Ordering Techniques for Two- and Three-Dimensional Convection-Dominated Elliptic Boundary Value Problems

The numerical solution of the diffusion equation in VLSI process simulation leads to large systems of nonlinear equations which have to be solved at every time step. For this purpose, a multigrid (MG) algorithm with locally refined grids is constructed. It is demonstrated that the method used here yields typical MG algorithm convergence rates, and reduces the amount of work considerably. The local refinements are controlled by an estimation of the discretization error which is calculated within the nonlinear MG method (FAS) and requires no extra computational work.

Modal retrieval problem can be addressed using sparse estimation techniques coupled with a multigrid approach. Selection of the initial grid, in the multigrid algorithm, is a critical problem that needs satisfactory solutions. In this paper we propose a strategy for selecting a coarse initial grid which guarantees convergence of the algorithm even starting with a single atom in the dictionary. The idea is to define the atoms so as their correlation with the signal modes is greater than a specified threshold. We show through simulations that the proposed solution provides accurate estimations even in presence of noise. Moreover, we investigate the convergence of the multigrid algorithm towards the true parameters.

We present a study of the effectiveness of asynchronous incomplete LU factorization preconditioners for the time-implicit solution of compressible flow problems while exploiting thread-parallelism within a compute node. A block variant of the asynchronous fine-grained parallel preconditioner adapted to a finite volume discretization of the compressible Navier--Stokes equations on unstructured grids is presented, and convergence theory is extended to the new variant. Experimental (numerical) results on the performance of these preconditioners on inviscid and viscous laminar two-dimensional steady-state test cases are reported. It is found, for these compressible flow problems, that the block variant performs much better in terms of convergence, parallel scalability, and reliability than the original scalar asynchronous ILU preconditioner. For viscous flow, it is found that the ordering of unknowns may determine the success or failure of asynchronous block-ILU preconditioning, and an ordering of grid cells suitable for solving viscous problems is presented.

Elliptic interface problems with multi-domains have wide applications in engineering and science. However, it is challenging for most existing methods to solve three-dimensional elliptic interface problems with multi-domains due to local geometric complexity, especially for problems with matrix coefficient and sharp-edged interface. There are some recent work in two dimensions for multi-domains and in three dimensions for two domains. However, the extension to three dimensional multi-domain elliptic interface problems is non-trivial. In this paper, we present an efficient non-traditional finite element method with non-body-fitting grids for three-dimensional elliptic interface problems with multi-domains. Numerical experiments show that this method achieves close to second order accurate in the L ∞ norm for piecewise smooth solutions.

Deep learning method is of great importance in solving partial differential equations. In this paper, inspired by the failure-informed idea proposed by Gao et al. (SIAM Journal on Scientific Computing 45(4) (2023)) and as an improvement, a new accurate adaptive deep learning method is proposed for solving elliptic problems, including interface problems and convection-dominated problems. Based on the failure probability framework, the piece-wise uniform distribution is used to approximate the optimal proposal distribution and a kernel-based method is proposed for efficient sampling. Together with the improved Levenberg-Marquardt optimization method, the proposed adaptive deep learning method shows great potential in improving solution accuracy. Numerical tests on the elliptic problems without interface conditions, on one elliptic interface problem, and on the convection-dominated problems demonstrate the effectiveness of the proposed method, as it reduces the relative errors by a factor varying from $10^2$ to $10^4$ for different cases.

1 Numerical Analysis Group, DIAM, Faculty EEMCS, Delft University of Technology, Mekelweg 4, 2628 CD, Delft, The Netherlands, Email: h.binzubair@tudelft.nl 2 Department of Mathematics, Tufts University, Bromfield-Pearson Building, 503 Boston Avenue, Medford, MA 02155, USA. scott.maclachlan@tufts.edu 3 ’Centrum voor Wiskunde en Informatica (CWI), Dutch national research centre for mathematics and computer science’ Modelling, Analysis and Simulation (MAS2) Kruislaan 413, Amsterdam Email: c.w.oosterlee@cwi.nl

For the efficient numerical solution of elliptic variational inequalities on closed convex sets, multigrid methods based on piecewise linear finite elements have been investigated over the past decades. Essential to their success is the appropriate approximation of the constraint set on coarser grids which is based on function values for piecewise linear finite elements. On the other hand, there are a number of problems which profit from higher order approximations. Among these are the problem of pricing American options, formulated as a parabolic boundary value problem involving Black–Scholes’ equation with a free boundary. In addition to computing the free boundary (the optimal exercise price of the option) of particular importance are accurate pointwise derivatives of the value of the stock option up to order two, the so‐called Greek letters. In this paper, we propose a monotone multigrid method for discretizations in terms of B‐splines of arbitrary order to solve elliptic variational inequalities on a closed convex set. In order to maintain monotonicity (upper bound) and quasi optimality (lower bound) of the coarse grid corrections, we propose an optimized coarse grid correction (OCGC) algorithm which is based on B‐spline expansion coefficients. We prove that the OCGC algorithm is of optimal complexity of the degrees of freedom of the coarse grid and, therefore, the resulting monotone multigrid method is of asymptotically optimal multigrid complexity. Finally, the method is applied to a standard model for the valuation of American options. In particular, it is shown that a discretization based on B‐splines of order four enables us to compute the second derivative of the value of the stock option to high precision.

A computational methodology combining overlapping grid techniques with multigrid methods has been developed for three‐dimensional flow calculations in or around complex geometries. The computational accuracy, efficiency and capability of the present approach are investigated in this paper. The incompressible Navier–Stokes equations are discretized using a finite volume method on a semi‐staggered grid. The discrete problem is solved by a multigrid algorithm. Some numerical examples are chosen for evaluating numerical accuracy: (a) a straight pipe for which the exact solution is known; (b) curved pipes where previous experimental and numerical data are available; (c) an axisymmetric sudden expansion. The performance of the multigrid method on overlapping grids is assessed. Several cases of flows in stationary and time‐dependent complex geometries are given to demonstrate the capability and the potential of the methods that we employ.

As HPC system sizes grow to millions of cores and chip feature sizes continue to decrease, HPC applications become increasingly exposed to transient hardware faults. These faults can cause aborts and performance degradation. Most importantly, they can corrupt results. Thus, we must evaluate the fault vulnerability of key HPC algorithms to develop cost-effective techniques to improve application resilience.We present an approach that analyzes the vulnerability of applications to faults, systematically reduces it by protecting the most vulnerable components and predicts application vulnerability at large scales. Weinitially focus on sparse scientific applications and apply our approachin this paper to the Algebraic Multi Grid (AMG) algorithm. We empirically analyze AMG's vulnerability to hardware faults in both sequential and parallel (hybrid MPI/OpenMP) executions on up to 1,600 cores and propose and evaluate the use of targeted pointer replication to reduce it. Our techniques increase AMG's resilience to transient hardware faults by 50-80% and improve its scalability on faulty computational environments by 35%. Further, we show how to model AMG's scalability in fault-prone environments to predict execution times of large-scale runs accurately.

The development of scalable robust solvers for unstructured finite element applications related to viscous flow problems in earth sciences is an active research area. Solving high‐resolution convection problems with order of magnitude 108 degrees of freedom requires solvers that scale well, with respect to both the number of degrees of freedom as well as having optimal parallel scaling characteristics on computer clusters. We investigate the use of a smoothed aggregation (SA) algebraic multigrid (AMG)‐type solution strategy to construct efficient preconditioners for the Stokes equation. We integrate AMG in our solver scheme as a preconditioner to the conjugate gradient method (CG) used during the construction of a block triangular preconditioner (BTR) to the Stokes equation, accelerating the convergence rate of the generalized conjugate residual method (GCR). We abbreviate this procedure as BTA‐GCR. For our experiments, we use unstructured grids with quadratic finite elements, making the model flexible with respect to geometry and topology and O(h3) accurate. We find that AMG‐type methods scale linearly (O(n)), with respect to the number of degrees of freedom, n. Although not all parts of AMG have preferred parallel scaling characteristics, we show that it is possible to tune AMG, resulting in parallel scaling characteristics that we consider optimal, for our experiments with up to 100 million degrees of freedom. Furthermore, AMG‐type methods are shown to be robust methods, allowing us to solve very ill‐conditioned systems resulting from strongly varying material properties over short distances in the model interior.

A full multigrid/full approximation storage (FMG/FAS) method is developed along with a unified pressure-based algorithm to handle the complex fluid flow problems encompassing both compressible and incompressible regimes. Besides highlighting the key elements of the present algorithm on a curvilinear staggered grid system, several flow problems, ranging from incompressible to hypersonic speeds, are computed, in conjunction with an adaptive grid method, to demonstrate its computational capability. With wide variations in geometry, fluid physics, and grid distribution, speedup between the single- and multigrid procedures can be obtained. Because different pressure boundary conditions are needed for incompressible, subsonic, and supersonic inlet and outlet, the number of grid levels that can be effectively used by the present multigrid method appears flow dependent; the implications of this observation are discussed.

Apparent horizon plays an important role in numerical relativity as it provides a tool to characterize the existence and properties of black holes on three-dimensional spatial slices in 3+1 numerical spacetimes. Apparent horizon finders based on different techniques have been developed. In this paper, we revisit the apparent horizon finding problem in numerical relativity using multigrid-based algorithms. We formulate the nonlinear elliptic apparent horizon equation as a linear Poisson-type equation with a nonlinear source, and solve it using a multigrid algorithm with Gauss–Seidel line relaxation. A fourth order compact finite difference scheme in spherical coordinates is derived and employed to reduce the complexity of the line relaxation operator to a tri-diagonal matrix inversion. The multigrid-based apparent horizon finder developed in this work is capable of locating apparent horizons in generic spatial hypersurfaces without any symmetries. The finder is tested with both analytic data, such as Brill–Lindquist multiple black hole data, and numerical data, including off-centered Kerr–Schild data and dynamical inspiraling binary black hole data. The obtained results are compared with those generated by the current fastest finder AHFinderDirect (Thornburg 2003 Class. Quantum Grav. 21 743), which is the default finder in the open source code Einstein Toolkit. Our finder performs comparatively in terms of accuracy, and starts to outperform AHFinderDirect at high angular resolutions ( ∼ 1 ∘ ) in terms of speed. Our finder is also more flexible to initial guess, as opposed to the Newton’s method used in AHFinderDirect. This suggests that the multigrid algorithm provides an alternative option for studying apparent horizons, especially when high resolutions are needed.

In this thesis, we discuss a robust preconditioner (the AGKS preconditioner) for solving linear systems arising from approximations of partial differential equations (PDEs) with high-contrast coefficients. The problems considered here include the standard second and higher order elliptic PDEs such as high-contrast diffusion equation, Stokes' equation and biharmonic-plate equation. The goal of this study is the development of robust and parallelizable preconditioners that can easily be integrated to treat large configurations. The construction of the preconditioner consists of two phases. The first one is an algebraic phase which partitions the degrees of freedom into high and low permeability regions which may be of arbitrary geometry. This yields a corresponding block partitioning of the stiffness matrix allowing us to use a formula for the action of its inverse involving the inverses of both the high permeability block and its Schur complement in the original matrix. Singular perturbation analysis plays a big role to analyze the structure of the required subblock inverses in the high contrast case which shows that for high enough contrast each of the subblock inverses can be approximated well by solving only systems with constant coefficients. The second phase involves an efficient multigrid approximation of this exact inverse. After applying singular perturbation theory to each of the sub-blocks, we obtain that inverses of each of the subblocks with high contrast entries can be approximated efficiently using geometric multigrid methods, and that this approximation is robust with respect to both the contrast and the mesh size. The result is a multigrid method for high contrast problems which is provably optimal to both contrast and mesh size. We demonstrate the advantageous properties of the AGKS preconditioner using experiments on model high-contrast problems. We examine its performance against multigrid method under varying discretizations of diffusion equation, Stokes equation and biharmonic-plate equation. Thus, we show that we accomplished a desirable preconditioning design goal by using the same family of preconditioners to solve the elliptic family of PDEs with varying discretizations.

In [L. Adams and Z. Li, SIAM J. Sci. Comput., 24 (2002), pp. 463--479], a multigrid method was designed specifically for interface problems that have been discretized using the methods described in [L. Adams and Z. Li, SIAM J. Sci. Comput., 24 (2002), pp. 463--479] and in [Z. Li and K. Ito, SIAM J. Sci. Comput., 23 (2001), pp. 339--361] for elliptic interface problems using the maximum principle preserving schemes. In [L. Adams and T. P. Chartier, SIAM J. Sci. Comput., 25 (2002), pp. 1516--1533], a new method was introduced that utilizes a new interpolator for grid points near the immersed interface and a new restrictor that guarantees the coarse-grid matrices are M-matrices. This paper compares the immersed interface multigrid methods introduced in [L. Adams and Z. Li, SIAM J. Sci. Comput., 24 (2002), pp. 463--479] and [L. Adams and T. P. Chartier, SIAM J. Sci. Comput., 25 (2002), pp. 1516--1533] with algebraic multigrid, which uses no geometric information to set up the multigrid components for coarse-grid correction. We show that algebraic multigrid is a robust solver for our test problems. It outperforms the method in [L. Adams and Z. Li, SIAM J. Sci. Comput., 24 (2002), pp. 463--479] and performs nearly as well as the method in [L. Adams and T. P. Chartier, SIAM J. Sci. Comput., 25 (2002), pp. 1516--1533] which is shown to be the most efficient for all problem parameters and sizes.

We introduce a new geometric multigrid algorithm to solve elliptic interface problems. First we discretize the problems by the usual P 1 {P_{1}} -conforming finite element methods on a semi-uniform grid which is obtained by refining a uniform grid. To solve the algebraic system, we adopt subspace correction methods for which we use uniform grids as the auxiliary spaces. To enhance the efficiency of the algorithms, we define a new transfer operator between a uniform grid and a semi-uniform grid so that the transferred functions satisfy the flux continuity along the interface. In the auxiliary space, the system is solved by the usual multigrid algorithm with a similarly modified prolongation operator. We show 𝒲 {\mathcal{W}} -cycle convergence for the proposed multigrid algorithm. We demonstrate the performance of our multigrid algorithm for problems having various ratios of parameters. We observe that the computational complexity of our algorithms are robust for all problems we tested.