R-3 Repository :: Browsing by Author "Gillman, Adrianna"

Browsing by Author "Gillman, Adrianna"

Now showing 1 - 8 of 8

A fast direct solver for boundary value problems with locally-perturbed geometries
(2017-05-17) Zhang, Yabin; Gillman, Adrianna
Many problems in science and engineering can be formulated as integral equations with elliptic kernels. In particular, in optimal control and design problems, the domain geometry evolves and results in a sequence of discretized linear systems to be constructed and inverted. While the systems can be constructed and inverted independently, the computational cost is relatively high. In the case where the change in the domain geometry for each new problem is only local, i.e. the geometry remains the same except within a small subdomain, we are able to reduce the cost of inverting the new system by reusing the pre-computed fast direct solvers of the original system. The resulting solver only requires inexpensive matrix-vector multiplications and matrix inversion of small size, thus dramatically reducing the cost of inverting the new linear system.
Accelerated PDE Constrained Optimization using Direct Solvers
(2018-04-17) Geldermans, Peter; Gillman, Adrianna; Heinkenschloss, Matthias
In this thesis, I propose a method to reduce the cost of computing solutions to optimization problems governed by partial differential equations (PDEs). Standard second order methods such as Newton require the solution of two PDEs per iteration of the Newton system, which can be prohibitively expensive for iterative solvers. In contrast, this work takes advantage a recently developed high order discretization method that comes with an efficient direct solver. The new technique precomputes a solution operator that can be reused for any body load, which is applied whenever a PDE solve is required. Thus the precomputation cost is amortized over many PDE solves. This approach will make second order optimization algorithms computationally affordable for practical applications such as photoacoustic tomography and optimal design problems.
An accelerated Poisson solver based on multidomain spectral discretization
(Springer, 2018) Babb, Tracy; Gillman, Adrianna; Hao, Sijia; Martinsson, Per-Gunnar
This paper presents a numerical method for variable coefficient elliptic PDEs with mostly smooth solutions on two dimensional domains. The method works best for domains that can readily be mapped onto a rectangle, or a collection of nonoverlapping rectangles. The PDE is discretized via a multi-domain spectral collocation method of high local order (order 30 and higher have been tested and work well). Local mesh refinement results in highly accurate solutions even in the presence of local irregular behavior due to corner singularities, localized loads, etc. The system of linear equations attained upon discretization is solved using a direct (as opposed to iterative) solver with O(N1.5)O(N1.5) complexity for the factorization stage and O(NlogN)O(Nlog⁡N) complexity for the solve. The scheme is ideally suited for executing the elliptic solve required when parabolic problems are discretized via time-implicit techniques. In situations where the geometry remains unchanged between time-steps, very fast execution speeds are obtained since the solution operator for each implicit solve can be pre-computed.
GPU-accelerated discontinuous Galerkin methods on hybrid meshes: applications in seismic imaging
(2017-04-20) Wang, Zheng; Warburton, Timothy; Gillman, Adrianna
Seismic imaging is a geophysical technique assisting in the understanding of subsurface structure on a regional and global scale. With the development of computer technology, computationally intensive seismic algorithms have begun to gain attention in both academia and industry. These algorithms typically produce high-quality subsurface images or models, but require intensive computations for solving wave equations. Achieving high-fidelity wave simulations is challenging: first, numerical wave solutions may suffer from dispersion and dissipation errors in long-distance propagations; second, the efficiency of wave simulators is crucial for many seismic applications. High-order methods have advantages of decreasing numerical errors efficiently and hence are ideal for wave modelings in seismic problems. Various high order wave solvers have been studied for seismic imaging. One of the most popular solvers is the finite difference time domain (FDTD) methods. The strengths of finite difference methods are the computational efficiency and ease of implementation, but the drawback of FDTD is the lack of geometric flexibility. It has been shown that standard finite difference methods suffer from first order numerical errors at sharp media interfaces. In contrast to finite difference methods, discontinuous Galerkin (DG) methods, a class of high-order numerical methods built on unstructured meshes, enjoy geometric flexibility and smaller interface errors. Additionally, DG methods are highly parallelizable and have explicit semi-discrete form, which makes DG suitable for large-scale wave simulations. In this dissertation, the discontinuous Galerkin methods on hybrid meshes are developed and applied to two seismic algorithms---reverse time migration (RTM) and full waveform inversion (FWI). This thesis describes in depth the steps taken to develop a forward DG solver for the framework that efficiently exploits the element specific structure of hexahedral, tetrahedral, prismatic and pyramidal elements. In particular, we describe how to exploit the tensor-product property of hexahedral elements, and propose the use of hex-dominant meshes to speed up the computation. The computational efficiency is further realized through a combination of graphics processing unit (GPU) acceleration and multi-rate time stepping. As DG methods are highly parallelizable, we build the DG solver on multiple GPUs with element-specific kernels. Implementation details of memory loading, workload assignment and latency hiding are discussed in the thesis. In addition, we employ a multi-rate time stepping scheme which allows different elements to take different time steps. This thesis applies DG schemes to RTM and FWI to highlight the strengths of the DG methods. For DG-RTM, we adopt the boundary value saving strategy to avoid data movement on GPUs and utilize the memory load in the temporal updating procedure to produce images of higher qualities without a significant extra cost. For DG-FWI, a derivation of the DG-specific adjoint-state method is presented for the fully discretized DG system. Finally, sharp media interfaces are inverted by specifying perturbations of element faces, edges and vertices.
Hermite Methods for the Simulation of Wave Propagation
(2017-04-20) Vargas, Arturo; Warburton, Timothy; Gillman, Adrianna
Simulations of wave propagation play a crucial role in science and engineering. In applications of geophysics, they are the engine of many seismic imaging algorithms. For electrical engineers, they can be a useful tool for the design of radars and antennas. In these applications achieving high fidelity, simulations are challenging due to the inherent issues in modeling highly oscillatory waves and the associated high computational cost of high-resolution simulations. Thus the ideal numerical method should be able to capture high-frequency waves and be suitable for parallel computing. In both seismic applications and computational electromagnetics the Yee scheme, a finite difference time domain (FDTD) method, is the method of choice for structured grids. The scheme has the benefit of being easy to implement but performs poorly in the presence of high-frequency waves. High order accurate FDTD methods may be derived but ultimately rely on neighboring grid points when approximating derivative. In contrast to FDTD methods, the Hermite methods of Goodrich and co-authors (2006) use Hermite interpolation and a staggered (dual) grid to construct high order accurate numerical methods for first order hyperbolic equations. These methods achieve high order approximations in both time and space by reconstructing local polynomials within cells of the computational domain and employing Hermite-Taylor time stepping. The resulting schemes are able to evolve the solution locally within a cell making them ideal for parallel computing. Building on the original Hermite methods this thesis focuses on two goals: (1) the development of new Hermite methods and (2) their implementation on modern computing architectures. To accomplish the first objective, this thesis presents two variations of Hermite methods which are designed to simplify the scheme while preserving the favorable features. The first variation is a family of Hermite methods which do not require a dual grid. These methods eliminate the need for storing dual coefficients while maintaining optimal convergence rates. The second type of variation are Hermite methods which use leapfrog time-stepping. These schemes propagate the solution with less computation than the original scheme and may be used for either first or second order equations. To address the second objective, this thesis presents algorithms which take advantage of the many-core architecture of graphics processing units (GPU). As three-dimensional simulations can easily exceed the memory of a single GPU, techniques for partitioning the data across multiple GPUs are presented. Finally, this thesis presents numerical results and performance studies which confirm the accuracy and efficiency of the proposed Hermite methods for linear and nonlinear wave equations.
Hitchin Components, Riemannian Metrics and Asymptotics
(2014-12-04) Li, Qiongling; Wolf, Michael; Hardt, Robert; Gillman, Adrianna
Higher Teichm\"uller spaces are deformation spaces arising from subsets of the space of representations of a surface group into a general Lie group, e.g., $$PSL(n,\RR)$$, which share some of the properties of classical Teichmueller space. By the non-abelian Hodge theory, such representation spaces correspond to the space of Higgs bundles. We focus on two aspects on the Higher Teichm\"uller space: Riemannian geometry and dynamics. First, we construct a new Riemannian metric on the deformation space for $$PSL(3,\RR)$$, and then prove Teichmueller space endowed with Weil-Petersson metric is totally geodesic in deformation space for $$PSL(3,\RR)$$ with the new metric. Secondly, in a joint work with Brian Collier, we are able to obtain asymptotic behaviors and related properties of representations for certain families of Higgs bundles of rank n.
Numerical methods for boundary integral equations
(2020-08-13) Zhang, Yabin; Gillman, Adrianna; Chan, Jesse; Riviere, Beatrice; Stanciulescu, Ilinca
The thesis focuses on numerical methods for boundary integral equation (BIE) formulations of partial differential equations (PDEs). The work contains three parts: the first two consider numerical solution methods for boundary integral equations in wave scattering and Stokes flow, respectively. The last part proposes an adaptive discretization technique for BIEs in 2D. The proposed work is based on previous developments in fast direct solution techniques for BIEs. Such methods exploit the rank deficiency in the off-diagonal blocks of the discretized system and build an approximation to the inverse with linear cost for two-dimensional problems. Once the inverse approximation is constructed, applying it to any given vector is very cheap, making the methods ideal for problems with lots of right-hand-sides. The two direct solvers presented in this thesis are driven by real-life applications. The scattering solver is built to assist practitioners in designing acoustic and optic devices to manipulate waves. Its efficiency in handling multiple incident angles and minor modifications in the structure will be handy in an optimal design setting. The Stokes solver is to help with numerical simulation of objects such as bacteria and vesicles in viscous flow. To accurately capture the interaction between the objects and the confining wall, the discretization of the wall often needs to be locally refined in the region approached by the objects. This makes standard fast direct solvers too expensive to be useful, as the linear system changes for each time step. The proposed approach avoids this by pre-constructing a fast direct solver for the wall independently of time and updating the original solver to accommodate any refinements in discretization. The last part of the thesis describes an adaptive discretization technique for two-dimensional BIEs. Standard adaptive discretization method often requires a sequence of global boundary density solves each on a finer grid and terminates with the last grid if the improvements obtained from the next finer level is very small. The global density solves make the cost of the standard approach relatively high. The proposed alternative reduces the cost by replacing global solves with local solves for an approximate of the true density.
Robust and efficient numerical algorithms for the discrete prolate spheroidal wave functions
(2019-04-16) Geldermans, Peter; Gillman, Adrianna
This thesis presents novel algorithms for the numerical evaluation of the discrete prolate spheroidal wave functions (DPSWFs) and their associated integral operator eigenvalues. The DPSWFs and associated eigenvalues arise in a variety of science and engineering applications including signal processing, communications technology, paleoclimatology, fluid dynamics, and wave phenomena. Existing algorithms compute the integral operator eigenvalues to high relative accuracy when the eigenvalues are not close to zero. However, the integral operator eigenvalues computed by these algorithms lose all digits of relative accuracy when the eigenvalues are small. The new numerical algorithms compute the eigenvalues to high relative accuracy independent of their mangitude. The proposed algorithms exploit the fact that the integral operator commutes with a second order linear differential operator. While this differential operator was identified in 1978, it has not been used in numerical algorithms to evaluate the DPSWFs nor the associated eigenvalues until this work. Numerical experiments demonstrate the accuracy of the proposed algorithms. The design of the proposed algorithms exploits several properties of the DPSWFs to reduce computational cost. Furthermore, the use of high order numerical methods ensures that the algorithms are efficient. In addition to algorithms for computing the eigenvalues to high relative accuracy, several new properties of the DPSWFs are derived.

Browsing by Author "Gillman, Adrianna"

Results Per Page

Sort Options