fundamental limitations in passive time delay estimation part i narrow band systems time delay estimation of a noise like random signal observed at two or more spatially separated receivers is a problem of considerable practical interest in passive radar sonar applications a new method is presented to analyze the mean square error performance of delay estimation schemes based on a modified improved version of the ziv zakai lower bound zzlb this technique is shown to yield the tightest results on the attainable system performance for a wide range of signal to noise ratio snr conditions for delay estimation using narrow band ambiguity prone signals the fundamental result of this study is illustrated in fig 3 the entire domain of snr is divided into several disjoint segments indicating several distinct modes of operation if the available snr does not exceed snr 1 signal observations from the receiver outputs are completely dominated by noise thus essentially useless for the delay estimation as a result the attainable mean square error bar epsilon 2 is bounded only by the a priori parameter domain if snr 1 2 the modified zzlb coincides with the barankin bound in this regime differential delay observations are subject to ambiguities if snr snr 3 the modified zzlb coincides with the cramer rao lower bound indicating that the ambiguity in the differential delay estimation can essentially be resolved the transition from the ambiguity dominated mode of operation to the ambiguity free mode of operation starts at snr 2 and ends at snr 3 this is the threshold phenomenon in time delay estimation the various deflection points snr i and the various segments of the bound fig 3 are given as functions of such important system parameters as time bandwidth product wt signal bandwidth to center frequency ratio w 0 and the number of half wavelengths of the signal center frequency contained in the spacing between receivers with this information the composite bound illustrated in fig 3 provides the most complete characterization of the attainable system performance under any prespecified snr conditions

a levenberg marquardt method based on sobolev gradients abstract we extend the theory of sobolev gradients to include variable metric methods such as newton s method and the levenberg marquardt method as gradient descent iterations associated with stepwise variable inner products in particular we obtain existence uniqueness and asymptotic convergence results for a gradient flow based on a variable inner product

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

some analytical consequences of the inverse relation for the potts model the authors present some analytical results for the potts model using the symmetry group acting on its parameters generated by the inverse relation and other symmetries of this model in particular they find the critical manifolds and study the relationship between this group and the lee yang singularities in the complex plane for those cases where the symmetry group is finite they look at the possible consequences for some colouring problems in graph theory and more specifically for chromatic polynomials

a comparison of finite element based inversion algorithms local frequency estimation and direct inversion approach used in mre in quantitative elastography maps of the mechanical properties of soft tissue or elastograms are calculated from the measured displacement data by solving an inverse problem the model assumptions have a significant effect on elastograms motivated by the high sensitivity of imaging results to the model assumptions for in vivo magnetic resonance elastography of the prostate we compared elastograms obtained with four different methods two finite element method fem based methods developed by our group were compared with two other commonly used methods local frequency estimator lfe and curl based direct inversion c di all the methods assume a linear isotropic elastic model but the methods vary in their assumptions such as local homogeneity or incompressibility and in the specific approach used we report results using simulations phantom and ex vivo and in vivo data the simulation and phantom studies show for regions with an inclusion that the contrast to noise ratio cnr for the fem methods is about three to five times higher than the cnr for the lfe and c di and the rms error is about half the lfe method produces very smooth results i e low cnr and is fast c di is faster than the fem methods but it is only accurate in areas where elasticity variations are small the artifacts resulting from the homogeneity assumption in c di is detrimental in regions with large variations the ex vivo and in vivo results also show similar trends as the simulation and phantom studies the c fem method is more sensitive to noise compared with the mixed fem due to higher orders derivatives this is especially evident at lower frequencies where the wave curvature is smaller and it is more prone to such error causing a discrepancy in the absolute values between the mixed fem and c fem in our in vivo results in general the proposed fems use fewer simplifying assumptions and outperform the other methods but they are computationally more expensive

generalized finite element method using proper orthogonal decomposition a methodology is presented for generating enrichment functions in generalized finite element methods gfem using experimental and or simulated data the approach is based on the proper orthogonal decomposition pod technique which is used to generate low order representations of data that contain general information about the solution of partial differential equations one of the main challenges in such enriched finite element methods is knowing how to choose priori enrichment functions that capture the nature of the solution of the governing equations pod produces low order subspaces that are optimal in some norm for approximating a given data set for most problems since the solution error in galerkin methods is bounded by the error in the best approximation it is expected that the optimal approximation properties of pod can be exploited to construct efficient enrichment functions we demonstrate the potential of this approach through three numerical examples best approximation studies are conducted that reveal the advantages of using pod modes as enrichment functions in gfem over a conventional pod basis

is gauss quadrature better than clenshaw curtis we compare the convergence behavior of gauss quadrature with that of its younger brother clenshaw curtis seven line matlab codes are presented that implement both methods and experiments show that the supposed factor of 2 advantage of gauss quadrature is rarely realized theorems are given to explain this effect first following o hara and smith in the 1960s the phenomenon is explained as a consequence of aliasing of coefficients in chebyshev expansions then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log z 1 z 1 in the complex plane gauss quadrature corresponds to pade approximation at z infty clenshaw curtis quadrature corresponds to an approximation whose order of accuracy at z infty is only half as high but which is nevertheless equally accurate near 1 1

an hp adaptive mixed discontinuous galerkin fem for nearly incompressible linear elasticity we develop the a posteriori error estimation of mixed hp version discontinuous galerkin finite element methods for nearly incompressible elasticity problems in two space dimensions computable upper and lower bounds on the error measured in terms of a natural mesh dependent energy norm are derived the bounds are explicit in the local mesh sizes and approximation orders and are independent of the locking parameter a series of numerical experiments are presented which demonstrate the performance of the proposed error estimator within an automatic hp adaptive refinement procedure

the failure of a conjectured s4 symmetry in the three state checkerboard potts model low temperature expansions are obtained for the three state potts model on the checkerboard lattice the four boltzmann weights were set to various integer multiples of a low temperature variable x the series were evaluated to order x19 but it was found that the conjectured s4 symmetry failed at order x14 for the zero field partition function the spontaneous magnetisation and the zero field susceptibility

solution of the nonlinear elasticity imaging inverse problem the incompressible case we have recently developed and tested an efficient algorithm for solving the nonlinear inverse elasticity problem for a compressible hyperelastic material the data for this problem are the quasi static deformation fields within the solid measured at two distinct overall strain levels the main ingredients of our algorithm are a gradient based quasi newton minimization strategy the use of adjoint equations and a novel strategy for continuation in the material parameters in this paper we present several extensions to this algorithm first we extend it to incompressible media thereby extending its applicability to tissues which are nearly incompressible under slow deformation we achieve this by solving the forward problem using a residual based stabilized mixed finite element formulation which circumvents the ladyzenskaya babuska brezzi condition second we demonstrate how the recovery of the spatial distribution of the nonlinear parameter can be improved either by preconditioning the system of equations for the material parameters or by splitting the problem into two distinct steps finally we present a new strain energy density function with an exponential stress strain behavior that yields a deviatoric stress tensor thereby simplifying the interpretation of pressure when compared with other exponential functions we test the overall approach by solving for the spatial distribution of material parameters from noisy synthetic deformation fields

ultrasonic texture motion analysis theory and simulation a theoretical model was previously developed to evaluate the relationship between the dynamics of ultrasonic speckle and its underlying tissue the model is divided into an instrumental part represented by the point spread function in the far field of the ultrasonic apparatus and a moving tissue component described by a collection of scatterers by computing the convolution of these terms and then the envelope one obtains a simulated ultrasonic speckle pattern sequence which shows speckle motions closely linked to the tissue dynamics when small motion amplitudes are involved here a theoretical study of the correlation between various linear transformations of the tissue and the corresponding ultrasonic speckle motions is performed based on a 2d extension of the envelope cross correlation analysis of a narrow band gaussian noise in the linear scan case obviously tissue translation generates an identical speckle translation however tissue speckle motion correlation decreases with increasing rotation and or biaxial deformation lateral deformation perpendicular to the beam propagation axis being much less sensitive with respect to the transducer frequency the rotation and the axial deformation of the tissue show a better relationship with their respective speckle motion at lower frequencies while lateral deformation correlation is independent of the pulse frequency with respect to beam pulse size parameters tissue speckle correlation decreases with rotation when a wide ultrasonic beam is used while the axial deformation correlation decreases with the axial duration of the pulse this study sets the ground for the development of an ultrasonic strain gauge particularly useful for the assessment of biomechanical soft tissue and fluid flow properties based on speckle tracking

local regularization for the nonlinear inverse autoconvolution problem we develop a local regularization theory for the nonlinear inverse autoconvolution problem unlike classical regularization techniques such as tikhonov regularization this theory provides regularization methods that preserve the causal nature of the autoconvolution problem allowing for fast sequential numerical solution cal o rn 2 r 2n flops where r ll n for the method discussed in this paper as applied to the nonlinear problem in comparison the cost for tikhonov regularization applied to a general linear problem is cal o n 3 flops we prove the convergence of the regularized solutions to the true solution as the noise level in the data shrinks to zero and supply convergence rates for the cases of both l 2 and continuous data we propose several regularization methods and provide a theoretical basis for their convergence of note is that this class of methods does not require an initial guess of the unknown solution our numerical results confirm the effectiveness of the methods with results comparing favorably to numerical examples found in the literature for the autoconvolution problem e g g fleischer r gorenflo and b hofmann zamm z angew math mech 79 1999 pp 149 159 for examples using tikhonov regularization with total variation constraints and j janno inverse problems 16 2000 pp 333 348 for examples using the method of lavrent ev this especially seems to be true when it comes to the recovery of sharp features in the unknown solution we also show the effectiveness of our method in cases not covered by the theory

exact results for a checkerboard ising model with crossing and four spin interactions a seven parameter ising model with crossing and four spin interactions on a checkerboard lattice is mapped by a duality transformation onto a general eight vertex model which has only five independent parameters in a six dimensional subspace of the ising model for which the corresponding eight vertex model satisfies the free fermion condition an exact solution is found

modified error in constitutive equations mece approach for ultrasound elastography a partial differential equation constrained optimization approach is presented for reconstructing mechanical properties e g elastic moduli the proposed method is based on the minimization of an error in constitutive equations functional augmented with a least squares data misfit term referred to as mece for modified error in constitutive equations the main theme of this paper is to demonstrate several key strengths of the proposed method on experimental data in addition some illustrative examples are provided where the proposed method is compared with a common shear wave elastography swe approach to this end both synthetic data generated with transient finite element simulations as well as ultrasonically tracked displacement data from an acoustic radiation force arf experiment are used in a standard elasticity phantom the results indicate that the mece approach can produce accurate shear modulus reconstructions with significantly less bias than swe

an augmented lagrangian method for identifying discontinuous parameters in elliptic systems the identification of discontinuous parameters in elliptic systems is formulated as a constrained minimization problem combining the output least squares and the equation error method the minimization problem is then proved to be equivalent to the saddle point problem of an augmented lagrangian the finite element method is used to discretize the saddle point problem and the convergence of the discretization is also proved finally an uzawa algorithm is suggested for solving the discrete saddle point problem and is shown to be globally convergent

adjoint weighted equation for inverse problems of incompressible plane stress elasticity the adjoint weighted equation awe formulation is applied to incompressible isotropic plane stress elasticity for the solution of inverse problems for the strong form of the problem to have a solution a restrictive strain compatibility condition must hold in the awe variational formulation on the other hand a much milder sufficient condition is that uniaxial stress states be precluded making the method more robust to noisy data we prove that the formulation leads to a stable converging numerical method and show through computations that the method performs well

determination of an optimal regularization factor in system identification with tikhonov regularization for linear elastic continua this paper presents a geometric mean scheme gms to determine an optimal regularization factor for tikhonov regularization technique in the system identification problems of linear elastic continua the characteristics of non linear inverse problems and the role of the regularization are investigated by the singular value decomposition of a sensitivity matrix of responses it is shown that the regularization results in a solution of a generalized average between the a priori estimates and the a posteriori solution based on this observation the optimal regularization factor is defined as the geometric mean between the maximum singular value and the minimum singular value of the sensitivity matrix of responses the validity of the gms is demonstrated through two numerical examples with measurement errors and modelling errors copyright 2001 john wiley sons ltd

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

the critical manifolds of inhomogeneous bond percolation on bow tie and checkerboard lattices we give a conditional derivation of the inhomogeneous critical percolation manifold of the bow tie lattice with five different probabilities a problem that does not appear at first to fall into any known solvable class although our argument is mathematically rigorous only on a region of the manifold we conjecture that the formula is correct over its entire domain and we provide a non rigorous argument for this that employs the negative probability regime of the triangular lattice critical surface we discuss how the rigorous portion of our result substantially broadens the range of lattices in the solvable class to include certain inhomogeneous and asymmetric bow tie lattices and that if it could be put on a firm foundation the negative probability portion of our method would extend this class to many further systems including f y wu s checkerboard formula for the square lattice we conclude by showing that this latter problem can in fact be proved using a recent result of grimmett and manolescu for isoradial graphs lending strong evidence in favor of our other conjectured results this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

adaptive reduced basis generation for reduced order modeling for the solution of stochastic nondestructive evaluation problems a novel algorithm for creating a computationally efficient approximation of a system response that is defined by a boundary value problem is presented more specifically the approach presented is focused on substantially reducing the computational expense required to approximate the solution of a stochastic partial differential equation particularly for the purpose of estimating the solution to an associated nondestructive evaluation problem with significant system uncertainty in order to achieve this computational efficiency the approach combines reduced basis reduced order modeling with a sparse grid collocation surrogate modeling technique to estimate the response of the system of interest with respect to any designated unknown parameters provided the distributions are known the reduced order modeling component includes a novel algorithm for adaptive generation of a data ensemble based on a nested grid technique to then create the reduced order basis the capabilities and potential applicability of the approach presented are displayed through two simulated case studies regarding inverse characterization of material properties for two different physical systems involving some amount of significant uncertainty the first case study considered characterization of an unknown localized reduction in stiffness of a structure from simulated frequency response function based nondestructive testing then the second case study considered characterization of an unknown temperature dependent thermal conductivity of a solid from simulated thermal testing overall the surrogate modeling approach was shown through both simulated examples to provide accurate solution estimates to inverse problems for systems represented by stochastic partial differential equations with a fraction of the typical computational cost

maximum likelihood approach to strain imaging using ultrasound a maximum likelihood ml strategy for strain estimation is presented as a framework for designing and evaluating bioelasticity imaging systems concepts from continuum mechanics signal analysis and acoustic scattering are combined to develop a mathematical model of the ultrasonic waveforms used to form strain images the model includes three dimensional 3 d object motion described by affine transformations rayleigh scattering from random media and 3 d system response functions the likelihood function for these waveforms is derived to express the fisher information matrix and variance bounds for displacement and strain estimation the ml estimator is a generalized cross correlator for pre and post compression echo waveforms that is realized by waveform warping and filtering prior to cross correlation and peak detection experiments involving soft tissuelike media show the ml estimator approaches the cramer rao error bound for small scaling deformations at 5 mhz and 1 2 compression the predicted

a robust numerical solution to reconstruct a globally relative shear modulus distribution from strain measurements to noninvasively quantify tissue elasticity for differentiating malignancy of soft tissue the authors previously proposed a two dimensional 2 d mechanical inverse problem in which simultaneous partial differential equations pde s represented the target distribution globally of relative shear moduli with respect to reference shear moduli such that the relative values could be determined from strain distributions obtained by conventional ultrasound us or nuclear magnetic resonance nmr imaging based analysis here the authors further consider the analytic solution in the region of interest subsequently demonstrating that the problem is inevitably ill conditioned in real world applications i e noise in measurement data and improper configurations of mechanical sources reference regions make it impossible to guarantee the existence of a stable and unique target global distribution next based on clarification of the inherent problematic conditions the authors describe a newly developed numerical based implicit integration approach that novelly incorporates a computationally efficient regularization method designed to solve this differential inverse problem using just low pass filtered spectra derived from strain measurements to evaluate method effectiveness reconstructions of the global distribution are carried out using intentionally created ill conditioned models the resultant reconstructions indicate the robust solution is highly suitable while also showing it has high potential to be applied in the development of an effective yet versatile diagnostic tool for quantifying the distribution of elasticity in various soft tissues

inverse viscoelastic material characterization using pod reduced order modeling in acoustic structure interaction abstract a strategy is presented for applying the proper orthogonal decomposition pod technique for model reduction in computational inverse solution strategies for viscoelastic material characterization pod is used to derive a basis of optimal dimension from a selection of possible solution fields which are generated through a traditional acoustic structure interaction finite element model for a given vibroacoustic experiment the pod bases are applied with the galerkin weak form finite element method to create a reduced order numerical model with decreased computational cost but which still maintains accuracy close to that of the original full order finite element model the reduced order model is then combined with a global optimization technique to identify estimates to the viscoelastic material properties of a fluid immersed solid from vibroacoustic tests a strategy is also presented to select the viscoelastic parameters of the initial full order analyses used to create the pod bases the selection process is shown through an example to maximize the generalization capabilities of the reduced order model over the material search space for a minimal number of full order analyses two examples are then presented in which the parameters of rheological viscoelastic models are identified for solids immersed in water which are subject to a steady state harmonic pressure while the acoustic response is measured at a point in the surrounding fluid the pod reduced order models were able to generalize over the material search domains for the inverse problems therefore the reduced order solution strategy was capable of identifying accurate estimates to the viscoelastic behavior of the solids with minimal computational expense

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

on an inverse problem from magnetic resonance elastic imaging the imaging problem of elastography is an inverse problem the nature of an inverse problem is that it is ill conditioned we consider properties of the mathematical map which describes how the elastic properties of the tissue being reconstructed vary with the field measured by magnetic resonance imaging mri this map is a nonlinear mapping and our interest is in proving certain conditioning and regularity results for this operator which occurs implicitly in this problem of imaging in elastography in this treatment we consider the tissue to be linearly elastic isotropic and spatially heterogeneous we determine the conditioning of this problem of function reconstruction in particular for the stiffness function we further examine the conditioning when determining both stiffness and density we examine the frechet derivative of the nonlinear mapping which enables us to describe the properties of how the field affects the individual maps to the stiffness and density functions we illustrate how use of the implicit function theorem can considerably simplify the analysis of frechet differentiability and regularity properties of this underlying operator we present new results which show that the stiffness map is mildly ill posed whereas the density map suffers from medium ill conditioning computational work has been done previously to study the sensitivity of these maps but our work here is analytical the validity of the newton kantorovich and optimization methods for the computational solution of this inverse problem is directly linked to the frechet differentiability of the appropriate nonlinear operator which we justify

aerodynamic data reconstruction and inverse design using proper orthogonal decomposition the application of proper orthogonal decomposition for incomplete gappy data for compressible external aerodynamic problems has been demonstrated successfully in this paper for the first time using this approach it is possible to construct entire aerodynamic flowfields from the knowledge of computed aerodynamic flow data or measured flow data specified on the aerodynamic surface thereby demonstrating a means to effectively combine experimental and computational data the sensitivity of flow reconstruction results to available measurements and to experimental error is analyzed another new extension of this approach allows one to cast the problem of inverse airfoil design as a gappy data problem the gappy methodology demonstrates a great simplification for the inverse airfoil design problem and is found to work well on a range of examples including both subsonic and transonic cases

a computationally efficient approach for inverse material characterization combining gappy pod with direct inversion abstract an approach for computationally efficient inverse material characterization from partial field response measurements that combines the gappy proper orthogonal decomposition pod machine learning technique with a physics based direct inversion strategy is presented and evaluated gappy pod is used to derive a data reconstruction tool from a set of potential system response fields that are generated from available a priori information regarding the potential distribution of the unknown material properties then the gappy pod technique is applied to reconstruct the full spatial distribution of the system response from whatever portion of the response field has been measured with the chosen system testing method lastly a direct inversion strategy is presented that is derived from the equations governing the system response i e physics of the system which utilizes the full field response reconstructed by gappy pod to produce an estimate of the spatial distribution of the unknown material properties the direct inversion technique is a particularly computationally efficient inversion technique requiring a cost equivalent to a single numerical analysis therefore the majority of the computational expense of the presented approach is the one time potential response generation for the gappy pod technique which leads to an approach that is substantially computationally efficient overall two numerically simulated examples are shown in which the elastic modulus distribution was characterized based on partial field displacement response measurements both static and dynamic the inversion procedure was shown to have the capability to efficiently provide accurate estimates to material property distributions from partial field response measurements the direct inversion with gappy pod response estimation was also shown to be substantially tolerant to noise in comparison to the direct inversion given measured full field response lastly a physical example regarding elastography of an arterial construct from ultrasound imaging response measurements is shown to validate the practical applicability of the direct inversion approach with gappy pod response reconstruction

physiologic parameter estimation using inverse problems the estimation of in vivo physiologic parameters is an important but difficult issue in bio medicine therefore the development of mathematical techniques predicting these parameter values is very relevant in a previous work we have proposed a convection diffusion shape model which correlates colonic crypt patterns with the cellular kinetics occurring inside the crypts this correlation is significant in the context of colorectal cancer this model involves several physiologic parameters for which only qualitative information is available in the literature such as the birth rate of proliferative cells in this paper we present a framework for estimating this birth rate parameter in a colonic crypt assuming that the distribution of proliferative cells is known in that crypt more precisely we resolve an inverse problem where the unknown coefficient field the birth rate is connected to the observed measurements the proliferative cell density through a partial differential equation this inver

a study of methods of choosing the smoothing parameter in image restoration by regularization the method of regularization is portrayed as providing a compromise between fidelity to the data and smoothness with the tradeoff being determined by a scalar smoothing parameter various ways of choosing this parameter are discussed in the case of quadratic regularization criteria they are compared algebraically and their statistical properties are comparatively assessed from the results of all extensive simulation study based on simple images

inversion relations and disorder solutions on potts models the inversion symmetry and the automorphy group generated by the latter when combined with the spatial symmetries are studied for the anisotropic potts models on the triangular and checkerboard lattices the exact expression of the partition function which is known on some particular disorder subvarieties of the triangular model is checked and a generalisation is proposed for the checkerboard lattice the automorphy group is then used to extend the disorder solutions to the infinity of transformed subvarieties

constitutive error based material parameter estimation procedure for hyperelastic material abstract we have explored hyperelastic material parameter identification problem via minimizing trace of the error in constitutive relation function from partial and corrupted displacement measurements under quasi static loading the inverse problem is posed as an optimization problem where the cost function measures the error in constitutive relation due to a constitutively incompatible pair of stress and strain field incompatibility of stress and strain arises due to dissimilar constraints is imposed in generating these fields an alternating approach is adopted to solve the optimization problem a simple yet effective penalty based approach is adopted to generate strain field that weakly satisfy available measurements while the regularization due to illposedness is achieved through the penalization parameter this also allows us to consider partial and corrupted data map in our reconstruction strategy after generating incompatible fields desired reconstruction is obtained via minimizing the constitutive disparity in the process we obtain explicit parameter update formula finally we show the applicability and computational capabilities of the proposed method via limited number of numerical experiments

operator theoretic and computational approaches to ill posed problems with applications to antenna theory a general framework for regularization and approximation methods for ill posed problems is developed three levels in the resolution processes are distinguished and emphasized philosophy of resolution regularization approximation schema and regularization algorithms dilemmas and methodologies of resolution of ill posed problems and their numerical implementations are examined in this framework with particular reference to the problem of finding numerically minimum weighted norm least squares solutions of first kind integral equations and more generally of linear operator equations with nonclosed range a common problem in all these methods is delineated each method reduces the problem of resolution to a nonstandard minimization problem involving an unknown critical parameter whose optimal value is crucial to the numerical realization and amenability of the method the nonstandardness results from the fact that one does not have explicitly or a priori the function to be minimized it has to built up using additional information convergence rate estimates and robustness conditions etc several results are developed that complement recent advances in numerical analysis and regularization of inverse and ill posed identification and pattern synthesis problems an emphasis is placed on the role of constraints function space methods the role of generalized inverses and reproducing kernels in the regularization and stable computational resolution of these problems the results will be applied specifically to problems of antenna synthesis and identification however the thrust of the paper is devoted to the interdisciplinary character of operator theoretic and numerical methods for ill posed problems

surrogate model accelerated random search algorithm for global optimization with applications to inverse material identification abstract an optimization algorithm is proposed which is applicable for the global optimization of computationally expensive functions with specific applications in material identification the methodology referred to as the surrogate model accelerated random search smars algorithm is a non gradient based iterative application of a random search algorithm and the surrogate model method for optimization the random search algorithm drives the global search portion of smars thoroughly probing the search space to find optimal regions the surrogate model method then applies an artificial neural network to map local regions of the search space and produce computationally inexpensive estimates to the solution thereby accelerating the search through simulated examples the smars algorithm is shown to be both robust and efficient first the minimization of a well known function with multiple local minima was considered to demonstrate the smars optimization capabilities with a known complex response surface then two examples were considered for the inverse characterization of material properties the identification of parameters of a rheological viscoelasticity model was considered first and shows the smars algorithm s tolerance to non uniqueness over a large search space lastly the identification of the distribution of thermal diffusivity for a functionally graded material was considered and displays the smars capabilities to solve high dimensional inverse problems in all three examples the performances of two traditional global search algorithms a genetic algorithm and a random search algorithm were compared to that of the smars algorithm in all cases the smars algorithm outperformed both traditional algorithms by attaining more accurate solutions with fewer function evaluations

disorder points of the irf and checkerboard potts models conditions are given for a general interactions round a face irf model in statistical mechanics to have a disorder point and expressions obtained for the free energy and intra low correlations these are applied to the checkerboard potts model thereby verifying a recent conjecture by jaeckel and maillard 1984

methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation the application of regularization to ill conditioned problems necessitates the choice of a regularization parameter which trades fidelity to the data with smoothness of the solution the value of the regularization parameter depends on the variance of the noise in the data the problem of choosing the regularization parameter and estimating the noise variance in image restoration is examined an error analysis based on an objective mean square error mse criterion is used to motivate regularization two approaches for choosing the regularization parameter and estimating the noise variance are proposed the proposed and existing methods are compared and their relationship to linear minimum mean square error filtering is examined experiments are presented that verify the theoretical results

a computational framework for infinite dimensional bayesian inverse problems part i the linearized case with application to global seismic inversion we present a computational framework for estimating the uncertainty in the numerical solution of linearized infinite dimensional statistical inverse problems we adopt the bayesian inference formulation given observational data and their uncertainty the governing forward problem and its uncertainty and a prior probability distribution describing uncertainty in the parameter field find the posterior probability distribution over the parameter field the prior must be chosen appropriately in order to guarantee well posedness of the infinite dimensional inverse problem and facilitate computation of the posterior furthermore straightforward discretizations may not lead to convergent approximations of the infinite dimensional problem and finally solution of the discretized inverse problem via explicit construction of the covariance matrix is prohibitive due to the need to solve the forward problem as many times as there are parameters our computational framework builds on the infinite dimensional form

a new convex inversion framework for parameter identification in saddle point problems with an application to the elasticity imaging inverse problem of predicting tumor location this work presents a thorough theoretical and numerical analysis of the elasticity imaging inverse problem of tumor identification in the soft tissue of the human body beyond the obvious merits of its applications this problem also presents significant mathematical challenges the near incompressibility inherent in the model of linear elasticity in the body gives rise to the locking effect and necessitates a unique treatment of both the direct and inverse problems a general optimization framework for the identification of parameters in saddle point problems is presented along with a new modified output least squares mols objective functional the mols functional is shown to be convex thus overcoming the nonconvexity of the classical output least squares ols functional and the new framework is shown to be capable of accommodating both smooth and discontinuous parameters generalized derivative formulas for the coefficient to solution map are also given along with a complete convergence analysis

least squares methods for linear elasticity this paper develops least squares methods for the solution of linear elastic problems in both two and three dimensions our main approach is defined by simply applying the l2 norm least squares principle to a stress displacement system the constitutive and the equilibrium equations it is shown that the homogeneous least squares functional is elliptic and continuous in the h rm div omega d times h 1 omega d norm this immediately implies optimal error estimates for finite element subspaces of h rm div omega d times h 1 omega d it admits optimal multigrid solution methods as well if raviart thomas finite element spaces are used to approximate the stress tensor our method does not degrade when the material properties approach the incompressible limit least squares methods that impose boundary conditions weakly and use an inverse norm are also considered numerical results for a benchmark test problem of planar elasticity are included in order to illustrate the robustness of our

q state potts model on the checkerboard lattice a large q expansion of the partition function of the checkerboard q state potts model is given up to sixth order in q 12 this expansion is used to discuss the following two points first on the critical manifold the large q expansion agrees up to this order with the minimal solution suggested by the group structure associated with this model secondly the latent heat and some correlation functions are discussed from the point of view of their functional dependence in the parameters

the depth of young subgroups of symmetric groups abstract let n and k be positive integers we consider the ordinary depth of several young subgroups of the symmetric group s n for k n s n k has depth 2 n 2 k 1 in s n moreover for n 2 k we show that the depth of s k 1 n s 1 k s n k s k in s n is 2 2 n 1 k 1 1 in case n k 2 2 2 and it is 2 log 2 k 1 if n 2 k for 2 k n k 2 2 2 we give upper and lower bounds for the depth furthermore we give the precise values for k 10 and n 39

image reconstruction and restoration overview of common estimation structures and problems developments in the theory of image reconstruction and restoration over the past 20 or 30 years are outlined particular attention is paid to common estimation structures and to practical problems not properly solved yet the problem of image reconstruction and restoration is first formulated some of the current regularization approaches used to solve the problem are then described the concepts of a priori information and compound criterion are introduced a bayesian interpretation of the regularization techniques is given which clarifies the role of the tuning parameters and indicates how they could be estimated the practical aspects of computing the solution first when the hyperparameters are known and second when they must be estimated are then considered conclusions are drawn and points that still need to be investigated are outlined

a scalable framework for the solution of stochastic inverse problems using a sparse grid collocation approach experimental evidence suggests that the dynamics of many physical phenomena are significantly affected by the underlying uncertainties associated with variations in properties and fluctuations in operating conditions recent developments in stochastic analysis have opened the possibility of realistic modeling of such systems in the presence of multiple sources of uncertainties these advances raise the possibility of solving the corresponding stochastic inverse problem the problem of designing estimating the evolution of a system in the presence of multiple sources of uncertainty given limited information a scalable parallel methodology for stochastic inverse design problems is developed in this article the representation of the underlying uncertainties and the resultant stochastic dependant variables is performed using a sparse grid collocation methodology a novel stochastic sensitivity method is introduced based on multiple solutions to deterministic sensitivity problems the stochastic inverse design problem is transformed to a deterministic optimization problem in a larger dimensional space that is subsequently solved using deterministic optimization algorithms the design framework relies entirely on deterministic direct and sensitivity analysis of the continuum systems thereby significantly enhancing the range of applicability of the framework for the design in the presence of uncertainty of many other systems usually analyzed with legacy codes various illustrative examples with multiple sources of uncertainty including inverse heat conduction problems in random heterogeneous media are provided to showcase the developed framework

the hamiltonian structure of nonlinear elasticity the material and convective representations of solids rods and plates it is our belief that a thorough understanding of the mathematical underpinnings of elasticity is crucial to its analytical and numerical implementation for example in the analysis of rotating structures the coupling of the equations for geometrically inexact models obtained by linearization or other approximations with those for rotating rigid bodies can easily lead to misleading artificial softening effects that can significantly alter numerical results see simo and vuquoc 1986c especially equations 3 and 5 in this paper we consider fully nonlinear geometrically exact models for rods plates and shells which take into account shear and torsion as well as the usual bending effects in traditional rod and plate models these models can be obtained either from the three dimensional theory by a systematic use of projection methods see e g antman 1972 and naghdi 1972 or by a direct approach within the context of cosserat continuum remarkably the two approaches lead to essentially the same form of the governing field equations in the present context we have chosen as a model problem a particular rod model which may be regarded as an extension of the classical kirchhoff love model see love 1944 to include shear deformations as in

series analysis of the q state checkerboard potts model the low temperature expansion of the checkerboard potts model in a magnetic field is obtained up to order twelve from the so called disorder solutions various expansions and results known in the literature it is shown that this expansion drastically simplifies on the dual of the disorder variety the low temperature expansion of the magnetisation is seen to become equal to one up to order twelve when it is restricted to the dual of the disorder variety these results have to be seen as exact formal constraints on the analytic continuation of the low temperature expansion of the partition function per site similar simplifications occur for the susceptibility and higher derivatives with respect to the magnetic field these expansions are also analysed in the vicinity of these particular varieties

a variational principle for domino tilings 1 1 description of results a domino is a 1 x 2 or 2 x 1 rectangle and a tiling of a region by dominos is a way of covering that region with dominos so that there are no gaps or overlaps in 1961 kasteleyn kal found a formula for the number of domino tilings of an m x n rectangle with mn even as shown in figure 1 for rm n 68 temperley and fisher tf used a differenit method and arrived at the same result at almost exactly the same time both lines of calculation showed that the logarithm of number of tilings divided by the number of dominos in a tiling that is mn 2 converges to 2g 7r 0 58 here g is catalan s constant on the other hand in 1992 elkies et al eklp studied domino tilings of regions they called aztec diamonds figure 2 shows an aztec diamond of order 48 and showed that the logarithm of the number of tilings divided by the number of dominos converges to the smaller number log 2 2 0 35 thus even though the region in figure 1 has slightly smaller area than the region in figure 2 the former has far more domino tilings for regions with other shapes neither of these asymptotic formulas may apply in the present paper we consider simply connected regions of arbitrary shape we give an exact formula for the limiting value of the logarithm of the number of tilings per unit area as a function of the shape of the boundary of the region as the size of the region goes to infinity in particular we show that computation of this limit is intimately linked with an understanding of long range variations in the local statistics of random domino tilings such variations can be seen by comparing figures 1 and 2 each of the two tilings is random in the sense that the algorithm pwi that was used to create it generates each of the possible tilings of the region being tiled with the same probability hence one can expect each tiling to be qualitatively typical of the overwhelming majority of tilings of the region in

s4 symmetry on the checkerboard potts model the large q expansion of the partition function z of the checkerboard q state potts model with a magnetic field is given up to sixth order in q 12 up to this order z is shown to exhibit an unexpected s4 symmetry using various arguments exact results expansion etc the authors suggest the general validity of this new symmetry property of the considered model an exact expression for the magnetisation jump is also proposed and checked on the large q expansion

anisotropic linear elastic parameter estimation using error in the constitutive equation functional a modified error in the constitutive equation based approach for identification of heterogeneous and linear anisotropic elastic parameters involving static measurements is proposed and explored following an alternating minimization procedure associated with the underlying optimization problem the new strategy results in an explicit material parameter update formula for general anisotropic material this immediately allows us to derive the necessary constraints on measured data and thus restrictions on physical experimentation to achieve the desired reconstruction we consider a few common materials to derive such conditions then we exploit the invariant relationships of the anisotropic constitutive tensor to propose an identification procedure for space dependent material orientations finally we assess the numerical efficacy of the developed tools against a few parameter identification problems of engineering interest

nonlinear dynamics and loop formation in kirchhoff rods with implications to the mechanics of dna and cables the paper contributes a general dynamical formulation and numerical solution procedure for studying nonlinear and three dimensional dynamics of kirchhoff rods target applications include the dynamic formation of dna loops and supercoils as well as loops hockles in marine cables the formulation accommodates non homogeneous and non isotropic inextensible rods both with and without coupling of tension and torsion the utility of this formulation is illustrated by studying the dynamics and quasi static response of a clamped clamped rod subject to compression and or twist for slow loading rates the computed quasi static responses converge to published equilibrium solutions for a benchmark problem as loading rates increase new behaviors are observed including hysteresis in the neighborhood of equilibrium bifurcations the addition of chirality in the form of tension torsion coupling has a pronounced influence on the computed looped geometries this finding has implications for dna loops formed by dna protein binding as well as loop formation in helically wound wire and synthetic cables

on the depth of subgroups and group algebra extensions we investigate notions of depth for inclusions of rings b a in particular for group algebra extensions rh rg for finite groups h g and a non zero commutative ring r a group theoretic or combinatorial notion of depth for h in g is defined and used to show that rh rg has always finite depth we compare the depths of h g and rh rg and investigate how the depth varies with r

tissue elasticity reconstruction using linear perturbation method a new method to reconstruct the elastic modulus of soft tissue subjected to an external static compression is presented in this approach the newton raphson method is used to vary a finite element fe model of the elasticity equations to fit in a least squared sense a set of axial tissue displacement fields estimated using a correlation technique applied to ultrasound signals the ill conditioning of the hessian matrix is eliminated using the tikhonov regularization technique this regularization provides a compromise between fidelity to the observed data and a priori information of the solution using an echographic image formation model it is shown that the method converges within a few iterations 8 10 and that strain images artifacts which are common in elastography are significantly reduced after the resolution of the inverse problem

depth two hopf subalgebras of a semisimple hopf algebra abstract a depth two hopf subalgebra k of a semisimple hopf algebra h is shown to be a normal hopf subalgebra on the one hand we prove this using galois theory of quantum groupoids on the other hand we give a second proof using character theory of semisimple hopf algebras

reduced order output feedback control design for pde systems using proper orthogonal decomposition and nonlinear semidefinite programming the design of an optimal output feedback reduced order control roc law for a dynamic control system is an important example of a difficult and in general non convex nonlinear optimal control problem in this paper we present a novel numerical strategy to the solution of the roc design problem if the control system is described by partial differential equations pde the discretization of the roc problem with pde constraints leads to a large scale non convex nonlinear semidefinite program nsdp for reducing the size of the high dimensional control system first we apply a proper orthogonal decomposition pod method to the discretized pde the pod approach leads to a low dimensional model of the control system thereafter we solve the corresponding small sized nsdp by a fully iterative interior point constraint trust region ipctr algorithm ipctr is designed to take advantage of the special structure of the nsdp finally the solution is a roc for the low dimensional approximation of the control system in our numerical examples we demonstrate that the reduced order controller computed from the small scaled problem can be used to control the large scale approximation of the pde system

an equation error approach for the elasticity imaging inverse problem for predicting tumor location the primary objective of this work is to study the elasticity imaging inverse problem of identifying cancerous tumors in the human body this nonlinear inverse problem not only represents an important and interesting application it also brings forth noteworthy mathematical challenges since the underlying model is a system of elasticity equations involving incompressibility due to the locking effect classical finite element methods are not effective for incompressible elasticity equations therefore special treatment is necessary for both the direct and inverse problems to study the inverse problem in an optimization framework we propose an extension of the equation error approach we focus on two cases namely when the material parameter is sufficiently smooth and when it is may be discontinuous for the latter case we extend the total variation regularization method to the elasticity imaging inverse problem we give the existence results for the proposed equation error approach and give the convergence analysis for the discretized problem we give sufficient details on the discrete formulas as well as on the implementation issues numerical examples for smooth and discontinuous coefficients are given

sparse grid collocation schemes for stochastic natural convection problems in recent years there has been an interest in analyzing and quantifying the effects of random inputs in the solution of partial differential equations that describe thermal and fluid flow problems spectral stochastic methods and monte carlo based sampling methods are two approaches that have been used to analyze these problems as the complexity of the problem or the number of random variables involved in describing the input uncertainties increases these approaches become highly impractical from implementation and convergence points of view this is especially true in the context of realistic thermal flow problems where uncertainties in the topology of the boundary domain boundary flux conditions and heterogeneous physical properties usually require high dimensional random descriptors the sparse grid collocation method based on the smolyak algorithm offers a viable alternate method for solving high dimensional stochastic partial differential equations an extension of the collocation approach to include adaptive refinement in important stochastic dimensions is utilized to further reduce the numerical effort necessary for simulation we show case the collocation based approach to efficiently solve natural convection problems involving large stochastic dimensions equilibrium jumps occurring due to surface roughness and heterogeneous porosity are captured comparison of the present method with the generalized polynomial chaos expansion and monte carlo methods are made

a stochastic collocation method for elliptic partial differential equations with random input data this work proposes and analyzes a stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms these input data are assumed to depend on a finite number of random variables the method consists of a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it treats easily a wide range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction of the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method finally we include a section with developments posterior to the original publication of this work there we review sparse grid stochastic collocation methods which are effective collocation strategies for problems that depend on a moderately large number of random variables

linearized bayesian inference for young s modulus parameter field in an elastic model of slender structures the deformations of several slender structures at nano scale are conceivably sensitive to their non homogenous elasticity owing to their small scale it is not feasible to discern their elasticity parameter fields accurately using observations from physical experiments molecular dynamics simulations can provide an alternative or additional source of data however the challenges still lie in developing computationally efficient and robust methods to solve inverse problems to infer the elasticity parameter field from the deformations in this paper we formulate an inverse problem governed by a linear elastic model in a bayesian inference framework to make the problem tractable we use a gaussian approximation of the posterior probability distribution that results from the bayesian solution of the inverse problem of inferring young s modulus parameter fields from available data the performance of the computational framework is demonstrated using two representative loading scenarios one involving cantilever bending and the other involving stretching of a helical rod an intrinsically curved structure the results show that smoothly varying parameter fields can be reconstructed satisfactorily from noisy data we also quantify the uncertainty in the inferred parameters and discuss the effect of the quality of the data on the reconstructions

theoretical limitations of the elastic wave equation inversion for tissue elastography this article examines the theoretical limitations of the local inversion techniques for the measurement of the tissue elasticity most of these techniques are based on the estimation of the phase speed or the algebraic inversion of a one dimensional wave equation to analyze these techniques the wave equation in an elastic continuum is revisited it is proven that in an infinite medium harmonic shear waves can travel at any phase speed greater than the classically known shear wave speed by demonstrating this for a special case with cylindrical symmetry hence in addition to the mechanical properties of the tissue the phase speed depends on the geometry of the wave as well the elastic waves in an infinite cylindrical rod are studied it is proven that multiple phase speeds can coexist for a harmonic wave at a single frequency this shows that the phase speed depends not only on the mechanical properties of the tissue but also on its shape the final conclusion is that the only way to avoid theore

dimensionality reduction and polynomial chaos acceleration of bayesian inference in inverse problems we consider a bayesian approach to nonlinear inverse problems in which the unknown quantity is a spatial or temporal field endowed with a hierarchical gaussian process prior computational challenges in this construction arise from the need for repeated evaluations of the forward model e g in the context of markov chain monte carlo and are compounded by high dimensionality of the posterior we address these challenges by introducing truncated karhunen loeve expansions based on the prior distribution to efficiently parameterize the unknown field and to specify a stochastic forward problem whose solution captures that of the deterministic forward model over the support of the prior we seek a solution of this problem using galerkin projection on a polynomial chaos basis and use the solution to construct a reduced dimensionality surrogate posterior density that is inexpensive to evaluate we demonstrate the formulation on a transient diffusion equation with prescribed source terms inferring the spatially varying diffusivity of the medium from limited and noisy data

estimating the non homogeneous elastic modulus distribution from surface deformations abstract we present a novel approach to solve the inverse problem in finite elasticity for the non homogeneous shear modulus distribution solely from known surface deformation fields the inverse problem is posed as a constrained optimization problem under regularization and solved utilizing the adjoint equations hypothetical measured surface displacement fields are created by inducing indentations on the exterior of the specimen these surface displacement fields are used to test the inverse strategy on a problem domain consisting of a stiff circular inclusion in a softer homogeneous background we observe that the shear modulus reconstruction as well as the shape of the circular inclusion improves with an increasing number of surface displacement fields furthermore with increasing noise level in the surface displacement field the contrast of the reconstructions decreases

on the inverse problem of identifying lame coefficients in linear elasticity an output least squares type functional is employed to identify the lame parameters in linear elasticity to be able to identify even the discontinuous lame parameters the regularization is performed by the bv seminorm finite element discretization is used and convergence analysis is given numerical examples are given to show the feasibility of the approach

adjoint weighted variational formulation for the direct solution of inverse problems of general linear elasticity with full interior data we describe a novel variational formulation of inverse elasticity problems given interior data the class of problems considered is rather general and includes as special cases plane deformations compressibility and incompressiblity in isotropic materials 3d deformations and anisotropy the strong form of this problem is governed by equations of pure advective transport the variational formulation is based on a generalization of the adjoint weighted variational equation awe formulation originally developed for flow of a passive scalar we describe how to apply awe to various cases and prove several properties we prove that the galerkin discretization of the awe formulation leads to a stable convergent numerical method and prove optimal rates of convergence the numerical examples demonstrate optimal convergence of the method with mesh refinement for multiple unknown material parameters graceful performance in the presence of noise and robust behavior of the method when the target solution is c c0 or discontinuous copyright 2009 john wiley sons ltd

detection of cracks using neural networks and computational mechanics an inverse analysis method is proposed to simulate the a scan ultrasonic nondestructive testing by means of back propagation neural networks and computational mechanics both direct problem and inverse problem are considered in this study in the direct problem the frequency responses of a cracked medium subjected to an impact loading are calculated by the computational mechanics combining the finite element method with the boundary integral equation the transient responses are obtained using fast fourier transform in the inverse problem the back propagation neural networks are trained by the characteristic parameters extracted from the various surface responses obtained from the direct problem these surface responses carry a great deal of information about the structure of the medium with or without cracks the trained neural networks are then utilized for the classification and identification of the crack in the medium to determine the type location and length of the crack

an inverse problem approach for elasticity imaging through vibroacoustics a methodology for estimating the spatial distribution of elastic moduli using the steady state dynamic response of solids immersed in fluids is presented the technique relies on the ensuing acoustic field from a remotely excited solid to inversely estimate the spatial distribution of young s modulus of biological structures e g breast tissue this work proposes the use of gaussian radial basis functions grbf to represent the spatial variation of elastic moduli grbf are shown to possess the advantage of representing smooth functions with quasi compact support and can efficiently represent elastic moduli distributions such as those that occur in soft biological tissue in the presence of unhealthy tissue e g tumors and calcifications the direct problem consists of a coupled acoustic structure interaction boundary value problem solved in the frequency domain using the finite element method the inverse problem is cast as an optimization problem in which the error functional is defined as a measure of discrepancy between an experimentally measured response and a finite element representation of the system nongradient based optimization algorithms are used to solve the resulting optimization problem the feasibility of the proposed approach is demonstrated through a series of simulations and an experiment for comparison purposes the surface velocity response was also used for the inverse characterization as the measured response in place of the acoustic pressure

modified constitutive relation error identification strategy for transient dynamics with corrupted data the elastic case abstract in this paper we present an identification strategy for transient dynamic tests with corrupted data and its first development for the identification of elastic properties this method is based on the modified constitutive relation error principle which was initially introduced for model updating in vibration analysis the governing principle of the method consists in splitting the equations of the problem into two groups a reliable group and an uncertain group and defining mechanical fields which verify the equations of the reliable group exactly and those of the uncertain group only approximately then these fields are used to estimate the quality of the model to be identified we tested our method on various examples and found it to be very robust with respect to high levels of perturbations furthermore we show that this good performance is due to the fact that the method takes into account all the experimental data in a single calculation and introduces a distance between the measurements and the calculation we also propose an alternative strategy dedicated to cases where there is no a priori knowledge of the perturbations

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

robust structural damage identification based on multi objective optimization structural damage identification is an inverse problem which often is formulated as an optimization problem the design variables are degrees of damage in computational model and the objective is the discrepancies between the computed responses and the measured responses conventional single objective optimization approach defines the objective function by combining multiple error terms into a single one which leads to a weaker constraint in solving the identification problem an alternative approach explained in this paper is a multi objective approach that simultaneously minimizes multiple error terms a stronger constraint from multiple objectives promotes the solutions to converge to the correct solution numerical examples based on static testing are provided to illustrate the multi objective approach also conceptual explanations are given to extend the approach to include both the static testing and the dynamic testing expected challenges will also be explained

stochastic isotropic hyperelastic materials constitutive calibration and model selection biological and synthetic materials often exhibit intrinsic variability in their elastic responses under large strains owing to microstructural inhomogeneity or when elastic data are extracted from viscoelastic mechanical tests for these materials although hyperelastic models calibrated to mean data are useful stochastic representations accounting also for data dispersion carry extra information about the variability of material properties found in practical applications we combine finite elasticity and information theories to construct homogeneous isotropic hyperelastic models with random field parameters calibrated to discrete mean values and standard deviations of either the stress strain function or the nonlinear shear modulus which is a function of the deformation estimated from experimental tests these quantities can take on different values corresponding to possible outcomes of the experiments as multiple models can be derived that adequately represent the observed phenomena we apply occam s razor by providing an explicit criterion for model selection based on bayesian statistics we then employ this criterion to select a model among competing models calibrated to experimental data for rubber and brain tissue under single or multiaxial loads

cellular automata and statistical mechanical models we elaborate on the analogy between the transfer matrix of usual lattice models and the master equation describing the time development of cellular automata transient and stationary properties of probabilistic automata are linked to surface and bulk properties respectively of restricted statistical mechanical systems it is demonstrated that methods of statistical physics can be successfully used to describe the dynamic and the stationary behavior of such automata some exact results are derived including duality transformations exact mappings disorder and linear solutions many examples are worked out in detail to demonstrate how to use statistical physics in order to construct cellular automata with desired properties this approach is considered to be a first step toward the design of fully parallel probabilistic systems whose computational abilities rely on the cooperative behavior of their components

domain decomposition of stochastic pdes theoretical formulations we present a novel theoretical framework for the domain decomposition of uncertain systems defined by stochastic partial differential equations the methodology involves a domain decomposition method in the geometric space and a functional decomposition in the probabilistic space the probabilistic decomposition is based on a version of stochastic finite elements based on orthogonal decompositions and projections of stochastic processes the spatial decomposition is achieved through a schur complement based domain decomposition the methodology aims to exploit the full potential of high performance computing platforms by reducing discretization errors with high resolution numerical model in conjunction to giving due regards to uncertainty in the system the mathematical formulation is numerically validated with an example of waves in random media copyright 2008 john wiley sons ltd

a reconstruction method for gappy and noisy arterial flow data proper orthogonal decomposition pod kriging interpolation and smoothing are applied to reconstruct gappy and noisy data of blood flow in a carotid artery while we have applied these techniques to clinical data in this paper in order to rigorously evaluate their effectiveness we rely on data obtained by computational fluid dynamics cfd specifically gappy data sets are generated by removing nodal values from high resolution 3 d cfd data at random or in a fixed area while noisy data sets are formed by superimposing speckle noise on the cfd results a combined pod kriging procedure is applied to planar data sets mimicking coarse resolution ultrasound like blood flow images a method for locating the vessel wall boundary and for calculating the wall shear stress wss is also proposed the results show good agreement with the original cfd data the combined pod kriging method enhanced by proper smoothing if needed holds great potential in dealing effectively with gappy and noisy data reconstruction of in vivo velocity measurements based on color doppler ultrasound cdus imaging or magnetic resonance angiography mra

a modified error in constitutive equation approach for frequency domain viscoelasticity imaging using interior data this paper presents a methodology for the inverse identification of linearly viscoelastic material parameters in the context of steady state dynamics using interior data the inverse problem of viscoelasticity imaging is solved by minimizing a modified error in constitutive equation mece functional subject to the conservation of linear momentum the treatment is applicable to configurations where boundary conditions may be partially or completely underspecified the mece functional measures the discrepancy in the constitutive equations that connect kinematically admissible strains and dynamically admissible stresses and also incorporates the measurement data in a quadratic penalty term regularization of the problem is achieved through a penalty parameter in combination with the discrepancy principle due to morozov numerical results demonstrate the robust performance of the method in situations where the available measurement data is incomplete and corrupted by noise of varying levels

alleviating mesh constraints model reduction parallel time integration and high resolution homogenization industrial processes involving composite materials need for efficient numerical simulations in order to optimize the process parameters even if the thermo mechanical models are nowadays well established efficient simulations need for further developments in this work we are addressing some of these issues in particular the one related to fast solutions combining model reduction and parallel time integration a separated representation will be also proposed in the context of material homogenization allowing to alleviate the usual mesh constraints