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All sourcesNASA NTRSDOE OSTI
NASA NTRS2002

Singular Vectors for Moisture-Measuring Norms

In dynamic meteorology, singular vectors (SVs) are the structures that maximize a given norm of a forecast perturbation given a tangent linear model and a quadratic constraint on the initial perturbation. That constraint is a prescription of the value of either the same or a different norm applied to the initial perturbations. In the sense that SVs maximize the forecast perturbations according to a specified measure, they may be considered as optimal perturbations. SVs are used to characterize predictability, to identify and correct possible initial errors given forecast errors, to create a set of significant perturbations for ensemble forecasting, or to determine locations for observation targeting. They are a specific application of generally defined singular vectors in mathematics.

Errico, Ronald M.↗
NASA NTRS1993

An accelerated training method for back propagation networks

The principal objective is to provide a training procedure for a feed forward, back propagation neural network which greatly accelerates the training process. A set of orthogonal singular vectors are determined from the input matrix such that the standard deviations of the projections of the input vectors along these singular vectors, as a set, are substantially maximized, thus providing an optimal means of presenting the input data. Novelty exists in the method of extracting from the set of input data, a set of features which can serve to represent the input data in a simplified manner, thus greatly reducing the time/expense to training the system.

Shelton, Robert O.↗
NASA NTRS1984

Singular value decomposition with systolic arrays

Systolic arrays for determining the singular value decomposition of a mxn, m n, matrix A of bandwidth w are presented. After A has been reduced to bidiagonal form B by means of Givens plane rotations, the singular values of B are computed by the Golub-Reinsch iteration. The products of plane rotations form the matrices of left and right singular vectors. Assuming each processor can compute or supply a plane rotation, O(wn) processors accomplish the reduction to bidiagonal form in O(np) steps, where p is the number of superdiagonals. A constant number of processors then determines each singular value in about 6n steps. The singular vectors are computed by rerouting the rotations through the arrays used for the reduction to bidiagonal form, or else along the way by employing another rectangular array of O(wm) processors.

Ipsen, I. C. F.↗
NASA NTRS1984

Singular value decomposition with systolic arrays

Systolic arrays for determining the singular value decomposition of a mxn, m greater than or equal to n, matrix A of bandwidth w are presented. After A has been reduced to bidiagonal form B by means of Givens plane rotations, the singular values of B are computed by the Golub-Reinsch iteration. The products of plane rotations form the matrices of left and right singular vectors. Assuming each processor can compute or supply a plane rotation, O(wn) processors accomplish the reduction to bidiagonal form in O(np) steps, where p is the number of superdiagonals. A constant number of processors then determines each singular value in about 6n steps. The singular vectors are computed by rerouting the rotations through the arrays used for the reduction to bidiagonal form, or else along the way by employing another rectangular array of O(wm) processors.

Ipsen, I.↗
NASA NTRS1992

Large-scale sparse singular value computations

Four numerical methods for computing the singular value decomposition (SVD) of large sparse matrices on a multiprocessor architecture are presented. Lanczos and subspace iteration-based methods for determining several of the largest singular triplets (singular values and corresponding left and right-singular vectors) for sparse matrices arising from two practical applications: information retrieval and seismic reflection tomography are emphasized. The target architectures for implementations are the CRAY-2S/4-128 and Alliant FX/80. The sparse SVD problem is well motivated by recent information-retrieval techniques in which dominant singular values and their corresponding singular vectors of large sparse term-document matrices are desired, and by nonlinear inverse problems from seismic tomography applications which require approximate pseudo-inverses of large sparse Jacobian matrices.

Berry, Michael W.↗
NASA NTRS1990

Reliable and Efficient Parallel Processing Algorithms and Architectures for Modern Signal Processing

Least-squares (LS) estimations and spectral decomposition algorithms constitute the heart of modern signal processing and communication problems. Implementations of recursive LS and spectral decomposition algorithms onto parallel processing architectures such as systolic arrays with efficient fault-tolerant schemes are the major concerns of this dissertation. There are four major results in this dissertation. First, we propose the systolic block Householder transformation with application to the recursive least-squares minimization. It is successfully implemented on a systolic array with a two-level pipelined implementation at the vector level as well as at the word level. Second, a real-time algorithm-based concurrent error detection scheme based on the residual method is proposed for the QRD RLS systolic array. The fault diagnosis, order degraded reconfiguration, and performance analysis are also considered. Third, the dynamic range, stability, error detection capability under finite-precision implementation, order degraded performance, and residual estimation under faulty situations for the QRD RLS systolic array are studied in details. Finally, we propose the use of multi-phase systolic algorithms for spectral decomposition based on the QR algorithm. Two systolic architectures, one based on triangular array and another based on rectangular array, are presented for the multiphase operations with fault-tolerant considerations. Eigenvectors and singular vectors can be easily obtained by using the multi-pase operations. Performance issues are also considered.

Liu, Kuojuey Ray↗
NASA NTRS1979

Robustness of multiloop linear feedback systems

This paper presents a new approach to the frequency-domain analysis of multiloop linear feedback systems. The properties of the return difference equation are examined using the concepts of singular values, singular vectors and the spectral norm of a matrix. A number of new tools for multiloop systems are developed which are analogous to those for scalar Nyquist and Bode analysis. These provide a generalization of the scalar frequency-domain notions such as gain, bandwidth, stability margins and M-circles, and provide considerable insight into system robustness.

Doyle, J. C.↗
NASA NTRS2003

Optimal Observations for Variational Data Assimilation

An important aspect of Ocean state estimation is the design of an observing system that allows the efficient study of climate aspects in the ocean. A solution of the design problem is presented here in terms of optimal observations that emerge as nondimensionalized singular vectors of the modified data resolution matrix. The actual computation is feasible only for scalar quantities in the limit of large observational errors. In the framework of a lo resolution North Atlantic primitive equation model it is demonstrated that such optimal observations when applied to determining the strength of the volume and heat transport across the Greenland-Scotland ridge, perform significantly better than traditional section data. On seasonal to inter-annual time-scales optimal observations are located primarily along the continental shelf and information about heat-transport, wind stress and stratification is being communicated via boundary waves and advective processes. On time-scales of about a month, sea surface height observations appear to be more efficient in reconstructing the cross-ridge heat transport than hydrographic observations. Optimal observations also provide a tool for understanding how the ocean state is effected by anomalies of integral quantities such as meridional heat transport.

Koehl, Armin↗
NASA NTRS2008

The Global Modeling and Assimilation Office (GMAO) 4d-Var and its Adjoint-based Tools

The fifth generation of the Goddard Earth Observing System (GEOS-5) Data Assimilation System (DAS) is a 3d-var system that uses the Grid-point Statistical Interpolation (GSI) system developed in collaboration with NCEP, and a general circulation model developed at Goddard, that includes the finite-volume hydrodynamics of GEOS-4 wrapped in the Earth System Modeling Framework and physical packages tuned to provide a reliable hydrological cycle for the integration of the Modern Era Retrospective-analysis for Research and Applications (MERRA). This MERRA system is essentially complete and the next generation GEOS is under intense development. A prototype next generation system is now complete and has been producing preliminary results. This prototype system replaces the GSI-based Incremental Analysis Update procedure with a GSI-based 4d-var which uses the adjoint of the finite-volume hydrodynamics of GEOS-4 together with a vertical diffusing scheme for simplified physics. As part of this development we have kept the GEOS-5 IAU procedure as an option and have added the capability to experiment with a First Guess at the Appropriate Time (FGAT) procedure, thus allowing for at least three modes of running the data assimilation experiments. The prototype system is a large extension of GEOS-5 as it also includes various adjoint-based tools, namely, a forecast sensitivity tool, a singular vector tool, and an observation impact tool, that combines the model sensitivity tool with a GSI-based adjoint tool. These features bring the global data assimilation effort at Goddard up to date with technologies used in data assimilation systems at major meteorological centers elsewhere. Various aspects of the next generation GEOS will be discussed during the presentation at the Workshop, and preliminary results will illustrate the discussion.

Todling, Ricardo↗
NASA NTRS2012

Optimal Initial Perturbations for Ensemble Prediction of the Madden-Julian Oscillation during Boreal Winter

An initialization strategy, tailored to the prediction of the Madden-Julian oscillation (MJO), is evaluated using the Goddard Earth Observing System Model, version 5 (GEOS-5), coupled general circulation model (CGCM). The approach is based on the empirical singular vectors (ESVs) of a reduced-space statistically determined linear approximation of the full nonlinear CGCM. The initial ESV, extracted using 10 years (1990-99) of boreal winter hindcast data, has zonal wind anomalies over the western Indian Ocean, while the final ESV (at a forecast lead time of 10 days) reflects a propagation of the zonal wind anomalies to the east over the Maritime Continent an evolution that is characteristic of the MJO. A new set of ensemble hindcasts are produced for the boreal winter season from 1990 to 1999 in which the leading ESV provides the initial perturbations. The results are compared with those from a set of control hindcasts generated using random perturbations. It is shown that the ESV-based predictions have a systematically higher bivariate correlation skill in predicting the MJO compared to those using the random perturbations. Furthermore, the improvement in the skill depends on the phase of the MJO. The ESV is particularly effective in increasing the forecast skill during those phases of the MJO in which the control has low skill (with correlations increasing by as much as 0.2 at 20 25-day lead times), as well as during those times in which the MJO is weak.

Ham, Yoo-Geun↗
NASA NTRS2015

Introduction to Adjoint Models

In this lecture, some fundamentals of adjoint models will be described. This includes a basic derivation of tangent linear and corresponding adjoint models from a parent nonlinear model, the interpretation of adjoint-derived sensitivity fields, a description of methods of automatic differentiation, and the use of adjoint models to solve various optimization problems, including singular vectors. Concluding remarks will attempt to correct common misconceptions about adjoint models and their utilization.

EnKF↗
NASA NTRS2017

Development and Applications of the FV3 GEOS-5 Adjoint Modeling System

GMAO has developed a highly sophisticated adjoint modeling system based on the most recent version of the finite volume cubed sphere (FV3) dynamical core. This provides a mechanism for investigating sensitivity to initial conditions and examining observation impacts. It also allows for the computation of singular vectors and for the implementation of hybrid 4DVAR. In this work we will present the scientific assessment of the new adjoint system and show results from a number of research application of the adjoint system.

Holdaway, Daniel↗
NASA NTRS2018

The Tangent Linear and Adjoint of the FV3 Dynamical Core: Development and Applications

GMAO (NASA's Global Modeling and Assimilation Office) has developed a highly sophisticated adjoint modeling system based on the most recent version of the finite volume cubed sphere (FV3) dynamical core. This provides a mechanism for investigating sensitivity to initial conditions and examining observation impacts. It also allows for the computation of singular vectors and for the implementation of hybrid 4DVAR (4-Dimensional Variational Assimilation). In this work we will present the scientific assessment of the new adjoint system and show results from a number of research application of the adjoint system.

FV3↗
NASA NTRS2018

Progress Towards Integrating the Finite-Volume Cubed-Sphere (FV3) Dynamical Core Tangent Linear and Adjoint Models into JEDI

The Joint Effort for Data assimilation Integration (JEDI) -- led by the Joint Center for Satellite Data Assimilation (JCSDA) -- is an inter-organizational endeavor to develop a common framework for performing data assimilation. This extensive framework will ultimately provide solvers, observation operators, interpolation and model interfaces using object oriented modeling. Two partners involved in JEDI use or plan to use the Finite Volume Cubed-Sphere (FV3) dynamical core to produce weather forecasts; these are NASA's Global Modeling and Assimilation Office and NOAA's National Center for Environment Prediction. In this work we present an update on ongoing efforts to integrate the FV3 tangent linear and adjoint models into the prototype JEDI framework. We setup and run a simple cycled data assimilation experiment using 4DVAR on the cubed sphere grid and with the FV3 tangent linear and adjoint models. Development of the observation operators for JEDI is separately underway. Instead of using real observations a simplified set of simulated observations will be used. We discuss the steps required to bring the FV3 linearized model into the object oriented framework and consider what would be the computational requirements of running this configuration for an operational system. FV3 uses a small time-step to ensure that small scales are well resolved, however this presents design challenges when running 4DVAR with the adjoint. An approach to storing the FV3 model trajectory has been developed that maintains the flexibility of using automatic differentiation. We discuss how this approach is incorporated into the framework. Other important uses of adjoint models include computing observation impacts and singular vectors, we consider how these tools can be included in JEDI.

Holdaway, Daniel↗
NASA NTRS1991

Singular control in minimum time spacecraft reorientation

Spacecraft reorientation is investigated numerically for an inertially symmetric rigid spacecraft with three bounded independent control torques aligned with the principal axes. The dynamical system of the spacecraft and the framework of the optimal-control problem are established in order to identify all of the potential strategies. The investigation lists bang-bang solutions and finite-order and infinite-order singular arcs, and the conditions for the finite-order singular arcs are given. Numerical examples are developed for all of the control-logic systems, and the suboptimality of the rest-to-rest maneuvers is proven for principal-axis rotations. The most efficient control technique is the singular control of infinite order, and the vector-valued singular control can be utilized in a derivative of the switching function.

Seywald, Hans↗
NASA NTRS1990

Classification of singular configurations for redundant manipulators

A general methodology is presented for the singularity analysis of kinematically redundant manipulators. The singular configurations are classified based on the possibility of reconfiguration into a nonsingular posture using self-motion. A procedure is presented to test for the possibility of self-motion at a singular configuration. Necessary and sufficient conditions for admissible tangent vectors at the singular configuration are presented, as well as a method to construct such vectors. Sufficient conditions are derived for instances when the singular system can be reconfigured into a nonsingular state by displacements along the admissible null vectors.

Bedrossian, Nazareth S.↗