multivariate gaussian process emulators with nonseparable covariance structures the gaussian process regression model is a popular type of emulator used as a fast surrogate for computationally expensive simulators deterministic computer models for simulators with multivariate output common practice is to specify a separable covariance structure for the gaussian process though computationally convenient this can be too restrictive leading to poor performance of the emulator particularly when the different simulator outputs represent different physical quantities also treating the simulator outputs as independent can lead to inappropriate representations of joint uncertainty we develop nonseparable covariance structures for gaussian process emulators based on the linear model of coregionalization and convolution methods using two case studies we compare the performance of these covariance structures both with standard separable covariance structures and with emulators that assume independence between the outputs in each case study we find that only emulators with nons

learning on dynamic statistical manifolds hyperbolic balance laws with uncertain random parameters and inputs are ubiquitous in science and engineering quantification of uncertainty in predictions derived from such laws and reduction of predictive uncertainty via data assimilation remain an open challenge that is due to nonlinearity of governing equations whose solutions are highly non gaussian and often discontinuous to ameliorate these issues in a computationally efficient way we use the method of distributions which here takes the form of a deterministic equation for spatio temporal evolution of the cumulative distribution function cdf of the random system state as a means of forward uncertainty propagation uncertainty reduction is achieved by recasting the standard loss function i e discrepancy between observations and model predictions in distributional terms this step exploits the equivalence between minimization of the square error discrepancy and the kullback leibler divergence the loss function is regularized by adding a lagrangian constraint enforcing fulfilment of the cdf equation minimization is performed sequentially progressively updating the parameters of the cdf equation as more measurements are assimilated

prediction for computer experiments having quantitative and qualitative input variables this article introduces a bayesian methodology for the prediction for computer experiments having quantitative and qualitative inputs the proposed model is a hierarchical bayesian model with conditional gaussian stochastic process components for each of the qualitative inputs our model assumes that the outputs corresponding to different levels of the qualitative input have similar functional behavior in the quantitative inputs the predictive accuracy of this method is compared with the predictive accuracies of alternative proposals in examples the method is illustrated in a biomechanical engineering application

kriging hyperparameter tuning strategies response surfaces have been extensively used as a method of building effective surrogate models of high fidelity computational simulations of the numerous types of response surface models kriging is perhaps one of the most effective due to its ability to model complicated responses through interpolation or regression of known data while providing an estimate of the error in its prediction there is however little information indicating the extent to which the hyperparameters of a kriging model need to be tuned for the resulting surrogate model to be effective the following paper addresses this issue by investigating how often and how well it is necessary to tune the hyperparameters of a kriging model as it is updated during an optimization process to this end an optimization benchmarking procedure is introduced and used to assess the performance of five different tuning strategies over a range of problem sizes the results of this benchmark demonstrate the performance gains that can be associated with reducing the complexity of the hyperparameter tuning process for complicated design problems the strategy of tuning hyperparameters only once after the initial design of experiments is shown to perform poorly

noncyclic geometric phase and its non abelian generalization we use the theory of dynamical invariants to yield a simple derivation of noncyclic analogues of the abelian and non abelian geometric phases this derivation relies only on the principle of gauge invariance and elucidates the existing definitions of the abelian noncyclic geometric phase we also discuss the adiabatic limit of the noncyclic geometric phase and compute the adiabatic non abelian noncyclic geometric phase for a spin 1 magnetic or electric quadrupole interacting with a precessing magnetic electric field

inferring solutions of differential equations using noisy multi fidelity data for more than two centuries solutions of differential equations have been obtained either analytically or numerically based on typically well behaved forcing and boundary conditions for well posed problems we are changing this paradigm in a fundamental way by establishing an interface between probabilistic machine learning and differential equations we develop data driven algorithms for general linear equations using gaussian process priors tailored to the corresponding integro differential operators the only observables are scarce noisy multi fidelity data for the forcing and solution that are not required to reside on the domain boundary the resulting predictive posterior distributions quantify uncertainty and naturally lead to adaptive solution refinement via active learning this general framework circumvents the tyranny of numerical discretization as well as the consistency and stability issues of time integration and is scalable to high dimensions

effects of measurement precision and finite numbers of electrodes on linear impedance imaging algorithms impedance imaging systems reconstruct approximations to the conductivity inside a body from electrical measurements made on the boundary the linearized problem was studied by calderon who showed that if the conductivity differs little from a constant then it can be reconstructed approximately in his work calderon assumed that arbitrary current densities can be applied anywhere on the body s surface and the resulting voltages perfectly measured in practice however imaging systems apply currents through a finite number of electrodes on the surface and measure the resulting voltages with a limited precision in this paper linear algorithms are described for approximating the electrical conductivity in a disk from finitely many limited precision measurements the effects of finite precision and electrode number on a two dimensional version of calderon s method are studied it is shown that as the measurement errors go to zero and the electrodes proliferate in a space filling manner the finite electro

multifidelity cokriging for high dimensional output functions with application to hypersonic airloads computation an accurate surrogate model is capable of reproducing high fidelity results at a fraction of the computational cost cokriging is an efficient method for constructing surrogate models when two leve

stability and resolution analysis for a topological derivative based imaging functional the aim of this paper is to study a topological derivative based anomaly detection algorithm we compare its performance with other imaging approaches such as multiple signal classification backpropagation and kirchhoff migration we also investigate its stability with respect to medium and measurement noises as well as its resolution a simple postprocessing of the data set is introduced and shown to be essential in order to obtain an efficient topological based imaging functional both in terms of resolution and stability

a music type algorithm for detecting internal corrosion from electrostatic boundary measurements we establish an asymptotic representation formula for the steady state current perturbations caused by internal corrosive boundary parts of small surface measure based on this formula we design a non iterative method of music multiple signal classification type for localizing the corrosive parts from voltage to current observations we perform numerical experiments to test the viability of the algorithm and the results clearly demonstrate that the algorithm works well even in the presence of relatively high noise ratios

information based objective functions for active data selection learning can be made more efficient if we can actively select particularly salient data points within a bayesian learning framework objective functions are discussed that measure the expected informativeness of candidate measurements three alternative specifications of what we want to gain information about lead to three different criteria for data selection all these criteria depend on the assumption that the hypothesis space is correct which may prove to be their main weakness

music imaging and electromagnetic inverse scattering of multiple scattering small anisotropic spheres the foldy lax equation is used to derive a multiple scattering model for the multiple scattering small anisotropic spheres by this model if the number of the non zero singular values of the multistatic response msr matrix is smaller than the number of the antennas the range space of the msr matrix is found to be spanned by the background green s function vectors corresponding to the and components of the electric and magnetic dipoles induced in each scatterer which indicates that the multiple signal classification music method could be implemented to obtain the locations of the scatterers after estimating the positions of the scatterers a non iterative analytical method is proposed for retrieving the polarization strength tensors as well as the orientations of the principle axes of each scatterer two numerical simulations show that the music method and the non iterative method are efficacious for the nonlinear inverse scattering problem of determining the locations and polarization strength tensors of multiple scattering small anisotropic spheres such methods could also be applied to the inversion of small isotropic spheres or extended to the inversion of small bianisotropic spheres

natural norm a posteriori error estimators for reduced basis approximations we present a technique for the rapid and reliable prediction of linear functional outputs of coercive and noncoercive linear elliptic partial differential equations with affine parameter dependence the essential components are i rapidly convergent global reduced basis approximations galerkin projection onto a space wn spanned by solutions of the governing partial differential equation at n judiciously selected points in parameter space ii a posteriori error estimation relaxations of the error residual equation that provide inexpensive yet sharp bounds for the error in the outputs of interest and iii offiine online computational procedures methods which decouple the generation and projection stages of the approximation process the operation count for the online stage in which given a new parameter value we calculate the output of interest and associated error bound depends only on n typically very small and the parametric complexity of the problem in this paper we propose a new natural norm formulation for our reduced basis error estimation framework that a greatly simplifies and improves our inf sup lower bound construction offline and evaluation online a critical ingredient of our a posteriori error estimators and b much better controls significantly sharpens our output error bounds in particular through deflation for parameter values corresponding to nearly singular solution behavior we apply the method to two illustrative problems a coercive laplacian heat conduction problem which becomes singular as the heat transfer coefficient tends to zero and a non coercive helmholtz acoustics problem which becomes singular as we approach resonance in both cases we observe very economical and sharp construction of the requisite natural norm inf sup lower bound rapid convergence of the reduced basis approximation reasonable effectivities even for near singular behavior for our deflated output error estimators and significant several order of magnitude online computational savings relative to standard finite element procedures

manifold learning for the emulation of spatial fields from computational models repeated evaluations of expensive computer models in applications such as design optimization and uncertainty quantification can be computationally infeasible for partial differential equation pde models the outputs of interest are often spatial fields leading to high dimensional output spaces although emulators can be used to find faithful and computationally inexpensive approximations of computer models there are few methods for handling high dimensional output spaces for gaussian process gp emulation approximations of the correlation structure and or dimensionality reduction are necessary linear dimensionality reduction will fail when the output space is not well approximated by a linear subspace of the ambient space in which it lies manifold learning can overcome the limitations of linear methods if an accurate inverse map is available in this paper we use kernel pca and diffusion maps to construct gp emulators for very high dimensional output spaces arising from pde model simulations for diffusion maps we develop a new inverse map approximation several examples are presented to demonstrate the accuracy of our approach

on the real and complex geometric phases some of the existing definitions of the real and complex geometric phases are unified and generalized the relationship of the real to the complex geometric phase is discussed it is shown why these two phases are in general completely independent in particular a simple explicit example is given which shows that the real geometric phase is not in general the real part of the complex geometric phase

multi frequency topological derivative for approximate shape acquisition of curve like thin electromagnetic inhomogeneities abstract in this paper we investigate a non iterative imaging algorithm based on the topological derivative in order to retrieve the shape of penetrable electromagnetic inclusions when their dielectric permittivity and or magnetic permeability differ from those in the embedding homogeneous space the main objective is the imaging of crack like thin inclusions but the algorithm can be applied to arbitrarily shaped inclusions for this purpose we apply multiple time harmonic frequencies and normalize the topological derivative imaging function by its maximum value in order to verify its validity we apply it for the imaging of two dimensional crack like thin electromagnetic inhomogeneities completely hidden in a homogeneous material corresponding numerical simulations with noisy data are performed for showing the efficacy of the proposed algorithm

discovering variable fractional orders of advection dispersion equations from field data using multi fidelity bayesian optimization abstract the fractional advection dispersion equation fade can describe accurately the solute transport in groundwater but its fractional order has to be determined a priori here we employ multi fidelity bayesian optimization to obtain the fractional order under various conditions and we obtain more accurate results compared to previously published data moreover the present method is very efficient as we use different levels of resolution to construct a stochastic surrogate model and quantify its uncertainty we consider two different problem set ups in the first set up we obtain variable fractional orders of one dimensional fade considering both synthetic and field data in the second set up we identify constant fractional orders of two dimensional fade using synthetic data we employ multi resolution simulations using two level and three level gaussian process regression models to construct the surrogates

non cyclic geometric phases in a proposed two photon interferometric experiment an experimental arrangement is proposed which combines the idea of two photon interferometry with that of geometric phases a pair of photons generated by parametric down conversion is fed into a pair of mach zehnder interferometers in each of which there is phase shift produced by the helically wound optical fibre in the manner of tomita and chiao 1986 if the winding angle is a multiple of 2 pi there is a closed path in parameter space in accordance with the pioneering analysis of berry 1984 more general windings are also considered in which the path in parameter space is not closed and the geometric phase for such open paths is calculated in accordance with the gauge invariant prescription of aitchison and wanelik if the proposed experiment is realized it will provide a test for the noncyclic geometric phase for photons and for the set of prescriptions equivalent to that of aitchison and wanelik 1992 furthermore the geometric phase in this arrangement will be unequivocally quantum mechanical because of the non classical character of two photon interferometry

topological derivative strategy for one step iteration imaging of arbitrary shaped thin curve like electromagnetic inclusions in this manuscript a fast imaging of thin curve like electromagnetic inclusions completely hidden in the homogeneous domain with smooth boundaries is considered by creating an electromagnetic inclusion of a small diameter and applying the asymptotic expansion formula in the existence of such an inclusion the topological derivative is successfully derived based on this derivative a one step iteration imaging algorithm is designed by solving an adjoint problem various numerical experiments of single and multiple inclusions demonstrate the viability and limitation of the designed algorithm

probabilistic non linear principal component analysis with gaussian process latent variable models summarising a high dimensional data set with a low dimensional embedding is a standard approach for exploring its structure in this paper we provide an overview of some existing techniques for discovering such embeddings we then introduce a novel probabilistic interpretation of principal component analysis pca that we term dual probabilistic pca dppca the dppca model has the additional advantage that the linear mappings from the embedded space can easily be non linearised through gaussian processes we refer to this model as a gaussian process latent variable model gp lvm through analysis of the gp lvm objective function we relate the model to popular spectral techniques such as kernel pca and multidimensional scaling we then review a practical algorithm for gp lvms in the context of large data sets and develop it to also handle discrete valued data and missing attributes we demonstrate the model on a range of real world and artificially generated data sets

asymmetric nested lattice samples we propose an approach for constructing a new class of space filling designs called asymmetric nested lattice samples as a stepping stone to the proposed approach we introduce two methods for generating asymmetric nested orthogonal arrays we then present an elaborate randomization method to shuffle the levels of an asymmetric nested orthogonal array to produce a pair of asymmetric nested lattice samples the constructed designs have a desirable nested structure possess attractive space filling properties and allow different axes to be divided at different scales of fineness for multifidelity computer experiments sequential evaluations and multistep functional fitting and linking parameters the asymmetric feature is appealing for situations where some factors are believed to be more important or deserve more attention than others the proposed designs are also useful in problems where different factors by nature require dividing their axes at different levels of fineness

electromagnetic music type imaging of perfectly conducting arc like cracks at single frequency we propose a non iterative music multiple signal classification type algorithm for the time harmonic electromagnetic imaging of one or more perfectly conducting arc like cracks found within a homogeneous space r 2 the algorithm is based on a factorization of the multi static response msr matrix collected in the far field at a single nonzero frequency in either transverse magnetic tm mode dirichlet boundary condition or transverse electric te mode neumann boundary condition followed by the calculation of a music cost functional expected to exhibit peaks along the crack curves each half a wavelength numerical experimentation from exact noiseless and noisy data shows that this is indeed the case and that the proposed algorithm behaves in robust manner with better results in the tm mode than in the te mode for which one would have to estimate the normal to the crack to get the most optimal results

computer experiments with both qualitative and quantitative variables computer experiments have received a great deal of attention in many fields of science and technology most literature assumes that all the input variables are quantitative however researchers often encounter computer experiments involving both qualitative and quantitative variables bqqv in this article a new interface on design and analysis for computer experiments with bqqv is proposed the new designs are one kind of sliced latin hypercube designs with points clustered in the design region and possess good uniformity for each slice for computer experiments with bqqv such designs help to measure the similarities among responses of different level combinations in the qualitative variables an adaptive analysis strategy intended for the proposed designs is developed the proposed strategy allows us to automatically extract information from useful auxiliary responses to increase the precision of prediction for the target response the interface between the proposed design and the analysis strategy

crack reconstruction using a level set strategy we present a level set based technique to recover key characteristics of a defect or crack e g location length and shape in a two dimensional material from boundary electrical measurements the key feature of this work is to extend the usual level set technique for modeling volumetric objects to very thin objects two level set functions are employed the first one models the location and form of the crack and the second one models its length and connectivity an efficient gradient based method is derived in order to define evolution laws for these two level set functions which minimize the least squares data misfit numerical experiments show the utility of this method even in the presence of a significant noise level in the measurements a finite element method is used to simulate the electric field behavior in the presence of very thin objects

imaging schemes for perfectly conducting cracks we consider the problem of locating perfectly conducting cracks and estimating their geometric features from multistatic response matrix measurements at a single or multiple frequencies a main objective is to design specific crack detection rules and to analyze their receiver operating characteristics and the associated signal to noise ratios in this paper we introduce an analytic framework that uses asymptotic expansions which are uniform with respect to the wavelength to crack size ratio in combination with a hypothesis test based formulation to construct specific procedures for detection of perfectly conducting cracks a central ingredient in our approach is the use of random matrix theory to characterize the signal space associated with the multistatic response matrix measurements we present numerical experiments to illustrate some of our main findings

on the limited memory bfgs method for large scale optimization we study the numerical performance of a limited memory quasi newton method for large scale optimization which we call the l bfgs method we compare its performance with that of the method developed by buckley and lenir 1985 which combines cycles of bfgs steps and conjugate direction steps our numerical tests indicate that the l bfgs method is faster than the method of buckley and lenir and is better able to use additional storage to accelerate convergence we show that the l bfgs method can be greatly accelerated by means of a simple scaling we then compare the l bfgs method with the partitioned quasi newton method of griewank and toint 1982a the results show that for some problems the partitioned quasi newton method is clearly superior to the l bfgs method however we find that for other problems the l bfgs method is very competitive due to its low iteration cost we also study the convergence properties of the l bfgs method and prove global convergence on uniformly convex problems

petrov galerkin and spectral collocation methods for distributed order differential equations distributed order fractional operators offer a rigorous tool for mathematical modelling of multiphysics phenomena where the differential orders are distributed over a range of values rather than being just a fixed integer fraction as in standard fractional odes pdes we develop two spectrally accurate schemes namely a petrov galerkin spectral method and a spectral collocation method for distributed order fractional differential equations these schemes are developed based on the fractional sturm liouville eigen problems fslps m zayernouri and g e karniadakis j comput phys 47 2013 pp 2108 2131 in the petrov galerkin method we employ fractional nonpolynomial basis functions called jacobi polyfractonomials which are the eigenfunctions of the fslp of first kind while we employ another space of test functions as the span of polyfractonomial eigenfunctions of the fslp of second kind we define the underlying distributed sobolev space and the associated norms where we carry out the

inverse scattering by a continuation method with initial guesses from a direct imaging algorithm a novel continuation method is presented for solving the inverse medium scattering problem of the helmholtz equation which is to reconstruct the shape of the inhomogeneous medium from boundary measurements of the scattered field the boundary data is assumed to be available at multiple frequencies initial guesses are chosen from a direct imaging algorithm multiple signal classification music along with a level set representation at a certain wavenumber where the born approximation may not be valid each update via recursive linearization on the wavenumbers is obtained by solving one forward and one adjoint problem of the helmholtz equation

an inverse source problem for maxwell s equations in magnetoencephalography consider an inverse source problem for maxwell s equations which arises in determining locations of epileptic foci in the living human brain the human brain is modeled as a heterogeneous medium where the electric permittivity magnetic permeability and conductivity may all be functions a current dipole is used to model the epilepsy the inverse source problem in this context is to determine the current dipole from boundary measurements of the fields in this paper a new computational method is introduced for solving the inverse problem the method is based on a low frequency asymptotic analysis of maxwell s equations our method is constructive and has good convergence properties a crucial step of the method is to construct special test functions uniqueness and stability results for the inverse problem are also established the method and results are expected to find applications particularly in magnetoencephalography

gaussian functional regression for output prediction in this paper we introduce a gaussian functional regression gfr technique that integrates multi fidelity models with model reduction to efficiently predict the input output relationship of a high fidelity model the gfr method combines the high fidelity model with a low fidelity model to provide an estimate of the output of the high fidelity model in the form of a posterior distribution that can characterize uncertainty in the prediction a reduced basis approximation is constructed upon the low fidelity model and incorporated into the gfr method to yield an inexpensive posterior distribution of the output estimate as this posterior distribution depends crucially on a set of training inputs at which the high fidelity models are simulated we develop a greedy sampling algorithm to select the training inputs our approach results in an output prediction model that inherits the fidelity of the high fidelity model and has the computational complexity of the reduced basis approximation numerical results are presented to demonstrate the proposed approach

incorporating topological derivatives into level set methods the aim of this paper is to investigate the use of topological derivatives in combination with the level set method for shape reconstruction and optimization problems we propose a new approach generalizing the standard speed method which is obtained by using a source term in the level set equation that depends on the topological derivative of the objective functional the resulting approach can be interpreted as a generalized fixed point iteration for the optimality system with respect to topological and shape variations moreover we apply the new approach for a simple model problem in shape reconstruction where the topological derivative can be computed without additional effort finally we present numerical tests related to this model problem which demonstrate that the new method based on shape and topological derivative successfully reconstructs obstacles in situations where the standard level set approach fails

imaging of location and geometry for extended targets using the response matrix in this paper we study how to image both the location and the shape of extended targets using the response matrix obtained from inter element response of an active array of transducers in particular the time reversal technique is used for efficient initial localization of the target and the level set method is used for shape reconstruction we then show how to couple the location estimation and shape reconstruction in a complementary way to improve accuracy for range estimation resolution analysis for active arrays in remote sensing regime is also presented we illustrate with numerical experiments which show that the method is capable of imaging objects with complicated shapes and that the method is robust with respect to noisy data

machine learning of linear differential equations using gaussian processes this work leverages recent advances in probabilistic machine learning to discover governing equations expressed by parametric linear operators such equations involve but are not limited to ordinary and partial differential integro differential and fractional order operators here gaussian process priors are modified according to the particular form of such operators and are employed to infer parameters of the linear equations from scarce and possibly noisy observations such observations may come from experiments or black box computer simulations as demonstrated in several synthetic examples and a realistic application in functional genomics employ probabilistic machine learning to discover governing equations expressed by parametric linear operators proper placement of gaussian process priors allows one to efficiently infer model parameters via maximum likelihood estimation a general treatment of inverse problems governed by linear operators leading to model discovery from just a handful of noisy measurements

asymptotic imaging of perfectly conducting cracks in this paper we consider cracks with dirichlet boundary conditions we first derive an asymptotic expansion of the boundary perturbations that are due to the presence of a small crack based on this formula we design a noniterative approach for locating a collection of small cracks in order to do so we construct a response matrix from the boundary measurements the location and the length of the crack are estimated respectively from the projection onto the noise space and the first significant singular value of the response matrix indeed the direction of the crack is estimated from the second singular vector we then consider an extended crack with dirichlet boundary conditions we rigorously derive an asymptotic expansion for the boundary perturbations that are due to a shape deformation of the crack to reconstruct an extended crack from many boundary measurements we develop two methods for obtaining a good guess several numerical experiments show how the proposed techniques for imaging small cracks as well as those for obtaining good initial guesses toward reconstructing an extended crack behave

music algorithm for two dimensional inverse problems with special characteristics of cylinders a multiple signal classification music algorithm is proposed to determine the positions of small cylinders which have some special characteristics that may lend traditional music unsuccessful the tm case is discussed in detail up to three independent currents are induced on each cylinder in the tm case however for a cylinder if the material property is the same as the background medium in one or two of the principal axes or if the geometry makes one or two of the induced currents very small with respect to the others so as to be negligible in noisy scenario the number of significant sources may be less than three and the traditional music may fail in locating the cylinder in such event knowledge of the direction of the induced source becomes mandatory to locate the cylinder the proposed method determines the direction of the induced source and uses the background green s function corresponding to it as music s test function the method uses minimization of the analytical expressions and gives good results in noise free as well as noisy scenarios

predicting the output from a complex computer code when fast approximations are available s we consider prediction and uncertainty analysis for complex computer codes which can be run at different levels of sophistication in particular we wish to improve efficiency by combining expensive runs of the most complex versions of the code with relatively cheap runs from one or more simpler approximations a bayesian approach is described in which prior beliefs about the codes are represented in terms of gaussian processes an example is presented using two versions of an oil reservoir simulator 1 c complex mathematical models implemented in large computer codes have been used to study real systems in many areas of scientific research sacks et al 1989 usually because physical experimentation is too costly and sometimes impossible as in the case of large environmental systems a computer experiment involves running the code with various input values for the purpose of learning something about the real system often a simulator can be run at different levels of complexity with versions ranging from the most sophisticated high level code to the most basic for example in 4 we consider two codes which simulate oil pressure at a well of a hydrocarbon reservoir both codes use finite element analysis in which the rocks comprising the reservoir are represented by small interacting grid blocks the flow of oil within the reservoir can be simulated by considering the interaction between the blocks the two codes differ in the resolution of the grid so that we have a very accurate slow version using many small blocks and a crude approximation using large blocks which runs much faster alternatively a mathematical model could be expanded to include more of the scientific laws underlying the physical processes simple fast versions of the code may well include the most important features and are useful for preliminary investigations in real time applications the number of runs from a high level simulator may be limited by expense then there is a need to trade off the complexity of the expensive code with the availability of the simpler approximations the purpose of the current paper is to explore ways in which runs from several levels of a code can be used to make inference about the output from the most complex code we may also have uncertainty about values for the input parameters which apply in any given application uncertainty analysis of computer codes describes how this uncertainty on the inputs affects our uncertainty about the output

analysis of multi frequency subspace migration weighted by natural logarithmic function for fast imaging of two dimensional thin arc like electromagnetic inhomogeneities the present study seeks to investigate mathematical structures of a multi frequency subspace migration weighted by the natural logarithmic function for imaging of thin electromagnetic inhomogeneities from measured far field pattern to this end we designed the algorithm and calculated the indefinite integration of square of bessel function of order zero of the first kind multiplied by the natural logarithmic function this is needed for mathematical analysis of improved algorithm to demonstrate the reason why proposed multi frequency subspace migration contributes to yielding better imaging performance compared to previously suggested subspace migration algorithm this analysis is based on the fact that the singular vectors of the collected multi static response msr matrix whose elements are the measured far field pattern can be represented as an asymptotic expansion formula in the presence of such inhomogeneities to support the main research results several numerical experiments with noisy data are illustrated

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

a simple approach to emulation for computer models with qualitative and quantitative factors we propose a flexible yet computationally efficient approach for building gaussian process models for computer experiments with both qualitative and quantitative factors this approach uses the hypersphere parameterization to model the correlations of the qualitative factors thus avoiding the need of directly solving optimization problems with positive definite constraints the effectiveness of the proposed method is successfully illustrated by several examples

neural net induced gaussian process regression for function approximation and pde solution abstract neural net induced gaussian process nngp regression inherits both the high expressivity of deep neural networks deep nns as well as the uncertainty quantification property of gaussian processes gps we generalize the current nngp to first include a larger number of hyperparameters and subsequently train the model by maximum likelihood estimation unlike previous works on nngp that targeted classification here we apply the generalized nngp to function approximation and to solving partial differential equations pdes specifically we develop an analytical iteration formula to compute the covariance function of gp induced by deep nn with an error function nonlinearity we compare the performance of the generalized nngp for function approximations and pde solutions with those of gps and fully connected nns we observe that for smooth functions the generalized nngp can yield the same order of accuracy with gp while both nngp and gp outperform deep nn for non smooth functions the generalized nngp is superior to gp and comparable or superior to deep nn

bayesian design and analysis of computer experiments use of derivatives in surface prediction this article is concerned with the problem of predicting a deterministic response function yo over a multidimensional domain t given values of yo and all of its first derivatives at a set of design sites points in t the intended application is to computer experiments in which yo is an output from a computer model of a physical system and each point in t represents a particular configuration of the input parameters it is assumed that the first derivatives are already available e g from a sensitivity analysis or can be produced by the code that implements the model a bayesian approach in which the random function that represents prior uncertainty about yo is taken to be a stationary gaussian stochastic process is used the calculations needed to update the prior given observations of yo and its first derivatives at the design sites are given and are illustrated in a small example the issue of experimental design is also discussed in particular the criterion of maximizing the reduction in entropy

solving partial differential equations by collocation using radial basis functions after a series of application papers have proven the approach to be numerically effective this paper gives the first theoretical foundation for methods solving partial differential equations by collocation with possibly radial basis functions

mechanistic hierarchical gaussian processes the statistics literature on functional data analysis focuses primarily on flexible black box approaches which are designed to allow individual curves to have essentially any shape while characterizing variability such methods typically cannot incorporate mechanistic information which is commonly expressed in terms of differential equations motivated by studies of muscle activation we propose a nonparametric bayesian approach that takes into account mechanistic understanding of muscle physiology a novel class of hierarchical gaussian processes is defined that favors curves consistent with differential equations defined on motor damper spring systems a gibbs sampler is proposed to sample from the posterior distribution and applied to a study of rats exposed to noninjurious muscle activation protocols although motivated by muscle force data a parallel approach can be used to include mechanistic information in broad functional data analysis applications

from structured data to evolution linear partial differential equations abstract this paper is devoted to the derivation of computational methods for constructing partial differential equations from data following some recent works 7 14 15 20 we propose a methodology based on symbolic calculus 8 9 13 pseudospectral methods 2 3 and stochastic processes 6 in order to determine non constant coefficients of linear evolution partial differential equations pdes from a set of structured data constituted by solutions at given times and positions of an unknown linear pde

a bi fidelity method for the multiscale boltzmann equation with random parameters abstract in this paper we study the multiscale boltzmann equation with multi dimensional random parameters by a bi fidelity stochastic collocation sc method developed in 52 70 71 by choosing the compressible euler system as the low fidelity model we adapt the bi fidelity sc method to combine computational efficiency of the low fidelity model with high accuracy of the high fidelity boltzmann model with only a small number of high fidelity asymptotic preserving solver runs for the boltzmann equation the bi fidelity approximation can capture well the macroscopic quantities of the solution to the boltzmann equation in the random space a priori estimate on the accuracy between the high and bi fidelity solutions together with a convergence analysis is established finally we present extensive numerical experiments to verify the efficiency and accuracy of our proposed method

bayesian calibration of computer models we consider prediction and uncertainty analysis for systems which are approximated using complex mathematical models such models implemented as computer codes are often generic in the sense that by a suitable choice of some of the model s input parameters the code can be used to predict the behaviour of the system in a variety of specific applications however in any specific application the values of necessary parameters may be unknown in this case physical observations of the system in the specific context are used to learn about the unknown parameters the process of fitting the model to the observed data by adjusting the parameters is known as calibration calibration is typically effected by ad hoc fitting and after calibration the model is used with the fitted input values to predict the future behaviour of the system we present a bayesian calibration technique which improves on this traditional approach in two respects first the predictions allow for all sources of uncertainty including the remaining uncertainty over the fitted parameters second they attempt to correct for any inadequacy of the model which is revealed by a discrepancy between the observed data and the model predictions from even the best fitting parameter values the method is illustrated by using data from a nuclear radiation release at tomsk and from a more complex simulated nuclear accident exercise

gaussian process modeling of a functional output with information from boundary and initial conditions and analytical approximations a partial differential equation pde models a physical quantity as a function of space and time these models are often solved numerically with the finite element fe method and the computer outp

accurate uncertainty quantification using inaccurate computational models this paper proposes a novel uncertainty quantification framework for computationally demanding systems characterized by a large vector of non gaussian uncertainties it combines state of the art techniques in advanced monte carlo sampling with bayesian formulations the key departure from existing works is the use of inexpensive approximate computational models in a rigorous manner such models can readily be derived by coarsening the discretization size in the solution of the governing pdes increasing the time step when integration of odes is performed using fewer iterations if a nonlinear solver is employed or making use of lower order models it is shown that even in cases where the inaccurate models provide very poor approximations of the high fidelity response statistics of the latter can be quantified accurately with significant reductions in the computational effort multiple approximate models can be used and rigorous confidence bounds of the estimates produced are provided at all stages

two numerical methods for an inverse problem for the 2 d helmholtz equation two solution methods for the inverse problem for the 2 d helmholtz equation are developed tested and compared the proposed approaches are based on a marching finite difference scheme which requires the solution of an overdetermined system at each step the preconditioned conjugate gradient method is used for rapid solutions of these systems and an efficient preconditioner has been developed for this class of problems underlying target applications include the imaging of land mines unexploded ordinance and pollutant plumes in environmental cleanup sites each formulated as an inverse problem for a 2 d helmholtz equation the images represent the electromagnetic properties of the respective underground regions extensive numerical results are presented

a phase and space coherent direct imaging method a direct imaging algorithm for point and extended targets is presented the algorithm is based on a physical factorization of the response matrix of a transducer array the factorization is used to transform a passive target problem to an active source problem and to extract principal components tones in a phase consistent way the multitone imaging function can superpose multiple tones spatial diversity aperture of the array and frequencies bandwidth of the probing signal based on phase coherence the method is a direct imaging algorithm that is simple and efficient since no forward solver or iteration is needed robustness of the algorithm with respect to noise is demonstrated via numerical examples