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King Abdullah University of Science and Technology

1.
Alharthi, Noha.
* Fast* High-order Integral Equation

Degree: Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division, 2019, King Abdullah University of Science and Technology

URL: http://hdl.handle.net/10754/660105

Acoustic and electromagnetic scattering from arbitrarily shaped structures can be numerically characterized by solving various surface integral equations (SIEs). One of the
most effective techniques to solve SIEs is the Nyström method. Compared to other existing methods,the Nyström method is easier to implement especially when the geometrical discretization is non-conforming and higher-order representations of the geometry and unknowns are desired. However,singularities of the Green’s function are more difﬁcult to”manage”since they are not ”smoothened” through the use of a testing function.
This dissertation describes purely numerical schemes to account for different orders of
singularities that appear in acoustic and electromagnetic SIEs when they are solved by a high-order Nyström method utilizing a mesh of curved discretization elements. These schemes make use of two sets of basis functions to smoothen singular integrals: the grid robust high-order Lagrange and the high-order Silvester-Lagrange interpolation basis functions. Numerical results comparing the convergence of two schemes are presented.
Moreover, an extremely scalable implementation of fast multipole method (FMM) is developed to efﬁciently (and iteratively) solve the linear system resulting from the discretization of the acoustic SIEs by the Nyström method. The implementation results in O(N log N) complexity for high-frequency scattering problems. This FMM-accelerated solver can handle N =2 billion on a 200,000-core Cray XC40 with 85% strong scaling efﬁciency.
Iterative solvers are often ineffective for ill-conditioned problems. Thus, a fast direct (LU)solver,which makes use of low-rank matrix approximations,is also developed. This solver relies on tile low rank (TLR) data compression format, as implemented in the hierarchical computations on many corearchitectures (HiCMA) library. This requires to taskify the underlying SIE kernels to expose ﬁne-grained computations. The resulting asynchronous execution permit to weaken the artifactual synchronization points,while mitigating the overhead of data motion. We compare the obtained performance results of our TLRLU factorization against the state-of-the-art dense factorizations on shared
memory systems. We achieve up to a fourfold performance speedup on a 3D acoustic problem with up to 150 K unknowns in double complex precision arithmetics.
*Advisors/Committee Members: Keyes, David E. (advisor), Hadwiger, Markus (committee member), Bagci, Hakan (committee member), Kressner, Daniel (committee member).*

Subjects/Keywords: Boundary Integral Equation; Acoustic Scattering; LU-Based Solver; Fast Solvers; Fast Multipole Solvers; Tile Low-Rank Approximations

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Alharthi, N. (2019). Fast High-order Integral Equation Solvers for Acoustic and Electromagnetic Scattering Problems. (Thesis). King Abdullah University of Science and Technology. Retrieved from http://hdl.handle.net/10754/660105

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Alharthi, Noha. “Fast High-order Integral Equation Solvers for Acoustic and Electromagnetic Scattering Problems.” 2019. Thesis, King Abdullah University of Science and Technology. Accessed May 08, 2021. http://hdl.handle.net/10754/660105.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Alharthi, Noha. “Fast High-order Integral Equation Solvers for Acoustic and Electromagnetic Scattering Problems.” 2019. Web. 08 May 2021.

Vancouver:

Alharthi N. Fast High-order Integral Equation Solvers for Acoustic and Electromagnetic Scattering Problems. [Internet] [Thesis]. King Abdullah University of Science and Technology; 2019. [cited 2021 May 08]. Available from: http://hdl.handle.net/10754/660105.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Alharthi N. Fast High-order Integral Equation Solvers for Acoustic and Electromagnetic Scattering Problems. [Thesis]. King Abdullah University of Science and Technology; 2019. Available from: http://hdl.handle.net/10754/660105

Not specified: Masters Thesis or Doctoral Dissertation

University of Texas – Austin

2.
-5494-1880.
* Fast* and scalable

Degree: PhD, Aerospace Engineering, 2019, University of Texas – Austin

URL: http://dx.doi.org/10.26153/tsw/5474

The hybridized discontinuous Galerkin methods (HDG) introduced a decade ago is a promising candidate for high-order spatial discretization combined with implicit/implicit-explicit time stepping. Roughly speaking, HDG methods combines the advantages of both discontinuous Galerkin (DG) methods and hybridized methods. In particular, it enjoys the benefits of equal order spaces, upwinding and ability to handle large gradients of DG methods as well as the smaller globally coupled linear system, adaptivity, and multinumeric capabilities of hybridized methods. However, the main bottleneck in HDG methods, limiting its use to small to moderate sized problems, is the lack of scalable linear solvers. In this thesis we develop fast and scalable solvers for HDG methods consisting of domain decomposition, multigrid and multilevel solvers/preconditioners with an ultimate focus on simulating large scale problems in fluid dynamics and magnetohydrodynamics (MHD). First, we propose a domain decomposition based solver namely iterative HDG for partial differential equations (PDEs). It is a fixed point iterative scheme, with each iteration consisting only of element-by-element and face-by-face embarrassingly parallel solves. Using energy analysis we prove the convergence of the schemes for scalar and system of hyperbolic PDEs and verify the results numerically. We then propose a novel geometric multigrid approach for HDG methods based on fine scale Dirichlet-to-Neumann maps. The algorithm combines the robustness of algebraic multigrid methods due to operator dependent intergrid transfer operators and at the same time has fixed coarse grid construction costs due to its geometric nature. For diffusion dominated PDEs such as the Poisson and the Stokes equations the algorithm gives almost perfect hp – scalability. Next, we propose a multilevel algorithm by combining the concepts of nested dissection, a fill-in reducing ordering strategy, variational structure and high-order properties of HDG, and domain decomposition. Thanks to its root in direct solver strategy the performance of the solver is almost independent of the nature of the PDEs and mostly depends on the smoothness of the solution. We demonstrate this numerically with several prototypical PDEs. Finally, we propose a block preconditioning strategy for HDG applied to incompressible visco-resistive MHD. We use a least squares commutator approximation for the inverse of the Schur complement and algebraic multigrid or the multilevel preconditioner for the approximate inverse of the nodal block. With several 2D and 3D transient examples we demonstrate the robustness and parallel scalability of the block preconditioner
*Advisors/Committee Members: Bui-Thanh, Tan (advisor), Demkowicz, Leszek F (committee member), Ghattas, Omar (committee member), Raja, Laxminarayan L (committee member), Shadid, John N (committee member), Waelbroeck, Francois L (committee member), Wheeler, Mary F (committee member).*

Subjects/Keywords: Hybridized discontinuous Galerkin; Fast solvers; Multigrid; Multilevel; MHD; Domain decomposition

Record Details Similar Records

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

-5494-1880. (2019). Fast and scalable solvers for high-order hybridized discontinuous Galerkin methods with applications to fluid dynamics and magnetohydrodynamics. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://dx.doi.org/10.26153/tsw/5474

Note: this citation may be lacking information needed for this citation format:

Author name may be incomplete

Chicago Manual of Style (16^{th} Edition):

-5494-1880. “Fast and scalable solvers for high-order hybridized discontinuous Galerkin methods with applications to fluid dynamics and magnetohydrodynamics.” 2019. Doctoral Dissertation, University of Texas – Austin. Accessed May 08, 2021. http://dx.doi.org/10.26153/tsw/5474.

Note: this citation may be lacking information needed for this citation format:

Author name may be incomplete

MLA Handbook (7^{th} Edition):

-5494-1880. “Fast and scalable solvers for high-order hybridized discontinuous Galerkin methods with applications to fluid dynamics and magnetohydrodynamics.” 2019. Web. 08 May 2021.

Note: this citation may be lacking information needed for this citation format:

Author name may be incomplete

Vancouver:

-5494-1880. Fast and scalable solvers for high-order hybridized discontinuous Galerkin methods with applications to fluid dynamics and magnetohydrodynamics. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2019. [cited 2021 May 08]. Available from: http://dx.doi.org/10.26153/tsw/5474.

Author name may be incomplete

Council of Science Editors:

-5494-1880. Fast and scalable solvers for high-order hybridized discontinuous Galerkin methods with applications to fluid dynamics and magnetohydrodynamics. [Doctoral Dissertation]. University of Texas – Austin; 2019. Available from: http://dx.doi.org/10.26153/tsw/5474

Author name may be incomplete