More Like this

1. 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. 

   more » « less
2. Algebraic multigrid for the nonlinear powerflow equations

   https://doi.org/10.1002/nla.2347  

   Lee, Barry; Pereira Batista, Enrique (November 2020, Numerical Linear Algebra with Applications)

   Abstract In a recent article, one of the authors developed a multigrid technique for coarse‐graining dynamic powergrid models. A key component in this technique is a relaxation‐based coarsening of the graph Laplacian given by the powergrid network and its weighted graph, which is represented by the admittance matrix. In this article, we use this coarsening strategy to develop a multigrid method for solving a static system of nonlinear equations that arises through Ohm's law, the so‐called powerflow equations. These static equations are tightly knitted to the dynamic model in that the full powergrid model is an algebraic‐differential system with the powerflow equations describing the algebraic constraints. We assume that the dynamic model corresponds to a stable operating powergrid, and thus, the powerflow equations are associated with a physically stable system. This stability permits the coarsening of the powerflow equations to be based on an approximate graph Laplacian, which is embedded in the powerflow system. By algebraically constructing a hierarchy of approximate weighted graph Laplacians, a hierarchy of nonlinear powerflow equations immediately becomes apparent. This latter hierarchy can then be used in a full approximation scheme (FAS) framework that leads to a nonlinear solver with generally a larger basin of attraction than Newton's method. Given the algebraic multigrid (AMG) coarsening of the approximate Laplacians, the solver is an AMG‐FAS scheme. Alternatively, using the coarse‐grid nodes and interpolation operators generated for the hierarchy of approximate graph Laplacians, a multiplicative‐correction scheme can be derived. The derivation of both schemes will be presented and analyzed, and numerical examples to demonstrate the performance of these schemes will be given. 

   more » « less
3. 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. 

   more » « less
4. 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. 

   more » « less
5. AMPS: REAL-TIME MESH CUTTING WITH AUGMENTED MATRICES FOR SURGICAL SIMULATIONS

   Yeung, Yu-Hong; Pothen, Alex; Crouch, Jessica (January 2020, Numerical linear algebra with applications)

   We present the augmented matrix for principal submatrix update (AMPS) algorithm, a finite element solution method that combines principal submatrix updates and Schur complement techniques, well-suited for interactive simulations of deformation and cutting of finite element meshes. Our approach features real-time solutions to the updated stiffness matrix systems to account for interactive changes in mesh connectivity and boundary conditions. Updates are accomplished by an augmented matrix formulation of the stiffness equations to maintain its consistency with changes to the underlying model without refactorization at each timestep. As changes accumulate over multiple simulation timesteps, the augmented solution algorithm enables tens or hundreds of updates per second. Acceleration schemes that exploit sparsity, memoization and parallelization lead to the updates being computed in real-time. The complexity analysis and experimental results for this method demonstrate that it scales linearly with the number of nonzeros of the factors of the stiffness matrix. Results for cutting and deformation of 3D elastic models are reported for meshes with up to 50,000 nodes, and involve models of surgery for astigmatism and the brain. 

   more » « less