Search NASA⌕ Search

SEARCH · Search NASA

Results for “Problem Solving”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 343 records · Page 19

Engineered Microorganisms for Enhanced Rare Earth Element Bio-mining and Separations (Final Technical Report)

Rare earth elements (REE) are critical ingredients of sustainable energy technologies, but their extraction from ore and separation from one another pose formidable challenges. To solve the challenge of REE supply, we used advanced genomics, high-throughput screening with synthetic REE minerals, and synthetic biology to engineer two sets of exotic microbes to (1) extract REE from ores, spent cracking catalysts, coal ash and electronic waste with high efficiency and selectivity, and (2) to purify REE into single element batches, all under benign conditions without the need of harsh solvents and high temperatures. This work integrated our expertise in systems and synthetic biology (Buz Barstow); rare-earth geochemistry (Esteban Gazel) and mineral synthesis (Megan Holycross); and microsystems engineering (Mingming Wu) by first elucidating the set of rules that predict an organism’s phenotype and then applying them to solve this critical problem in sustainable energy. These new technologies could help to revitalize the US rare earth industry and provide a new source of these critical elements for future energy technologies. We have already had some big success in tech transfer. Two of our team members (postdoctoral fellow Alexa Schmitz and graduate student Sean Medin) were able to study the supply chain for REE in the United States, and identify an opportunity to commercialize our REE mineral-dissolution technology. Alexa and Sean recently founded REEgen, Inc., an REE biomining company. Dr. Schmitz was recently awarded a fellowship from the Activate Foundation to support the first two years of REEgen. Cornell showed its support for this technology and company, and Dr. Schmitz was awarded the Rising Women Innovator’s award. These two awards unlocked support from Cornell’s Praxis Incubator.

58 GEOSCIENCES↗

NEML2: A High Performance Library for Constitutive Modeling

NEML2, the New Engineering Material model Library, version 2, is an offshoot of NEML, an earlier material modeling code developed at Argonne National Laboratory. NEML2 extends the key philosophy of its predecessor, i.e., material models are flexible, modular, and can be built from smaller blocks. It also provides modern features that do not exist in the framework of its predecessor such as material model vectorization, automatic differentiation, device-portable just-in-time compilation, operator fusion, lazy tensor evaluation, etc. Moreover, NEML2 can seamlessly integrate with the popular machine learning package PyTorch to take advantage of modern and fast-growing machine learning techniques. In this fiscal year, the development of core library features and capabilities are complete. The purpose of this report is not to serve as a verbatim copy of the software API reference (which is available online at https://reverendbedford.github.io/neml2/). Instead, this report documents the motivation, implementation, design choices, and usage of each core capability as well as their applications in solving practical engineering problems. This report is compiled based on the NEML2 major release 2.0.0.

36 MATERIALS SCIENCE↗

Quantum Computing Technology Roadmaps and Capability Assessment for Scientific Computing - An analysis of use cases from the NERSC workload

The National Energy Research Scientific Computing Center (NERSC), as the high-performance computing (HPC) facility for the Department of Energy’s Office of Science, recognizes the essential role of quantum computing in its future mission. In this report, we analyze the NERSC workload and identify materials science, quantum chemistry, and high-energy physics as the science domains and application areas that stand to benefit most from quantum computers. These domains jointly make up over 50% of the current NERSC production workload, which is illustrative of the impact quantum computing could have on NERSC’s mission going forward. We perform an extensive literature review and determine the quantum resources required to solve classically intractable problems within these science domains. This review also shows that the quantum resources required have consistently decreased over time due to algorithmic improvements and a deeper understanding of the problems. At the same time, public technology roadmaps from a collection of ten quantum computing companies predict a dramatic increase in capabilities over the next five to ten years. Our analysis reveals a significant overlap emerging in this time frame between the technological capabilities and the algorithmic requirements in these three scientific domains. We anticipate that the execution time of large-scale quantum workflows will become a major performance parameter and propose a simple metric, the Sustained Quantum System Performance (SQSP), to compare system-level performance and throughput for a heterogeneous workload.

97 MATHEMATICS AND COMPUTING↗

Multiphysics Modeling of Microreactors with NEAMS codes, and Validation Based on KRUSTY Reactivity Insertion

The NEAMS Multiphysics Applications team continues to assess code usability and functionality for microreactor design and safety analyses, while demonstrating that NEAMS tools capture both steady-state and transient behavior across distinct microreactor concepts. In FY2025, the team advanced full-core, high-fidelity, multiphysics models that solve more complex problems and strengthen verification/validation for several microreactor systems: heat-pipe microreactor (HPMR), gas-cooled microreactor (GCMR), and the KRUSTY experiment. These models employ the MOOSE MultiApp/Transfers architecture with Griffin for neutronics, BISON for heat conduction/thermomechanics, Sockeye for heat pipes, SAM/THM for coolant channels and loops, and SWIFT for hydride behavior, with meshes generated via the MOOSE Reactor Module. The graphite models available in the Grizzly code were also investigated for future analyses. For the HPMR, a Na-HPMR variant was constructed to align with recently validated heat-pipe experiments and Sockeye’s LCVF capability, enabling mechanistic heat-pipe transients and startup modeling. The Na-HPMR will serve as the primary model for HPMR investigations in upcoming tasks. The load-following and single heat-pipe failure scenarios (Griffin/BISON/Sockeye), which were previously modeled for the K-HPMR, were replicated for the Na-HPMR, showing strong negative temperature feedback and highly localized thermal effects, respectively, while the startup case captured vapor-front progression and heat-removal activation. Solid mechanics was added to the previously built K-HPMR full-core model in BISON, showing minimal impact on steady-state reactivity yet enabling stress-field predictions that prepare the path for full-core TRISO performance analyses. For the GCMR, automated steady-state and four transient scenarios were executed using Griffin/BISON/SAM/SWIFT. Results confirm robust inherent safety: power collapses promptly in loss-of-cooling events, the inlet-temperature drop settles to a new equilibrium, and a single-channel blockage yields only a ~30 K local fuel-temperature rise with <0.4% power decrease. SWIFT-predicted hydrogen redistribution affects reactivity during both steady-state and transient conditions, underscoring its importance. A Brayton-cycle balance of plant (BOP) model in SAM/THM demonstrated stable startup behavior, and xenon-driven reactivity during load following was analyzed. To improve TRISO-compact temperature fidelity, a fast multiscale Heat Source Decomposition (HSD) treatment was implemented. Against heterogeneous benchmarks, HSD reduces underprediction of kernel temperatures and lowers predicted peak powers in reactivity-insertion transients compared to previous homogenized models. KRUSTY warm-critical validation progressed from FY2024 baselines: the 15Ȼ insertion shows excellent agreement in peak power (~2% high) and temperature trends, and the 30Ȼ case was automated via a feedback controller that maintained power near 3 kW for ~150 s with close agreement to data. The successful modeling of the warm critical tests has laid a strong foundation for simulating more complex nuclear system tests in the years ahead. Throughout FY2025, developer feedback was provided (e.g., MOOSE batch mesh generation, distributed pre-split meshes, Griffin sweeper on displaced meshes), several new models were contributed to the Virtual Test Bed, and an OECD-NEA WPRS multiphysics benchmark based on the HPMR was initiated to enable broader cross-comparison and best-practice development with the nuclear community at large.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

Abstract for CRADA between NETL and Rivalia Chemical Co

The National Energy Technology Laboratory (NETL) will assist Rivalia Chemical Co (Participant) with system analyses to aid the Participant with setting priorities and securing funding. The Participant is an early-stage startup pioneering new chemical extraction technology to solve two important problems: critical mineral scarcity and coal ash waste management. The U.S. generates coal fly ash from coal combustion for power generation and has amassed over two billion metric tons of ash, which contains valuable rare earth elements. The ash is often stored in unlined ponds and frequently contaminates the local environment. The Participant’s patent-pending process harvests rare earths from ash, then transforms the residual ash for use in green concrete, providing an economic pathway for utilities to empty and remediate the ash ponds. Founder Laura Stoy created the company in July 2022 after developing the core intellectual property during her Ph.D. at the Georgia Institute of Technology. Stoy is currently participating in the Chain Reactions Innovations (CRI) program at the U.S. Department of Energy (DOE) Argonne National Laboratory (ANL) where she will be optimizing the core technology and scaling from bench scale to demonstration scale. The technology has been validated at bench scale (TRL3); the goal is to approach TRL5 by the conclusion of the CRI program and prepare to raise capital investment for a pilot facility. NETL and the Participant will develop an updated technoeconomic analysis (TEA) and identify key sustainability performance parameters to inform ’s research and development priorities and de-risk the technology. NETL will also provide the Participant with guidance to prepare a more detailed life cycle assessment (LCA), and a narrative on the Participant’s product markets. These tools will aid the Participant with the next steps of commercialization.

01 COAL, LIGNITE, AND PEAT↗

Building a quantum computing architecture using 3D superconducting cavities

Quantum computers promise advantages over classical machines for solving certain complex problems, but building processors that truly deliver this advantage remains a central challenge, particularly due to limited coherence times. Three-dimensional superconducting radio-frequency (SRF) cavities offer an attractive platform due to their exceptionally long lifetimes. However, since these harmonic systems require nonlinear elements, such as transmons, for control, additional losses are often introduced. In this talk, I will present a multimode quantum system based on an elliptical SRF cavity hosting two cavity modes weakly coupled to an ancillary transmon circuit. This architecture is carefully engineered to preserve coherence while enabling efficient control. By optimizing the design to mitigate transmon-induced decoherence, we realize single-photon lifetimes of 20.6 ms and 15.6 ms in the two modes, with pure dephasing times exceeding 40 ms. Using sideband interactions and error-resilient protocols, such as measurement-based correction and post-selection, we demonstrate high-fidelity state control, including preparation of Fock states up to N=20 with fidelities above 95% (to our knowledge, the highest reported to date), as well as high-fidelity two-mode entanglement. These results highlight 3D SRF cavities as a robust foundation for qudit-based quantum information processing, harnessing the large Hilbert space of cavity modes. I will conclude by outlining strategies to further enhance coherence in both cavities and ancilla qubits, and discuss pathways toward scaling this architecture into a larger quantum computing platform.

Roy, Tanay [Fermilab]↗

PowerAmerica (Final Technical Report)

The U.S. Department of Energy’s Advanced Manufacturing Office (predecessor to AMMTO) established PowerAmerica in December 2014 to develop and accelerate the adoption of wide bandgap (WBG) semiconductor chips and power electronics in manufacturing. The objective was to spark early commercialization of energy efficient products; educate and train the workforce; create high-tech jobs; and nurture the growth of the U.S. WBG semiconductor manufacturing industry. PowerAmerica isled by North Carolina State University by way of a five-year, $140M cooperative agreement with DoE. This private-public partnership with DoE includes member companies ranging from startups and small-medium enterprises to large system integrators, world-class universities, and national labs. The WBG power electronic ecosystem formed by our diverse membership is focused on using advanced manufacturing to 1) lower the cost of silicon carbide and gallium nitride semiconductor devices to be comparable to silicon devices; 2) demonstrate the system benefits and energy efficiency advantages of WBG semiconductor power electronics through system demonstrations that validate their effectiveness; and 3) build an education pipeline for a skilled workforce to meet the future demand for emerging WBG semiconductor markets — and enhance U.S. economic competitiveness globally. PowerAmerica was initially funded in six budget periods, each lasting 12 to 18 months. By the end of Budget Period 6 (August 2023), PowerAmerica had achieved its major objectives in technology development, semiconductor device cost reduction, ecosystem growth and engagement, and education and workforce development. Through strategic foundry investments, we helped to establish the first U.S. SiC foundry (X-FAB) and supported a lab (Microchip) to start SiC volume production. We’ve helped bring together companies from different parts of the supply chain, resulting in several successful new partnerships and business relationships. Through projects with some of the largest manufacturers of energy-intensive equipment in the U.S. — John Deere, GE, United Technologies, Raytheon, Carrier, Toshiba, and others — we have successfully demonstrated the energy and system benefits of WBG technology. We have also harnessed the unique capabilities and facilities of national labs — the Naval Research Laboratory, National Renewable Energy Laboratory, and Argonne National Laboratory — to help solve challenging technical problems for industry. Thanks to our many webinars, tutorials, annual events, and presence at major energy and electronics conferences around the world, the PowerAmerica name has become synonymous with WBG technology advancement. The commercial interest in WBG technology is higher than ever; companies and governments around the world have announced hundreds of millions of dollarsin future investment to build capacity — and compete with silicon in many markets and applications. We have trained hundreds of engineering students, from universities across the U.S., through hands-on projects and WBG coursework. Working professionals have also benefited through the years from our many targeted short courses, tutorials, and technical webinars. In short, PowerAmerica has made great strides in each of our key objectives, and the organization has been operating without government or NC State

14 SOLAR ENERGY↗

Fundamental Algorithmic Research for Quantum Computing (FAR-QC) (Final Technical Report)

Anticipation of the noisy intermediate‐scale quantum (NISQ) era has sparked unprecedented interest in quantum computing, yet we still lack a clear understanding of how NISQ‐era applications will perform relative to the best classical algorithms solving the same problems. The goals of this project include: (1) Developing better tools for characterizing the performance of NISQ devices and for assessing whether such devices can achieve a quantum advantage. (2) Conceiving and analyzing potential applications of quantum computing technology in the NISQ era and beyond.

97 MATHEMATICS AND COMPUTING↗

Blueprint for DOE Quantum Supercomputing: Ensuring U.S. Leadership in the Quantum Decade

Quantum computing stands at the threshold of a transformative decade, where the field will evolve from small-scale demonstrations toward practical scientific computing at scale. This Blueprint identifies fault-tolerant quantum computers (FTQCs) as a viable, scalable, and broadly applicable path to achieving “quantum scientific utility,” defined as solving scientifically valuable problems beyond the reach of conventional, classical computers. This capability is expected to show scientific demonstrations in the late 2020s and to mature in the early-to-mid 2030s. This Blueprint outlines a strategy to prepare the U.S. Department of Energy (DOE) for FTQCs and their integration into the U.S. national scientific computing infrastructure. Its purpose is to identify the steps, milestones, and research directions necessary for DOE to enable initial deployment of FTQCs in 2028 as a scientific tool for the nation and mature this capability into the 2030s. DOE has a long history of supporting quantum information science and technology, contributing significantly to research advancements, training a quantum-ready workforce, and providing access to early small-scale quantum hardware. Given recent demonstrations of logical operations on error-corrected logical qubits and the advancement of commercial hardware roadmaps, DOE should begin preparations for large-scale, fault-tolerant quantum computing deployment for DOE science missions. This Blueprint proposes that DOE focus on (1) deploying first-generation scientifically relevant quantum computers with at least 100 logical qubits and performing at least 10,000 to 100,000 hard logical operations in scientifically relevant calculations; (2) developing essential FTQC programming competencies, system software, and facility readiness; and (3) investing in cutting edge focused R&D that fosters breakthroughs in scientific applications, algorithms, and logical architectures needed to accelerate the advent of scientific utility. This effort will position DOE to transition to larger systems: production-scale quantum computers that comprise 1,000 to 10,000 logical qubits, perform 1 to 10 billion hard logical operations, and execute scientifically useful computations at scale. Achieving these goals will require DOE facilities to evolve with urgency to support scientific campaigns that integrate quantum and classical computing resources into efficient workflows, novel software and firmware environments for compiling and routing quantum programs on FTQC machines, and suitable infrastructure for quantum hardware. It will also require further development and optimization of scientific applications from the fields of materials science, quantum chemistry, and high-energy and nuclear physics. The Blueprint calls for transformative R&D and collective action to accelerate the advent of scientific quantum utility and bring it within reach by 2028.

97 MATHEMATICS AND COMPUTING↗

Why Seeding Works When Nucleation Barriers Vanish

This paper explains why seeds can speed up crystallization even when the formation of new crystal nuclei is not the slow step. In zeolite synthesis, local crystalline order can form relatively early, but the material may still appear poorly crystalline by X-ray diffraction because many small crystallites are misoriented and lack long-range coherence. Using coarse-grained molecular simulations, the study showed that seeds help solve this coherence problem. Rather than primarily lowering a nucleation barrier, seeds impose a common orientation on nearby growing crystallites, allowing them to merge into larger, better aligned domains. This accelerates the development of X-ray-detectable crystallinity even if the total amount of locally ordered material has not changed much. The work changes how induction times in zeolite synthesis should be interpreted: the delay before X-ray crystallinity appears can reflect the time needed to build long-range structural coherence, not simply the time needed to form crystalline material. This provides a new explanation for why seeding is effective in growth-limited crystallization and offers design principles for controlling crystallization rate and crystal quality.

36 MATERIALS SCIENCE↗

A Novel Approach to Investigate Thermal Protection Systems Materials

The Koo Research Group (KRG) at The University of Texas at Austin (UT) and KAI has specialized in “Ablation Research” for more than fifteen years. Recently, the group has developed several incredibly unique capabilities that can advance “Thermal Protection Systems (TPS) Materials Research & Development” using an integrated experimental and numerical approach. The paper aims to introduce the methodology KRG has developed to solve this challenging problem. It will discuss how the KRG develops “Process-Properties-Performance” relationships of novel TPS materials in a systematical approach using (a) processing and fabrication, (b) thermal characterization of properties, (c) aerothermal testing, (d) microstructures characterization and analysis, and (e) numerical modeling. Progress and challenges of this research will also be discussed.

Engineering↗

Joint Modeling of Quasar Variability and Accretion Disk Reprocessing Using Latent Stochastic Differential Equations

Quasars are bright active galactic nuclei powered by the accretion of matter around supermassive black holes at the center of galaxies. Their stochastic brightness variability depends on the physical properties of the accretion disk and black hole. The upcoming Rubin Observatory Legacy Survey of Space and Time (LSST) is expected to observe tens of millions of quasars, so there is a need for efficient techniques like machine learning that can handle the large volume of data. Quasar variability is believed to be driven by an X-ray corona, which is reprocessed by the accretion disk and emitted as UV/optical variability. We are the first to introduce an auto-differentiable simulation of the accretion disk and reprocessing. We use the simulation as a direct component of our neural network to jointly model the driving variability and reprocessing, trained with supervised learning on simulated LSST-like 10 yr quasar light curves. We encode the light curves using a transformer encoder, and the driving variability is reconstructed using latent stochastic differential equations, a physically motivated generative deep learning method that can model continuous-time stochastic dynamics. By embedding the physical processes of the driving signal and reprocessing into our network, we achieve a model that is more robust and interpretable. We demonstrate that our model outperforms a Gaussian process regression baseline and can infer accretion disk parameters and time delays between wave bands, even for out-of-distribution driving signals. Our approach provides a powerful framework that can be adapted to solve other inverse problems in multivariate time series.

Fagin, Joshua [City Univ. of New York (CUNY), NY (↗

Stochastic Error Cancellation in Analog Quantum Simulation

Analog quantum simulation is a promising path towards solving classically intractable problems in many-body physics on near-term quantum devices. However, the presence of noise limits the size of the system and the length of time that can be simulated. In our work, we consider an error model in which the actual Hamiltonian of the simulator differs from the target Hamiltonian we want to simulate by small local perturbations, which are assumed to be random and unbiased. We analyze the error accumulated in observables in this setting and show that, due to stochastic error cancellation, with high probability the error scales as the square root of the number of qubits instead of linearly. We explore the concentration phenomenon of this error as well as its implications for local observables in the thermodynamic limit. Moreover, we show that stochastic error cancellation also manifests in the fidelity between the target state at the end of time-evolution and the actual state we obtain in the presence of noise. This indicates that, to reach a certain fidelity, more noise can be tolerated than implied by the worst-case bound if the noise comes from many statistically independent sources.

Analog quantum simulation↗

Streaming Matching and Edge Cover in Practice

Graph algorithms with polynomial space and time requirements often become infeasible for massive graphs with billions of edges or more. State-of-the-art approaches therefore employ approximate serial, parallel, and distributed algorithms to tackle these challenges. However, such approaches require storing the entire graph in memory and thus need access to costly computing resources such as clusters and supercomputers. In this paper, we present practical streaming approaches for solving massive graph problems using limited memory for two prototypical graph problems: maximum weighted matching and minimum weighted edge cover. For matching, we conduct a thorough computational study on two of the semi-streaming algorithms including a recent breakthrough result that achieves a $1/(2+\varepsilon)$-approximation of the weight while using $O( n \log W /\epsilon)$ memory (here $n$ is the number of vertices and $W$ is the maximum edge weight), designed by Paz and Schwartzman [SODA, 2017]. Empirically, we show that the semi-streaming algorithms produce matchings whose weight is close to the best $1/2$-approximate offline algorithm while requiring less time and an order-of-magnitude less memory. For minimum weighted edge cover, we develop three novel semi-streaming algorithms. Two of these algorithms require a single pass through the input graph, require $O(n \log n)$ memory, and provide a 2-approximation guarantee on the objective. We also leverage a relationship between approximate maximum weighted matching and approximate minimum weighted edge cover to develop a two-pass $3/2+\epsilon$-approximate algorithm with the memory requirement of Paz and Schwartzman's semi-streaming matching algorithm. These streaming approaches are compared against the state-of-the-art 3/2-approximate offline algorithm. The semi-streaming matching and the novel edge cover algorithms proposed in this paper can process graphs with several billions of edges in under 30 minutes using 6 GB of memory, which is at least an order of magnitude improvement from the offline (non-streaming) algorithms. For the largest graph, the best alternative offline parallel approximation algorithm (GPA+ROMA) could not finish in three hours even while employing hundreds of processors and 1 TB of memory. We also demonstrate an application of the semi-streaming algorithm by computing a matching using linearly bounded memory on item intersection graphs derived from three machine learning datasets, whereas the existing offline algorithms could not complete on one of these datasets since their memory requirements exceeded 1TB.

Ferdous, S M.↗

Nonperturbative and perturbative dynamics of a light QCD axion: Dark matter and the strong 𝐶⁢𝑃 problem

Considerable theoretical efforts have gone into expanding the reach of the quantum chromodynamics (QCD) axion beyond its canonical mass–decay-constant relation. The 𝑍 𝒩 QCD axion model reduces the QCD axion mass naturally, by invoking a discrete 𝑍 𝒩 symmetry through which the axion field is coupled to 𝒩 copies of the Standard Model. Before the QCD phase transition at temperature 𝑇 QCD , the 𝑍 𝒩 potential has a minimum at misalignment angle 𝜃 = 𝜋. At 𝑇 QCD , 𝜃 = 𝜋 becomes a maximum; the axion potential becomes exponentially suppressed and develops 𝒩 minima—only one of which actually solves the strong 𝐶⁢𝑃 problem. Before 𝑇 QCD , 𝜃 relaxes toward 𝜋. After 𝑇 QCD , the axion field starts from around the hilltop and may have sufficient kinetic energy to overcome the newly suppressed potential barriers. Such a field evolution leads to nonperturbative effects via the self-interactions near the hilltop, which can cause the exponential growth of fluctuations and backreaction on the coherent motion. This behavior can influence the relic density of the field and the minimum in which it settles. We conduct the first lattice simulations of the 𝑍 𝒩 QCD axion using 𝒞osmoℒattice to accurately calculate dark matter abundances and find nonperturbative dynamics reduce the abundance by up to a factor of two. We furthermore find that the probability of solving the strong 𝐶⁢𝑃 problem tends to diverge considerably from the naïve expectation of 1/𝒩.

Axions↗

Deep Point Cloud Building Envelope Segmentation (DeeP-CuBES) using Deep Learning

Building Information Modeling (BIM) plays an important role in building design and construction, particularly for achieving energy-efficient retrofits. Building envelope retrofits using panelized prefabricated system, such as those popularized by the Energiesprong program, need accurate as-built dimensions of facade features (windows, doors, etc.) to achieve the desired thermal and air tightness. Traditionally, building surveying is done manually, resulting in a time-consuming and labor-intensive process. Recently, 3D point clouds from terrestrial LiDAR have been used to automate the generation of as-built dimensions of existing buildings. However, automated BIM using LiDAR relies on solving the point cloud semantic segmentation (PCSS) problem. In this work, we propose a robust pipeline for solving the PCSS problem using deep neural networks, focusing on overcoming challenges posed by imbalanced datasets and complex architectural features. We introduce the first high-density, labeled, and validated building envelope point cloud dataset derived from multiple building scans, specifically curated to tackle challenges in facade-level segmentation. Results from the trained neural networks show that advanced attention-based architectures and incorporating radiometry (light intensity and RGB) features significantly boost segmentation accuracy for windows and doors.

Selvakumar, Balaji [ORNL]↗

Efficient shallow Ritz method for 1D diffusion problems

This paper studies the shallow Ritz method for solving the one-dimensional diffusion problem. It is shown that the shallow Ritz method improves the order of approximation dramatically for non-smooth problems. To realize this optimal or nearly optimal order of the shallow Ritz approximation, we develop a damped block Newton (dBN) method that alternates between updates of the linear and non-linear parameters. Per each iteration, the linear and the non-linear parameters are updated by exact inversion and one step of a modified, damped Newton method applied to a reduced non-linear system, respectively. The computational cost of each dBN iteration is $\mathcal{O}$(n). Starting with the non-linear parameters as a uniform partition of the interval, numerical experiments show that the dBN is capable of efficiently moving mesh points to nearly optimal locations. In conclusion, to improve the efficiency of the dBN further, we propose an adaptive damped block Newton (AdBN) method by combining the dBN with the adaptive neuron enhancement (ANE) method [28].

Diffusion problems↗