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All sourcesNASA NTRSDOE OSTI
NASA NTRS1981

A variational approach to the question of temporal stability of equilibrium models of solar prominences. I - The formal theory

A Lagrangian approach to the equations describing small amplitude departures from equilibrium of solar prominences is presented. Seven quantities are derived which, by analogy to the concepts of energy, momentum, and angular momentum, are conserved under conditions corresponding to ignorable coordinates of classical mechanics. It is expected that these conservation laws will be useful as criteria of accuracy in obtaining eigenfrequencies for the perturbation equations when numerical techniques are used. The method is capable of handling complicated spatial dependences and has internal criteria for assessing the accuracy of any given numerical convergence scheme.

Lerche, I.↗
NASA NTRS1982

A variational approach to the study of capillary phenomena

The problem of determining the free surface of a liquid in a capillary tube, and of a liquid drop, sitting first on a horizontal plane and then on more general surfaces is considered. With some modifications, the method applies to the study of pendent drops and of rotating drops as well. The standard capillary problem, i.e. the determination of the free surface of a liquid in a thin tube of general cross section, which resuls from the simultaneous action of surface tension, boundary adhesion and gravity is discussed. It turns out that in this case the existence of the solution surface depends heavily on the validity of a simple geometric condition about the mean curvature of the boundary curve of the cross section of the capillary tube. Some particular examples of physical interest are also be discussed. Liquid drops sitting on or hanging from a fixed horizontal plane are discussed. The symmetry of the solutions (which can actually be proved, as consequence of a general symmetrization argument) now plays the chief role in deriving both the existence and the regularity of energy-minimizing configurations. When symmetry fails (this is the case, for example, when the contact angle between the drop and the plate is not constant, or when the supporting surface is not itself symmetric), then more sophisticated methods must be used. Extensions in this direction are outlined.

Emmer, M.↗
NASA NTRS1987

Variational approach to probabilistic finite elements

Probabilistic finite element method (PFEM), synthesizing the power of finite element methods with second-moment techniques, are formulated for various classes of problems in structural and solid mechanics. Time-invariant random materials, geometric properties, and loads are incorporated in terms of their fundamental statistics viz. second-moments. Analogous to the discretization of the displacement field in finite element methods, the random fields are also discretized. Preserving the conceptual simplicity, the response moments are calculated with minimal computations. By incorporating certain computational techniques, these methods are shown to be capable of handling large systems with many sources of uncertainties. By construction, these methods are applicable when the scale of randomness is not very large and when the probabilistic density functions have decaying tails. The accuracy and efficiency of these methods, along with their limitations, are demonstrated by various applications. Results obtained are compared with those of Monte Carlo simulation and it is shown that good accuracy can be obtained for both linear and nonlinear problems. The methods are amenable to implementation in deterministic FEM based computer codes.

Belytschko, T.↗
NASA NTRS1992

A variational approach to three-dimensional elasticity solutions of laminated composite plates

The displacements in a laminated composite are represented as products of two sets of unknown functions, one of which is only a function of the thickness coordinate and the other is a function of the in-plane coordinates (i.e., separation of variables approach), and the minimization of the total potential energy is reduced to a sequence of iterative linear problems. Analytical solutions are developed for cross-ply and angle-ply laminated composite rectangular plates. The solution for simply-supported cross-ply plates under sinusoidal transverse load reduces to that of Pagano. Numerical results for stresses and is placements for antisymmetric angle-ply laminates are presented. The three-dimensional elasticity solutions developed are important because they can be used to study the behavior of composite laminates, in addition to serving as reference for approximate solutions by numerical methods and two-dimensional theories.

Savoia, M.↗
NASA NTRS1991

Variational approach to probabilistic finite elements

Probabilistic finite element methods (PFEM), synthesizing the power of finite element methods with second-moment techniques, are formulated for various classes of problems in structural and solid mechanics. Time-invariant random materials, geometric properties and loads are incorporated in terms of their fundamental statistics viz. second-moments. Analogous to the discretization of the displacement field in finite element methods, the random fields are also discretized. Preserving the conceptual simplicity, the response moments are calculated with minimal computations. By incorporating certain computational techniques, these methods are shown to be capable of handling large systems with many sources of uncertainties. By construction, these methods are applicable when the scale of randomness is not very large and when the probabilistic density functions have decaying tails. The accuracy and efficiency of these methods, along with their limitations, are demonstrated by various applications. Results obtained are compared with those of Monte Carlo simulation and it is shown that good accuracy can be obtained for both linear and nonlinear problems. The methods are amenable to implementation in deterministic FEM based computer codes.

Belytschko, T.↗
NASA NTRS2013

Downscaling Satellite Precipitation with Emphasis on Extremes: A Variational 1-Norm Regularization in the Derivative Domain

The increasing availability of precipitation observations from space, e.g., from the Tropical Rainfall Measuring Mission (TRMM) and the forthcoming Global Precipitation Measuring (GPM) Mission, has fueled renewed interest in developing frameworks for downscaling and multi-sensor data fusion that can handle large data sets in computationally efficient ways while optimally reproducing desired properties of the underlying rainfall fields. Of special interest is the reproduction of extreme precipitation intensities and gradients, as these are directly relevant to hazard prediction. In this paper, we present a new formalism for downscaling satellite precipitation observations, which explicitly allows for the preservation of some key geometrical and statistical properties of spatial precipitation. These include sharp intensity gradients (due to high-intensity regions embedded within lower-intensity areas), coherent spatial structures (due to regions of slowly varying rainfall),and thicker-than-Gaussian tails of precipitation gradients and intensities. Specifically, we pose the downscaling problem as a discrete inverse problem and solve it via a regularized variational approach (variational downscaling) where the regularization term is selected to impose the desired smoothness in the solution while allowing for some steep gradients(called 1-norm or total variation regularization). We demonstrate the duality between this geometrically inspired solution and its Bayesian statistical interpretation, which is equivalent to assuming a Laplace prior distribution for the precipitation intensities in the derivative (wavelet) space. When the observation operator is not known, we discuss the effect of its misspecification and explore a previously proposed dictionary-based sparse inverse downscaling methodology to indirectly learn the observation operator from a database of coincidental high- and low-resolution observations. The proposed method and ideas are illustrated in case studies featuring the downscaling of a hurricane precipitation field.

Hurricanes↗
DOE OSTI2024

Scalable semidefinite programming approach to variational embedding for quantum many-body problems

In quantum embedding theories, a quantum many-body system is divided into localized clusters of sites which are treated with an accurate ‘high-level’ theory and glued together self-consistently by a less accurate ‘low-level’ theory at the global scale. The recently introduced variational embedding approach for quantum many-body problems combines the insights of semidefinite relaxation and quantum embedding theory to provide a lower bound on the ground-state energy that improves as the cluster size is increased. The variational embedding method is formulated as a semidefinite program (SDP), which can suffer from poor computational scaling when treated with black-box solvers. Here, we exploit the interpretation of this SDP as an embedding method to develop an algorithm which alternates parallelizable local updates of the high-level quantities with updates that enforce the low-level global constraints. Moreover, we show how translation invariance in lattice systems can be exploited to reduce the complexity of projecting a key matrix to the positive semidefinite cone.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1986

Variational calculations of subbands in a quantum well with uniform electric field - Gram-Schmidt orthogonalization approach

Variational calculations of subband eigenstates in an infinite quantum well with an applied electric field using Gram-Schmidt orthogonalized trial wave functions are presented. The results agree very well with the exact numerical solutions even up to 1200 kV/cm. It is also shown that, for increasing electric fields, the energy of the ground state decreases, while that of higher subband states increases slightly up to 1000 kV/cm and then decreases for a well size of 100 A.

Ahn, Doyeol↗
DOE OSTI2022

A Variational Inference Approach to Inverse Problems with Gamma Hyperpriors

Hierarchical models with gamma hyperpriors provide a flexible, sparse-promoting framework to bridge L 1 and L 2 regularizations in Bayesian formulations to inverse problems. Despite the Bayesian motivation for these models, existing methodologies are limited to maximum a posteriori estimation. The potential to perform uncertainty quantification has not yet been realized. This paper introduces a variational iterative alternating scheme for hierarchical inverse problems with gamma hyperpriors. The proposed variational inference approach yields accurate reconstruction, provides meaningful uncertainty quantification, and is easy to implement. In addition, it lends itself naturally to conduct model selection for the choice of hyperparameters. Here, we illustrate the performance of our methodology in several computed examples, including a deconvolution problem and sparse identification of dynamical systems from time series data.

Bayesian shrinkage↗
DOE OSTI2024

The interplay of excitonic delocalization and vibrational localization in optical lineshapes: A variational polaron approach

The dynamics of molecular excitonic systems are complicated by a competition between electronic coupling (which drives delocalization) and vibrational-electronic (vibronic) interactions (which tend to encourage electronic localization). A particular challenge of molecular systems is that they typically possess a large number of independent vibrations, with frequencies often spanning the entire spectrum of relevant electronic energy gaps. Recent spectroscopic observations and numerical simulations on a water-soluble chlorophyll-binding protein (WSCP) reveal a transition between two regimes of vibronic behavior, a Redfield-like regime in which low-frequency vibrations respond to a delocalized excitonic state, and a Förster-like regime where high-frequency vibrations act as incoherent excitations on individual pigments. Although numerical simulations can reproduce these effects, there is a need for a simple, systematic theory that accurately describes the smooth transition between these two regimes in experimental spectra. Here we address this challenge by generalizing the variational polaron transform approach of [Bloemsma et al., Chem. Phys. 481, 250 (2016)] to include arbitrary bath densities for systems with or without symmetry. We benchmark this theory against both numerical matrix-diagonalization methods and experimental 77 K fluorescence spectra for two WSCP variants, obtaining quite satisfactory agreement in both cases. Here, we apply this theory to offer an explanation for the large loss in apparent electronic coupling in the WSCP Q57K mutant and to examine the likely impact of the interplay between excitonic delocalization and vibrational localization on vibrational sideband shapes and apparent coupling strengths in high-resolution optical spectra for chlorophyll-protein complexes such as WSCP.

14 SOLAR ENERGY↗