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DOE OSTI2026

Scalable Computation of Topological Abstractions for Scalar Data

Topological data analysis has become an important tool for large scale scalar data analysis and visualization, efficiently extracting the inherent structure and features of interest of the data. However, with growing dataset sizes and complexity, it is increasingly becoming infeasible to compute topological abstractions of interest in serial and on single machines. This paper presents the state of the art in the scalable computation of topological abstractions on scalar data, in shared memory parallel on single machines, and in distributed memory parallel on multiple machines. We highlight results for set‐based, graph‐based and complex‐based abstractions and organize the state of the art based on this taxonomy. The paper identifies parallelization and distribution techniques common in topological algorithms and highlights further areas of interest with underdeveloped efforts.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2013

Light refraction in sapphire plates with a variable angle of crystal optical axis to the surface

The modification of sapphire by inhomogeneous plastic deformation makes it possible to obtain plates with a variable angle of inclination of the crystal optical axis to the plate surface. The refraction of light in this plate at perpendicular and oblique incidence of a parallel beam of rays is considered. The algorithm of calculating the refractive index of extraordinary ray and the birefringence is proposed.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2022

PETSc TSAdjoint: A Discrete Adjoint ODE Solver for First-Order and Second-Order Sensitivity Analysis

Here, we present a new software system PETSc TSAdjoint for first-order and second order adjoint sensitivity analysis of time-dependent nonlinear differential equations. The derivative calculation in PETSc TSAdjoint is essentially a high-level algorithmic differentiation process. The adjoint models are derived by differentiating the timestepping algorithms and implementing them based on the parallel infrastructure in PETSc. Full differentiation of the library code, including MPI routines, is avoided, and users do not need to derive their own adjoint models for their specific applications. PETSc TSAdjoint can compute the first-order derivative, that is, the gradient of a scalar functional, and the Hessian-vector product, which carries second-order derivative information, while requiring minimal input (a few callbacks) from the users. The adjoint model employs optimal checkpointing schemes in a manner that is transparent to users. Finally, usability, efficiency, and scalability are demonstrated through examples from a variety of applications.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2022

kynema-fmb [SWR-23-07]

Kynema-FMB (FKA: Kynema) is an open-source performance portable flexible multibody (FMB) dynamics solver designed for time-domain simulations. While originally tailored for wind turbine structural dynamics, the formulation and implementation are those of a general flexible-multidbody dynamics solver that can readily be applied to a wide range of systems. Kynema was designed with a narrow focus, namely to provide a lightweight, fast, accurate FMD solver for coupling to computational-fluid-dynamics (CFD) codes, especially the CFD codes in the Kynema suite, for fluid-structure-interaction (FSI) simulations. Kynema-FMB is equipped to model systems that can be represented as a collection of beams and rigid bodies that are connected through constraints. Degrees of freedom are defined in the inertial/global frame of reference and include displacements and rotations (formally as rotation matrices, but stored as quaternions). The underlying formulation is built on a Lie-group time integrator designed for index-3 differential-algebraic equations, which is second-order accurate in time (Bruls et al., 2012). Beam models are based on geometrically exact beam theory and are discretized as high-order spectral finite elements similar to those in BeamDyn (Wang et al., 2017). The governing equations for a FMD system like a wind turbine constitute a highly nonlinear system of constrained partial-differential equations. Kynema-FMB uses analytical Jacobians in the nonlinear-system solves in each time step. Linear systems use sparse storage and several third-party sparse-linear-system solvers are enabled. Ill conditioning of linear systems is mitigated with preconditioning described in Bottasso et al, 2008. Kynema-FMB is integrated with a simple open-source controller (ROSCO). There is an application programming interface (API) for coupling to geometry-resolved CFD (like that in Sharma et al., 2023) and actuator-force CFD (like that in Kuhn et al., 2025). In the latter, for actuator-line models, Kynema-FMB includes an internal blade-element solver that depends on user-provided lookup tables for coefficients of lift and drag, i.e., aerodynamic polars. Kynema-FMB is written in C++ and leverages Kokkos and Kokkos-Kernels (KokkosEcosystem) as its performance portability layer enabling simulations on both CPU and GPU systems. The repository is equipped with extensive automated testing at the unit and regression/system levels. The following describes the high-level development objectives conceived for Kynema: *Kynema will follow modern software development best practices, including test-driven development (TDD), version control, hierarchical automated testing, and continuous integration (CI) for a robust development environment. *The core data structures are memory efficient and enable vectorization and parallelization at multiple levels. *Data structures are data-oriented to exploit methods for accelerated computing including high utilization of chip resources (e.g., single instruction multiple data (SIMD) instruction sets) and parallelization using GP-GPUs. *The computational algorithms incorporate robust open-source libraries for mathematical operations, resource allocation, and data management. *The API design considers multiple stakeholder needs and ensure integration with existing and future ecosystems for data science, machine learning, and AI. *Kynema-FMB is written in modern C++ and leverages Kokkos as its performance-portability library with inspiration from the kynema stack.

Sprague, MichaelA.↗

DOE OSTI2017

SCIDAC Center for simulation of wave particle interactions CompX participation

Harnessing the energy that is released in fusion reactions would provide a safe and abundant source of power to meet the growing energy needs of the world population. The next step toward the development of fusion as a practical energy source is the construction of ITER, a device capable of producing and controlling the high performance plasma required for self-sustaining fusion reactions, or “burning” plasma. The input power required to drive the ITER plasma into the burning regime will be supplied primarily with a combination of external power from radio frequency waves in the ion cyclotron range of frequencies and energetic ions from neutral beam injection sources, in addition to internally generated Ohmic heating from the induced plasma current that also serves to create the magnetic equilibrium for the discharge. The ITER project is a large multi-billion dollar international project in which the US participates. The success of the ITER project depends critically on the ability to create and maintain burning plasma conditions, it is absolutely necessary to have physics-based models that can accurately simulate the RF processes that affect the dynamical evolution of the ITER discharge. The Center for Simulation of WavePlasma Interactions (CSWPI), also known as RF-SciDAC, is a multi-institutional collaboration that has conducted ongoing research aimed at developing: (1) Coupled core-to-edge simulations that will lead to an increased understanding of parasitic losses of the applied RF power in the boundary plasma between the RF antenna and the core plasma; (2) Development of models for core interactions of RF waves with energetic electrons and ions (including fusion alpha particles and fast neutral beam ions) that include a more accurate representation of the particle dynamics in the combined equilibrium and wave fields; and (3) Development of improved algorithms that will take advantage of massively parallel computing platforms at the petascale level and beyond to achieve the needed physics, resolution, and/or statistics to address these issues. CompX provides computer codes and analysis for the calculation of the electron and ion distributions in velocity-space and plasma radius which are necessary for reliable calculations of power deposition and toroidal current drive due to combined radiofrequency and neutral beam at high injected powers. It has also contributed to ray tracing modeling of injected radiofrequency powers, and to coupling between full-wave radiofrequency wave models and the distribution function calculations. In the course of this research, the Fokker-Planck distribution function calculation was made substantially more realistic by inclusion of finite-width drift-orbit effects (FOW). FOW effects were also implemented in a calculation of the phase-space diffusion resulting from radiofrequency full-wave models. Average level of funding for CompX was approximately three man-months per year.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2018

Pararhombi: Parallel Implementation of 3-D Seismic Wave Propagation in Orthorhombic Media

Many earth materials and minerals are seismically anisotropic; however, due to the weakness of anisotropy and for simplicity, the earth is often approximated as an isotropic medium. Specific circumstances, such as in shales, tectonic fabrics, or oriented fractures, for example, require the use of anisotropic simulations in order to accurately model the earth. This report details the development of a new massively parallel 3-D full seismic waveform simulation algorithm within the principle coordinate system of an orthorhombic material, which is a specific form of anisotropy common in layered, fractured media. The theory and implementation of Pararhombi is described along with verification of the code against other solutions.

58 GEOSCIENCES↗

DOE OSTI2018

Final Technical Report for Grant No. DE-FC02-01ER54648, SciDAC Center for Simulation of Wave-Plasma Interactions

Harnessing the energy that is released in fusion reactions would provide a safe and abundant source of power to meet the growing energy needs of the world population. The next step toward the development of fusion as a practical energy source is the construction of ITER, a device capable of producing and controlling the high performance plasma required for self-sustaining fusion reactions, or “burning” plasma. The input power required to drive the ITER plasma into the burning regime will be supplied primarily with a combination of external power from radio frequency waves in the ion cyclotron range of frequencies and energetic ions from neutral beam injection sources, in addition to internally-generated Ohmic heating from the induced plasma current that also serves to create the magnetic equilibrium for the discharge. Since the success of the ITER project depends critically on the ability to create and maintain burning plasma conditions, it is absolutely necessary to have physics-based models that can accurately simulate the RF processes that affect the dynamical evolution of the ITER discharge. The Center for Simulation of Wave-Plasma Interactions (CSWPI) is a multi-institutional collaboration that has conducted ongoing research aimed at developing: (1) Coupled core-to-edge simulations that will lead to an increased understanding of parasitic losses of the applied RF power in the boundary plasma between the RF antenna and the core plasma; (2) Development of models for core interactions of RF waves with energetic electrons and ions (including fusion alpha particles and fast neutral beam ions) that include a more accurate representation of the particle dynamics in the combined equilibrium and wave fields; and (3) Development of improved algorithms that will take advantage of massively parallel computing platforms at the petascale level and beyond to achieve the needed physics, resolution, and/or statistics to address these issues. This research reported in this final technical report is focused on further verification and validation of simulation models for interactions of RF waves with energetic particles in the core with a long term goal of assessing their readiness for inclusion in integrated modeling frameworks and utilizing the world’s largest supercomputers to quantitatively predict the detailed electric field structure in the vicinity of the launching structures.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2018

Pararhombi: Parallel Implementation of 3-D Seismic Wave Propagation in Orthorhombic Media

58 GEOSCIENCES↗

DOE OSTI2019

An Adaptive Particle Tracking Algorithm for Lagrangian-Eulerian Simulations of Dispersed Multiphase Flows

The Lagrangian-Eulerian method is broadly used for simulations of dispersed multiphase flows by solving the continuous phase in the Eulerian framework while treating the dispersed phase as point particles in a Lagrangian framework. The accuracy of the Lagrangian-Eulerian method largely depends on the number of computational particles tracked for the Lagrangian phase. In this study, an adaptive Lagrangian particle tracking algorithm is proposed to balance the statistical error and the computational cost by dividing and merging the particles according to the local particle statistics in the computational domain. The computational particles are considered as weighted sampling points of the particle probability density functions (PDF) which represents a statistical equivalence of the Lagrangian phase. The algorithm is implemented as a part of a C++ library for Lagrangian particles, Grit, with performance portability to multi-core or many-core CPUs and Graphic Processing Unit (GPU) architectures. The accuracy of the proposed algorithm with two different schemes are studied through a test problem with analytical solutions for both the particle number density and momentum source term. The results are used to evaluate the proposed algorithm as well as to validate the parallel implementation.

Ge, Wenjun↗

DOE OSTI2014

ISMB Conference Funding to Support Attendance of Early Researchers and Students

ISMB Conference Funding for Students and Young Scientists Historical Description The Intelligent Systems for Molecular Biology (ISMB) conference has provided a general forum for disseminating the latest developments in bioinformatics on an annual basis for the past 22 years. ISMB is a multidisciplinary conference that brings together scientists from computer science, molecular biology, mathematics and statistics. The goal of the ISMB meeting is to bring together biologists and computational scientists in a focus on actual biological problems, i.e., not simply theoretical calculations. The combined focus on “intelligent systems” and actual biological data makes ISMB a unique and highly important meeting. 21 years of experience in holding the conference has resulted in a consistently well-organized, well attended, and highly respected annual conference. "Intelligent systems" include any software which goes beyond straightforward, closed-form algorithms or standard database technologies, and encompasses those that view data in a symbolic fashion, learn from examples, consolidate multiple levels of abstraction, or synthesize results to be cognitively tractable to a human, including the development and application of advanced computational methods for biological problems. Relevant computational techniques include, but are not limited to: machine learning, pattern recognition, knowledge representation, databases, combinatorics, stochastic modeling, string and graph algorithms, linguistic methods, robotics, constraint satisfaction, and parallel computation. Biological areas of interest include molecular structure, genomics, molecular sequence analysis, evolution and phylogenetics, molecular interactions, metabolic pathways, regulatory networks, developmental control, and molecular biology generally. Emphasis is placed on the validation of methods using real data sets, on practical applications in the biological sciences, and on development of novel computational techniques. The ISMB conferences are distinguished from many other conferences in computational biology or artificial intelligence by an insistence that the researchers work with real molecular biology data, not theoretical or toy examples; and from many other biological conferences by providing a forum for technical advances as they occur, which otherwise may be shunned until a firm experimental result is published. The resulting intellectual richness and cross-disciplinary diversity provides an important opportunity for both students and senior researchers. ISMB has become the premier conference series in this field with refereed, published proceedings, establishing an infrastructure to promote the growing body of research.

60 APPLIED LIFE SCIENCES↗

DOE OSTI2016

Matrix Factorizations at Scale: a Comparison of Scientific Data Analytics in Spark and C+MPI Using Three Case Studies

We explore the trade-offs of performing linear algebra using Apache Spark, compared to traditional C and MPI implementations on HPC platforms. Spark is designed for data analytics on cluster computing platforms with access to local disks and is optimized for data-parallel tasks. We examine three widely-used and important matrix factorizations: NMF (for physical plausibility), PCA (for its ubiquity) and CX (for data interpretability). We apply these methods to 1.6TB particle physics, 2.2TB and 16TB climate modeling and 1.1TB bioimaging data. The data matrices are tall-and-skinny which enable the algorithms to map conveniently into Spark’s data parallel model. We perform scaling experiments on up to 1600 Cray XC40 nodes, describe the sources of slowdowns, and provide tuning guidance to obtain high performance.

Gittens, Alex↗

DOE OSTI2021

Progress on Optimizing Wind Farms and Rotor Designs Using Adjoints

Modern wind plants are increasingly tasked with multiple performance objectives. In addition to designing plants that maximize power output and minimize the levelized cost of energy (LCOE), the design and operation of wind plants is increasingly influenced by challenges regarding grid integration of variable generation renewables. This places a growing emphasis on making wind plants more controllable and predictable. WindSE is a Reynolds-averaged Navier-Stokes (RANS) model designed around analytical gradient and adjoint methods, with the ability to capture terrain-induced effects, as shown in Figure 1. The recent addition of an unsteady solver with an actuator line method (ALM) and ongoing work to enable massively parallel optimizations gives it a unique niche to explore coupled plant-level controls and design problems. This code is an open source python package built on the FEniCS framework that utilizes fast, parallel PETSc solvers to model fluid flow throughout wind-farm scale domains. Two recent studies performed using WindSE demonstrate the capability to optimize under a wide variety of flow conditions and objective functions. In the first, we present an optimization focused on modifying the layout of a wind farm with a fixed number of turbines for maximum total power output [1]. This study highlights the ability to quickly perform simulations using the steady Navier-Stokes solver combined with rotors represented as actuator disks while also stressing the importance of capturing terrain-induced effects. Gradient-based optimization using the RANS equations is viable due to the inclusion of efficiently computed adjoint derivatives. We interpret the physical results of the optimal layout and also discuss the computational cost of scaling to larger problems. In the second study, we present the capabilities of the unsteady Navier-Stokes solver, where rotor-blade profiles represented by actuator lines are optimized to enhance wake steering effects and overall power production [2]. We quantify the wind plant performance gains obtained from this type of simultaneous control co-design optimization as compared to optimizing the blade design and yaw independently. Figure 2 shows the differences between a baseline two-turbine system and an optimized system where we fine-tune the blade chord profile. Results and challenges from each study are quickly summarized and used to motivate the current development efforts within WindSE. Current and future work is focused on enabling higher-resolution studies with more degrees of freedom through parallelization of both the simulation and optimization algorithms. We present benchmarking results to show that WindSE performs well in both weak- and strong-scaling tests and further demonstrate that the optimizer obtains the same convergence rates in both shared- and distributed-memory environments. Using larger wind farms, we can study deep-array effects within an optimization context, allowing the use of objective functions that have been previously unstudied. As an example, we present ongoing work on a blockage metric which characterizes the loss of available kinetic energy due to wake effects from multiple upstream turbines.

adjoint optimization↗

DOE OSTI2018

Communication-avoiding optimization methods for distributed massive-scale sparse inverse covariance estimation

Across a variety of scientific disciplines, sparse inverse covariance estimation is a popular tool for capturing the underlying dependency relationships in multivariate data. Unfortunately, most estimators are not scalable enough to handle the sizes of modern high-dimensional data sets (often on the order of terabytes), and assume Gaussian samples. To address these deficiencies, we introduce HP-CONCORD, a highly scalable optimization method for estimating a sparse inverse covariance matrix based on a regularized pseudolikelihood framework, without assuming Gaussianity. Our parallel proximal gradient method uses a novel communication-avoiding linear algebra algorithm and runs across a multi-node cluster with up to 1k nodes (24k cores), achieving parallel scalability on problems with up to ≈819 billion parameters (1.28 million dimensions); even on a single node, HP-CONCORD demonstrates scalability, outperforming a state-of-the-art method. We also use HP-CONCORD to estimate the underlying dependency structure of the brain from fMRI data, and use the result to identify functional regions automatically. The results show good agreement with a clustering from the neuroscience literature.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2018

Communication-Avoiding Optimization Methods for Distributed Massive-Scale Sparse Inverse Covariance Estimation.

Across a variety of scientific disciplines, sparse inverse covariance estimation is a popular tool for capturing the underlying dependency relationships in multivariate data. Unfortunately, most estimators are not scalable enough to handle the sizes of modern high-dimensional data sets (often on the order of terabytes), and assume Gaussian samples. To address these deficiencies, we introduce HP-CONCORD, a highly scalable optimization method for estimating a sparse inverse covariance matrix based on a regularized pseudolikelihood framework, without assuming Gaussianity. Our parallel proximal gradient method uses a novel communication-avoiding linear algebra algorithm and runs across a multi-node cluster with up to 1k nodes (24k cores), achieving parallel scalability on problems with up to ~819 billion parameters (1.28 million dimensions); even on a single node, HP-CONCORD demonstrates scalability, outperforming a state-of-the-art method. We also use HP-CONCORD to estimate the underlying dependency structure of the brain from fMRI data, and use the result to identify functional regions automatically. The results show good agreement with a clustering from the neuroscience literature.

Koanantakool, Penporn↗

DOE OSTI2025

Track reconstruction as a service for collider physics

Optimizing charged-particle track reconstruction algorithms is crucial for efficient event reconstruction in Large Hadron Collider (LHC) experiments due to their significant computational demands. Existing track reconstruction algorithms have been adapted to run on massively parallel coprocessors, such as graphics processing units (GPUs), to reduce processing time. Nevertheless, challenges remain in fully harnessing the computational capacity of coprocessors in a scalable and non-disruptive manner. This paper proposes an inference-as-a-service approach for particle tracking in high energy physics experiments. To evaluate the efficacy of this approach, two distinct tracking algorithms are tested: Patatrack, a rule-based algorithm, and Exa.TrkX, a machine learning-based algorithm. The as-a-service implementations show enhanced GPU utilization and can process requests from multiple CPU cores concurrently without increasing per-request latency. The impact of data transfer is minimal and insignificant compared to running on local coprocessors. This approach greatly improves the computational efficiency of charged particle tracking, providing a solution to the computing challenges anticipated in the High-Luminosity LHC era.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

DOE OSTI2015

SU-D-207-04: GPU-Based 4D Cone-Beam CT Reconstruction Using Adaptive Meshing Method

Purpose: Due to the limited number of projections at each phase, the image quality of a four-dimensional cone-beam CT (4D-CBCT) is often degraded, which decreases the accuracy of subsequent motion modeling. One of the promising methods is the simultaneous motion estimation and image reconstruction (SMEIR) approach. The objective of this work is to enhance the computational speed of the SMEIR algorithm using adaptive feature-based tetrahedral meshing and GPU-based parallelization. Methods: The first step is to generate the tetrahedral mesh based on the features of a reference phase 4D-CBCT, so that the deformation can be well captured and accurately diffused from the mesh vertices to voxels of the image volume. After the mesh generation, the updated motion model and other phases of 4D-CBCT can be obtained by matching the 4D-CBCT projection images at each phase with the corresponding forward projections of the deformed reference phase of 4D-CBCT. The entire process of this 4D-CBCT reconstruction method is implemented on GPU, resulting in significantly increasing the computational efficiency due to its tremendous parallel computing ability. Results: A 4D XCAT digital phantom was used to test the proposed mesh-based image reconstruction algorithm. The image Result shows both bone structures and inside of the lung are well-preserved and the tumor position can be well captured. Compared to the previous voxel-based CPU implementation of SMEIR, the proposed method is about 157 times faster for reconstructing a 10 -phase 4D-CBCT with dimension 256×256×150. Conclusion: The GPU-based parallel 4D CBCT reconstruction method uses the feature-based mesh for estimating motion model and demonstrates equivalent image Result with previous voxel-based SMEIR approach, with significantly improved computational speed.

60 APPLIED LIFE SCIENCES↗

DOE OSTI2021

Resiliency in numerical algorithm design for extreme scale simulations

Here this work is based on the seminar titled ‘Resiliency in Numerical Algorithm Design for Extreme Scale Simulations’ held March 1–6, 2020, at Schloss Dagstuhl, that was attended by all the authors. Advanced supercomputing is characterized by very high computation speeds at the cost of involving an enormous amount of resources and costs. A typical large-scale computation running for 48 h on a system consuming 20 MW, as predicted for exascale systems, would consume a million kWh, corresponding to about 100k Euro in energy cost for executing 10 23 floating-point operations. It is clearly unacceptable to lose the whole computation if any of the several million parallel processes fails during the execution. Moreover, if a single operation suffers from a bit-flip error, should the whole computation be declared invalid? What about the notion of reproducibility itself: should this core paradigm of science be revised and refined for results that are obtained by large-scale simulation? Naive versions of conventional resilience techniques will not scale to the exascale regime: with a main memory footprint of tens of Petabytes, synchronously writing checkpoint data all the way to background storage at frequent intervals will create intolerable overheads in runtime and energy consumption. Forecasts show that the mean time between failures could be lower than the time to recover from such a checkpoint, so that large calculations at scale might not make any progress if robust alternatives are not investigated. More advanced resilience techniques must be devised. The key may lie in exploiting both advanced system features as well as specific application knowledge. Research will face two essential questions: (1) what are the reliability requirements for a particular computation and (2) how do we best design the algorithms and software to meet these requirements? While the analysis of use cases can help understand the particular reliability requirements, the construction of remedies is currently wide open. One avenue would be to refine and improve on system- or application-level checkpointing and rollback strategies in the case an error is detected. Developers might use fault notification interfaces and flexible runtime systems to respond to node failures in an application-dependent fashion. Novel numerical algorithms or more stochastic computational approaches may be required to meet accuracy requirements in the face of undetectable soft errors. These ideas constituted an essential topic of the seminar. The goal of this Dagstuhl Seminar was to bring together a diverse group of scientists with expertise in exascale computing to discuss novel ways to make applications resilient against detected and undetected faults. In particular, participants explored the role that algorithms and applications play in the holistic approach needed to tackle this challenge. This article gathers a broad range of perspectives on the role of algorithms, applications and systems in achieving resilience for extreme scale simulations. The ultimate goal is to spark novel ideas and encourage the development of concrete solutions for achieving such resilience holistically.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2015

A parallel domain decomposition-based implicit method for the Cahn–Hilliard–Cook phase-field equation in 3D

We present a numerical algorithm for simulating the spinodal decomposition described by the three dimensional Cahn–Hilliard–Cook (CHC) equation, which is a fourth-order stochastic partial differential equation with a noise term. The equation is discretized in space and time based on a fully implicit, cell-centered finite difference scheme, with an adaptive time-stepping strategy designed to accelerate the progress to equilibrium. At each time step, a parallel Newton–Krylov–Schwarz algorithm is used to solve the nonlinear system. We discuss various numerical and computational challenges associated with the method. The numerical scheme is validated by a comparison with an explicit scheme of high accuracy (and unreasonably high cost). We present steady state solutions of the CHC equation in two and three dimensions. The effect of the thermal fluctuation on the spinodal decomposition process is studied. We show that the existence of the thermal fluctuation accelerates the spinodal decomposition process and that the final steady morphology is sensitive to the stochastic noise. We also show the evolution of the energies and statistical moments. In terms of the parallel performance, it is found that the implicit domain decomposition approach scales well on supercomputers with a large number of processors.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗