Stochastic Collocation Method Research Articles

Quantile-based sequential optimization and reliability assessment method under random and interval hybrid uncertainty

Under random and interval hybrid uncertainties, solving hybrid reliability based design optimization (HRBDO) can acquire an optimal balance between structural performance and reliability. Since solving HRBDO includes a triple nested framework involving minimum analysis of performance function (PF), failure probability constraint analysis and design parameter optimization, the computational complexity of HRBDO is high, especially for dealing with complex structures. Therefore, a quantile-based sequential optimization and reliability assessment method (QSORA) is proposed for reducing the computational complexity of HRBDO. In the proposed QSORA for HRBDO, failure probability constraint is firstly transformed into minimum PF (MPF) quantile one corresponding to target failure probability. Then, approximating the difference between PF and its target quantile at current iteration by that at previous one, the failure probability constraint analysis is decoupled from the design parameter optimization. Moreover, by approximating the minimum point of the PF with respect to the interval input in the current iteration by that in the previous one, the minimum analysis of PF is separated from the design parameter optimization. By the separation of minimum analysis and failure probability constraint analysis from the design parameter optimization in the proposed QSORA, the triple nested framework of HRBDO is decoupled sequentially as the deterministic design optimization, the minimum analysis of the PF and the target MPF quantile estimation, and this way of reconstructing the HRBDO from the triple nested framework to three single-loop frameworks can significantly enhance the efficiency of solving HRBDO. Furthermore, the MPF quantile at the current design parameter is estimated by stochastic collocation based statistical moment method, in which the stochastic collocation method is employed to efficiently estimate the MPF moment to approximate the probability density function of MPF. The efficiency and accuracy of the QSORA are validated by four numerical and engineering examples finally.

Numerical methods for reinterpreted discrete fracture models with random inputs

This paper investigates the impact of uncertainty on the reinterpreted discrete fracture model (RDFM) for flow in porous media featuring fractures and barriers. The stochastic RDMF is formulated by parameterizing the uncertainty in the permeability, as well as the width of the fracture and barrier networks as random variables. To discretize these random variable, we adopt a stochastic collocation method based on a special choice of nodes. This collocation scheme offers optimal accuracy and easy implementation. The resulting collocation scheme is then combined with the local discontinuous Galerkin (LDG) method to accommodate the hybrid-dimensional Darcy’s law and the nature of the pressure/flux discontinuity. Additionally, a slope limiter is utilized to enhance stability. Numerical experiments are carried out to illustrate the accuracy and stability of the proposed scheme.

A novel and efficient stochastic collocation method for estimating failure probability function in one-dimensional reduced space

Failure probability function (FPF) is an important index when measuring the effect of random input distribution parameters on the safety of structures. It can decouple reliability-based design optimization. However, efficient estimation of the FPF is challenging. This study proposes a novel stochastic collocation method for estimating the FPF. The novelty of the proposed method lies in three aspects: First, an equivalent integral expression is proposed for the FPF in one-dimensional reduced space, enabling FPF estimation using an efficient stochastic collocation method. Secondly, a unified sampling density function is constructed to remove the coupling between the computational cost of FPF estimation and the number of distribution parameter realizations. Thirdly, a stochastic collocation point set is derived and shared to estimate the whole FPF using arbitrary distribution parameter realization. The proposed method is particularly suitable for the strength-stress and resistance-load models used in the field of engineering. The efficiency and accuracy of the proposed method are verified by several examples.

A deep learning-based Monte Carlo simulation scheme for stochastic differential equations driven by fractional Brownian motion

Stochastic differential equations (SDEs) are widely used models to describe the evolution of stochastic processes. Among them, SDEs driven by fractional Brownian motion (fBm) have been shown to be capable of describing systems with temporal dependencies. In this paper, we develop a neural network based Monte Carlo methodology in which we can efficiently simulate SDEs that are governed by fBm. Particularly, we focus on large time step simulations. A property of fBm that complicates the development of such Monte Carlo schemes is the long-range temporal correlation. To this end, we build the network based on the encoder–decoder framework and employ the attention mechanism to learn the temporal relationships in the historical paths of such SDEs. In addition, a loss function based on the quantile loss is used, where the quantile levels to be predicted are determined by means of the stochastic collocation method. Experimental results show that this kind of loss function is superior to conventional loss functions in terms of solution accuracy, and the resulting scheme can learn and simulate SDEs driven by fBm accurately and highly efficiently.

Failure Mechanism Analysis of the Stochastic Galerkin Method in EMC Simulation Considering Geometric Randomness

By virtue of its high calculational accuracy and efficiency, the stochastic Galerkin method (SGM) has been successfully applied many times in electromagnetic compatibility (EMC) simulation in recent years. This paper proposes a calculating example taking geometric uncertainty factors into consideration. As is proved in the paper, there is a relatively large error when using the SGM to solve the example mentioned above. According to failure mechanism, the fundamental reason of the failure of the simulation lies in the additional error caused by using numerical integration to solve the inner product formula. Meanwhile, it is proved that no additional errors are introduced when using the stochastic collocation method (SCM), so the SCM is better than the SGM in stability. In the end, the paper revised the general selective strategy for uncertainty analysis methods, thus providing theoretical basis for their universal application in EMC field.

Unraveling pine wilt disease: Comparative study of stochastic and deterministic model using spectral method

The pinewood nematode, which causes a disease known as pine wilt, is a serious threat to pine forests all over the world. In this paper, we analyze the dynamics of pine wilt disease and its effects on forest ecosystems using mathematical modeling of pine wilt disease, its numerical simulations, and analytical techniques. Understanding the nematode’s spread, evaluating the efficiency of management efforts, and foreseeing the long-term effects on pine populations are some of our research’s goals. To gain a comprehensive understanding of the dynamics of the disease, we categorize pine trees and vectors systematically into susceptible, exposed, and infected groups. We first presented disease dynamics through a flow diagram which is then transformed to system of ordinary differential equations along with non-negative initial conditions. The deterministic model is then transformed to a stochastic model. Using both deterministic and stochastic modeling approaches, this study explores the complex transmission dynamics of Pine Wilt disease. Basic reproduction number has been calculated via next generation technique and some cases of stability analysis on the basis of the values of reproduction number has been discussed. For the solution of stochastic model Legendre Spectral collocation method has been used as it is known for its high accuracy in approximating solutions to complex stochastic models. It can achieve exponential convergence rates, making it suitable for problems where high precision is required. The results of both deterministic and stochastic models are compared via different figures. We find crucial thresholds for disease transmission and highlight significant factors affecting the spread of diseases through simulations. Our research shows that preventing the severe impacts of pine wilt disease requires early detection and intervention. Additionally, our mathematical model is a useful tool for enhancing disease management plans. This research advances our knowledge of the dynamics of the pine wilt disease and provides useful advice for managing forests and promoting conservation. Our research highlights the need for quick action to protect pine forests from this harmful virus.

Uncertainty quantification in timber-like beams using sparse grids: Theory and examples with off-the-shelf software utilization

When dealing with timber structures, the characteristic strength and stiffness of the material are made highly variable and uncertain by the unavoidable, yet hardly predictable, presence of knots and other defects. In this work, we apply the sparse grids stochastic collocation method to perform uncertainty quantification for structural engineering in the scenario described above. Sparse grids have been developed by the mathematical community in the last decades, and their theoretical background has been rigorously and extensively studied. The document proposes a brief practice-oriented introduction with minimal theoretical background, provides detailed instructions for the use of the off-the-shelf Sparse Grid Matlab kit (freely available online and straightforward to use) and discusses two preliminary examples inspired from timber engineering problems that highlight how sparse grids exhibit superior performances compared to the plain Monte Carlo method.

Effect of uncertain operating conditions on the aerodynamic performance of high-pressure axial turbomachinery blades

The objective existence of uncertain operating conditions can cause significant variations in the aerodynamic performance of turbomachinery. This study presents a comprehensive investigation into the impact of six uncertain operating conditions on the aerodynamic performance of a turbine. Additionally, a global sensitivity analysis was performed to identify the most important operating condition parameters. To reduce the computational cost of uncertainty quantification (UQ), a new UQ framework, the Nested Sparse-grid Stochastic Collocation Method (NSSCM), is used. It has been observed that uncertain operating conditions can lead to significant variations in energy loss, which in turn directly impacts the operating efficiency of the turbine. This research finds that the suction surface and trailing edge of the blade are particularly sensitive to uncertain operating conditions. The impact of uncertainty on the static pressure coefficient decreases as the flow develops, while the uncertainty of the Mach number and energy loss coefficient increases. Inlet total pressure and outlet static pressure are the primary factors impacting turbomachinery aerodynamics, as determined by sensitivity analysis. This study provides new insights into the impact of uncertain operating conditions on turbine efficiency and can guide the development of more robust turbine designs in the future.

Collocation method for stochastic delay differential equations

In this work, we present a collocation-based numerical approach for handling stochastic delay differential equations. We approximate the solution function, and after that, we carry out integrations between the predefined collocation points to achieve a mapping from the delayed state to the present state. We build the first and second moment mapping matrices based on the mapping, and we utilize the matrices to approximate the stationary first and second moments and their stability. Numerical studies of a first and second-order stochastic delay differential equation show the convergence and time complexity of the stochastic collocation method. The last section covers the issues and possible further improvements of the method.

Stochastic collocation for optimal control problems with stochastic PDE constraints by meshless techniques

In this paper, we investigate the stochastic collocation method with meshless techniques for an optimal control problem of elliptic equations with random inputs. The proposed method does not depend on the mesh partition and only requires the solution of uncoupled deterministic problem at each interpolation point, which can be competitive for high-dimensional random inputs and handle easily a wider range of situations such as random variables that are unbounded and input data that depend nonlinearly on the random variables. A priori error estimates are provided for the control, state and co-state variables. The stochastic collocation method based on gradient projection is presented, and some numerical examples are carried out to illustrate the performance of the new scheme.

Stochastic collocation method for thermal buckling and vibration analysis of functionally graded sandwich microplates

Thermal buckling and vibration behaviors of functionally graded (FG) sandwich microplates are investigated by using a unified higher-order shear deformation theory and stochastic collocation (SC) method. Uniform and linear distributions are considered for thermal effect and lognormal distributions are used to characterize the variability of the materials properties. The governing equations are derived by using Hamilton’s principle and solved by Ritz’s approach. To demonstrate the effectiveness and accuracy of the current model, the results from SC are compared with those from Monte Carlo Simulation. The effects of boundary conditions, temperature distribution, thickness-to-length ratio, material scale characteristics and power-law index on the fundamental frequencies and critical buckling temperature of the FG sandwich microplates are investigated. The FG sandwich microplates’ stochastic analysis provides some new findings that may be utilized as references in the future.

Adaptive density-based robust topology optimization under uncertain loads using parallel computing

This work presents an efficient parallel implementation of density-based robust topology optimization (RTO) using adaptive mesh refinement (AMR) schemes permitting us to address the problem with modest computational resources. We use sparse grid stochastic collocation methods (SCMs) for transforming the RTO problem into a weighted multiple-loading deterministic problem at the collocation points. The calculation of these deterministic problems and the functional sensitivity is computationally expensive. We combine distributed-memory parallel computing and AMR techniques to address the problem efficiently. The former allows us to exploit the computational resources available, whereas the latter permits us to increase performance significantly. We propose the parallel incremental calculation of the deterministic problems and the contribution to the functional sensitivity maintaining a similar memory allocation to the one used in the deterministic counterpart. The cumulative computing uses buffers to adapt the evaluation at the collocation points to the parallel computing resources permitting the exploitation of the embarrassing parallelism of SCMs. We evaluate the deterministic problems in a coarse mesh generated for each topology optimization iteration to increase the performance. We perform the regularization and design variable update in a fine mesh to obtain an equivalent design to the one generated in such a mesh. We evaluate the proposal in two- and three-dimensional problems to test its feasibility and scalability. We also check the performance improvement using computational buffers in parallel computing nodes. Finally, we compare the proposal to the same approach using different preconditioners without AMR schemes showing significant performance improvements.

An hp‐adaptive multi‐element stochastic collocation method for surrogate modeling with information re‐use

AbstractThis article introduces an ‐adaptive multi‐element stochastic collocation method, which additionally allows to re‐use existing model evaluations during either ‐ or ‐refinement. The collocation method is based on weighted Leja nodes. After ‐refinement, local interpolations are stabilized by adding and sorting Leja nodes on each newly created sub‐element in a hierarchical manner. For ‐refinement, the local polynomial approximations are based on total‐degree or dimension‐adaptive bases. The method is applied in the context of forward and inverse uncertainty quantification to handle non‐smooth or strongly localized response surfaces. The performance of the proposed method is assessed in several test cases, also in comparison to competing methods.

Deterministic–stochastic modeling of transcranial magnetic stimulation featuring the use of method of moments and stochastic collocation

This work examines the influence of the human brain tissue parameters’ uncertainty and the stimulation coil positioning variations within the framework of transcranial magnetic stimulation (TMS). A combination of deterministic modeling and the stochastic collocation method (SCM) is used for the assessment of the input parameter uncertainties’ effects on several outputs of interest: the induced electric field and the related electric current density in the homogeneous human brain model, as well as the maximum values of electric field, magnetic flux density and the induced electric current density. The deterministic model features the formulation based on the surface integral equation approach whose numerical solution is carried out using the method of moments. The SCM shows satisfactory convergence in the assessment of the stochastic mean, variance and standard deviation for the output parameters of interest. Confidence intervals are computed, and the impact of input permittivity, conductivity and coil positioning is assessed. It is found that due to a non-symmetric nature of the utilized brain model the highest electric field variance is shifted from the expected location directly under the coil geometric center.

A Study of The Stochastic Burgers’ Equation Using The Dynamical Orthogonal Method

In the current work, the stochastic Burgers’ equation is studied using the Dynamically Orthogonal (DO) method. The DO presents a low-dimensional representation for the stochastic fields. Unlike many other methods, it has a time-dependent property on both the spatial basis and stochastic coefficients, with flexible representation especially in the strongly transient and nonstationary problems. We consider a computational study and application of the DO method and compare it with the Polynomial Chaos (PC) method. For comparison, both the stochastic viscous and inviscid Burgers’ equations are considered. A hybrid approach, combining the DO and PC is proposed in case of deterministic initial conditions to overcome the singularities that occur in the DO method. The results are verified with the stochastic collocation method. Overall, we observe that the DO method has a higher rate of convergence as the number of modes increases. The DO method is found to be more efficient than PC for the same level of accuracy, especially for the case of high-dimensional parametric spaces. The inviscid Burgers’ equation is analyzed to study the shock wave formation when using the DO after suitable handling of the convective term. The results show that the sinusoidal wave shape is distorted and sharpened as the time evolves till the shock wave occurs.

Sc-Pinns: Integrating the Stochastic Collocation Method with Physics Informed Neural Networks for Uncertainty Propagation

Sc-Pinns: Integrating the Stochastic Collocation Method with Physics Informed Neural Networks for Uncertainty Propagation

Stochastic collocation method for computing eigenspaces of parameter-dependent operators

We consider computing eigenspaces of an elliptic self-adjoint operator depending on a countable number of parameters in an affine fashion. The eigenspaces of interest are assumed to be isolated in the sense that the corresponding eigenvalues are separated from the rest of the spectrum for all values of the parameters. We show that such eigenspaces can in fact be extended to complex-analytic functions of the parameters and quantify this analytic dependence in a way that leads to convergence of sparse polynomial approximations. A stochastic collocation method on an anisoptropic sparse grid in the parameter domain is proposed for computing a basis for the eigenspace of interest. The convergence of this method is verified in a series of numerical examples based on the eigenvalue problem of a stochastic diffusion operator.

A Multilevel Stochastic Collocation Method for Schrödinger Equations with a Random Potential

A Multilevel Stochastic Collocation Method for Schrödinger Equations with a Random Potential

Stochastic Collocation Method for Stochastic Optimal Boundary Control of the Navier–Stokes Equations

Stochastic Collocation Method for Stochastic Optimal Boundary Control of the Navier–Stokes Equations

Preconditioners for robust optimal control problems under uncertainty

AbstractThe discretization of robust quadratic optimal control problems under uncertainty using the finite element method and the stochastic collocation method leads to large saddle‐point systems, which are fully coupled across the random realizations. Despite its relevance for numerous engineering problems, the solution of such systems is notoriously challenging. In this manuscript, we study efficient preconditioners for all‐at‐once approaches using both an algebraic and an operator preconditioning framework. We show in particular that for values of the regularization parameter not too small, the saddle‐point system can be efficiently solved by preconditioning in parallel all the state and adjoint equations. For small values of the regularization parameter, robustness can be recovered by the additional solution of a small linear system, which however couples all realizations. A mean approximation and a Chebyshev semi‐iterative method are proposed to solve this reduced system. We consider a random elliptic partial differential equation whose diffusion coefficient is modeled as an almost surely continuous and positive random field, though not necessarily uniformly bounded and coercive. We further provide estimates of the dependence of the spectrum of the preconditioned system matrix on the statistical properties of the random field and on the discretization of the probability space. Such estimates involve either the first or second moment of the random variables and , where is the spatial domain. The theoretical results are confirmed by numerical experiments, and implementation details are further addressed.