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1. Scalable DPG multigrid solver with applications in high-frequency wave propagation

   https://doi.org/10.26153/tsw/55685  

   Badger, Jacob (January 2024, The University of Texas at Austin)

   Wave propagation is fundamental to applications including natural resource exploration, nuclear fusion research, and military defense, among others. However, developing accurate and efficient numerical algorithms for solving time-harmonic wave propagation problems is notoriously difficult. One difficulty is that classical discretization techniques (e.g., Galerkin finite elements, finite difference, etc.) yield indefinite discrete systems that preclude the use of many scalable solution algorithms. Significant progress has been made to develop specialized preconditioners for high-frequency wave propagation problems but robust and scalable solvers for general problems, including non-homogenous media and complex geometries, remain elusive. An alternative approach is to use minimum residual discretization methods—that yield Hermitian positive-definite discrete systems—and may be amenable to more standard preconditioners. Indeed, popularization of the first-order system least-squares methodology (FOSLS) was driven by the applicability of geometric and algebraic multigrid to otherwise indefinite problems. However, for wave propagation problems, FOSLS is known to be highly dissipative and is thus less competitive in the high-frequency regime. The discontinuous Petrov–Galerkin (DPG) method of Demkowicz and Gopalakrishnan is a minimum residual finite element method with several additional attractive properties: mesh-independent stability, a built-in error indicator, and applicability to a number of variational formulations. In the context of high-frequency wave propagation, the ultraweak DPG formulation has been observed to produce pollution error roughly commensurate to Galerkin discretizations. DPG discretizations may thus deliver accuracy typical of classical discretization techniques, but result in Hermitian positive-definite discrete systems that are often more amenable to preconditioning. A multigrid preconditioner for DPG systems, developed in the dissertation work of S. Petrides, was shown to scale efficiently in a shared-memory implementation. The primary objective of this dissertation is development of an efficient, distributed implementation of the DPG multigrid solver (DPG-MG). The distributed DPG-MG solver developed in this work will be demonstrated to be massively scalable, enabling solution of three-dimensional problems with O(10¹²) degrees of freedom on up to 460 000 CPU cores, an unprecedented scale for high-frequency wave propagation. The scalability of the DPG-MG solver will be further combined with hp-adaptivity to enable efficient solution of challenging real-world high-frequency wave propagation problems including optical fiber modeling, simulation of RF heating in tokamak devices, and seismic simulation. These applications include complex three-dimensional geometries, heterogeneous and anisotropic media, and localized features; demonstrating the robustness and versatility of the solver and tools developed in this dissertation. 

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2. Multigrid for model reduction of power grid networks

   https://doi.org/10.1002/nla.2201  

   Lee, B. (June 2018, Numerical Linear Algebra with Applications)

   Summary This paper presents a method for determining the relevant buses for reduced models of power grid networks described by systems of differential‐algebraic equations and for constructing the coarse‐grain dynamical power grid systems. To determine these buses, time integration of differential equations is not needed, but rather, a stationary system is analyzed. However, unlike stationary‐system approaches that determine only coarse generator buses by approximating the coherency of the generators, the proposed method analyzes the graph Laplacian associated with the admittance matrix. The buses for the reduced model are chosen to ensure that the graph Laplacian of the reduced model is an accurate approximation to the graph Laplacian of the full system. Both load and generator buses can be selected by this procedure since the Laplacian is defined on all the buses. The basis of this proposed approach lies in the close relationship between the synchrony of the system and the spectral properties of this Laplacian, that is, conditions on the spectrum of this Laplacian that almost surely guarantee the synchrony of the system. Thus, assuming that the full system is in synchrony, our strategy is to coarsen the full‐system Laplacian such that the coarse Laplacian possesses good approximation to these spectral conditions. Accurate approximation to these conditions then can better lead to synchronous reduced models. The coarsened Laplacian is defined on coarse degrees of freedom (DOFs), which are associated with the relevant buses to include in the reduced model. To realize this coarse DOF selection, we use multigrid coarsening techniques based on compatible relaxation. Multigrid is the natural choice since it has been extensively used to coarsen Laplacians arising from discretizations of elliptic partial differential equations and is actively being extended to graph Laplacians. With the selection of the buses for the reduced model, the reduced model is completed by constructing the coarse admittance matrix values and other physical parameters using standard power grid techniques or by using the intergrid operators constructed in the coarse DOFs selection process. Unfortunately, the selection of the coarse buses and the coarsening of the admittance matrix and physical parameters are not sufficient by themselves to produce a stable reduced system. To achieve a stable system, system structures of the fine‐grain model must be preserved in the reduced model. We analyze this to develop a multigrid methodology for constructing stable reduced models of power grid systems. Numerical examples are presented to validate this methodology. 

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3. A multigrid method for kernel functions acting on interacting structures with applications to biofluids

   https://doi.org/10.1016/j.jcp.2023.112506  

   Liu, Weifan; Rostami, Minghao W (December 2023, Journal of computational physics)

   Simulating the dynamics of discretized interacting structures whose relationship is dictated by a kernel function gives rise to a large dense matrix. We propose a multigrid solver for such a matrix that exploits not only its data-sparsity resulting from the decay of the kernel function but also the regularity of the geometry of the structures and the quantities of interest distributed on them. Like the well-known multigrid method for large sparse matrices arising from boundary-value problems, our method requires a smoother for removing high-frequency terms in solution errors, a strategy for coarsening a grid, and a pair of transfer operators for exchanging information between two grids. We develop new techniques for these processes that are tailored to a kernel function acting on discretized interacting structures. They are matrix-free in the sense that there is no need to construct the large dense matrix. Numerical experiments on a variety of bio-inspired microswimmers immersed in a Stokes flow demonstrate the effectiveness and efficiency of the proposed multigrid solver. In the case of free swimmers that must maintain force and torque balance, additional sparse rows and columns need to be appended to the dense matrix above. We develop a matrix-free fast solver for this bordered matrix as well, in which the multigrid method is a key component. 

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4. Improving Performance and Scalability of Algebraic Multigrid through a Specialized MATVEC

   Majid Rasouli, Vidhi Zala (September 2018, IEEE High Performance Extreme Computing Conference)

   Algebraic Multigrid (AMG) is an extremely popular linear system solver and/or preconditioner approach for matrices obtained from the discretization of elliptic operators. However, its performance and scalability for large systems obtained from unstructured discretizations seem less consistent than for geometric multigrid (GMG). To a large extent, this is due to loss of sparsity at the coarser grids and the resulting increased cost and poor scalability of the matrix-vector multiplication. While there have been attempts to address this concern by designing sparsification algorithms, these affect the overall convergence. In this work, we focus on designing a specialized matrix-vector multiplication (matvec) that achieves high performance and scalability for a large variation in the levels of sparsity. We evaluate distributed and shared memory implementations of our matvec operator and demonstrate the improvements to its scalability and performance in AMG hierarchy and finally, we compare it with PETSc. 

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5. Security of a Power System Under High Penetration of Wind Energy Considering Contingencies and Stability Margins

   https://doi.org/10.1109/SCOReD53546.2021.9652753  

   Dorile, Pierre O.; Jagessar, Daniel R.; McCann, Roy A. (November 2021, IEEE 19th Student Conference on Research and Development (SCOReD))

   Security is a well-known function to any transmission operator and system planner. As the world is moving toward the decarbonization of the power industry, it is more complicated for the system operators to maintain an acceptable level of security in the power system operation. More large-scale wind farms are being incorporated into the grid, and thus, the voltage stability concern is increasing. In practice, several contingencies are imagined by the system operators to assess the reliability of the grid. Since voltage stability is one of the major menaces that can trigger voltage instability in a power system, this paper is attempting to present to the transmission system planners and operators a dedicated methodology to facilitate the incorporation of large-scale wind farms into a transmission grid under high penetration of wind power. the stability of a wind-dominated power system is discussed based on Q-V and P-V methodologies and some N-1 contingencies with the Remedial Action Schemes (RAS). Furthermore, a methodology to rank the worst contingencies and to predict the voltage collapse during the highest wind penetration level is presented. Simulations have been, extensively, carried out to examine the methodology and have provided valuable information about the static security of the wind-dominated power system. The results can be used by the transmission system operator to anticipate voltage instability or voltage collapse in the power system during high wind penetration levels. 

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