Evaluating the uncertainty of Darcy velocity with sparse grid collocation method

This paper presents non-intrusive, efficient stochastic approaches for predicting uncertainties associated with petroleum reservoir simulations. The Monte Carlo simulation method, which is the most common and straightforward approach for uncertainty quantification in the industry, requires to perform a large number of reservoir simulations and is thus computationally expensive especially for large-scale problems. We propose an efficient and accurate alternative through the collocation-based stochastic approaches. The reservoirs are considered to exhibit randomly heterogeneous flow properties. The underlying random permeability field can be represented by the Karhunen-Loeve expansion (or principal component analysis), which reduces the dimensionality of random space. Two different collocation-based methods are introduced to propagate uncertainty of the reservoir response. The first one is the probabilistic collocation method that deals with the random reservoir responses by employing the orthogonal polynomial functions as the bases of the random space and utilizing the collocation technique in the random space. The second one is the sparse grid collocation method that is based on the multi-dimensional interpolation and high-dimensional quadrature techniques. They are non-intrusive in that the resulting equations have the exactly the same form as the original equations and can thus be solved with existing reservoir simulators. These methods are efficient since only a small number of simulations are required and the statistical moments and probability density functions of the quantities of interest in the oil reservoirs can be accurately estimated. The proposed approaches are demonstrated with a 3D reservoir model originating from the 9th SPE comparative project. The accuracy, efficiency, and compatibility are compared against Monte Carlo simulations. This study reveals that, compared to traditional Monte Carlo simulations, the collocation-based stochastic approaches can accurately quantify uncertainty in petroleum reservoirs and greatly reduce the computational cost.

Uncertainty propagation (UP) of the frequency response is essential for the robust design of viscoelastic damping structures. One challenge in solving this problem is enormous computation cost from the UP analysis. This paper aims to develop a novel computational effective method based on the combination of the adaptive sparse grid collocation (ASGC) method and the high dimensional model representation (HDMR) technique to evaluate the variability of the frequency response of viscoelastic damping structures. First, a well-validated layer-wise finite element method is employed to model the viscoelastic damping structures. The direct frequency response (DFR) method is utilized to calculate the response. Then, a modified adaptive strategy using expectation increments as the sampling indicator is proposed for cost-effective sparse grid construction. Lastly, high dimensional model representation (HDMR) technique is introduced to address the difficulty in moderate and high random dimensional situations. Two numerical examples are provided to assess the performances of the proposed method. Variations in constitutive parameters of viscoelastic material and thicknesses of the viscoelastic layer are considered. Numerical results show that, compared to the original high-dimensional ASGC method and Monte Carlo Simulation (MCS), the proposed method accurately predicts the variability of frequency responses and significantly improves the computational efficiency.

In this article, we propose and analyse a sparse grid collocation method to solve an optimal control problem involving an elliptic partial differential equation with random coefficients and forcing terms. The input data are assumed to be dependent on a finite number of random variables. We prove that an optimal solution exists, and derive an optimality system. A Galerkin approximation in physical space and a sparse grid collocation in the probability space is used. Error estimates for a fully discrete solution using an appropriate norm are provided, and we analyse the computational efficiency. Computational evidence complements the present theory, to show the effectiveness of our stochastic collocation method.

Initial orbit determination is highly enabling for operational awareness and safety in space. Reliable orbit determination in non-Keplerian systems, such as multi-body orbital mechanics, is challenging due to chaotic and unpredictable motion of these systems over long time scales. Typical approaches to non-Keplerian orbit determination use explicit integration with numerical shooting methods to propagate candidate trajectories. These methods often break down for measurement gaps on the order of days or even hours. This work presents a novel algorithm for non-Keplerian orbit determination using sparse grid collocation methods. The approach provides many benefits, including a large region of convergence without a close initial guess, as well as increased stability and performance over long observing time scales. A further advancement is made for estimating periodic orbits, in which a repeating orbit and its natural period are determined simultaneously from angles-only measurements. Orbit determination using collocation may significantly reduce sensor tasking requirements and enhance capabilities in complex orbital environments such as cislunar space.

In this paper we use a splitting technique to develop new multiscale basis functions for the multiscale finite element method (MsFEM). The multiscale basis functions are iteratively generated using a Green's kernel. The Green's kernel is based on the first differential operator of the splitting. The proposed MsFEM is applied to deterministic elliptic equations and stochastic elliptic equations, and we show that the proposed MsFEM can considerably reduce the dimension of the random parameter space for stochastic problems. By combining the method with sparse grid collocation methods, the need for a prohibitive number of deterministic solves is alleviated. We rigorously analyze the convergence of the proposed method for both the deterministic and stochastic elliptic equations. Computational complexity discussions are also offered to supplement the convergence analysis. A number of numerical results are presented to confirm the theoretical findings.

We address the tension between software generality and performance in the domain of simulations based on Monte-Carlo methods. We simultaneously achieve generality and high performance by a novel development methodology and software architecture centred around the concept of a specialising simulator generator. Our approach combines and extends methods from functional programming, generative programming, partial evaluation, and runtime code generation. We also show how to generate parallelised simulators. We evaluated our approach by implementing a simulator for advanced forms of polymerisation kinetics. We achieved unprecedented performance, making Monte-Carlo methods practically useful in an area that was previously dominated by deterministic PDE solvers. This is of high practical relevance, as Monte-Carlo simulations can provide detailed microscopic information that cannot be obtained with deterministic solvers.

This work proposes and analyzes an anisotropic sparse grid stochastic collocation
\nmethod for solving partial differential equations with random coefficients and forcing terms (input
\ndata of the model). The method consists of a Galerkin approximation in the space variables and a
\ncollocation, in probability space, on sparse tensor product grids utilizing either Clenshaw–Curtis or
\nGaussian knots. Even in the presence of nonlinearities, the collocation approach leads to the solution
\nof uncoupled deterministic problems, just as in the Monte Carlo method. This work includes a priori
\nand a posteriori procedures to adapt the anisotropy of the sparse grids to each given problem.
\nThese procedures seem to be very effective for the problems under study. The proposed method
\ncombines the advantages of isotropic sparse collocation with those of anisotropic full tensor product
\ncollocation: the first approach is effective for problems depending on random variables which weigh
\napproximately equally in the solution, while the benefits of the latter approach become apparent when
\nsolving highly anisotropic problems depending on a relatively small number of random variables, as
\nin the case where input random variables are Karhunen–Lo`eve truncations of “smooth” random
\nfields. This work also provides a rigorous convergence analysis of the fully discrete problem and
\ndemonstrates (sub)exponential convergence in the asymptotic regime and algebraic convergence in
\nthe preasymptotic regime, with respect to the total number of collocation points. It also shows
\nthat the anisotropic approximation breaks the curse of dimensionality for a wide set of problems.
\nNumerical examples illustrate the theoretical results and are used to compare this approach with
\nseveral others, including the standard Monte Carlo. In particular, for moderately large-dimensional
\nproblems, the sparse grid approach with a properly chosen anisotropy seems to be very efficient and
\nsuperior to all examined methods.

Numerical approximation of poroelasticity with random coefficients using Polynomial Chaos and Hybrid High-Order methods

In this thesis a new method for the analysis of power transients in a nuclear reactor is developed, which is more accurate than the present state-of-the-art methods. Transient analysis is important tool when designing nuclear reactors, since they predict the behaviour of a reactor during changing conditions, such as a control-rod movement, induced by an operator, or an accident scenario. The current methods for transient analysis apply deterministic solvers to calculate the neutronic response, but not only do these deterministic solvers always need to discretise the problem, they also often apply more fundamental approximations such as homogenization or diffusion. Therefore a stochastic method is needed, which only has a statistical uncertainty. A challenge for applying the Monte Carlo method on transient analysis, is the different types of particles which must be simulated in a single simulation. It is common for a neutronics calculation to simulate the prompt neutrons, however the simulation of precursors is new. The main challenge when simulating precursors lies in their long lifetime, which is seconds, whereas the prompt neutrons have a lifetime of microseconds. This has been solved by dividing the transient problem into time intervals and forcing the precursors to produce a delayed neutron is all time intervals. This will ensure prompt-neutron chains in all intervals. The prompt neutrons will form prompt-neutron chains, which poses another challenge for the Monte Carlo method. The chain length of these prompt neutrons varies a lot and the can also split into many branches, which averages out in a real power reactor, but is difficult to simulate on a computer. To improve the chain-length statistics, a new variance reduction technique is developed, which dictates that a single neutron will emerge from each interaction. This emerging neutron can be the result of a scattering event or a fission event. Finally, a dynamic simulation scheme, which can run in parallel, has been devised, simulating all time intervals consecutively. Also, a method for sampling the initial conditions is created and this scheme is implemented in a purpose-built Monte Carlo code and in a general-purpose Monte Carlo code. The two codes have been tested in several cases and they behave as expected, agreeing with deterministic method where expected, but deviating when the deterministic methods are no longer valid. In order to simulate realistic transients in a power reactor, feedback has to be taken into account. To achieve this, the dynamic Monte Carlo method has been coupled to a thermal-hydraulics code, using an explicit scheme. The results of this coupled simulation are compared to a state-of-the-art coupled diffusion calculation in a NURISP-benchmark calculation and the results agree well, except for a small deviation towards the end. The exact source of this deviation should be further investigated, but the difference is small, when realising that there is already a deviation in the steady-state results. The results demonstrate the feasibility of performing a fully dynamic transient analysis, using only Monte Carlo for the neutronics part of the calculation. Transients can now be calculated, with detailed modelling of any complex geometry and with continuous energy, which is especially useful for newly developed reactor types and one-of-a-kind research reactors.

Risk-averse structural topology optimization under random fields using stochastic expansion methods

Monte Carlo simulations are performed to investigate the statistical properties of electromagnetic scattering from 2D random rough surfaces in 3D space,. We develop a strategy to solve this high-frequency problem. The surfaces are characterized by perfectly conducting Gaussian random surfaces on a finite plate. The scattering problem is studied for a single realization of a random profile on which the radar cross-section depends. Making a comparison between the multilevel fast multipole algorithm and the proposed high-frequency techniques based on the small perturbation method (SPM), we have confirmed that SPM is effective for the case of small height and small correlation length.

Bioelectromagnetism focuses on the study of electromagnetic fields in biological tissues from direct current (DC) to optical frequencies. It is challenging to develop an electromagnetics (EM) simulation method to cover this entire frequency band due to the electrically small/large scattering problem at extremely low/high frequencies. This paper focuses on the band from DC to microwave frequencies in bioelectromagnetism. Its main research objective is to develop a method that can overcome the low frequency breakdown problem at low frequencies (practically DC) and still stay stable at microwave frequencies. Based on the scattered field vector Helmholtz equation, the mixed finite element method (mixed FEM) is developed for the broadband electromagnetic field simulation in biological tissues. By imposing Gauss' law as the constraint condition, the mixed FEM overcomes the low frequency breakdown problem without resorting to the quasi-static approximation and remains effective and accurate at high frequencies. Extremely low frequency and high frequency numerical results are demonstrated to verify that the mixed FEM is a stable full-wave electromagnetic field simulation method for the full-bandwidth bioelectromagnetism.

Scalable simulation of coupled adsorption and transport of methane in confined complex porous media with density preconditioning

Hierarchical matrix approximation for the uncertainty quantification of potentials on random domains

Continuum limit at small correlation lengths