A Study of Multiscale Kinetic Models with Uncertainties

Modeling and characterization of complex composite structures is challenging due to uncertainties inherent in these materials. Uncertainty is present at each length scale in composites and must be quantified in order to accurately model the mechanical response and damage progression of this material. The ability to pass information between length scales permits multiscale models to transport uncertainties from one scale to the next. Limitations in the physics and errors in numerical methods pose additional challenges for composite models. By replacing deterministic inputs with random inputs, stochastic methods can be implemented within these multiscale models making them more robust. This work focuses on understanding the sensitivity of multiscale models and damage progression variations to stochastic input parameters as well as quantifying these uncertainties within a modeling framework. A multiscale, sectional model is used due to its efficiency and capacity to incorporate stochastic methods with little difficulty. The sectional micromechanics in this model are similar to that of the Generalized Method of Cells with the difference being the discretization techniques of the unit cell and the continuity conditions. A Latin Hypercube sampling technique is used due to its reported computational savings over other methods such as a fully random Monte Carlo simulation. Specifically in the sectional model, the Latin Hypercube sampling method provides an approximate 35 % reduction in computations compared to the fully random Monte Carlo method. The Latin Hypercube sampling is a stratified technique which discretizes the distribution function and randomizes the input parameters within those discretized fields. Within this multiscale modeling framework, a progressive failure theory is implemented using these stochastic methods and a modified Hashin failure theory. With a combined stochastic method and progressive failure theory, this multiscale model is capable of modeling the uncertainty and material property variations for composite materials.

Stochastic collocation and stochastic Galerkin methods for linear differential algebraic equations

In [L. Liu and S. Jin, Multiscale Model. Simult., 16, 1085-1114, 2018], spectral convergence and long-time decay of the numerical solution towards the global equilibrium of the stochastic Galerkin approximation for the Boltzmann equation with random inputs in the initial data and collision kernel for hard potentials and Maxwellian molecules under Grad's angular cutoff were established using the hypocoercive properties of the collisional kinetic model. One assumption for the random perturbation of the collision kernel is that the perturbation is in the order of the Knudsen number, which can be very small in the fluid dynamical regime. In this article, we remove this smallness assumption, and establish the same results but now for random perturbations of the collision kernel that can be of order one. The new analysis relies on the establishment of a spectral gap for the numerical collision operator.

Over the last few years there have been dramatic advances in the area of uncertainty quantification. In particular, we have seen a surge of interest in developing efficient, scalable, stable, and convergent computational methods for solving differential equations with random inputs. Stochastic collocation (SC) methods, which inherit both the ease of implementation of sampling methods like Monte Carlo and the robustness of nonsampling ones like stochastic Galerkin to a great deal, have proved extremely useful in dealing with differential equations driven by random inputs. In this work we propose a novel enhancement to stochastic collocation methods using deterministic model reduction techniques. Linear parabolic partial differential equations with random forcing terms are analysed. The input data are assumed to be represented by a finite number of random variables. A rigorous convergence analysis, supported by numerical results, shows that the proposed technique is not only reliable and robust but also efficient.

Most epidemiological models are rooted in the pioneering work proposed by Kermack and McKendrick and are based on systems of deterministic ODEs, which describe the temporal evolution of the spread of an infectious disease assuming population and territorial homogeneity. Generally, the concept of the average behavior of a population is sufficient to have a first reliable description of an epidemic development, but the inclusion of the spatial component becomes crucial when it is necessary to consider spatially heterogeneous interventions, as in the case of the COVID-19 pandemic. Moreover, any realistic data-driven model must take into account the large uncertainty in the values reported by official sources such as the amount of infectious individuals. In this work, drawing inspiration from kinetic theory, recent advances on the development of stochastic multiscale kinetic transport models for the spread of epidemics under uncertain data are presented. The propagation of the infectious disease is described by the spatial movement and interactions of individuals divided into commuters moving in the territory on a wide scale and non-commuters acting only on urban scales. The resulting models are solved numerically through a suitable stochastic Asymptotic-Preserving IMEX Runge-Kutta Finite Volume Collocation Method, which ensures a consistent treatment of the system of equations, without loss of accuracy when entering in the stiff, diffusive regime. Application studies concerning the spread of the COVID-19 pandemic in Italy assess the validity of the proposed methodology.

The stochastic collocation method (Babuska et al. in SIAM J Numer Anal 45(3):1005---1034, 2007; Nobile et al. in SIAM J Numer Anal 46(5):2411---2442, 2008a; SIAM J Numer Anal 46(5):2309---2345, 2008b; Xiu and Hesthaven in SIAM J Sci Comput 27(3):1118---1139, 2005) has recently been applied to stochastic problems that can be transformed into parametric systems. Meanwhile, the reduced basis method (Maday et al. in Comptes Rendus Mathematique 335(3):289---294, 2002; Patera and Rozza in Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations Version 1.0. Copyright MIT, http://augustine.mit.edu , 2007; Rozza et al. in Arch Comput Methods Eng 15(3):229---275, 2008), primarily developed for solving parametric systems, has been recently used to deal with stochastic problems (Boyaval et al. in Comput Methods Appl Mech Eng 198(41---44):3187---3206, 2009; Arch Comput Methods Eng 17:435---454, 2010). In this work, we aim at comparing the performance of the two methods when applied to the solution of linear stochastic elliptic problems. Two important comparison criteria are considered: (1), convergence results of the approximation error; (2), computational costs for both offline construction and online evaluation. Numerical experiments are performed for problems from low dimensions $$O(1)$$ O ( 1 ) to moderate dimensions $$O(10)$$ O ( 10 ) and to high dimensions $$O(100)$$ O ( 100 ) . The main result stemming from our comparison is that the reduced basis method converges better in theory and faster in practice than the stochastic collocation method for smooth problems, and is more suitable for large scale and high dimensional stochastic problems when considering computational costs.

Reduced basis ANOVA methods for partial differential equations with high-dimensional random inputs

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.

In this paper, the Stochastic Collocation Method is introduced into EMC uncertainty analysis. It is demonstrated to be accurate, with high computational efficiency, compared with the Method of Moments, the perturbation method and the Monte Carlo method. The Stochastic Collocation Method requires no modification to the original solver, and it is fit for being introduced into EMC simulation software.

Fast and accurate evaluation of system failure rate is performed using stochastic collocation methods. First, we demonstrate that variability in I/O performance, such as driven voltage and slew rate, will impact failure probability of the link. For instance, higher slew rates will lead to increased levels of crosstalk between signals. Crosstalk above a certain threshold can be an indicator of system failure. Then, we demonstrate that by defining an upper bound for failure probability, an optimally relaxed set of I/O performance metrics can be generated corresponding to a stochastic interconnect model. This can be achieved by minimizing a cost function formulated with some performance metric, if these metrics are random variables. Lastly, we demonstrate that, if more information about the packaging design becomes available, the randomness of the interconnect model can be reduced, leading to a more relaxed set of I/O performance metrics. The method proposed in [1] is then be used to verify that the packaging design will meet the tolerated failure probability.

We consider a stochastic analysis of non-linear viscous fluid flow problems with smooth and sharp gradients in stochastic space. As a representative example we consider the viscous Burgers’ equation and compare two typical intrusive and non-intrusive uncertainty quantification methods. The specific intrusive approach uses a combination of polynomial chaos and stochastic Galerkin projection. The specific non-intrusive method uses numerical integration by combining quadrature rules and the probability density functions of the prescribed uncertainties. The two methods are compared in terms of error in the estimated variance, computational efficiency and accuracy. This comparison, although not general, provide insight into uncertainty quantification of problems with a combination of sharp and smooth variations in stochastic space. It suggests that combining intrusive and non-intrusive methods could be advantageous.

In this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms (input data of the model). The input data are assumed to depend on a finite number of random variables. The method consists in a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It can be seen as a generalization of the stochastic Galerkin method proposed in [I. Babuška, R. Tempone, and G. E. Zouraris, SIAM J. Numer. Anal., 42 (2004), pp. 800–825] and allows one to treat easily a wider range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the “probability error” with respect to the number of Gauss points in each direction in the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method.

An asymptotic-preserving stochastic Galerkin method for the radiative heat transfer equations with random inputs and diffusive scalings

In this paper we study the stochastic Galerkin approximation for the linear transport equation with random inputs and diffusive scaling. We first establish uniform (in the Knudsen number) stability results in the random space for the transport equation with uncertain scattering coefficients and then prove the uniform spectral convergence (and consequently the sharp stochastic asymptotic-preserving property) of the stochastic Galerkin method. A micro–macro decomposition-based fully discrete scheme is adopted for the problem and proved to have a uniform stability. Numerical experiments are conducted to demonstrate the stability and asymptotic properties of the method.

Our ultimate goal is a topology optimization for a permanent-magnet (PM) machine, while including material uncertainties. The uncertainties in the output data are, e.g., due to measurement errors in the non-/linear material laws. In the resulting stochastic forward problem, these uncertainties are stochastically modeled by random fields. The solution of the underlying PDE, which describes magnetostatics, is represented using the generalized polynomial chaos expansion. As crucial ingredient we exploit the stochastic collocation method (SCM). Eventually, this leads to a random-dependent bi-objective cost functional, which is comprised of the expectation and the variance. Subject to the optimization of the PM machine are the shapes of the rotor poles, which are described by zero-level sets. Thus, the optimization will be done by redistributing the iron and magnet material over the design domain, which allows to attain an innovative low cogging torque design of an electric machine. For this purpose, the gradient directions are evaluated by using the continuous design sensitivity analysis in conjunction with the SCM. In the end, our numerical result for the optimization of a two-dimensional model demonstrates that the proposed approach is robust and effective.