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Results for “ORTHOGONAL FUNCTION”

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.

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At least 253 records · Page 14
All sourcesNASA NTRSDOE OSTI
NASA NTRS1980

Investigations of medium wavelength magnetic anomalies in the eastern Pacific using MAGSAT data

The computer program package for dealing with MAGSAT data was refined and a program listing and deck of cards was forwarded for use by interested investigators. Instead of doing equivalent source calculations, it is proposed to use spherical harmonics to model the observed data since spherical harmonic functions are orthogonal and addition of higher degree and order harmonics can be done without recalculation of the harmonics of lower degree and order already calculated. These functions are much more easy to manipulate in that the height of the term of the observation can be taken care of with great facility. Downward calculation (after suitable filtering) can also be performed very simply.

Harrison, C. G. A.↗
DOE OSTI2018

Basis adaptive sample efficient polynomial chaos (BASE-PC)

For a large class of orthogonal basis functions, there has been a recent identification of expansion methods for computing accurate, stable approximations of a quantity of interest. Here this paper presents, within the context of uncertainty quantification, a practical implementation using basis adaptation, and coherence motivated sampling, which under assumptions has satisfying guarantees. This implementation is referred to as Basis Adaptive Sample Efficient Polynomial Chaos (BASE-PC). A key component of this is the use of anisotropic polynomial order which admits evolving global bases for approximation in an efficient manner, leading to consistently stable approximation for a practical class of smooth functionals. This fully adaptive, non-intrusive method, requires no a priori information of the solution, and has satisfying theoretical guarantees of recovery. A key contribution to stability is the use of a presented correction sampling for coherence-optimal sampling in order to improve stability and accuracy within the adaptive basis scheme. Theoretically, the method may dramatically reduce the impact of dimensionality in function approximation, and numerically the method is demonstrated to perform well on problems with dimension up to 1000.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Weak Form Scientific Machine Learning: Test Function Construction for System Identification

Weak form Scientific Machine Learning (WSciML) is a recently developed framework for data-driven modeling and scientific discovery. It leverages the weak form of equation error residuals to provide enhanced noise robustness in system identification via convolving model equations with test functions, reformulating the problem to avoid direct differentiation of data. The performance, however, relies on wisely choosing a set of compactly supported test functions. In this work, we mathematically motivate a novel data-driven method for constructing Single-scale-Local reference functions for creating the set of test functions. Our approach numerically approximates the integration error introduced by the quadrature and identifies the support size for which the error is minimal, without requiring access to the model parameter values. Through numerical experiments across various models, noise levels, and temporal resolutions, we demonstrate that the selected supports consistently align with regions of minimal parameter estimation error. We also compare the proposed method against the strategy for constructing Multi-scale-Global (and orthogonal) test functions introduced in our prior work, demonstrating the improved computational efficiency.

FOS: Computer and information sciences↗
DOE OSTI2025

The effective number of parameters in kernel density estimation

We devise a new formula for measuring the effective degrees of freedom (EDoF) in kernel density estimation (KDE). Starting from the orthogonal polynomial sequence (OPS) expansion for the ratio of the empirical to the oracle density, we show how convolution with the kernel leads to a new OPS with respect to which one may express the resulting KDE. The expansion coefficients of the two OPS systems can then be related via a kernel sensitivity matrix, which leads to a natural oracle definition of EDoF through the trace operator. Asymptotic properties of the (empirical) plug-in EDoF are worked out through influence functions, and connections with other empirical EDoFs are established. Minimization of Kullback-Leibler divergence is investigated as an alternative to integrated squared error based bandwidth selection rules, yielding a new normal scale rule. The methodology, which arises from a proper oracle formulation and is not restricted to convolution kernels, suggests the possibility of a new bandwidth selection rule based on an information criterion such as AIC.

bandwidth selection↗
NASA NTRS1998

Radiance Covariance and Climate Models

Spectral Empirical Orhtogonal Functions (EOFs) derived from the covariance of satellite radiance spectra may be interpreted in terms of the vertical distribution of the covariance of temperature, water vapor, and clouds. The purpose of the investigation is to demonstrate the important constraints that resolved spectral radiances can place upon climate models.

Spectral Empirical Orthogonal Functions EOFs radia↗
DOE OSTI2021

A fourth-order Lagrangian discontinuous Galerkin method using a hierarchical orthogonal basis on curvilinear grids

The existing high-order Lagrangian discontinuous Galerkin (DG) hydrodynamic methods are restricted to using quadratic meshes with quadratic polynomials (P2), which in turn, yield up to third-order accuracy. Additionally, these existing DG hydrodynamic schemes, when extended to work with cubic meshes and cubic polynomials (P3), can be unstable on strong-shock problems. Therefore, this paper presents a new Lagrangian discontinuous Galerkin (DG) hydrodynamic method to simulate compressible material dynamics ( e.g., gasses, fluids, and solids) with strong-shocks using cubic meshes and cubic polynomials, and delivers up to fourth-order accuracy on smooth flows. The stability on shock problems is achieved using new hierarchical orthogonal basis functions and a new subcell mesh stabilization (SMS) scheme for cubic meshes. The accuracy and robustness of the new high-order accurate Lagrangian DG hydrodynamic method is demonstrated by simulating a diverse suite of challenging test problems covering gas and solid dynamic problems on curvilinear meshes.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1967

Convergence in the mean

Theorem proving for orthogonal systems of functions relative to convergence in mean of series

SERIES EXPANSION↗
DOE OSTI2023

A parallel p ‐adaptive discontinuous Galerkin method for the Euler equations with dynamic load‐balancing on tetrahedral grids

Abstract A novel p ‐adaptive discontinuous Galerkin (DG) method has been developed to solve the Euler equations on three‐dimensional tetrahedral grids. Hierarchical orthogonal basis functions are adopted for the DG spatial discretization while a third order TVD Runge‐Kutta method is used for the time integration. A vertex‐based limiter is applied to the numerical solution in order to eliminate oscillations in the high order method. An error indicator constructed from the solution of order and is used to adapt degrees of freedom in each computational element, which remarkably reduces the computational cost while still maintaining an accurate solution. The developed method is implemented with under the Charm++ parallel computing framework. Charm++ is a parallel computing framework that includes various load‐balancing strategies. Implementing the numerical solver under Charm++ system provides us with access to a suite of dynamic load balancing strategies. This can be efficiently used to alleviate the load imbalances created by p ‐adaptation. A number of numerical experiments are performed to demonstrate both the numerical accuracy and parallel performance of the developed p ‐adaptive DG method. It is observed that the unbalanced load distribution caused by the parallel p ‐adaptive DG method can be alleviated by the dynamic load balancing from Charm++ system. Due to this, high performance gain can be achieved. For the testcases studied in the current work, the parallel performance gain ranged from 1.5× to 3.7×. Therefore, the developed p ‐adaptive DG method can significantly reduce the total simulation time in comparison to the standard DG method without p ‐adaptation.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Robust 3D multi-material hydrodynamics using discontinuous Galerkin methods

A high-order discontinuous Galerkin (DG) method is presented for nonequilibrium multi-material (m ≥ 2) flow with sharp interfaces. Material interfaces are reconstructed using the algebraic THINC approach, resulting in a sharp interface resolution. The system assumes stiff velocity relaxation and pressure nonequilibrium. The presented DG method uses Dubiner's orthogonal basis functions on tetrahedral elements. This results in a unique combination of sharp multimaterial interfaces and high-order accurate solutions in smooth single-material regions. A novel shock indicator based on the interface conservation condition is introduced to mark regions with discontinuities. Slope limiting techniques are applied only in these regions so that nonphysical oscillations are eliminated while maintaining high-order accuracy in smooth regions. A local projection is applied on the limited solution to ensure discrete closure law preservation. The effectiveness of this novel limiting strategy is demonstrated for complex three-dimensional multi-material problems, where robustness of the method is critical. The presented numerical problems demonstrate that more accurate and efficient multi-material solutions can be obtained by the DG method, as compared to second-order finite volume methods.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

High-order fully implicit solver for all-speed fluid dynamics: AUSM ride from nearly incompressible variable-density flows to shock dynamics

We introduce a novel Newton–Krylov (NK)-based fully implicit algorithm for solving fluid flows in a wide range of flow conditions—from variable density nearly incompressible to supersonic shock dynamics. The key enabling feature of our all-speed solver is the ability to efficiently solve conservation laws by choosing a set of independent variables that produce a well-conditioned Jacobian matrix for the linear iterations of the global nonlinear iterative solver. In particular, instead of choosing to discretize the conservative variables (density, momentum, total energy), which is traditionally used in Eulerian high-speed compressible fluid dynamics, we demonstrate superior performance by discretizing the primitive variables—pressure–velocity–temperature in the very low-Mach flow limits or density–velocity–temperature/entropy in the shock dynamics range. Moreover, our method allows us to avoid direct inversion of the mass matrix in discrete time derivatives, which is usually an additional source for stiffness, especially pronounced when going to very high-order schemes with non-orthogonal basis functions. Here, we show robust solutions obtained for discontinuous finite element discretization up to seventh-order accuracy. Another important aspect of the solution algorithm is the Advection Upstream Splitting Method (AUSM), adopted to compute numerical fluxes within our reconstructed discontinuous Galerkin (rDG) spatial discretization scheme. The use of the low-Mach modification of the hyperbolic flux operator is found to be necessary for enabling robust simulations of very stiff liquids and metals for Mach numbers below M=10 -5 , which is well known to be very computationally challenging for compressible solvers. We demonstrate that our fully implicit rDG-NK solver with the AUSM + -up flux treatment produces efficient and high-resolution numerical solutions at all speeds, ranging from vanishing Mach numbers to transonic and supersonic, without substantial modifications of the solution procedures. (At high speed, we add limiting and use a simpler preconditioning of the Krylov solver.) Numerical examples include nearly incompressible constant-property flow past a backward-facing step with heat transfer, low-Mach variable-property channel flow of water at supercritical state, phase change and melt pool dynamics for laser spot welding and selective laser melting in additive manufacturing, and Mach 3 flow in a wind tunnel with a step.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2015

Multi-element least square HDMR methods and their applications for stochastic multiscale model reduction

Stochastic multiscale modeling has become a necessary approach to quantify uncertainty and characterize multiscale phenomena for many practical problems such as flows in stochastic porous media. The numerical treatment of the stochastic multiscale models can be very challengeable as the existence of complex uncertainty and multiple physical scales in the models. To efficiently take care of the difficulty, we construct a computational reduced model. To this end, we propose a multi-element least square high-dimensional model representation (HDMR) method, through which the random domain is adaptively decomposed into a few subdomains, and a local least square HDMR is constructed in each subdomain. These local HDMRs are represented by a finite number of orthogonal basis functions defined in low-dimensional random spaces. The coefficients in the local HDMRs are determined using least square methods. We paste all the local HDMR approximations together to form a global HDMR approximation. To further reduce computational cost, we present a multi-element reduced least-square HDMR, which improves both efficiency and approximation accuracy in certain conditions. To effectively treat heterogeneity properties and multiscale features in the models, we integrate multiscale finite element methods with multi-element least-square HDMR for stochastic multiscale model reduction. This approach significantly reduces the original model's complexity in both the resolution of the physical space and the high-dimensional stochastic space. We analyze the proposed approach, and provide a set of numerical experiments to demonstrate the performance of the presented model reduction techniques. - Highlights: • Multi-element least square HDMR is proposed to treat stochastic models. • Random domain is adaptively decomposed into some subdomains to obtain adaptive multi-element HDMR. • Least-square reduced HDMR is proposed to enhance computation efficiency and approximation accuracy in certain conditions. • Integrating MsFEM and multi-element least square HDMR can significantly reduce computation complexity.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Compound Meta-Optics for Complete and Loss-Less Field Control

Optical metasurfaces offer a compact platform for manipulation of the amplitude, phase, and polarization state of light. Independent control over these properties, however, is hindered by the symmetric transmission matrix associated with single-layer metasurfaces. Here, we utilize multilayer birefringent meta-optics to realize high-efficiency, independent control over the amplitude, phase, and polarization state of light. In this work, high-efficiency control is enabled by redistributing the wavefront between cascaded metasurfaces, while end-to-end inverse design is used to realize independent complex-valued functions for orthogonal polarization states. Based on this platform, we demonstrate spatial mode division multiplexing, optical mode conversion, and universal vectorial holograms, all with diffraction efficiencies over 80%. This meta-optic platform expands the design space of flat optics and could lead to advances in optical communications, quantum entanglement, and information encryption.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2019

Designing molecular building blocks for the self-assembly of complex porous networks

The creation of molecular or colloidal building blocks which can self-assemble into complex, ordered porous structures has been long sought-after, and so are the guiding principles behind this creation. The pursuit of this goal has led to the creation of novel classes of materials like metal organic frameworks (MOFs) and covalent organic frameworks (COFs). In theory, a tremendous number of structures can be formed by these materials due to the variety of geometries available to their building blocks. However, most realized crystal structures tend to be simple or homoporous and typically assemble from building blocks with high degrees of symmetry. Building blocks with low degrees of symmetry suitable for assembly into the more complex structures tend to assemble into polymorphous or disordered structures instead. Here, we use Monte Carlo simulations of patchy vertex-like building blocks to show how the addition of chemical specificity via orthogonally reacting functional sites can allow vertex-like building blocks with even asymmetric geometries to self-assemble into ordered crystallites of various complex structures. In addition to demonstrating the utility of such a strategy in creating ordered, heteroporous structures, we also demonstrate that it can be used as a means for tuning specific features of the crystal structure, accomplishing such aims as the control of relative pore sizes. We also discuss heuristics for properly designing molecules so that they can assemble into target structures.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Engineering modular enzyme assembly: synthetic interface strategies for natural products biosynthesis applications

Covering: 2020 to 2025Natural products remain indispensable sources of therapeutic and bioactive compounds, yet traditional discovery strategies are constrained by compound rediscovery. Modular biosynthetic enzymes, such as type I polyketide synthases (PKSs) and type A non-ribosomal peptide synthetases (NRPSs), offer promising platforms for combinatorial biosynthesis owing to their programmable architectures. However, practical implementation is frequently limited by inter-modular incompatibility and domain-specific interactions. This review highlights recent advances in modular enzyme assembly enabled by synthetic interfaces-including cognate docking domains, synthetic coiled-coils, SpyTag/SpyCatcher, and split inteins-which function as orthogonal, standardized connectors to facilitate post-translational complex formation. These interfaces support rational investigations into substrate specificity, module compatibility, and pathway derivatization as well as general enzyme clustering applications beyond PKS and NRPS systems. Synthetic interfaces can be integrated with computational tools to support a more systematic and scalable framework for modular enzyme engineering by providing predictive insights into domain compatibility and interface design. These approaches within iterative design-build-test-learn workflows can accelerate the programmable assembly of biosynthetic systems and expand the accessible chemical space for natural products.

Kim, Gahyeon↗
DOE OSTI2014

A Gaussian graphical model approach to climate networks

Distinguishing between direct and indirect connections is essential when interpreting network structures in terms of dynamical interactions and stability. When constructing networks from climate data the nodes are usually defined on a spatial grid. The edges are usually derived from a bivariate dependency measure, such as Pearson correlation coefficients or mutual information. Thus, the edges indistinguishably represent direct and indirect dependencies. Interpreting climate data fields as realizations of Gaussian Random Fields (GRFs), we have constructed networks according to the Gaussian Graphical Model (GGM) approach. In contrast to the widely used method, the edges of GGM networks are based on partial correlations denoting direct dependencies. Furthermore, GRFs can be represented not only on points in space, but also by expansion coefficients of orthogonal basis functions, such as spherical harmonics. This leads to a modified definition of network nodes and edges in spectral space, which is motivated from an atmospheric dynamics perspective. We construct and analyze networks from climate data in grid point space as well as in spectral space, and derive the edges from both Pearson and partial correlations. Network characteristics, such as mean degree, average shortest path length, and clustering coefficient, reveal that the networks posses an ordered and strongly locally interconnected structure rather than small-world properties. Despite this, the network structures differ strongly depending on the construction method. Straightforward approaches to infer networks from climate data while not regarding any physical processes may contain too strong simplifications to describe the dynamics of the climate system appropriately.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2018

Spatially Dependent Multiple Testing Under Model Misspecification, With Application to Detection of Anthropogenic Influence on Extreme Climate Events

The Weather Risk Attribution Forecast (WRAF) is a forecasting tool that uses output from global climate models to make simultaneous attribution statements about whether and how greenhouse gas emissions have contributed to extreme weather across the globe. However, in conducting a large number of simultaneous hypothesis tests, the WRAF is prone to identifying false “discoveries.” A common technique for addressing this multiple testing problem is to adjust the procedure in a way that controls the proportion of true null hypotheses that are incorrectly rejected, or the false discovery rate (FDR). Unfortunately, generic FDR procedures suffer from low power when the hypotheses are dependent, and techniques designed to account for dependence are sensitive to misspecification of the underlying statistical model. In this paper, we develop a Bayesian decision-theoretical approach for dependent multiple testing and a nonparametric hierarchical statistical model that flexibly controls false discovery and is robust to model misspecification. We illustrate the robustness of our procedure to model error with a simulation study, using a framework that accounts for generic spatial dependence and allows the practitioner to flexibly specify the decision criteria. Finally, we apply our procedure to several seasonal forecasts and discuss implementation for the WRAF workflow. Lastly, supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2014

SU-E-T-140: Dynamic Wave Arc Trajectory Verification Using KV X-Ray Fluoroscopy

Purpose: Purpose: This study investigates the geometric accuracy of simultaneous Gantry/Ring rotation during Dynamic Wave Arc (DWA) delivery. Methods: The Vero SBRT system consists of a 6MV LINAC mounted on an O-ring gantry that can rotate around the vertical axis (±60°), similar to couch rotation on C-arm gantries. To provide CBCT and fluoroscopy imaging functionalities, two orthogonal kV imaging units are attached to the O-ring at −45°/+45° from the beam axis.Dynamic Wave Arc maximizes Vero's motion capabilities by employing synchronized gantry and ring motion on a complex non-coplanar trajectory in combination with aperture based optimized MLC segments.Four wave arc trajectories (T1-4) were delivered using a cubic phantom with a configuration of five lead beads. O-ring gantry position information was retrieved through continuous dual-source kV X-ray image acquisition during DWA. An in-house algorithm read in the image set, extracted the projected marker positions and determined the angulation through reconstruction of the beam source position. The geometric error was quantified as the distance between the independently detected positions from kV-images and reference trajectory derived from the treatment plan in the Ring-Gantry space. Results: The average displacement between the 3D gantry/ring positions reconstructed from the fluoroscopy images and the reference trajectory was 0.346 mm (SD 0,171) for T1. A mean offset of 0.348 mm (SD 0,182) and 0.357 mm (SD 0.194) was observed for trajectory T2(2segmens) and T3(4segments), respectively. The saw shape T4 presented a mean geometric error of 0.363 (SD 0.156). The overall systematic error of 0.350 was caused by the difference between planned reference trajectory created by linear interpolation between CP, and the machine delivery following a spline curve. Conclusion: An independent geometric QA approach has been developed for DWA delivery verification, successfully applied on diverse trajectories and disclosed that the Vero system is capable of accurately follow complex trajectories.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2022

Infrared plasmons propagate through a hyperbolic nodal metal

Metals are canonical plasmonic media at infrared and optical wavelengths, allowing one to guide and manipulate light at the nanoscale. A special form of optical waveguiding is afforded by highly anisotropic crystals revealing the opposite signs of the dielectric functions along orthogonal directions. These media are classified as hyperbolic and include crystalline insulators, semiconductors, and artificial metamaterials. Layered anisotropic metals are also anticipated to support hyperbolic waveguiding. However, this behavior remains elusive, primarily because interband losses arrest the propagation of infrared modes. Here, we report on the observation of propagating hyperbolic waves in a prototypical layered nodal-line semimetal ZrSiSe. The observed waveguiding originates from polaritonic hybridization between near-infrared light and nodal-line plasmons. Unique nodal electronic structures simultaneously suppress interband loss and boost the plasmonic response, ultimately enabling the propagation of infrared modes through the bulk of the crystal.

77 NANOSCIENCE AND NANOTECHNOLOGY↗