Scalable Execution of Legacy Scientific Codes

Flash memory-based computing-in-memory (CIM) architectures have gained popularity due to their remarkable performance in various computation tasks of data processing, including machine learning, neuron networks, and scientific calculations. Especially in the partial differential equation (PDE) solver that has been widely utilized in scientific calculations, high accuracy, processing speed, and low power consumption are the key requirements. This work proposes a novel flash memory-based PDE solver to implement PDE with high accuracy, low power consumption, and fast iterative convergence. Moreover, considering the increasing current noise in nanoscale devices, we investigate the robustness against the noise in the proposed PDE solver. The results show that the noise tolerance limit of the solver can reach more than five times that of the conventional Jacobi CIM solver. Overall, the proposed flash memory-based PDE solver offers a promising solution for scientific calculations that require high accuracy, low power consumption, and good noise immunity, which could help to develop flash-based general computing.

The GALerkin Method Of Lines code package, GALMOL, is designed to solve parabolic partial differential equations having a single space dimension. It uses either the finite element method or the Gaussian collocation method for space discretizations. A robust ordinary differential equation solver is used as the time integrator; it is very efficient and achieves high accuracy. After introductory material, some theoretical background about the semidiscrete Galerkin method is given. Next, the features and implementation of GALMOL are described, including the structures and different options of the program. Although the input language is largely self-explanatory, a description and a user's guide are provided. The performance and ease of use of the package are illustrated for the heat equation, Burger's equation, and the wave equation. Some typical input and prototype CYBER JCL sequences are given. (RWR)

In this study, a nonvolatile memory (NVM)-based mixed-precision partial differential equation (PDE) solver is developed to achieve both high computational accuracy and high energy efficiency in a non-Von Neumann manner. The mixed-precision architecture consists of an outer solver utilizing CPU/GPU for a small amount of high-precision computing and an inner solver using NVM array for a large number of approximate iterations by in-memory computing. The impacts of inner iteration number and inner precision on time and energy cost are analyzed to configure the optimum algorithm. Based on this, we investigate the impacts of device reliability properties on the performance of mixed-precision computing system, including device variation, read noise, ON/OFF ratio, and read disturb. The results indicate that the computational accuracy is highly tolerant to the device variation. However, the time and energy cost are increased due to the device reliability, especially the ON/OFF ratio. To suppress the impact of ON/OFF ratio, we present a circuit compensation method, which subtracts the current of reference string with all devices OFF before quantization. To improve the hardware utilization and reduce the number of iterations, a matrix slicing method combined with hybrid layer update is proposed. By employing the two optimization methods, a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$57.3\times $ </tex-math></inline-formula> reduction at energy cost with the matrix size of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1600\times1600$ </tex-math></inline-formula> is achieved.

Label-free learning of elliptic partial differential equation solvers with generalizability across boundary value problems

In prior-research the authors have demonstrated that, for stencil-based numerical solvers for Partial Differential Equations (PDEs), the parallel performance can be significantly improved by selecting sub-domains that are not cubic in shape (Saxena et. al., HPCS 2016, pp. 875-885). This is achieved through accounting for cache utilization in both the message passing and the computational kernel, where it is demonstrated that the optimal domain decompositions not only depend on the communication and load balance but also on the cache-misses, amongst other factors. In this work we demonstrate that those conclusions may also be extended to more advanced numerical discretizations, based upon Adaptive Mesh Refinement (AMR). In particular, we show that when basing our AMR strategy on the local refinement of patches of the mesh, the optimal patch shape is not typically cubic. We provide specific examples, with accompanying explanation, to show that communication minimizing strategies are not necessarily the best choice when applying AMR in parallel. All numerical tests undertaken in this work are based upon the open source BoxLib library.

Efficient Global Element Indexing for Parallel Adaptive Flow Solvers

Whenever we develop software we know it will evolve over time, and it will need to be adapted to changing requirements. A variation point is where a program has been designed for change, or where it easily can be accommodated to change. Here we discuss variation points in numerical solvers for Partial Differential Equations (PDEs). Our experience is that an abstraction oriented approach based on coordinate-free numerics naturally provides most, if not all, variation points needed.

The increasing adoption of GPUs as mainstream computing devices, coupled with the imminent availability of large high-bandwidth caches based on die-stacked memory makes it important to analyze and understand modern GPU compute applications from the perspective of their memory access and data reuse characteristics. This paper presents detailed workload characterization studies on four GPU compute applications that process large data sets. The applications studied include tree traversal and search algorithms, a partial differential equation (PDE) solver, and a synthetic array processing application. Our studies indicate that while the memory footprint consumed by these applications can be very large, the effectiveness of several GB worth of cache may vary significantly across workloads. This suggests that provisioning cache resources in a die-stacked memory based system needs to be done very carefully, through detailed characterization of target workloads. An added benefit of our work was the discovery that accurate memory characterization data can lead to a significantly more optimized strategy for scheduling GPU threads by taking advantage of a workload's access characteristics. In particular, for the PDE solver, our analysis led to an optimization that achieved 30% measured gain in application performance. This paper also describes our analysis methodology for conducting these types of studies. The methodology is based on trace analysis, where the traces capture memory traffic and calls to the GPU compute API. For each application we highlight the characterization metrics and analysis techniques that were most useful in generating insights about their memory access patterns.

Presents a paradigm that combines the speed of single address spaces with the traditional protection mechanisms found in multiple address space systems such as UNIX. Multiple address spaces are common today for providing safety and security in operating systems. A process in one address space cannot corrupt or compromise memory locations an another process's address space. However, inter-process communication is slow because it is expensive to context switch between address spaces. Single address space systems provide a solution to the problem of slow context switches. In a single address space system, which is typically implemented in a 64-bit address space, all processes shape a common address space. Hence, the need to move between address spaces during inter-process communication is avoided. It is our position that it is not strictly necessary to use a single address space in order to achieve fast inter-process communication. >

Deep learning surrogate models aim to accelerate the solving of partial differential equations (PDEs) and have achieved certain promising results. Although several main-stream models through neural operator learning have been applied to delve into PDEs on varying geometries, they were designed to map the complex geometry to a latent uniform grid, which is still challenging to learn by the networks with general architectures. In this work, we rethink the critical factors of PDE solutions and propose a novel model-agnostic framework, called 3D Geometry-Guided Conditional adaptation (3D-GeoCA), for solving PDEs on arbitrary 3D geometries. Starting with a 3D point cloud geometry encoder, 3D-GeoCA can extract the essential and robust representations of any kind of geometric shapes, which conditionally guides the adaptation of hidden features in the surrogate model. We conduct experiments on two public computational fluid dynamics datasets, the Shape-Net Car and Ahmed-Body dataset, using several surrogate models as the backbones with various point cloud geometry encoders to simulate corresponding large-scale Reynolds Average Navier-Stokes equations. Equipped with 3D-GeoCA, these backbone models can reduce the L-2 error by a large margin. Moreover, this 3D-GeoCA is model-agnostic so that it can be applied to any surrogate model.

This work explores the numerical translation of the weak or integral solution of nonlinear partial differential equations into a numerically efficient, time-evolving scheme. Specifically, we focus on partial differential equations separable into a quasilinear term and a nonlinear one, with the former defining the Green function of the problem. Utilizing the Green function under a short-time approximation, it becomes possible to derive the integral solution of the problem by breaking it into three integral terms: the propagation of initial conditions and the contributions of the nonlinear and boundary terms. Accordingly, we follow this division to describe and separately analyze the resulting algorithm. To ensure low interpolation error and accurate numerical Green functions, we adapt a piecewise interpolation collocation method to the integral scheme, optimizing the positioning of grid points near the boundary region. At the same time, we employ a second-order quadrature method in time to efficiently implement the nonlinear terms. Validation of both adapted methodologies is conducted by applying them to problems with known analytical solution, as well as to more challenging, norm-preserving problems such as the Burgers equation and the soliton solution of the nonlinear Schrödinger equation. Finally, the boundary term is derived and validated using a series of test cases that cover the range of possible scenarios for boundary problems within the introduced methodology.

We present a parallel finite element solver for mesh-connected multicomputers (ie., message-passing, distributed-memory multiprocessors) and provide a theoretical proof tht the algorithms used are highly scalable and close to optimal. Further, the algorithms are written in an object-oriented language (ESP C++) to enhance reusability of code. Performance results are presented for a Motorola multicomputer and a network of Sun workstations.

An automatic differentiation tool is used to develop the adjoint code for a three-dimensional computational fluid dynamics solver. Rather than using automatic differentiation to differentiate the entire source code of the computational fluid dynamics solver, we have applied it selectively to produce code that computes the flux Jacobian matrix and the other partial derivatives that are necessary to compute total derivatives using an adjoint method. The resulting linear discrete adjoint system is then solved using the portable, extensible toolkit for scientific computation. This selective application of automatic differentiation is the central idea behind the automatic differentiation adjoint (ADjoint) approach. This approach has the advantage that it is applicable to arbitrary sets of governing equations and cost functions, and that it is exactly consistent with the gradients that would be computed by exact numerical differentiation of the original solver. Furthermore, the approach is largely automatic, thus avoiding the lengthy development times usually required to develop adjoint solvers for partial differential equations. These significant advantages come at the cost of increased memory requirements for the adjoint solver. Derivatives of drag and lift coefficients are validated, and the low computational cost and ease of implementation of the method are shown.

In engineering design, an optimized solution often turns out to be suboptimal when errors are encountered. Although the theory of robust convex optimization has taken significant strides over the past decade, all approaches fail if the underlying cost function is not explicitly given; it is even worse if the cost function is nonconvex. In this work, we present a robust optimization method that is suited for unconstrained problems with a nonconvex cost function as well as for problems based on simulations, such as large partial differential equations (PDE) solver, response surface, and Kriging metamodels. Moreover, this technique can be employed for most real-world problems because it operates directly on the response surface and does not assume any specific structure of the problem. We present this algorithm along with the application to an actual engineering problem in electromagnetic multiple scattering of aperiodically arranged dielectrics, relevant to nanophotonic design. The corresponding objective function is highly nonconvex and resides in a 100-dimensional design space. Starting from an “optimized” design, we report a robust solution with a significantly lower worst-case cost, while maintaining optimality. We further generalize this algorithm to address a nonconvex optimization problem under both implementation errors and parameter uncertainties.

A simple numerical approach for assessing coupled transport processes in partitioning systems