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DOE OSTI2026

Integral Kernel Methods for Nonlinear Parabolic-Elliptic Systems

Nonlinear parabolic-elliptic systems arise in many physical, biological, and chemical phenomena such as chemotaxis, ion transport, self-gravitating particles, and Brownian vortices. Existing methods struggle with the strong coupling and high nonlinearity and nonlocality of some of these systems, especially the ill-conditioned, convection-dominated problems. To overcome numerical difficulties, current approaches rely on initial guesses, preconditioning, or iterative techniques with no convergence guarantees. They might suffer from poor scalability, large memory usage, and difficulty to parallelize. Inspired by the connection of parabolic-elliptic systems to stochastic processes, we introduce a novel meshless, monolithic, and fully explicit method that naturally encapsulates the elliptic and parabolic operators into a single step which updates each node deterministically with global information. By being fully quadrature-based, it avoids solving systems of discretized equations and does not utilize initial guesses or preconditioning, while requiring little memory and being easy to parallelize. We first derive the method in an integral kernel formulation with quadratic complexity in the number of integration nodes and then leverage kernel-independent fast multipole methods (FMM) to present a scalable algorithm with linear complexity. We provide numerical examples for the Poisson-Nernst-Planck equations in one, two, and three dimensions, together with the derivation of the integral kernel for each case. Furthermore, the examples demonstrate the fast convergence and scalability of the FMM-accelerated algorithm, as well as its suitability for convection-dominated problems, making it competitive against traditional PDE solvers.

PDE systems↗

DOE OSTI2024

A Contextually-Aware Sensitivity Analysis to Guide the Design of Randomized Least Squares Solvers in Applications

Our work on the DOE-sponsored project “A Contextually-Aware Sensitivity Analysis to Guide the Design of Randomized Least Squares Solvers in Applications,” was an effort to address critical challenges in nu merical computing and its applications to optimization. The increasing demand for robust and scalable solutions to large-scale linear algebra problems has highlighted the limitations of traditional approaches, particularly in heterogeneous and extreme-scale computing environments. Randomized Numerical Linear Algebra (RandNLA) offers a promising framework to address these challenges, and this proposal builds on this foundation by introducing innovations in sensitivity analysis and computational adaptability.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2026

A Co-Simulation Framework for Steady-State Analyses of Multiple Droop-based MTdc Grids in Continental-Scale Systems

The growing scale and complexity of planning continental hybrid ac and multi-terminal dc (MTdc) systems require scalable steady-state modeling and analysis approaches not currently available in commercial tools. This paper presents a comprehensive multi-fidelity model-conversion framework that enables the efficient transition of MTdc grid models from production cost modeling (PCM) and approximated ac power flow to detailed ac–MTdc power flow for large-scale planning studies. The core of this framework is a scalable co-simulation approach that, for the first time, enables power flow analysis in continental-scale ac–MTdc systems. It seamlessly couples commercial ac solvers with a detailed MTdc grid model that incorporates droop-based control and current-limiting strategies of multiple meshed MTdc grids. Leveraging this capability, an evaluation framework to systematically assess and compare different MTdc power redispatch strategies under ac and dc contingencies is introduced. The proposed framework and algorithm are evaluated using a combined Western and Eastern Interconnection system with 11 MTdc grids of various sizes, showing a coherent transition from PCM to detailed ac–MTdc power flow and improved system performance in voltage regulation and line overload mitigation following typical contingencies.

Nguyen, Quan H.↗

DOE OSTI2025

Scalable freeform optimization of wide-aperture 3D metalenses by zoned discrete axisymmetry

We introduce a novel framework for design and optimization of 3D freeform metalenses that attains nearly linear scaling of computational cost with diameter, by breaking the lens into a sequence of radial “zones” with 𝑛-fold discrete axisymmetry, where 𝑛 increases with radius. This allows vastly more design freedom than imposing continuous axisymmetry, while avoiding the compromises of the locally periodic approximation (LPA) or scalar diffraction theory. Using a GPU-accelerated finite-difference time-domain (FDTD) solver in cylindrical coordinates, we perform full-wave simulation and topology optimization within each supra-wavelength zone. We validate our approach by designing millimeter and centimeter-scale, poly-achromatic, 3D freeform metalenses which outperform the state of the art. By demonstrating the scalability and resulting optical performance enabled by our “zoned discrete axisymmetry” (ZDA) and supra-wavelength domain decomposition, we highlight the potential of our framework to advance large-scale meta-optics and next-generation photonic technologies.

Sun, Mengdi [Wesleyan University]↗

NASA NTRSDate not supplied

A Study of Parallel Scalability and Dynamic Workload Balancing in GlennICE

The Glenn Icing Computational Environment (GlennICE) is a computational tool designed to calculate ice growth on complex three-dimensional geometries. It utilizes user-supplied computational fluid dynamics solutions for the geometry of interest. Key developments include advancements in convergence of collection efficiency, trajectory optimization, and refinement methodology. These improvements have significantly enhanced GlennICE’s efficiency for practical engineering applications. A recent study focused on benchmarking GlennICE’s scalability in a parallel environment using static scheduling. Findings indicated a potential twofold increase in efficiency through workload balance enhancements. This paper presents an analysis of the solver’s new workload balancing improvements, incorporating shared memory and dynamic scheduling routines. Results demonstrate a highly efficient and consistent algorithm across high-performance computing clusters.

Computational Icing↗

NASA NTRSDate not supplied

A Study of Parallel Scalability and Dynamic Workload Balancing in GlennICE

Computational Icing↗

DOE OSTI2024

Sparse Cholesky factorization for solving nonlinear PDEs via Gaussian processes

In recent years, there has been widespread adoption of machine learning-based approaches to automate the solving of partial differential equations (PDEs). Among these approaches, Gaussian processes (GPs) and kernel methods have garnered considerable interest due to their flexibility, robust theoretical guarantees, and close ties to traditional methods. They can transform the solving of general nonlinear PDEs into solving quadratic optimization problems with nonlinear, PDE-induced constraints. However, the complexity bottleneck lies in computing with dense kernel matrices obtained from pointwise evaluations of the covariance kernel, and its partial derivatives, a result of the PDE constraint and for which fast algorithms are scarce. The primary goal of this paper is to provide a near-linear complexity algorithm for working with such kernel matrices. We present a sparse Cholesky factorization algorithm for these matrices based on the near-sparsity of the Cholesky factor under a novel ordering of pointwise and derivative measurements. The near-sparsity is rigorously justified by directly connecting the factor to GP regression and exponential decay of basis functions in numerical homogenization. We then employ the Vecchia approximation of GPs, which is optimal in the Kullback-Leibler divergence, to compute the approximate factor. This enables us to compute ϵ-approximate inverse Cholesky factors of the kernel matrices with complexity O(N log d (N/ϵ)) in space and O(N log 2d (N/ϵ)) in time. We integrate sparse Cholesky factorizations into optimization algorithms to obtain fast solvers of the nonlinear PDE. We numerically illustrate our algorithm’s near-linear space/time complexity for a broad class of nonlinear PDEs such as the nonlinear elliptic, Burgers, and Monge-Ampère equations. In summary, we provide a fast, scalable, and accurate method for solving general PDEs with GPs and kernel methods.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1996

Unstructured Adaptive Grid Computations on an Array of SMPs

Dynamic load balancing is necessary for parallel adaptive methods to solve unsteady CFD problems on unstructured grids. We have presented such a dynamic load balancing framework called JOVE, in this paper. Results on a four-POWERnode POWER CHALLENGEarray demonstrated that load balancing gives significant performance improvements over no load balancing for such adaptive computations. The parallel speedup of JOVE, implemented using MPI on the POWER CHALLENCEarray, was significant, being as high as 31 for 32 processors. An implementation of JOVE that exploits 'an array of SMPS' architecture was also studied; this hybrid JOVE outperformed flat JOVE by up to 28% on the meshes and adaption models tested. With large, realistic meshes and actual flow-solver and adaption phases incorporated into JOVE, hybrid JOVE can be expected to yield significant advantage over flat JOVE, especially as the number of processors is increased, thus demonstrating the scalability of an array of SMPs architecture.

Biswas, Rupak↗

NASA NTRS1995

Runtime volume visualization for parallel CFD

This paper discusses some aspects of design of a data distributed, massively parallel volume rendering library for runtime visualization of parallel computational fluid dynamics simulations in a message-passing environment. Unlike the traditional scheme in which visualization is a postprocessing step, the rendering is done in place on each node processor. Computational scientists who run large-scale simulations on a massively parallel computer can thus perform interactive monitoring of their simulations. The current library provides an interface to handle volume data on rectilinear grids. The same design principles can be generalized to handle other types of grids. For demonstration, we run a parallel Navier-Stokes solver making use of this rendering library on the Intel Paragon XP/S. The interactive visual response achieved is found to be very useful. Performance studies show that the parallel rendering process is scalable with the size of the simulation as well as with the parallel computer.

Ma, Kwan-Liu↗

NASA NTRS1996

Climate Data Assimilation on a Massively Parallel Supercomputer

We have designed and implemented a set of highly efficient and highly scalable algorithms for an unstructured computational package, the PSAS data assimilation package, as demonstrated by detailed performance analysis of systematic runs on up to 512-nodes of an Intel Paragon. The preconditioned Conjugate Gradient solver achieves a sustained 18 Gflops performance. Consequently, we achieve an unprecedented 100-fold reduction in time to solution on the Intel Paragon over a single head of a Cray C90. This not only exceeds the daily performance requirement of the Data Assimilation Office at NASA's Goddard Space Flight Center, but also makes it possible to explore much larger and challenging data assimilation problems which are unthinkable on a traditional computer platform such as the Cray C90.

supercomputer↗

NASA NTRSDate not supplied

Validation of Gas-Liquid Sharp Interface Model in Loci-Stream for Propellant Tank Self-pressurization under Normal Gravity

Deep space missions require advances in cryogenic fluid management (CFM) for long term storage of propellants. One of the important phenomena is self-pressurization due to heat leakages into the tank. Managing self-pressurization is one of the key technologies for deep space exploration and long-term space missions. The complex interactions involving natural convection, thermal gradients, turbulence, and phase change near the gas-liquid interface cannot be modeled using reduced order or nodal analysis models, and 3-D CFD analyses are necessary to fully characterize the dynamics. CFD analyses pose their own difficulties. The requisite CFD tool to tackle this problem need to be modular with the ability to incorporate various physics models, efficient, and computationally scalable for simulating flight size tanks. In this paper, we discuss modeling approach for self-pressurization in situations where the liquid interface is static and validate NASA MSFC’s Loci-Stream CFD solver for predicting self-pressurization in a test tank in normal gravity. This tank was designed to capture quantifiable and accurate data for understanding various two-phase fluid phenomena and to validate modeling tools. Specifically, we demonstrate the capability of the Loci-Stream solver with a two-phase sharp-interface treatment to predict self-pressurization of a tank with an unperturbed gas-liquid interface. This validation highlights the reliability of our modeling approach and our solver to serve as a design and analysis tool for NASA’s CFM application needs.

cryogenic fluid management↗

NASA NTRSDate not supplied

Validation of Gas-Liquid Sharp Interface Model in Loci-Stream for Propellant Tank Self-pressurization under Normal Gravity

Deep space missions require advances in cryogenic fluid management (CFM) for long term storage of propellants. One of the important phenomena is self-pressurization due to heat leakages into the tank. Managing self-pressurization is one of the key technologies for deep space exploration and long-term space missions. The complex interactions involving natural convection, thermal gradients, turbulence, and phase change near the gas-liquid interface cannot be modeled using reduced order or nodal analysis models, and 3-D CFD analyses are necessary to fully characterize the dynamics. CFD analyses pose their own difficulties. The requisite CFD tool to tackle this problem need to be modular with the ability to incorporate various physics models, efficient, and computationally scalable for simulating flight size tanks. In this paper, we discuss modeling approach for self-pressurization in situations where the liquid interface is static and validate NASA MSFC's Loci-Stream CFD solver for predicting self-pressurization in a test tank in normal gravity. This tank was designed to capture quantifiable and accurate data for understanding various two-phase fluid phenomena and to validate modeling tools. Specifically, we demonstrate the capability of the Loci-Stream solver with a two-phase sharp-interface treatment to predict self-pressurization of a tank with an unperturbed gas-liquid interface. This validation highlights the reliability of our modeling approach and our solver to serve as a design and analysis tool for NASA's CFM application needs.

cryogenic fluid management↗

NASA NTRS2000

Parallel performance investigations of an unstructured mesh Navier-Stokes solver

A Reynolds-averaged Navier-Stokes solver based on unstructured mesh techniques for analysis of high-lift configurations is described. The method makes use of an agglomeration multigrid solver for convergence acceleration. Implicit line-smoothing is employed to relieve the stiffness associated with highly stretched meshes. A GMRES technique is also implemented to speed convergence at the expense of additional memory usage. The solver is cache efficient and fully vectorizable, and is parallelized using a two-level hybrid MPI-OpenMP implementation suitable for shared and/or distributed memory architectures, as well as clusters of shared memory machines. Convergence and scalability results are illustrated for various high-lift cases.

Mavriplis, Dimitri J.↗

NASA NTRS2013

ISSM: Ice Sheet System Model

In order to have the capability to use satellite data from its own missions to inform future sea-level rise projections, JPL needed a full-fledged ice-sheet/iceshelf flow model, capable of modeling the mass balance of Antarctica and Greenland into the near future. ISSM was developed with such a goal in mind, as a massively parallelized, multi-purpose finite-element framework dedicated to ice-sheet modeling. ISSM features unstructured meshes (Tria in 2D, and Penta in 3D) along with corresponding finite elements for both types of meshes. Each finite element can carry out diagnostic, prognostic, transient, thermal 3D, surface, and bed slope simulations. Anisotropic meshing enables adaptation of meshes to a certain metric, and the 2D Shelfy-Stream, 3D Blatter/Pattyn, and 3D Full-Stokes formulations capture the bulk of the ice-flow physics. These elements can be coupled together, based on the Arlequin method, so that on a large scale model such as Antarctica, each type of finite element is used in the most efficient manner. For each finite element referenced above, ISSM implements an adjoint. This adjoint can be used to carry out model inversions of unknown model parameters, typically ice rheology and basal drag at the ice/bedrock interface, using a metric such as the observed InSAR surface velocity. This data assimilation capability is crucial to allow spinning up of ice flow models using available satellite data. ISSM relies on the PETSc library for its vectors, matrices, and solvers. This allows ISSM to run efficiently on any parallel platform, whether shared or distrib- ISSM: Ice Sheet System Model NASA's Jet Propulsion Laboratory, Pasadena, California uted. It can run on the largest clusters, and is fully scalable. This allows ISSM to tackle models the size of continents. ISSM is embedded into MATLAB and Python, both open scientific platforms. This improves its outreach within the science community. It is entirely written in C/C++, which gives it flexibility in its design, and the power/speed that C/C++ allows. ISSM is svn (subversion) hosted, on a JPL repository, to facilitate its development and maintenance. ISSM can also model propagation of rifts using contact mechanics and mesh splitting, and can interface to the Dakota software. To carry out sensitivity analysis, mesh partitioning algorithms are available, based on the Scotch, Chaco, and Metis partitioners that ensure equal area mesh partitions can be done, which are then usable for sampling and local reliability methods.

Larour, Eric↗

DOE OSTI2025

CUDO: closed-form universal dwell-time optimization for computer-controlled optical surfacing

Precision optical figuring demands fast and accurate dwell time optimization to reach nanometer- and sub-nanometer-level accuracy in next-generation optical systems. We introduce CUDO (closed-form universal dwell-time optimization), the first, to the best of our knowledge, unified closed-form analytical framework that supports both function-form and matrix-form dwell time models in computer-controlled optical surfacing (CCOS). In contrast to traditional methods, which rely on iterative optimization and hyperparameter tuning, our framework derives direct analytical solutions with no adjustable parameters. This approach unifies the solution principles of existing methods within a single mathematical model, delivering three key advantages: (1) accuracy on par with, or superior to, iterative solvers, (2) substantial reduction in computation time, and (3) numerical robustness. Comparative studies with prior art confirm that closed-form solutions achieve equivalent residual error while removing runtime bottlenecks. By simplifying the implementation and enabling real-time, scalable deployment, CUDO establishes a practical foundation for future deterministic fabrication of large-aperture and high-performance optics.

36 MATERIALS SCIENCE↗

NASA NTRS2018

PURE: Scalable Phase Unwrapping with Spatial Redundant Arcs

Phase unwrapping is a key problem in many coherent imaging systems, such as syntheticapertureradar(SAR)interferometry. Ageneralformulationforredundant integration of ﬁnite differences for phase unwrapping (Costantini et al., 2010) was shown to produce a more reliable solution by exploiting redundant differential estimates. However, this technique requires a commercial linear programming solver for large-scale problems. For a linear cost function, we propose a method based on Dual Decomposition that breaks the given problem deﬁned over a nonplanar graph into tractable sub-problems over planar subgraphs. We also propose a decomposition technique that exploits the underlying graph structure for solving thesub-problemsefﬁcientlyandguaranteesasymptoticconvergencetotheglobally optimal solution. The experimental results demonstrate that the proposed approach is comparable to the existing state-of-the-art methods in terms of the estimate with a better runtime and memory footprint.

Lanka, Ravi↗

NASA NTRS1997

A Multi-Level Parallelization Concept for High-Fidelity Multi-Block Solvers

The integration of high-fidelity Computational Fluid Dynamics (CFD) analysis tools with the industrial design process benefits greatly from the robust implementations that are transportable across a wide range of computer architectures. In the present work, a hybrid domain-decomposition and parallelization concept was developed and implemented into the widely-used NASA multi-block Computational Fluid Dynamics (CFD) packages implemented in ENSAERO and OVERFLOW. The new parallel solver concept, PENS (Parallel Euler Navier-Stokes Solver), employs both fine and coarse granularity in data partitioning as well as data coalescing to obtain the desired load-balance characteristics on the available computer platforms. This multi-level parallelism implementation itself introduces no changes to the numerical results, hence the original fidelity of the packages are identically preserved. The present implementation uses the Message Passing Interface (MPI) library for interprocessor message passing and memory accessing. By choosing an appropriate combination of the available partitioning and coalescing capabilities only during the execution stage, the PENS solver becomes adaptable to different computer architectures from shared-memory to distributed-memory platforms with varying degrees of parallelism. The PENS implementation on the IBM SP2 distributed memory environment at the NASA Ames Research Center obtains 85 percent scalable parallel performance using fine-grain partitioning of single-block CFD domains using up to 128 wide computational nodes. Multi-block CFD simulations of complete aircraft simulations achieve 75 percent perfect load-balanced executions using data coalescing and the two levels of parallelism. SGI PowerChallenge, SGI Origin 2000, and a cluster of workstations are the other platforms where the robustness of the implementation is tested. The performance behavior on the other computer platforms with a variety of realistic problems will be included as this on-going study progresses.

Hatay, Ferhat F.↗

NASA NTRS2014

Using SpF to Achieve Petascale for Legacy Pseudospectral Applications

Pseudospectral (PS) methods possess a number of characteristics (e.g., efficiency, accuracy, natural boundary conditions) that are extremely desirable for dynamo models. Unfortunately, dynamo models based upon PS methods face a number of daunting challenges, which include exposing additional parallelism, leveraging hardware accelerators, exploiting hybrid parallelism, and improving the scalability of global memory transposes. Although these issues are a concern for most models, solutions for PS methods tend to require far more pervasive changes to underlying data and control structures. Further, improvements in performance in one model are difficult to transfer to other models, resulting in significant duplication of effort across the research community. We have developed an extensible software framework for pseudospectral methods called SpF that is intended to enable extreme scalability and optimal performance. Highlevel abstractions provided by SpF unburden applications of the responsibility of managing domain decomposition and load balance while reducing the changes in code required to adapt to new computing architectures. The key design concept in SpF is that each phase of the numerical calculation is partitioned into disjoint numerical kernels that can be performed entirely inprocessor. The granularity of domain decomposition provided by SpF is only constrained by the datalocality requirements of these kernels. SpF builds on top of optimized vendor libraries for common numerical operations such as transforms, matrix solvers, etc., but can also be configured to use open source alternatives for portability. SpF includes several alternative schemes for global data redistribution and is expected to serve as an ideal testbed for further research into optimal approaches for different network architectures. In this presentation, we will describe our experience in porting legacy pseudospectral models, MoSST and DYNAMO, to use SpF as well as present preliminary performance results provided by the improved scalability.

DYNAMO↗