Large-scale Partial Differential Equation Research Articles

Parallel Computing Techniques for Solving Large-Scale Partial Differential Equation

Two-layered partial differential equations (PDEs) will be structured in this investigation using a variety of parallel mathematical techniques, such as Red Dull Gauss Seidel and Gauss Seidel. For the solution of this study, the two-layered PDEs of the elliptic and illustrated sorts were selected. Parallel Virtual Machine (PVM) is used on the correspondence among all chips of the Parallel Handling System. PVM is well-known for its item system, which enables several diverse computers to be used as flexible and easy-to-use concurrent handling resources. An evaluation of the parallel exhibition assessment including the Gauss Seidel and Red Gauss Seidel techniques in terms of execution time, speed, capability, ampleness, and transient execution will follow the graphical presentation of the mathematical results. Since the goal of this work was to create a successful Two-Dimensional PDE Solver (TDPDES), performance evaluations are simple. Numerous design and mathematical sectors will see an improvement in their examination and investigation systems thanks to this new, effective TDPDES technology..

A Fast and Scalable Computational Framework for Large-Scale High-Dimensional Bayesian Optimal Experimental Design

We develop a fast and scalable computational framework to solve Bayesian optimal experimental design problems governed by partial differential equations (PDEs) with application to optimal sensor placement by maximizing expected information gain (EIG). Such problems are particularly challenging due to the curse of dimensionality for high-dimensional parameters and the expensive solution of large-scale PDEs. To address these challenges, we exploit two fundamental properties: (1) the low-rank structure of the Jacobian of the parameter-to-observable map, to extract the intrinsically low-dimensional data-informed subspace, and (2) a series of approximations of the EIG that reduce the number of PDE solves while retaining high correlation with the true EIG. Based on these properties, we propose an efficient offline-online decomposition for the optimization problem. The offline stage dominates the cost and entails precomputing all components that require PDE solves. The online stage optimizes sensor placement and does not require any PDE solves. For the online stage, we propose a new greedy algorithm that first places an initial set of sensors using leverage scores and then swaps the selected sensors with other candidates until certain convergence criteria are met, which we call a swapping greedy algorithm. We demonstrate the efficiency and scalability of the proposed method by both linear and nonlinear inverse problems. In particular, we show that the number of required PDE solves is small, independent of the parameter dimension, and only weakly dependent on the data dimension for both problems.

Localized boundary knot method and its application to large-scale acoustic problems

We present a new localized boundary knot method (LBKM) for the solution of large-scale partial differential equations based on the non-singular general solutions. The proposed local algorithm only requires several discrete nodes inside the physical domain and along its boundary without any mesh. For every node, a local subdomain with a simple geometry can be firstly determined via Euclidean distance between nodes. And then the unknown variable at every node can be expressed as a linear combination of function values at nodes inside its corresponding local subdomain. Finally, a sparse linear system can be formed by using the governing equation and the corresponding boundary conditions. In our computations, two different-types of LBKM formulations have been proposed for deriving the final sparse linear system. One is based on the inverse matrix calculation and the other is based on the moving least square (MLS) technique. Unlike the traditional boundary knot method (BKM) with the “global” discretization, the present LBKM is a local discretization method and requires less computer time and storage due to its feature of sparse and banded matrix, which makes it more suitable for solving large-scale problems. Preliminary numerical experiments are carried out in the large-scale acoustics models and bimaterial problems. The comparisons of proposed two types of LBKM formulations are also investigated. The results show the accuracy, validity and great promising applications of the proposed LBKM for the simulation of large-scale problems in complicated geometries.

Compression Challenges in Large Scale Partial Differential Equation Solvers

Solvers for partial differential equations (PDEs) are one of the cornerstones of computational science. For large problems, they involve huge amounts of data that need to be stored and transmitted on all levels of the memory hierarchy. Often, bandwidth is the limiting factor due to the relatively small arithmetic intensity, and increasingly due to the growing disparity between computing power and bandwidth. Consequently, data compression techniques have been investigated and tailored towards the specific requirements of PDE solvers over the recent decades. This paper surveys data compression challenges and discusses examples of corresponding solution approaches for PDE problems, covering all levels of the memory hierarchy from mass storage up to the main memory. We illustrate concepts for particular methods, with examples, and give references to alternatives.

Introduction: Big data and partial differential equations

Partial differential equations (PDEs) are expressions involving an unknown function in many independent variables and their partial derivatives up to a certain order. Since PDEs express continuous change, they have long been used to formulate a myriad of dynamical physical and biological phenomena: heat flow, optics, electrostatics and -dynamics, elasticity, fluid flow and many more. Many of these PDEs can be derived in a variational way, i.e. via minimization of an ‘energy’ functional. In this globalised and technologically advanced age, PDEs are also extensively used for modelling social situations (e.g. models for opinion formation, mathematical finance, crowd motion) and tasks in engineering (such as models for semiconductors, networks, and signal and image processing tasks). In particular, in recent years, there has been increasing interest from applied analysts in applying the models and techniques from variational methods and PDEs to tackle problems in data science. This issue of the European Journal of Applied Mathematics highlights some recent developments in this young and growing area. It gives a taste of endeavours in this realm in two exemplary contributions on PDEs on graphs [1, 2] and one on probabilistic domain decomposition for numerically solving large-scale PDEs [3].

Scalable domain decomposition preconditioners for heterogeneous elliptic problems

Domain decomposition methods are, alongside multigrid methods, one of the dominant paradigms in contemporary large-scale partial differential equation simulation. In this paper, a lightweight implementation of a theoretically and numerically scalable preconditioner is presented in the context of overlapping methods. The performance of this work is assessed by numerical simulations executed on thousands of cores, for solving various highly heterogeneous elliptic problems in both 2D and 3D with billions of degrees of freedom. Such problems arise in computational science and engineering, in solid and fluid mechanics. While focusing on overlapping domain decomposition methods might seem too restrictive, it will be shown how this work can be applied to a variety of other methods, such as non-overlapping methods and abstract deflation based preconditioners. It is also presented how multilevel preconditioners can be used to avoid communication during an iterative process such as a Krylov method.

Scalable Domain Decomposition Preconditioners for Heterogeneous Elliptic Problems

Domain decomposition methods are, alongside multigrid methods, one of the dominant paradigms in contemporary large-scale partial differential equation simulation. In this paper, a lightweight implementation of a theoretically and numerically scalable preconditioner is presented in the context of overlapping methods. The performance of this work is assessed by numerical simulations executed on thousands of cores, for solving various highly heterogeneous elliptic problems in both 2D and 3D with billions of degrees of freedom. Such problems arise in computational science and engineering, in solid and fluid mechanics. While focusing on overlapping domain decomposition methods might seem too restrictive, it will be shown how this work can be applied to a variety of other methods, such as non-overlapping methods and abstract deflation based preconditioners. It is also presented how multilevel preconditioners can be used to avoid communication during an iterative process such as a Krylov method.

An adjoint-based scalable algorithm for time-parallel integration

As parallel architectures evolve the number of available cores continues to increase. Applications need to display a high degree of concurrency in order to effectively utilize the available resources. Large scale partial differential equations mainly rely on a spatial domain decomposition approach, where the number of parallel tasks is limited by the size of the spatial domain. Time parallelism offers a promising approach to increase the degree of concurrency. ‘Parareal’ is an iterative parallel in time algorithm that uses both low and high accuracy numerical solvers. Though the high accuracy solvers are computed in parallel, the low accuracy ones are in serial.This paper revisits the parallel in time algorithm [11] using a nonlinear optimization approach. Like in the traditional ‘Parareal’ method, the time interval is partitioned into subintervals, and local time integrations are carried out in parallel. The objective cost function quantifies the mismatch of local solutions between adjacent subintervals. The optimization problem is solved iteratively using gradient-based methods. All the computational steps – forward solutions, gradients, and Hessian-vector products – involve only ideally parallel computations and therefore are highly scalable.The feasibility of the proposed algorithm is studied on three different model problems, namely, heat equation, Arenstorf's orbit, and the Lorenz model.

Multigrid Methods on Adaptively Refined Grids

Using multigrid solvers in the adaptive finite element method yields a powerful tool for solving large-scale partial differential equations that exhibit localized features such as singularities or shocks. In addition to describing the basic method and related theory, this article numerically demonstrates the method's performance and utility on 2D and 3D problems

Efficient Parallel Implementation of Hybrid Optimization Approaches for Solving Groundwater Inverse Problems

Inverse problems that are constrained by large-scale partial differential equation (PDE) systems demand very large computational resources. Solutions to these problems generally require the solution of a large number of complex PDE systems. Three-dimensional groundwater inverse problems fall under this category. In this paper, we describe the implementation of a parallel simulation-optimization framework for solving PDE-based inverse problems and demonstrate it for the solution of groundwater contaminant source release history reconstruction problems that are of practical importance. The optimization component employs several optimization algorithms, including genetic algorithms (GAs) and several local search (LS) approaches that can be used in a hybrid mode. This hybrid GA-LS optimizer is used to drive a parallel finite-element (FEM) groundwater forward transport simulator. Parallelism is exploited within the transport simulator (fine grained parallelism) as well as the optimizer (coarse grained parallelism) through the exclusive use of the Message Passing Interface (MPI) communication library. Algorithmic and parallel performance results are presented for an IBM SP3 supercomputer. Simulation and performance results presented in this paper illustrate that an effective combination of efficient optimization algorithms and parallel computing can enable solution to three-dimensional groundwater inverse problems of a size and complexity not attempted before.

Higher order impedance and absorbing boundary conditions

Traditionally, generalized impedance boundary conditions (GIBCs) have been used to model dielectrics and coated surfaces, and absorbing boundary conditions (ABCs) have been used to simulate nonreflecting surfaces. The two types have the same mathematical form and, in most instances, a higher order condition involving higher order field derivatives has a better accuracy. We demonstrate that there is a close connection between the two and this enables us to use a systematic method which is available for generating GIBCs of any desired order to derive new two- and three-dimensional ABCs. The method is applicable to curvilinear/doubly-curved surfaces and examples are given. Finally, curves are presented that quantify the accuracy of two-dimensional ABCs up to the fourth order, and show how higher order ABCs can improve the efficiency of large scale partial differential equation (PDE) solutions.