Browsing by Subject "Averaging"

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Numerical methods for averaging and homogenization
(2020-05-08) Dussinger, Milica; Engquist, Björn, 1945-; Gamba, Irene M; Arbogast, Todd; Elber, Ron
Science and engineering are full of examples of multiscale problems, which pose severe challenges to numerical simulations. In multiscale problems, processes interact on different scales in space and time. Numerical methods, which by direct simulation fully resolves this interaction demands a tremendous amount of computational time as well as memory resources. The smallest scale should be well approximated over the full computational domain. This thesis is concerned with developing and studying numerical algorithms following the framework of the heterogeneous multiscale methods (HMM). We will focus on two numerical methods that mimic the analytical techniques of averaging and homogenization respectively. The goal is to approximate the effective or averaged solution even when the explicit analytic form may not be available. The computational challenge is to include the effects of the small scales without the cost of resolving them over the full domain. In the first part of the thesis, we focus on a class of methods for the numerical averaging of highly oscillatory ordinary differential equations. The algorithms will represent an extension to the previous work done by Tao, Owhadi and Marsden. We present analysis and apply the technique to model equations. In the second part of the thesis, we focus on methods for numerical computing the effective or homogenized form of multiscale elliptic equations. We present a procedure that reduces the effect from boundary conditions, or the so-called cell resonance error. This has been an active field of research during the last few years. We use averaging kernels that have special regularity and vanishing "negative" moment properties in order to average and thereby reduce the boundary error.
Numerical multiscale methods: from homogenization to milestoning
(2024-05) Chen, Ziheng; Engquist, Björn, 1945-; Martinsson, Per Gunnar; Arbogast, Todd J.; Tsai, Richard Yen-Hsi; Bajaj, Chandrajit
The dissertation focuses on addressing the challenges posed by multiscale problems in applied mathematics, which stem from the intricate interplay between microscales and the computational demands of resolving fine details. To alleviate this burden, numerical homogenization and averaging methods are favored. This study explores three interconnected topics related to numerical techniques for handling multiscale problems in both spatial and temporal domains. In the first part, we establish the equivalence principle between time averaging and space homogenization. This principle facilitates the application of various numerical averaging techniques, such as FLAVORS, Seamless, and HMM, to boundary value problems. Moreover, we introduce the dilation operator as a decomposition-free approach for numerical homogenization in higher dimensions. Additionally, we utilize the Synchrosqueezing transform as a preprocessing step to extract oscillatory components, crucial for the structure-aware dilation method. The second part extends the Deep Ritz method to multiscale problems. We delve into the scale convergence theory to derive the [Gamma]-limit of energy functionals exhibiting oscillatory behavior. The resulting limit object, formulated as a minimization problem, captures spatial oscillations and can be tackled using existing neural network architectures. In the third part, we lay the groundwork for the milestoning algorithm, a successful tool in computational chemistry for molecular dynamics simulations. We adapt this algorithm to a domain-decomposition-based framework for coarse-grained descriptions and establish the well-posedness of primal and dual PDEs. Additionally, we investigate the convergence rate and optimal milestone placements. We illustrate this framework through the understanding of the Forward Flux algorithm as a specific example.