training process of multi-scale deep neural network (DNN) approximations of the NS solution. Four forms of linearization are considered. We solve highly oscillating stationary flows in complex domains utilizing the proposed linearized learning with multiscale neural networks. The theorem of universal approximations to nonlinear operators proposed by Chen et al. [11] is extended to operators with causalities, and the proposed Causality-DeepONet implements the physical causality in its framework. The proposed Causality-DeepONet considers causality (the state of the system at the current time is not affected by that of the future, but only by its current state and past history) and uses a convolution-type weight in its design. To demonstrate its effectiveness in handling the causal response of a physical system, the Causality-DeepONet is applied to learn the operator representing the response of a building due to earthquake ground accelerations. Finally, we proposed a deep neural network approximation to the evolution operator for time dependent PDE systems over long time period by recursively using one single neural network propagator, in the form of POD-DeepONet with built-in causality feature, for a small-time interval.

]]>than surface diffusion on a crystal lattice.

Observing the viscous character of the amorphous layer, it is natural to consider whether stress-based continuum models might help explain pattern formation under ion bombardment and the observations described above. Indeed, there are early indications from the experimental literature that this may be the case, and, at low energies (∼ 1keV), at least one experimental-theoretical study has shown that they may even dominate erosive and redistributive effects in their contribution to surface evolution.

In this thesis, we develop a continuum model based on viscous thin-film flow and ion-

induced stresses within the amorphous layer. This model is a composite of, and significant generalization of, a previously-studied “anisotropic plastic flow” (APF) mechanism and a previously-studied “ion-induced isotropic swelling” (IIS) mechanism. Previous work has shown that, with certain simplifying assumptions about the amorphous-crystalline interface and spatial homogeneity of anisotropic plastic flow, this mechanism produces an instability capable of predicting pattern formation beginning at 45◦ angle of incidence against the macroscopically-flat substrate, consistent with some experimental systems. Under similar simplifying assumptions, ion-induced swelling has been shown to be capable of suppressing pattern formation. Our generalizations allow the use of simulation data to inform both linear and nonlinear surface evolution due to the spatial localization of APF and IIS to certain regions of the bulk, improved treatment of the amorphous-crystalline geometry, and

boundary conditions suitable to the physical systems of interest. We are then able to provide insight into several phenomena that have previously been difficult to explain, but seem to emerge naturally from a more detailed treatment of the physical system.

We develop the FMM in layered media, by first studying analytical and numerical properties of the Green's functions in layered media for the 2-D and 3-D Helmholtz equation, the linearized Poisson--Boltzmann equation, the Laplace's equation, and the tensor Green's functions for the time-harmonic Maxwell's equations and the elastic wave equation. Then, we propose the far-field expansions in layered media as the natural extension of the Graf's addition theorems for the free-space problems. We define a modified far-field polarization distance in layered structure according to the exponential convergence behavior of these far-field expansions, and design the FMM framework for layered media based on the modified polarization distance. Numerical tests are conducted for a massive number of particles in layered media, verifying both the efficiency and the accuracy of the developed FMM algorithms.

Numerical algorithms using the deep neural network (DNN) for the purpose of finding a remedy to address the curse of dimensionality are proposed for the solution to high-dimensional quasi-linear parabolic PDEs based on the Pardoux--Peng theory of the FBSDEs. The algorithms are shown to generate DNN solutions for a 100-dimensional Black--Scholes--Barenblatt equation that are accurate in a finite region in the solution space, and have a convergence rate similar to that of the Euler--Maruyama discretization scheme used for the FBSDEs.

]]>energy sources are leading to changes in the dynamics of modern power grid,

which have brought renewed attention to the solution of the AC power flow equations.

In particular, development of fast and robust solvers for the power flow problem

continues to be actively investigated. A novel multigrid technique for coarse-graining

dynamic power grid models has been developed recently. This technique uses an

algebraic multigrid (AMG) coarsening strategy applied to the weighted

graph Laplacian that arises from the power network's topology for the construction

of coarse-grain approximations to the original model. Motivated by this technique,

a new multigrid method for the AC power flow equations is developed using this

coarsening procedure. The AMG coarsening procedure is used to build a multilevel

hierarchy of admittance matrices, which automatically leads to a hierarchy of

nonlinear power flow equations. The hierarchy of power flow equations is then used

in a full approximation scheme (FAS) and a multiplicative correction multigrid

framework to produce multilevel solvers for the power flow equations. ]]>

Similar to the parabolic FMM, this method is based on a subdivision of the matrices into an appropriate hierarchical block structure. However, instead of an expansion of the kernel in both space and time we interpolate kernel in the temporal variables and use of the adaptive cross approximation (ACA) in the spatial variables.

The second objective of this dissertation is to extend the software package BEM++, which was written for elliptic operators, to handle thermal layer potentials. To that end, we use the package's ACA implementation for the space variables and develop a python interface to handle time dependence.

The validity of our implementation is tested for several problems with known solution and for problems with geometries that are more close to realistic engineering applications. The results demonstrate that the fast method can reproduce the theoretical convergence rates of the direct method while improving the computational cost to nearly linear complexity in the number of discretization parameters.

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