed and difficult to solve in order to address the ill posedness of this parameter estimation problem we use mollification techniques combined with finite differences to get a numerical method which is easy to implement in rectangular domains some numerical examples of interest for n 2 are presented

a fast multipole method for the rotne prager yamakawa tensor and its applications we present a fast multipole method fmm for computing sums involving the rotne prager yamakawa tensor the method similar to the approach in tornberg and greengard 2008 26 for the stokeslet decomposes the tensor vector product into a sum of harmonic potentials and fields induced by four different charge and dipole distributions unlike the approach based on the kernel independent fast multipole method ying et al 2004 31 which requires nine scalar fmm calls the method presented here requires only four we discuss its applications to brownian dynamics simulation with hydrodynamic interactions and present some timing results

the determination of the leading coefficient in the monotone potential sturm liouville operator from boundary measurements this paper deals with the problem of determining the leading coefficient k k u 2 of the nonlinear monotone potential sturm liouville operator au k u 2 u x q x u x x a b as an additional condition only two measured data at the boundary x a x b are used solvability and linearization of the corresponding nonlinear direct problem are given an existence of a quasi solution of the inverse problem is obtained in a suitable compact class of admissible coefficients in the second part of the paper an approximate analytical solution for the inverse problem is derived the approach presented permits to analyze well posed as well as all ill posed situations for the inverse coefficient problem numerical examples corresponding to the all considered situations are presented

regular article a fast adaptive multipole algorithm in three dimensions we present an adaptive fast multipole method for the laplace equation in three dimensions it uses both new compression techniques and diagonal forms for translation operators to achieve high accuracy at a reasonable cost

recovery of the transient heat transfer coefficient in the nonlinear boundary value problem for the heat equation the problem of numerical identification of the transient heat boundary coefficient for the one dimensional heat equation in a finite slab under nonlinear boundary conditions is examined on the basis of additional information regarding the solution after measuring the temperature and the heat flux at the non active boundary the inverse heat conduction problem is solved using a fully explicit and stable space marching scheme based on a finite difference implementation of the mollification method and the transient heat boundary coefficient is then approximately determined stability bounds are derived an several numerical examples to validate the technique provided

a fluctuating boundary integral method for brownian suspensions abstract we present a fluctuating boundary integral method fbim for overdamped brownian dynamics bd of two dimensional periodic suspensions of rigid particles of complex shape immersed in a stokes fluid we develop a novel approach for generating brownian displacements that arise in response to the thermal fluctuations in the fluid our approach relies on a first kind boundary integral formulation of a mobility problem in which a random surface velocity is prescribed on the particle surface with zero mean and covariance proportional to the green s function for stokes flow stokeslet this approach yields an algorithm that scales linearly in the number of particles for both deterministic and stochastic dynamics handles particles of complex shape achieves high order of accuracy and can be generalized to three dimensions and other boundary conditions we show that brownian displacements generated by our method obey the discrete fluctuation dissipation balance relation dfdb based on a recently developed positively split ewald method fiore et al 2017 24 near field contributions to the brownian displacements are efficiently approximated by iterative methods in real space while far field contributions are rapidly generated by fast fourier space methods based on fluctuating hydrodynamics fbim provides the key ingredient for time integration of the overdamped langevin equations for brownian suspensions of rigid particles we demonstrate that fbim obeys dfdb by performing equilibrium bd simulations of suspensions of starfish shaped bodies using a random finite difference temporal integrator

error analysis of a multisingular inverse coefficient problem for the sturm liouville operator based on boundary measurement in this paper we study the inverse problem of determining the unknown leading coefficient k k x of the linear sturm liouville operator au k x u x q x u x x 0 1 from boundary measurements when u x or and u x vanishes at several called singular points of the interval 0 1 as a result the considered inverse problem has simultaneously different types moderate or severely of ill conditioned situations in different parts of the interval 0 1 the presented inverse polynomial method permits use of a priori information about singular points either to increase the order of the polynomial approximation in each subinterval or to obtain an artificial cauchy data for the unknown coefficient error estimations for the polynomial approximations are presented for well conditioned as well as for ill conditioned situations the behaviour of the inverse problem solution with respect to both types dirichlet and neumann of noisy data is analyzed the obtained results are illustrated by various numerical examples

on the fast matrix multiplication in the boundary element method by panel clustering the boundary element method bem leads to a system of linear equations with a full matrix while fem yields sparse matrices this fact seems to require much computational work for the definition of the matrix for the solution of the system and in particular for the matrix vector multiplication which always occurs as an elementary in this paper a method for the approximate matrix vector multiplication is described which requires much less arithmetical work in addition the storage requirements are strongly reduced

a review of some recent results on the output least squares formulation of parameter estimation problems abstract in this paper recent advances in the analysis of the output least squares formulation of parameter estimation problems for distributed parameter systems are summarized we address problems of identifiability discuss the notion of output least squares stability involving tikhonov regularization in a nonlinear environment give results on the approximation of these infinite dimensional problems by finite dimensional ones and review numerical techniques

the optimal projection equations with petersen hollot bounds robust stability and performance via fixed order dynamic compensation for systems with structured real valued parameter uncertainty a feedback control design problem involving structured real valued plant parameter uncertainties is considered a quadratic lyapunov bound suggested by recent work of i r petersen and c v hollot 1986 is utilized in conjunction with the guaranteed cost approach of s s l chang and t k c peng 1972 to guarantee robust stability with robust performance bound necessary conditions that generalize the optimal projection equations for fixed order dynamic compensation are used to characterize the controller that minimizes the performance bound the design equations thus effectively serve as sufficient conditions for synthesizing dynamic output feedback controllers that provide robust stability and performance

the optimal projection equations for model reduction and the relationships among the methods of wilson skelton and moore first order necessary conditions for quadratically optimal reduced order modeling of linear time invariant systems are derived in the form of a pair of modified lyapunov equations coupled by an oblique projection which determines the optimal reduced order model this form of the necessary conditions considerably simplifies previous results of wilson 1 and clearly demonstrates the quadratic extremality and nonoptimality of the balancing method of moore 2 the possible existence of multiple solutions of the optimal projection equations is demonstrated and a relaxation type algorithm is proposed for computing these local extrema a component cost analysis of the model error criterion similar to the approach of skelton 3 is utilized at each iteration to direct the algorithm to the global minimum

identifiablity under approximation for an elliptic boundary value problem necessary and sufficient conditions for identifiability of the diffusion coefficient in galerkin approximations to a two point boundary value problem are derived for various choices of galerkin subspaces the results are further used to investigate output least squares identifiability and output least squares stability of the diffusion coefficient

robust reduced order nonstrictly proper state estimation via the optimal projection equations with petersen hollot bounds a state estimation design problem involving parametric plant uncertainties is considered an error bound suggested by recent work of petersen and hollot is utilized for guaranteeing robust estimation necessary conditions which generalize the optimal projection equations for reduced order state estimation are used to characterize the estimator which minimizes the error bound the design equations thus effectively serve as sufficient conditions for synthesizing robust estimators an additional feature is the presence of a static estimation gain in conjunction with the dynamic kalman estimator i e a nonstrictly proper estimator

determination of leading coefficients in sturm liouville operator from boundary measurements ii unicity and an engineering approach we study the problem of determining the leading coefficient k k x of the sturm liouville operator au k x u x q x u x x a b from a measured data given in the form of dirichlet or and neumann type additional boundary conditions the contribution of the each measured data to the unicity of the inverse problem solution is analysed based on the analysis the problem is treated as a cauchy problem for the first order ordinary differential equation with respect to k x for the case when the direct problem solution u u x belongs to the class of the second order polynomials and u x 0 we present an analytical formula for an approximate solution of the inverse problem numerical examples related to accuracy of the solution and to essential features of the inverse problem are demonstrated

inherent identifiability of parameters in elliptic differential equations together with appropriate boundary conditions for u on the boundary r of q for given a and f this is a well known equation in the unknown u we shall consider the inverse problem which givenf consists in determining a from u more precisely we study the question of uniqueness and continuous dependence of a in terms of u the uniqueness problem is generally referred to as the identifiability problem well known contributions are the borg and the gelfand levitan theories which require knowledge of spectral data of 1 1 see 5 61 for recent work an identifiability of a from neumann boundary data for all g in the elliptic equation v avu o 1 f g we refer to s the problem of continuous dependence of a with respect to u is of equal importance as identifiability but has received much less attention so far in 121 the study of the inverse problem associated with 1 1 is based on the observation that 1 1 is a hyperbolic equation in a employing the method of characteristics a continuous dependence analysis is given under combinations of the assumptions vul 0 and ldul 0 in this paper we concentrate on uniqueness and continuous dependence results without a priori assumptions on the sign of vul the basic and very simple idea for which we give an analytical justification is that a in 1 1

a fast multipole method for the three dimensional stokes equations many problems in stokes flow and linear elasticity require the evaluation of vector fields defined in terms of sums involving large numbers of fundamental solutions in the fluid mechanics setting these are typically the stokeslet the kernel of the single layer potential or the stresslet the kernel of the double layer potential in this paper we present a simple and efficient method for the rapid evaluation of such fields using a decomposition into a small number of coulombic n body problems following an approach similar to that of fu and rodin y fu g j rodin fast solution methods for three dimensional stokesian many particle problems commun numer meth en 16 2000 145 149 while any fast summation algorithm for coulombic interactions can be employed we present numerical results from a scheme based on the most modern version of the fast multipole method h cheng l greengard v rokhlin a fast adaptive multipole algorithm in three dimensions j comput phys 155 1999 468 498 this approach should be of value in both the solution of boundary integral equations and multiparticle dynamics

a first order lyapunov robustness method for linear systems with uncertain parameters a method for stability robustness analysis based on a quadratic lyapunov function that varies linearly with uncertainty parameters is derived linear time invariant systems with structured uncertainties are discussed the lyapunov function is optimized numerically to maximize the robustness region in parameter space numerical results are given for four examples in which the first order method is compared to previous lyapunov methods for robustness analysis while the zero order method is slightly better than the first order method for one example the first order method is clearly superior in the other three more realistic examples the first order method is especially superior for applications to active control of flexible structures where robustness with respect to unmodeled coupling between modeled modes are important issues for such applications the first order method is much better at detecting the increased robustness associated with increased separation between frequencies

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

model reduction via balanced realizations an extension and frequency weighting techniques two model reduction methods for discrete systems related to balanced realizations are described the first is a technique which utilizes the least controllable and observable subsystem in deriving a balanced discrete reduced order model for this technique as l sup infinity norm bound on the reduction error is given the second method is a frequency weighting technique for discrete and continuous time systems where the input normal or output normal realizations are modified to include a simple frequency weighting for this technique l sup infinity norm bounds on the weighted reduction errors are obtained

a lyapunov robustness bound for linear systems with periodic uncertainties abstract for linear systems involving uncertain parameters with known constant nominal values and uncertain perturbations that vary sinusoidally with time lyapunov robustness anslysis is used to determine a stability bound or margin for the amplitudes of the parameter perturbations this bound is the size of a hypercube in parameter space for which asymptotic stability is guaranteed the bound which is based on a quadratic lyapunov function that depends linearly on parameter perturbations varies with the frequency of the uncertain parameter perturbations the bound is asymptotically proportional to the square root of this frequency as it becomes large

uniqueness for an elliptic inverse problem for the elliptic equation nabla cdot p x nabla v lambda q x v f x in omega subset bbr n the problem of determining when one or more of the coefficient functions p q and f are defined uniquely by a knowledge of one or more of the solution functions v v p q f lambda is considered

a c0 finite element method for an inverse problem a c 0 finite element method is presented for an inverse problem in which the coefficient in the differential operator is to be determined from the measurement of the solution of a boundary value problem the unknown in the inverse problem is approximated by a minimizer of a cost function that includes both the output error and equation error error estimates in a weighted h 1 norm and l 2 are given numerical examples are presented to show features of the method

parameter identification by regularization for surface representation via the moving grid approach we consider the identification of a diffusion parameter in a second order elliptic equation in two dimensions by interior measurements the diffusion parameter is assumed to have discontinuities for its reconstruction we propose regularization algorithms with an adaptive grid the grid is adapted according to a measure of the smoothness of the regularized solution for the numerical computation we compare several iterative methods such as the minimal error method the steepest descent method and an inexact iteratively regularized gauss newton method the computations show that these algorithms can effectively identify the discontinuities

parameter estimation by spectral approximation in this paper chebyshev and legendre approximations are proposed for estimating parameters in differential equations which are easy to be performed the convergence and the spectral accuracy are proved even without some conditions as imposed in other papers the numerical results show the advantages of this new approach

numerical identification of a spatially varying diffusion coefficient we consider the inverse problem of identifying a spatially varying diffusion coefficient on the basis of an observed solution to the forward problem under appropriate conditions this inverse problem can be solved as a first order hyperbolic problem in the unknown coefficient we provide a modified upwind difference scheme for this hyperbolic problem and prove that its convergence rate is 0 h when certain conditions are met

an inverse polynomial method for the identification of the leading coefficient in the sturm liouville operator from boundary measurements an inverse polynomial method of determining the unknown leading coefficient k k x of the linear sturm liouville operator au k x u x q x u x x 0 1 is presented as an additional condition only two measured data at the boundary x 0 x 1 are used in absence of a singular point u x 0 u x 0 x 0 1 the inverse problem is classified as a well conditioned if there exists at least one singular point then the inverse problem is classified as moderately ill conditioned u x 0 0 x 0 0 1 u x 0 x x 0 u x 0 x 0 1 and severely ill conditioned u x 0 u x 0 0 x 0 0 1 u x 0 u x 0 x x 0 for each of the cases direct problem solution is approximated by corresponding polynomials and the inverse problem is reformulated as a cauchy problem for to the first order differential equation with respect the unknown function k k x an approximate analytical solution of the each cauchy problems are derived in explicit form numerical simulations all the above cases are given for noise free and noisy data an accuracy of the presented approach is demonstrated on numerical test solutions

variational method for multiple parameter identification in elliptic pdes abstract in the present paper we investigate the inverse problem of identifying simultaneously the diffusion matrix source term and boundary condition in the neumann boundary value problem for an elliptic partial differential equation pde from a measurement data which is weaker than required of the exact state a variational method based on energy functions with tikhonov regularization is here proposed to treat the identification problem we discretize the pde with the finite element method and prove the convergence as well as analyze error bounds of this approach

parameter identification for elliptic problems abstract the problem of computing a principal coefficient function p in the differential equation p x u f x r m m 1 on a bounded region from a knowledge of the solution function u and the right hand side f where u f are known only approximately and p may have mild discontinuities is solved by minimization of an associated functional

a kernel independent adaptive fast multipole algorithm in two and three dimensions we present a new fast multipole method for particle simulations the main feature of our algorithm is that it does not require the implementation of multipole expansions of the underlying kernel and it is based only on kernel evaluations instead of using analytic expansions to represent the potential generated by sources inside a box of the hierarchical fmm tree we use a continuous distribution of an equivalent density on a surface enclosing the box to find this equivalent density we match its potential to the potential of the original sources at a surface in the far field by solving local dirichlet type boundary value problems the far field evaluations are sparsified with singular value decomposition in 2d or fast fourier transforms in 3d we have tested the new method on the single and double layer operators for the laplacian the modified laplacian the stokes the modified stokes the navier and the modified navier operators in two and three dimensions our numerical results indicate that our method compares very well with the best known implementations of the analytic fmm method for both the laplacian and modified laplacian kernels its advantage is the relative simplicity of the implementation and its immediate extension to more general kernels

the optimal projection equations for fixed order dynamic compensation first order necessary conditions for quadratically optimal steady state fixed order dynamic compensation of a linear time invariant plant in the presence of disturbance and observation noise are derived in a new and highly simplified form in contrast to the pair of matrix riccati equations for the full order lqg case the optimal steady state fixed order dynamic compensator is characterized by four matrix equations two modified riccati equations and two modified lyapunov equations coupled by a projection whose rank is precisely equal to the order of the compensator and which determines the optimal compensator gains the coupling represents a graphic portrayal of the demise of the classical separation principle for the reduced order controller case

a riccati equation approach to the stabilization of uncertain linear systems this paper presents a method for designing a feedback control law to stabilize a class of uncertain linear systems the systems under consideration contain uncertain parameters whose values are known only to within a given compact bounding set furthermore these uncertain parameters may be time varying the method used to establish asymptotic stability of the closed loop system obtained when the feedback control is applied involves the use of a quadratic lyapunov function the main contribution of this paper involves the development of a computationally feasible algorithm for the construction of a suitable quadratic lyapunov function once the lyapunov function has been obtained it is used to construct the stabilizing feedback control law the fundamental idea behind the algorithm presented involves constructing an upper bound for the lyapunov derivative corresponding to the closed loop system this upper bound is a quadratic form by using this upper bounding procedure a suitable lyapunov function can be found by solving a certain matrix riccati equation another major contribution of this paper is the identification of classes of systems for which the success of the algorithm is both necessary and sufficient for the existence of a suitable quadratic lyapunov function

perturbation method for abstract second order inverse problems abstract we are concerned with an abstract second order inverse problem we restrict ourselves here to abstract inverse problems in which in order to solve the differential equation for u t it is necessary to know the value of some operator or functional u t g t as a function of the argument t a new approach based on the perturbation theory of linear operators is introduced among the results obtained we point out sufficient conditions of existence and uniqueness of a solution as a possible application of the abstract theorems some applications from elasticity theory are given

an inverse problem for the steady state diffusion equation we consider the problem of identifying a spatially varying diffusion coefficient using an observed solution to the forward problem under appropriate conditions this problem can be treated as a first order hyperbolic equation in the unknown coefficient we develop some continuous dependence results for this problem and also propose a particularly advantageous set of test conditions for observing the forward solution

multilevel matrix multiplication and fast solution of integral equations abstract a fast multigrid approach is described for the task of calculating k x y u y dy for each x r d discretizing by an equidistant grid with n points and meshsize h and approximating the integrations to o h2s accuracy it is shown that the complexity of this calculation can be reduced from o n2 to o sn provided the kernel k is sufficiently smooth for potential type kernels the complexity is reduced to o sn log n corresponding integral equations can be solved to a similar accuracy in basically the same amount of work using a special kind of distributed relaxation in a multigrid algorithm one and two dimensional numerical tests and theoretical derivations of optimal strategies are reported the method is applicable to the task of multiplying by any matrix with appropriate smoothness properties including most types of many body interactions

robust reduced order modeling via the optimal projection equations with peterson hollot bounds an optimal reduced order modeling problem with parametric plant uncertainty is considered a model reduction bound suggested by recent work of i r petersen and c v hollot is utilized for guaranteeing robust reduced order modeling over a specified range of uncertain parameters necessary conditions which generalize the optimal projection equations for model reduction are used to characterize the reduced order model which minimizes the model reduction bound the optimality equations thus effectively serve as sufficient conditions for characterizing robust reduced order models