Multi-element Probabilistic Collocation Method Research Articles

Density estimation for time-dependent PDE with random input by a Legendre-based multi-element probabilistic collocation method

This paper proposed a Legendre-based multi-element probabilistic collocation method for time-dependent stochastic differential equations, used for density estimation of unknown functions. This method involves discretizing the stochastic space, and on each element, constructing Lagrange interpolation basis functions based on Legendre–Gauss–Lobatto collocation/quadrature nodes. The proposed method is applied to approximate one-dimensional/two-dimensional smooth/non-smooth functions and is tested for accuracy in approximating random function values, density estimations, and mathematical expectations. This method is applied to stochastic nonlinear Schrödinger equations and coupled stochastic nonlinear Schrödinger equations, and all numerical results are compared with Monte Carlo simulation.

A Computational Stochastic Methodology for the Design of Random Meta-materials under Geometric Constraints

We present a computational stochastic methodology for generating and optimizing random meta-material (MM) configurations with nonoverlapping geometric constraints subject to various types of covariances and distributions characterizing the randomness of the MM configurations. The methodology developed consists of three main components: (1) a deterministic solver for electromagnetic scattering from multiple objects, (2) the Karhunen--Loève (K--L) expansion to represent the correlated configurations of the scattering objects, and (3) the multi-element probabilistic collocation method (ME-PCM) to provide flexibility in achieving desired distributions of the MMs. In the current work, we employ random fields from a Spartan family that includes covariance functions with damped oscillatory behavior. The algorithm is applied to study light propagation through random layered heterojunctions and random 3-D MMs. We found that greater transmission and reflection, compared to the uniformly spaced structures, can be achieved for a structure with an oscillatory spacing profile along the propagation direction. Optimized configurations of the heterojunctions and 3-D MMs have been found with larger or smaller transmission coefficients for different wave numbers of the incoming wave and different correlations.

Adaptive multi-element polynomial chaos with discrete measure: Algorithms and application to SPDEs

We develop a multi-element probabilistic collocation method (ME-PCM) for arbitrary discrete probability measures with finite moments and apply it to solve partial differential equations with random parameters. The method is based on numerical construction of orthogonal polynomial bases in terms of a discrete probability measure. To this end, we compare the accuracy and efficiency of five different constructions. We develop an adaptive procedure for decomposition of the parametric space using the local variance criterion. We then couple the ME-PCM with sparse grids to study the Korteweg–de Vries (KdV) equation subject to random excitation, where the random parameters are associated with either a discrete or a continuous probability measure. Numerical experiments demonstrate that the proposed algorithms lead to high accuracy and efficiency for hybrid (discrete–continuous) random inputs.

Uncertainty Quantification in Transcranial Magnetic Stimulation via High-Dimensional Model Representation

A computational framework for uncertainty quantification in transcranial magnetic stimulation (TMS) is presented. The framework leverages high-dimensional model representations (HDMRs), which approximate observables (i.e., quantities of interest such as electric (E) fields induced inside targeted cortical regions) via series of iteratively constructed component functions involving only the most significant random variables (i.e., parameters that characterize the uncertainty in a TMS setup such as the position and orientation of TMS coils, as well as the size, shape, and conductivity of the head tissue). The component functions of HDMR expansions are approximated via a multielement probabilistic collocation (ME-PC) method. While approximating each component function, a quasi-static finite-difference simulator is used to compute observables at integration/collocation points dictated by the ME-PC method. The proposed framework requires far fewer simulations than traditional Monte Carlo methods for providing highly accurate statistical information (e.g., the mean and standard deviation) about the observables. The efficiency and accuracy of the proposed framework are demonstrated via its application to the statistical characterization of E-fields generated by TMS inside cortical regions of an MRI-derived realistic head model. Numerical results show that while uncertainties in tissue conductivities have negligible effects on TMS operation, variations in coil position/orientation and brain size significantly affect the induced E-fields. Our numerical results have several implications for the use of TMS during depression therapy: 1) uncertainty in the coil position and orientation may reduce the response rates of patients; 2) practitioners should favor targets on the crest of a gyrus to obtain maximal stimulation; and 3) an increasing scalp-to-cortex distance reduces the magnitude of E-fields on the surface and inside the cortex.

An Adaptive Multi-Element Probabilistic Collocation Method for Statistical EMC/EMI Characterization

An adaptive multi-element probabilistic collocation (ME-PC) method for quantifying uncertainties in electromagnetic compatibility and interference phenomena involving electrically large, multi-scale, and complex platforms is presented. The method permits the efficient and accurate statistical characterization of observables (i.e., quantities of interest such as coupled voltages) that potentially vary rapidly and/or are discontinuous in the random variables (i.e., parameters that characterize uncertainty in a system's geometry, configuration, or excitation). The method achieves its efficiency and accuracy by recursively and adaptively dividing the domain of the random variables into subdomains using as a guide the decay rate of relative error in a polynomial chaos expansion of the observables. While constructing local polynomial expansions on each subdomain, a fast integral-equation-based deterministic field-cable-circuit simulator is used to compute the observable values at the collocation/integration points determined by the adaptive ME-PC scheme. The adaptive ME-PC scheme requires far fewer (computationally costly) deterministic simulations than traditional polynomial chaos collocation and Monte Carlo methods for computing averages, standard deviations, and probability density functions of rapidly varying observables. The efficiency and accuracy of the method are demonstrated via its applications to the statistical characterization of voltages in shielded/unshielded microwave amplifiers and magnetic fields induced on car tire pressure sensors.

Stochastic smoothed profile method for modeling random roughness in flow problems

We present an efficient computational method to model fluid flow in the presence of random wall roughness. A random flow domain is represented by a stochastic indicator function having a smoothed profile perpendicular to roughness, and the random domain is discretized with a fixed non-conformal grid. This procedure introduces a stochastic force into the Navier–Stokes equations, and modifies the boundary conditions at the fluid–solid interface. We employ a high-order semi-implicit splitting scheme implemented in the context of a spectral/hp element method in order to discretize the physical domain. The stochastic roughness is treated as a second-order autoregressive process that is represented by a Karhunen–Loève expansion. A multi-element probabilistic collocation method is employed to solve the resulting stochastic Navier–Stokes equations. This method is applied to simulate external flow past a rough cylinder and internal Stokes flow between two parallel plates with random wall roughness. In the first problem, we develop an analytical solution for the asymptotic behavior of the lift coefficient CL to verify the results. In the second test-case, we compare the mean and the standard deviation of the velocity field to those obtained from a different method called stochastic mapping approach (SMA), developed by Tartakovsky and Xiu (2006).

Statistical Characterization of Electromagnetic Wave Propagation in Mine Environments

A computational framework for statistically characterizing electromagnetic (EM) wave propagation through mine tunnels and galleries is presented. The framework combines a multi-element probabilistic collocation method with a full-wave fast Fourier transform and fast multipole method accelerated surface integral equation-based EM simulator to statistically characterize fields from wireless transmitters in complex mine environments.

Multi-element probabilistic collocation method in high dimensions

We combine multi-element polynomial chaos with analysis of variance (ANOVA) functional decomposition to enhance the convergence rate of polynomial chaos in high dimensions and in problems with low stochastic regularity. Specifically, we employ the multi-element probabilistic collocation method MEPCM [1] and so we refer to the new method as MEPCM-A. We investigate the dependence of the convergence of MEPCM-A on two decomposition parameters, the polynomial order μ and the effective dimension ν, with ν ≪ N , and N the nominal dimension. Numerical tests for multi-dimensional integration and for stochastic elliptic problems suggest that ν ⩾ μ for monotonic convergence of the method. We also employ MEPCM-A to obtain error bars for the piezometric head at the Hanford nuclear waste site under stochastic hydraulic conductivity conditions. Finally, we compare the cost of MEPCM-A against Monte Carlo in several hundred dimensions, and we find MEPCM-A to be more efficient for up to 600 dimensions for a specific multi-dimensional integration problem involving a discontinuous function.

Multi-element probabilistic collocation for sensitivity analysis in cellular signalling networks

The multi-element probabilistic collocation method (ME-PCM) as a tool for sensitivity analysis of differential equation models as applied to cellular signalling networks is formulated. This method utilises a simple, efficient sampling algorithm to quantify local sensitivities throughout the parameter space. The application of the ME-PCM to a previously published ordinary differential equation model of the apoptosis signalling network is presented. The authors verify agreement with the previously identified regions of sensitivity and then go on to analyse this region in greater detail with the ME-PCM. The authors demonstrate the generality of the ME-PCM by studying sensitivity of the system using a variety of biologically relevant markers in the system such as variation in one (or many) chemical species as a function of time, and total exposure of a single chemical species.

The multi-element probabilistic collocation method (ME-PCM): Error analysis and applications

Stochastic spectral methods are numerical techniques for approximating solutions to partial differential equations with random parameters. In this work, we present and examine the multi-element probabilistic collocation method (ME-PCM), which is a generalized form of the probabilistic collocation method. In the ME-PCM, the parametric space is discretized and a collocation/cubature grid is prescribed on each element. Both full and sparse tensor product grids based on Gauss and Clenshaw-Curtis quadrature rules are considered. We prove analytically and observe in numerical tests that as the parameter space mesh is refined, the convergence rate of the solution depends on the quadrature rule of each element only through its degree of exactness. In addition, the L 2 error of the tensor product interpolant is examined and an adaptivity algorithm is provided. Numerical examples demonstrating adaptive ME-PCM are shown, including low-regularity problems and long-time integration. We test the ME-PCM on two-dimensional Navier-Stokes examples and a stochastic diffusion problem with various random input distributions and up to 50 dimensions. While the convergence rate of ME-PCM deteriorates in 50 dimensions, the error in the mean and variance is two orders of magnitude lower than the error obtained with the Monte Carlo method using only a small number of samples (e.g., 100). The computational cost of ME-PCM is found to be favorable when compared to the cost of other methods including stochastic Galerkin, Monte Carlo and quasi-random sequence methods.

Stochastic modeling of random roughness in shock scattering problems: Theory and simulations

Random roughness is omnipresent in engineering applications and may often affect performance in unexpected way. Here, we employ synergistically stochastic simulations and second-order stochastic perturbation analysis to study supersonic flow past a wedge with random rough surface. The roughness (of length d) starting at the wedge apex is modeled as stochastic process (with zero mean and correlation length A) obtained from a new stochastic differential equation. A multi-element probabilistic collocation method (ME-PCM) on sparse grids is employed to solve the stochastic Euler equations while a WENO scheme is used to discretize the equations in two spatial dimensions. The perturbation analysis is used to verify the stochastic simulations and to provide insight for small values of A, where stochastic simulations become prohibitively expensive. We show that the random roughness enhances the lift and drag forces on the wedge beyond the rough region, and this enhancement is proportional to ( d / A ) 2 . The effects become more pronounced as the Mach number increases. These results can be used in designing smart rough skins for airfoils for maximum lift enhancement at a minimum drag penalty.