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Results for “truncation error matching”

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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12 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Foundations for high-order, conservative cut-cell methods: Stable discretizations on degenerate meshes

Cut-cell methods for unsteady flow problems can greatly simplify the grid generation process and allow for high-fidelity simulations on complex geometries. However, cut-cell methods have been limited to low orders of accuracy. This is driven, largely, by the variety of procedures typically introduced to evaluate derivatives in a stable manner near the highly irregular embedded geometry. Here, a completely new approach, termed TEMO (truncation error matching and optimization), is taken to solve this problem. The approach is based on two simple and intuitive design principles. These principles directly allow for the construction of stable 8th To the best of the authors' knowledge, these are the highest orders ever achieved for a cut-cell discretization by a significant margin. This is done for both explicit and compact finite differences and is accomplished without any geometric transformations or artificial stabilization procedures.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1999

Backus Effect on a Perpendicular Errors in Harmonic Models of Real vs. Synthetic Data

Measurements of geomagnetic scalar intensity on a thin spherical shell alone are not enough to separate internal from external source fields; moreover, such scalar data are not enough for accurate modeling of the vector field from internal sources because of unmodeled fields and small data errors. Spherical harmonic models of the geomagnetic potential fitted to scalar data alone therefore suffer from well-understood Backus effect and perpendicular errors. Curiously, errors in some models of simulated 'data' are very much less than those in models of real data. We analyze select Magsat vector and scalar measurements separately to illustrate Backus effect and perpendicular errors in models of real scalar data. By using a model to synthesize 'data' at the observation points, and by adding various types of 'noise', we illustrate such errors in models of synthetic 'data'. Perpendicular errors prove quite sensitive to the maximum degree in the spherical harmonic expansion of the potential field model fitted to the scalar data. Small errors in models of synthetic 'data' are found to be an artifact of matched truncation levels. For example, consider scalar synthetic 'data' computed from a degree 14 model. A degree 14 model fitted to such synthetic 'data' yields negligible error, but amplifies 4 nT (rmss) added noise into a 60 nT error (rmss); however, a degree 12 model fitted to the noisy 'data' suffers a 492 nT error (rmms through degree 12). Geomagnetic measurements remain unaware of model truncation, so the small errors indicated by some simulations cannot be realized in practice. Errors in models fitted to scalar data alone approach 1000 nT (rmss) and several thousand nT (maximum).

Voorhies, C. V.↗
DOE OSTI2021

Finite and infinite matrix product states for Gutzwiller projected mean-field wave functions

Matrix product states (MPS) and “dressed” ground states of quadratic mean fields (e.g., Gutzwiller projected Slater determinants) are both important classes of variational wave functions. This latter class has played important roles in understanding superconductivity and quantum spin liquids. We present a method to obtain both the finite and infinite MPS (iMPS) representation of the ground state of an arbitrary fermionic quadratic mean-field Hamiltonian (which in the simplest case is a Slater determinant and in the most general case is a Pfaffian). We also show how to represent products of such states (e.g., determinants times Pfaffians). From this representation one can project to single occupancy and evaluate the entanglement spectra after Gutzwiller projection. We then obtain the MPS and iMPS representation of Gutzwiller projected mean-field states that arise from the variational slave-fermion approach to the S = 1 bilinear-biquadratic quantum spin chain. To accomplish this, we develop an approach to orthogonalize degenerate iMPS to find all the states in the degenerate ground-state manifold. We find the energies of the MPS and iMPS states match the variational energies closely, indicating the method is accurate and there is minimal loss due to truncation error. We then present an exploration of the entanglement spectra of projected slave-fermion states, exploring their qualitative features and finding good qualitative agreement with the respective exact ground-state spectra found from density matrix renormalization group.

1-dimensional spin chains↗
NASA NTRS1992

Eigenvalue error analysis of viscously damped structures using a Ritz reduction method

The efficient solution of the eigenvalue problem that results from inserting passive dampers with variable stiffness and damping coefficients into a structure is addressed. Eigenanalysis of reduced models obtained by retaining a number of normal modes augmented with Ritz vectors corresponding to the static solutions resulting from the load patterns introduced by the dampers has been empirically shown to yield excellent approximations to the full eigenvalue problem. An analysis of this technique in the case of a single damper is presented. A priori and a posteriori error estimates are generated and tested on numerical examples. Comparison theorems with modally truncated models and a Markov parameter matching reduced-order model are derived. These theorems corroborate the heuristic that residual flexibility methods improve low-frequency approximation of the system. The analysis leads to other techniques for eigenvalue approximation. Approximate closed-form solutions are derived that include a refinement to eigenvalue derivative methods for approximation. An efficient Newton scheme is also developed. A numerical example is presented demonstrating the effectiveness of each of these methods.

Chu, Cheng-Chih↗
DOE OSTI2021

Theory of Trotter Error with Commutator Scaling

The Lie-Trotter formula, together with its higher-order generalizations, provides a simple approach to decomposing the exponential of a sum of operators. Despite significant effort, the error scaling of such product formulas remains poorly understood. We develop a theory of Trotter error that overcomes the limitations of truncating the Baker-Campbell-Hausdorff expansion. Our analysis directly exploits the commutativity of operator summands, producing tighter error bounds for both real- and imaginary-time evolutions. Whereas previous work achieves similar goals for systems with geometric locality or Lie-algebraic structure, our approach holds in general. We give a host of improved algorithms for digital quantum simulation and quantum Monte Carlo methods, nearly matching or even outperforming the best previous results. Our applications include: (i) a simulation of second-quantized plane-wave electronic structure, nearly matching the interaction-picture algorithm of Low and Wiebe; (ii) a simulation of $k$-local Hamiltonians almost with induced one-norm scaling, faster than the qubitization algorithm of Low and Chuang; (iii) a simulation of rapidly decaying power-law interactions, outperforming the Lieb-Robinson-based approach of Tran et al.; (iv) a hybrid simulation of clustered Hamiltonians, dramatically improving the result of Peng, Harrow, Ozols, and Wu; and (v) quantum Monte Carlo simulations of the transverse field Ising model and quantum ferromagnets, tightening previous analyses of Bravyi and Gosset. We obtain further speedups using the fact that product formulas can preserve the locality of the simulated system. Specifically, we show that local observables can be simulated with complexity independent of the system size for power-law interacting systems, which implies a Lieb-Robinson bound nearly matching a recent result of Tran et al. Our analysis reproduces known tight bounds for first- and second-order formulas. We further investigate the tightness of our bounds for higher-order formulas. For quantum simulation of a one-dimensional Heisenberg model with an even-odd ordering of terms, our result overestimates the complexity by only a factor of $5$. Our bound is also close to tight for power-law interactions and other orderings of terms. This suggests that our theory can accurately characterize Trotter error in terms of both the asymptotic scaling and the constant prefactor.

quantum computing, numerical analysis↗
DOE OSTI2020

A Performance Comparison of Low-Cost Near-Infrared (NIR) Spectrometers to a Conventional Laboratory Spectrometer for Rapid Biomass Compositional Analysis

The performance of a conventional laboratory near-infrared (NIR) spectrometer and two NIR spectrometer prototypes (a Texas Instruments NIRSCAN Nano evaluation model (EVM) and an InnoSpectra NIR-M-R2 spectrometer) are compared by collecting reflectance spectra of 270 well-characterized herbaceous biomass samples, building calibration models using the partial least squares (PLS-2) algorithm to predict five constituents of the samples from the reflectance spectra, and comparing the resulting model statistics. The prediction models developed using spectra from the Foss XDS spectrometer were slightly better than the prediction models developed using spectra from either the TI NIRSCAN Nano EVM and the InnoSpectra NIR-M-R2 as measured by the root mean square error (RMSECV) and the correlation coefficient (R 2 _cv) for “leave-one-out” cross-validation (CV). The models built from the two prototype units were not statistically significantly different from each other (p = 0.05). The Foss spectrometer has a larger wavelength range (400–2500 nm) compared with the two prototypes (900–1700 nm). When the spectra from the Foss XDS spectrometer were truncated so their wavelength range matched the wavelength range of the two prototype units, the resulting model was not statistically significantly different from the models from either prototype.

47 OTHER INSTRUMENTATION↗
DOE OSTI2020

Properties of the new $ \mathcal{N} $ = 1 AdS 4 vacuum of maximal supergravity

The recent comprehensive numerical study of critical points of the scalar potential of four-dimensional N = 8, SO(8) gauged supergravity using Machine Learning software in [1] has led to a discovery of a new N = 1 vacuum with a triality-invariant SO(3) symmetry. Guided by the numerical data for that point, we obtain a consistent SO(3) × Z 2 -invariant truncation of the N = 8 theory to an N = 1 supergravity with three chiral multiplets. Critical points of the truncated scalar potential include both the N = 1 point as well as two new non-supersymmetric and perturbatively unstable points not found by previous searches. Studying the structure of the submanifold of SO(3) × Z 2 -invariant supergravity scalars, we find that it has a simple interpretation as a submanifold of the 14-dimensional Z $3\atop{2}$ -invariant scalar manifold (SU(1, 1)/U(1)) 7 , for which we find a rather remarkable superpotential whose structure matches the single bit error correcting (7, 4) Hamming code. This 14-dimensional scalar manifold contains approximately one quarter of the known critical points. We also show that there exists a smooth supersymmetric domain wall which interpolates between the new N = 1 AdS4 solution and the maximally supersymmetric AdS4 vacuum. Using holography, this result indicates the existence of an N = 1 RG flow from the ABJM SCFT to a new strongly interacting conformal fixed point in the IR.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS2002

Finite-Difference Time Domain Solution of Light Scattering and Absorption by Particles in an Absorbing Medium

The three-dimensional (3-D) finite-difference time-domain (FDTD) technique has been extended to simulate light scattering and absorption by nonspherical particles embedded in an absorbing dielectric medium. A uniaxial perfectly matched layer (UPML) absorbing boundary condition is used to truncate the computational domain. When computing the single-scattering properties of a particle in an absorbing dielectric medium, we derive the single-scattering properties including scattering phase functions, extinction, and absorption efficiencies using a volume integration of the internal field. A Mie solution for light scattering and absorption by spherical particles in an absorbing medium is used to examine the accuracy of the 3-D UPML FDTD code. It is found that the errors in the extinction and absorption efficiencies from the 3-D UPML FDTD are less than similar to 2%. The errors in the scattering phase functions are typically less than similar to 5%. The errors in the asymmetry factors are less than similar to 0.l%. For light scattering by particles in free space, the accuracy of the 3-D UPML FDTD scheme is similar to a previous model.

Sun, W.↗
NASA NTRS1988

Infinite-dimensional approach to system identification of Space Control Laboratory Experiment (SCOLE)

The identification of a unique set of system parameters in large space structures poses a significant new problem in control technology. Presented is an infinite-dimensional identification scheme to determine system parameters in large flexible structures in space. The method retains the distributed nature of the structure throughout the development of the algorithm and a finite-element approximation is used only to implement the algorithm. This approach eliminates many problems associated with model truncation used in other methods of identification. The identification is formulated in Hilbert space and an optimal control technique is used to minimize weighted least squares of error between the actual and the model data. A variational approach is used to solve the problem. A costate equation, gradients of parameter variations and conditions for optimal estimates are obtained. Computer simulation studies are conducted using a shuttle-attached antenna configuration, more popularly known as the Space Control Laboratory Experiment (SCOLE) as an example. Numerical results show a close match between the estimated and true values of the parameters.

Hossain, S. A.↗
NASA NTRS2004

2004: Finite-Difference Time Domain Solution of Light Scattering by an Infinite Dielectric Column Immersed in an Absorbing Medium

The two-dimensional (2-D) finite-difference time domain (FDTD) method is applied to calculate light scattering and absorption by an arbitrarily shaped infinite column embedded in an absorbing dielectric medium. A uniaxial perfectly matched layer (UPML) absorbing boundary condition (ABC) is used to truncate the computational domain. The single-scattering properties of the infinite column embedded in the absorbing medium, including scattering phase functions, extinction and absorption efficiencies, are derived using an area integration of the internal field. An exact solution for light scattering and absorption by a circular cylinder in an absorbing medium is used to examine the accuracy of the 2-D UPML FDTD code. With use of a cell size of 1/120 incident wavelength in the FDTD calculations, the errors in the extinction and absorption efficiencies and asymmetry factors from the 2-D UPML FDTD are generally smaller than approx .1%. The errors in the scattering phase functions are typically smaller than approx .4%. Using the 2-D UPML FDTD technique, light scattering and absorption by long noncircular columns embedded in absorbing media can be accurately solved.

Sun, W.↗
DOE OSTI2022

Hamiltonian Truncation Effective Theory

Hamiltonian truncation is a non-perturbative numerical method for calculating observables of a quantum field theory. The starting point for this method is to truncate the interacting Hamiltonian to a finite-dimensional space of states spanned by the eigenvectors of the free Hamiltonian H_0 H 0 with eigenvalues below some energy cutoff E_\text{max} E max . In this work, we show how to treat Hamiltonian truncation systematically using effective field theory methodology. We define the finite-dimensional effective Hamiltonian by integrating out the states above E_\text{max} E max . The effective Hamiltonian can be computed by matching a transition amplitude to the full theory, and gives corrections order by order as an expansion in powers of 1/E_\text{max} 1 / E max . The effective Hamiltonian is non-local, with the non-locality controlled in an expansion in powers of H_0/E_\text{max} H 0 / E max . The effective Hamiltonian is also non-Hermitian, and we discuss whether this is a necessary feature or an artifact of our definition. We apply our formalism to 2D \lambda\phi^4 λ ϕ 4 theory, and compute the the leading 1/E_\text{max}^2 1 / E max 2 corrections to the effective Hamiltonian. We show that these corrections nontrivially satisfy the crucial property of separation of scales. Numerical diagonalization of the effective Hamiltonian gives residual errors of order 1/E_\text{max}^3 1 / E max 3 , as expected by our power counting. We also present the power counting for 3D \lambda \phi^4 λ ϕ 4 theory and perform calculations that demonstrate the separation of scales in this theory.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS2011

Sensitivity of C-Band Polarimetric Radar-Based Drop Size Distribution Measurements to Maximum Diameter Assumptions

The estimation of rain drop size distribution (DSD) parameters from polarimetric radar observations is accomplished by first establishing a relationship between differential reflectivity (Z(sub dr)) and the central tendency of the rain DSD such as the median volume diameter (D0). Since Z(sub dr) does not provide a direct measurement of DSD central tendency, the relationship is typically derived empirically from rain drop and radar scattering models (e.g., D0 = F[Z (sub dr)] ). Past studies have explored the general sensitivity of these models to temperature, radar wavelength, the drop shape vs. size relation, and DSD variability. Much progress has been made in recent years in measuring the drop shape and DSD variability using surface-based disdrometers, such as the 2D Video disdrometer (2DVD), and documenting their impact on polarimetric radar techniques. In addition to measuring drop shape, another advantage of the 2DVD over earlier impact type disdrometers is its ability to resolve drop diameters in excess of 5 mm. Despite this improvement, the sampling limitations of a disdrometer, including the 2DVD, make it very difficult to adequately measure the maximum drop diameter (D(sub max)) present in a typical radar resolution volume. As a result, D(sub max) must still be assumed in the drop and radar models from which D0 = F[Z(sub dr)] is derived. Since scattering resonance at C-band wavelengths begins to occur in drop diameters larger than about 5 mm, modeled C-band radar parameters, particularly Z(sub dr), can be sensitive to D(sub max) assumptions. In past C-band radar studies, a variety of D(sub max) assumptions have been made, including the actual disdrometer estimate of D(sub max) during a typical sampling period (e.g., 1-3 minutes), D(sub max) = C (where C is constant at values from 5 to 8 mm), and D(sub max) = M*D0 (where the constant multiple, M, is fixed at values ranging from 2.5 to 3.5). The overall objective of this NASA Global Precipitation Measurement Mission (GPM/PMM Science Team)-funded study is to document the sensitivity of DSD measurements, including estimates of D0, from C-band Z(sub dr) and reflectivity to this range of D(sub max) assumptions. For this study, GPM Ground Validation 2DVD's were operated under the scanning domain of the UAHuntsville ARMOR C-band dual-polarimetric radar. Approximately 7500 minutes of DSD data were collected and processed to create gamma size distribution parameters using a truncated method of moments approach. After creating the gamma parameter datasets the DSD's were then used as input to a T-matrix model for computation of polarimetric radar moments at C-band. All necessary model parameterizations, such as temperature, drop shape, and drop fall mode, were fixed at typically accepted values while the D(sub max) assumption was allowed to vary in sensitivity tests. By hypothesizing a DSD model with D(sub max) (fit) from which the empirical fit to D0 = F[Z(sub dr)] was derived via non-linear least squares regression and a separate reference DSD model with D(sub max) (truth), bias and standard error in D0 retrievals were estimated in the presence of Z(sub dr) measurement error and hypothesized mismatch in D(sub max) assumptions. Although the normalized standard error for D0 = F[Z(sub dr)r] can increase slightly (as much as from 11% to 16% for all 7500 DSDs) when the D(sub max) (fit) does not match D(sub max) (truth), the primary impact of uncertainty in D(sub max) is a potential increase in normalized bias error in D0 (from 0% to as much as 10% over all 7500 DSDs, depending on the extent of the mismatch between D(sub max) (fit) and D(sub max) (truth)). For DSDs characterized by large Z(sub dr) (Z(sub dr) > 1.5 to 2.0 dB), the normalized bias error for D0 estimation at C-band is sometimes unacceptably large (> 10%), again depending on the extent of the hypothesized D(sub max) mismatch. Modeled errors in D0 retrievals from Z(sub dr) at C-band are demonstrated in detail and comparedo similar modeled retrieval errors at S-band and X-band where the sensitivity to D(sub max) is expected to be less. The impact of D(sub max) assumptions to the retrieval of other DSD parameters such as Nw, the liquid water content normalized intercept parameter, are also explored. Likely implications for DSD retrievals using C-band polarimetric radar for GPM are assessed by considering current community knowledge regarding D(sub max) and quantifying the statistical distribution of Z(sub dr) from ARMOR over a large variety of meteorological conditions. Based on these results and the prevalence of C-band polarimetric radars worldwide, a call for more emphasis on constraining our observational estimate of D(sub max) within a typical radar resolution volume is made

Carey, Lawrence D.↗