Loeve Expansion Research Articles

Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability

This paper investigates structural wave propagation in waveguides with randomly varying material and geometrical properties along the axis of propagation. More specifically, it is assumed that the properties vary slowly enough such that there is no or negligible backscattering due to any changes in the propagation medium. This variability plays a significant role in the so called mid-frequency region for dynamics and vibration, but wave-based methods are typically only applicable to homogeneous and uniform waveguides. The WKB approximation is used to find a suitable generalization of the wave solutions for finite waveguides undergoing longitudinal and flexural motion. An alternative wave formulation approximation with piecewise constant properties is also derived and included, so that the internal reflections are taken into account, but this requires a discretization of the waveguide. Moreover, a Fourier like series, the Karhunen–Loeve expansion, is used to represent homogeneous and spatially correlated randomness and subsequently the wave propagation approach allows the statistics of the natural frequencies and the forced response to be derived. Experimental validation is presented using a cantilever beam whose mass per unit length is randomized by adding small discrete masses to an otherwise uniform beam. It is shown how the correlation length of the random material properties affects the natural frequency statistics and comparison with the predictions using the WKB approach is given.

Joint Channel Estimation and Symbol Detection for OFDM Systems in Rapidly Time-Varying Sparse Multipath Channels

In this paper, we propose a space-alternating generalized expectation maximization (SAGE) based joint channel estimation and data detection algorithm in compressive sensing (CS) framework for orthogonal frequency-division multiplexing (OFDM) systems in rapidly time-varying sparse multipath channels. Using dynamic parametric channel model, the sparse multipath channel is parameterized by a small number of distinct paths, each represented by the path delays and path gains. In our model, we assume that the path gains rapidly vary within the OFDM symbol duration while the number of paths and path delays vary symbol by symbol. Since the convergency of the SAGE algorithm needs statistically independent parameter set of interest to be estimated, we specifically choose the discrete orthonormal Karhunen---Loeve basis expansion model (DKL-BEM) to provide statistically independent BEM coefficients within one OFDM symbol duration and use just a few significant BEM coefficients to represent the rapidly time-varying path gains. The resulting SAGE algorithm that also incorporates inter-channel interference cancellation updates the data sequences and the channel parameters serially. The computer simulations show that our proposed algorithm has better channel estimation and symbol error rate performance than that of the orthogonal matching pursuit algorithm that is commonly proposed in the CS literature.

Error Analysis of the Dynamically Orthogonal Approximation of Time Dependent Random PDEs

In this work we discuss the dynamically orthogonal (DO) approximation of time dependent partial differential equations with random data. The approximate solution is expanded at each time instant on a time dependent orthonormal basis in the physical domain with a fixed and small number of terms. Dynamic equations are written for the evolution of the basis as well as the evolution of the stochastic coefficients of the expansion. We analyze the case of a linear parabolic equation with random data and derive a theoretical bound for the approximation error of the $S$-terms DO solution by the corresponding $S$-terms best approximation, i.e., the truncated $S$-terms Karhunen--Loeve expansion at each time instant. The bound is applicable on the largest time interval in which the best $S$-terms approximation is continuously time differentiable. Properties of the DO approximations are analyzed on simple cases of deterministic equations with random initial data. Numerical tests are presented that confirm the theoret...

Low-Rank Tensor Approximation for High-Order Correlation Functions of Gaussian Random Fields

Gaussian random fields are widely used as building blocks for modeling stochastic processes. This paper is concerned with the efficient representation of d-point correlations for such fields, which in turn enables the representation of more general stochastic processes that can be expressed as a function of one (or several) Gaussian random fields. Our representation consists of two ingredients. In the first step, we replace the random field by a truncated Karhunen--Loeve expansion and analyze the resulting error. The parameters describing the d-point correlation can be arranged in a tensor, but its storage grows exponentially in d. To avoid this, the second step consists of approximating the tensor in a low-rank tensor format, the so-called tensor train decomposition. By exploiting the particular structure of the tensor, an approximation algorithm is derived that does not need to form this tensor explicitly and allows processing correlations of order as high as d=20. The resulting representation is very c...

Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients

In this paper we analyze the numerical approximation of diffusion problems over polyhedral domains in $$\mathbb {R}^d$$Rd ($$d = 1, 2,3$$d=1,2,3), with diffusion coefficient $$a({\varvec{x}},\omega )$$a(x,?) given as a lognormal random field, i.e., $$a({\varvec{x}},\omega ) = \exp (Z({\varvec{x}},\omega ))$$a(x,?)=exp(Z(x,?)) where $${\varvec{x}}$$x is the spatial variable and $$Z({\varvec{x}}, \cdot )$$Z(x,·) is a Gaussian random field. The analysis presents particular challenges since the corresponding bilinear form is not uniformly bounded away from $$0$$0 or $$\infty $$? over all possible realizations of $$a$$a. Focusing on the problem of computing the expected value of linear functionals of the solution of the diffusion problem, we give a rigorous error analysis for methods constructed from (1) standard continuous and piecewise linear finite element approximation in physical space; (2) truncated Karhunen---Loeve expansion for computing realizations of $$a$$a (leading to a possibly high-dimensional parametrized deterministic diffusion problem); and (3) lattice-based quasi-Monte Carlo (QMC) quadrature rules for computing integrals over parameter space which define the expected values. The paper contains novel error analysis which accounts for the effect of all three types of approximation. The QMC analysis is based on a recent result on randomly shifted lattice rules for high-dimensional integrals over the unbounded domain of Euclidean space, which shows that (under suitable conditions) the quadrature error decays with $$\mathcal {O}(n^{-1+\delta })$$O(n-1+?) with respect to the number of quadrature points $$n$$n, where $$\delta >0$$?>0 is arbitrarily small and where the implied constant in the asymptotic error bound is independent of the dimension of the domain of integration.

Probabilistic analysis of consolidation that considers spatial variability using the stochastic response surface method

To obtain more accurate and reasonable results in the analyses of soil consolidation, the spatial variability of the soil properties should be considered. In this study, we analyzed the consolidation by vertical drains for soil improvement considering the spatial variability of the coefficients of consolidation. The coefficients for the variation in the vertical and horizontal coefficients of consolidation in Yeonjongdo, South Korea were evaluated, and the probability density function (PDF) was assumed by the Anderson–Darling goodness-of-fit test. Standard Gaussian random fields were generated based on a Karhunen–Loeve expansion, and then transformed using Hermite polynomials in the random field with the log-Gaussian PDF of the coefficient of consolidation. The average degree of consolidation was subsequently calculated using the finite difference method coupled with log-Gaussian random fields. In addition, the stochastic response surface method (SRSM) was applied for the efficient probabilistic uncertainty propagation. A sensitivity analysis was performed for the input parameters of the random field, and the spatial variability was considered using random variables from the Karhunen–Loeve expansion as the input data for the SRSM. The results indicated that when considering the spatial variability of soil properties, the probability of failure for the target degree of consolidation was smaller when the correlation distance was taken into account than when it was not. Additionally, the probability of failure decreased when the correlation distance decreased. Compared with the Monte Carlo simulation (MCS) results, the SRSM analysis can achieve results of similar accuracy to those obtained using the MCS analysis with a sample size of 100,000 (numerical runs), and a third-order SRSM expansion with only 333 numerical runs is sufficient for obtaining the probability with errors less than 0.01.

Independent Component Analysis to enhance performances of Karhunen–Loeve expansions for non-Gaussian stochastic processes: Application to uncertain systems

In the context of engineering systems, an essential step in uncertainty quantification is the development of accurate and efficient representation of the random input parameters. For such input parameters modeled as stochastic processes, Karhunen–Loeve expansion is a classical approach providing efficient representations using a set of uncorrelated, but generally statistically dependent random variables. The dependence structure among these random variables may be difficult to estimate statistically and is thus ignored in many practical applications. This simplifying assumption of independence may lead to considerable errors in estimating the variability in the system state, thus limiting the effectiveness of Karhunen–Loeve expansion in certain cases. In this paper, Independent Component Analysis is exploited to linearly transform the random variables used in Karhunen–Loeve expansion resulting into a set of random variables exhibiting higher order decorrelation. The stochastic wave equation is investigated for numerical illustration whereby the random stiffness coefficient is modeled as a non-Gaussian stochastic process. Under the assumption of independence among the random variables used in the Karhunen–Loeve expansion and Independent Component Analysis representations, the latter provides more accurate statistical characterization of the output process for the specific cases examined.

Variational Theory and Computations in Stochastic Plasticity

In this paper the irreversible behaviour of solids and structures in terms of rate-independent elastoplastic constitutive models in the presence of uncertainty in both material description and loading is studied. The mathematical background in convex analysis of deterministic elastoplasticity is extended to the stochastic domain, and numerical algorithms are formulated and explored. Computationally these problems—in analogy to the deterministic closest point return map—lead to the stochastic minimisation of a convex energy functional on tensor product spaces. The material parameters describing the problem are modelled as tensor-valued random fields whose numerical representation can be achieved in different ways, such as sampling, or with functional approximations using the Karhunen–Loeve and polynomial chaos expansions. Various numerical solution strategies such as direct integration, stochastic Galerkin and collocation approaches are formulated, discussed, and compared on computational examples.

Tolerance range selection of topologically optimized structures with the effects of uncertainties of manufacturing process

Robust design methods for topology optimization have received significant attention from researchers in recent years. There are various attempts in past to handle the manufacturing uncertainty of the topologically optimized components. In the present work, same issue is dealt by introducing a method of probabilistic distribution of material. Here, help from Karhunen–Loeve expansion of stochastic process, coupled with Monte Carlo method is taken. The proposed method retains the uncertainty characteristic of the specific manufacturing processes such as etching, e-beam lithography, laser micro machining, and milling. Hence, this method offers larger flexibility to the designer. The simulation for manufacturing uncertainty is utilized to determine the optimal tolerance range of the factors for robust and targeted performance, which is almost nonexistent in the literature. The methodology for tolerance range selection is capable to incorporate uncertainty of multiple factors simultaneously. In this method cross array design of experiment approach is used to analyze the effect of tolerance of each factor. The overall process for manufacturing uncertainty and tolerance range selection is illustrated using four benchmark problems. The chosen factors for considered structural problems are force, elasticity, volume fraction, and aspect ratio. Various combinations of tolerance ranges are used to simulate the performance of the optimized structure, which is expressed in terms of robustness and targeted values of compliance, and maximum deflection. Based on the simulated results of signal-to-noise ratio and mean values of performance, the combinations of tolerance range are suggested that gives a high level of robustness or targeted performance accuracy. To indicate the uniqueness of proposed approach, the obtained response for performances is compared with already available response for performance in literature for generalized approach. Current work is advantageous compared to usual robust design, and provides the performance for a specific scenario at each possible combination of tolerance ranges.

Karhunen–Loève expansion for additive Slepian processes

Abstract For Gaussian random fields defined as additive processes based on Slepian processes, we study their Karhunen–Loeve expansions and obtain the Pythagorean type distribution identities. As applications, the corresponding small deviation estimates are given.

Automatic history matching considering surrogate-based optimization and Karhunen–Loève expansions

In oil reservoir engineering one stage of great interest is the production history matching process, in which the numerical model is adjusted to reproduce the observed field production. The objective is to find the model parameters which minimize the difference between calculated and observed fluid production rates. However, the consideration of the rock properties along every cell of the numerical model that represents the reservoir as design variables would lead to an unfeasible large number of variables and consequently, a very difficult problem to be solved. To overcome that, we will assume that the permeability field will be stochastic and represented by a spectral decomposition through the Karhunen–Loeve expansion (KLE). In this work both linear and nonlinear forms for KLE expansions will be investigated. The mathematical formulation of this problems leads to nonsmooth and high-cost objective functions leading to a difficult problem to be tackled. To overcome that, surrogate models are built and incorporated in a sequential optimization strategy. Results are presented and the potential of the developed tool is discussed.

Robust Topology Optimization of Structures Under Loading Uncertainty

A new efficient and accurate approach to robust structural topology optimization is developed in this work. The objective is to minimize the expected compliance of structures under loading uncertainty. The new approach is akin to a deterministic multiple load case problem where the load cases and weights are derived by numerical methods. Efficient and accurate methods to compute expected compliance and its derivatives are developed for both concentrated and distributed uncertain loads. The Monte Carlo method and matrix decomposition are employed for concentrated loads, and the Karhunen–Loeve expansion and its orthogonal properties of random variables are used for distributed loads. Numerical examples using the modified solid isotropic material with penalization approach are provided to demonstrate the accuracy and efficiency of the proposed approach.

Uncertainty quantification of Delft catamaran resistance, sinkage and trim for variable Froude number and geometry using metamodels, quadrature and Karhunen–Loève expansion

A framework for assessing convergence and validation of non-intrusive uncertainty quantification (UQ) methods is studied and applied to a complex industrial problem in ship design, namely the high-speed Delft Catamaran advancing in calm water, with variable Froude number and geometry. Relationship between UQ studies and deterministic verification and validation is discussed. Computations are performed using high- (URANS) and low- (potential flow) fidelity simulations. Froude number has expected value and standard deviation equal to 0.5 and 0.05, respectively, on a truncated normal distribution. Geometric uncertainty is related to the research space of a simulation-based design optimization, and assessed through the Karhunen–Loeve expansion (KLE). Monte Carlo method with Latin hypercube sampling (MC-LHS) is used to compute expected value, standard deviation, distribution and uncertainty intervals for resistance, sinkage and trim. MC-LHS with CFD is used as a benchmark for validating less costly UQ methods, including MC-LHS with metamodels and standard quadrature formulas. Gaussian quadrature is found the most efficient method; however, MC-LHS with metamodels is preferred since provides with confidence intervals and distributions in a straightforward way and at reasonably small computational cost. UQ results are compared to earlier deterministic single- and multi-objective optimization; reduced-dimensional KLE studies for geometric variability indicate that stochastic optimization would not be of great benefit for the present problem.

Experimental Characterization of Transition Noise in HAMR

Due to the sensitivity of the write process to temperature variations and laser position alignment, transition noise in heat-assisted magnetic recording (HAMR) may exhibit characteristics that are significantly different from those of conventional magnetic recording. In this work, statistical properties of transition noise in HAMR are investigated using real data taken off a spin stand. The Karhunen Loeve (K-L) expansion method is applied to the read waveforms corresponding to a pseudorandom bit pattern written at varying locations of a disk.

A general analytical solution for the stochastic Milne problem using Karhunen–Loeve (K–L) expansion

This paper considers the solution of the stochastic integro-differential equation of Milne problem with random operator. The Pomraning–Eddington method is implemented to get a closed form solution deterministically. Relying on the spectral properties of the covariance function, the Karhunen–Loeve (K–L) expansion is used to represent the input stochastic process in the deterministic solution. This leads to an explicit expression for the solution process as a multivariate functional of a set of uncorrelated random variables. By using different distributions for these variables, the work is realized through computing the mean and the variance of the solution. The numerical results are found in agreement with those obtained in the literature.

A dynamically bi-orthogonal method for time-dependent stochastic partial differential equations I: Derivation and algorithms

We propose a dynamically bi-orthogonal method (DyBO) to solve time dependent stochastic partial differential equations (SPDEs). The objective of our method is to exploit some intrinsic sparse structure in the stochastic solution by constructing the sparsest representation of the stochastic solution via a bi-orthogonal basis. It is well-known that the Karhunen–Loeve expansion (KLE) minimizes the total mean squared error and gives the sparsest representation of stochastic solutions. However, the computation of the KL expansion could be quite expensive since we need to form a covariance matrix and solve a large-scale eigenvalue problem. The main contribution of this paper is that we derive an equivalent system that governs the evolution of the spatial and stochastic basis in the KL expansion. Unlike other reduced model methods, our method constructs the reduced basis on-the-fly without the need to form the covariance matrix or to compute its eigendecomposition. In the first part of our paper, we introduce the derivation of the dynamically bi-orthogonal formulation for SPDEs, discuss several theoretical issues, such as the dynamic bi-orthogonality preservation and some preliminary error analysis of the DyBO method. We also give some numerical implementation details of the DyBO methods, including the representation of stochastic basis and techniques to deal with eigenvalue crossing. In the second part of our paper [11], we will present an adaptive strategy to dynamically remove or add modes, perform a detailed complexity analysis, and discuss various generalizations of this approach. An extensive range of numerical experiments will be provided in both parts to demonstrate the effectiveness of the DyBO method.

Funclust: A curves clustering method using functional random variables density approximation

A new method for clustering functional data is proposed under the name Funclust. This method relies on the approximation of the notion of probability density for functional random variables, which generally does not exist. Using the Karhunen–Loeve expansion of a stochastic process, this approximation leads to define an approximation for the density of functional variables. Based on this density approximation, a parametric mixture model is proposed. The parameter estimation is carried out by an EM-like algorithm, and the maximum a posteriori rule provides the clusters. The efficiency of Funclust is illustrated on several real datasets, as well as for the characterization of the Mars surface.

Uncertainty quantification of contaminant transport and risk assessment with conditional stochastic collocation method

Solute transport prediction is always subject to uncertainty due to the scarcity of observation data. The data worth of limited measurements can be explored by conditional simulation. This paper presents an efficient approach for the conditional simulation of solute transport in a randomly heterogeneous aquifer. The conditioning conductivity field is parameterized by the Karhunen–Loeve (KL) expansion, and the concentration field is represented by Lagrange polynomials of random variables in the KL expansion. After employing the stochastic collocation method (SCM), stochastic governing advection–dispersion equations are reduced to a series of uncoupled deterministic equations. The concentration realizations can be obtained by sampling the established Lagrange polynomials instead of solving governing equations repeatedly. We assess the accuracy and computational efficiency of this method in comparison to the conditional Monte Carlo simulation. The influence of conditioning to hydraulic conductivity measurements on transport is analyzed. Numerical results demonstrate that the SCM can efficiently derive the conditional statistics of concentration as well as the probability of the aquifer to be contaminated. It is shown that the contamination risk is significantly influenced by measurements conditioning.

Coarse-Grid Sampling Interpolatory Methods for Approximating Gaussian Random Fields

Random fields can be approximated using grid-based discretizations of their covariance functions followed by, e.g., an eigendecomposition (i.e., a Karhunen--Loeve expansion) or a Cholesky factorization of the resulting covariance matrix. In this paper, we consider Gaussian random fields and we analyze the efficiency gains obtained by using low-rank approximations based on constructing a coarse grid covariance matrix, followed by either an eigendecomposition or a Cholesky factorization of that matrix, followed by interpolation from the coarse grid onto the fine grid. The result is coarser sampling and smaller decomposition or factorization problems than that for full-rank approximations. With one-dimensional experiments we examine the relative merits, with respect to accuracy achieved for the same computational complexity, of the different low-rank approaches. We find that interpolation from the coarse grid combined with the Cholesky factorization of the coarse grid covariance matrix yields the most effici...

Reduced Basis Methods for Parameterized Partial Differential Equations with Stochastic Influences Using the Karhunen--Loève Expansion

We consider parametric partial differential equations (PPDEs) with stochastic influences, e.g., in terms of random coefficients. Using standard discretizations such as finite elements, this often amounts to high-dimensional problems. In a many-query context, the PPDE has to be solved for various instances of the deterministic parameter as well as the stochastic influences. To decrease computational complexity, we derive a reduced basis method (RBM), where the uncertainty in the coefficients is modeled using Karhunen--Loeve (KL) expansions. We restrict ourselves to linear coercive problems with linear and quadratic output functionals. A new a posteriori error analysis is presented that generalizes and extends some of the results by Boyaval et al. [Comput. Methods Appl. Mech. Engrg., 198 (2009), pp. 3187--3206]. The additional KL-truncation error is analyzed for the state, output functionals, and also for statistical outputs such as mean and variance. Error estimates for quadratic outputs are obtained using...