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Full-Text Articles in Applied Mathematics
Neural Network Learning For Pdes With Oscillatory Solutions And Causal Operators, Lizuo Liu
Neural Network Learning For Pdes With Oscillatory Solutions And Causal Operators, Lizuo Liu
Mathematics Theses and Dissertations
In this thesis, we focus on developing neural networks algorithms for scientific computing. First, we proposed a phase shift deep neural network (PhaseDNN), which provides a uniform wideband convergence in approximating high frequency functions and solutions of wave equations. Several linearized learning schemes have been proposed for neural networks solving nonlinear Navier-Stokes equations. We also proposed a causality deep neural network (Causality-DeepONet) to learn the causal response of a physical system. An extension of the Causality-DeepONet to time-dependent PDE systems is also proposed. The PhaseDNN makes use of the fact that common DNNs often achieve convergence in the low frequency …
Adaptive Multirate Infinitesimal Time Integration, Alex Fish
Adaptive Multirate Infinitesimal Time Integration, Alex Fish
Mathematics Theses and Dissertations
As multiphysics simulations grow in complexity and application scientists desire more accurate results, computational costs increase greatly. Time integrators typically cater to the most restrictive physical processes of a given simulation\add{,} which can be unnecessarily computationally expensive for the less restrictive physical processes. Multirate time integrators are a way to combat this increase in computational costs by efficiently solving systems of ordinary differential equations that contain physical processes which evolve at different rates by assigning different time step sizes to the different processes. Adaptivity is a technique for further increasing efficiency in time integration by automatically growing and shrinking the …
A Node Elimination Algorithm For Cubatures Of High-Dimensional Polytopes, Arkadijs Slobodkins
A Node Elimination Algorithm For Cubatures Of High-Dimensional Polytopes, Arkadijs Slobodkins
Mathematics Theses and Dissertations
Node elimination is a numerical approach for obtaining cubature rules for the approximation of multivariate integrals over domains in Rn. Beginning with a known cubature, nodes are selected for elimination, and a new, more efficient rule is constructed by iteratively solving the moment equations. In this work, a new node elimination criterion is introduced that is based on linearization of the moment equations. In addition, a penalized iterative solver is introduced that ensures positivity of weights and interiority of nodes. We aim to construct a universal algorithm for convex polytopes that produces efficient cubature rules without any user …
Practical Implementation Of The Immersed Interface Method With Triangular Meshes For 3d Rigid Solids In A Fluid Flow, Norah Hakami
Practical Implementation Of The Immersed Interface Method With Triangular Meshes For 3d Rigid Solids In A Fluid Flow, Norah Hakami
Mathematics Theses and Dissertations
When employing the immersed interface method (IIM) to simulate a fluid flow around a moving rigid object, the immersed object can be replaced by a virtual fluid enclosed by singular forces on the interface between the real and virtual fluids. These forces represent the impact of the rigid motion on the fluid flow and cause jump discontinuities across the interface in the whole flow field. Then, the IIM resolves the fluid flow on a fixed computational domain by directly incorporating the jump conditions across the interface into numerical schemes. Previous development of the method is limited to simple smooth boundaries. …
Viscous Thin-Film Models Of Nanoscale Self-Organization Under Ion Bombardment, Tyler Evans
Viscous Thin-Film Models Of Nanoscale Self-Organization Under Ion Bombardment, Tyler Evans
Mathematics Theses and Dissertations
For decades, it has been observed that broad-beam irradiation of semiconductor surfaces can lead to spontaneous self-organization into highly regular patterns, sometimes at length scales of only a few nanometers. Initial theory was largely based on erosion and redistribution of material occurring on fast time scales, which are able to produce good agreement with certain aspects of surface evolution. However, further experimental and theoretical work eventually led to the realization that numerous effects are active in the irradiated target, including stresses associated with ion-implantation and the accumulation of damage leading to the development of a disordered, amorphous layer atop the …