A unified convergence theory for a class of iterative processes.
Unified theory for convergence results based on nonlinear estimates for iteration function and on majorizing sequences concept
Rheinboldt, W. C.↗
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UHMLE: Program description user guide
A program which computes maximum likelihood estimates for the general normal mixture is introduced. The program allows the user to fix any subsets of the mixture parameters; this fixed set may be redefined at various times in the iteration process. There is considerable time saved in computing the likelihoods if the diagonal form is specified. Hence, in the early iterations the diagonal assumption might be used, changing over to the full covariance mode later in the iteration process for a more refined solution. This flexibility allows the user to choose the sequence of parameter configurations in the iteration process which he feels will optimize the computation time required as well as possibly avoid convergence to suboptimal local maxima of the likelihood function.
Cobely, W. A.↗
H Lyman-alpha emission from photodissociation of H2O
Lyman-alpha fluorescence measurements were used to study the photodissociation of water vapor to produce H(n equals 2) atoms from threshold to 500 A. The experimental threshold of 813.0 A nearly coincides with the thermodynamic limit. Excitation cross sections rise slowly from threshold to a maximum value of (1.55 plus or minus 0.12) x 10 to the -18th sq cm at about 709 A and decrease to a negligible value below 550 A. Structure in the excitation function is attributed to predissociation of Rydberg series converging to the B(overbar)B2 state of H2O(plus).
Mentall, J. E.↗
On the rates of convergence of the finite element method
The rate of convergence of the finite element method is a function of the strategy by which the number of degrees-of-freedom are increased. Alternative strategies are examined in the light of recent theoretical results and computational experience.
Babuska, I.↗
Effects of photon noise on speckle image reconstruction with the Knox-Thompson algorithm
An analysis of the effects of photon noise on astronomical speckle image reconstruction using the Knox-Thompson algorithm is presented. It is shown that the quantities resulting from the speckle average arre biased, but that the biases are easily estimated and compensated. Calculations are also made of the convergence rate for the speckle average as a function of the source brightness. An illustration of the effects of photon noise on the image recovery process is included.
Nisenson, P.↗
Artificial dissipation and central difference schemes for the Euler and Navier-Stokes equations
An artificial dissipation model, including boundary treatment, that is employed in many central difference schemes for solving the Euler and Navier-Stokes equations is discussed. Modifications of this model such as the eigenvalue scaling suggested by upwind differencing are examined. Multistage time stepping schemes with and without a multigrid method are used to investigate the effects of changes in the dissipation model on accuracy and convergence. Improved accuracy for inviscid and viscous airfoil flow is obtained with the modified eigenvalue scaling. Slower convergence rates are experienced with the multigrid method using such scaling. The rate of convergence is improved by applying a dissipation scaling function that depends on mesh cell aspect ratio.
Swanson, R. C.↗
Artificial dissipation and central difference schemes for the Euler and Navier-Stokes equations
An artificial dissipation model, including boundary treatment, that is employed in many central difference schemes for solving the Euler and Navier-Stokes equations is discussed. Modifications of this model such as the eigenvalue scaling suggested by upwind differencing are examined.Multistage time stepping schemes with and without a multigrid method are used to investigate the effects of changes in the dissipation model on accuracy and convergence. Improved accuracy for inviscid and viscous airfoil flows is obtained with the modified eigenvalue scaling. Slower convergence rates are experienced with the multigrid method using such scaling. The rate of convergence is improved by applying a dissipation scaling function that depends on mesh cell aspect ratio.
Swanson, R. C.↗
On the Khinchin Constant
We prove known identities for the Khinchin constant and develop new identities for the more general Hoelder mean limits of continued fractions. Any of these constants can be developed as a rapidly converging series involving values of the Riemann zeta function and rational coefficients. Such identities allow for efficient numerical evaluation of the relevant constants. We present free-parameter, optimizable versions of the identities, and report numerical results.
Bailey, David H.↗
Calculation of the Full Scattering Amplitude without Partial Wave Decomposition II: Inclusion of Exchange
As is well known, the full scattering amplitude can be expressed as an integral involving the complete scattering wave function. We have shown that the integral can be simplified and used in a practical way. Initial application to electron-hydrogen scattering without exchange was highly successful. The Schrodinger equation (SE), which can be reduced to a 2d partial differential equation (pde), was solved using the finite element method. We have now included exchange by solving the resultant SE, in the static exchange approximation, which is reducible to a pair of coupled pde's. The resultant scattering amplitudes, both singlet and triplet, calculated as a function of energy are in excellent agreement with converged partial wave results.
Shertzer, Janine↗
Verification of a Byzantine-Fault-Tolerant Self-stabilizing Protocol for Clock Synchronization
This paper presents the mechanical verification of a simplified model of a rapid Byzantine-fault-tolerant self-stabilizing protocol for distributed clock synchronization systems. This protocol does not rely on any assumptions about the initial state of the system except for the presence of sufficient good nodes, thus making the weakest possible assumptions and producing the strongest results. This protocol tolerates bursts of transient failures, and deterministically converges within a time bound that is a linear function of the self-stabilization period. A simplified model of the protocol is verified using the Symbolic Model Verifier (SMV). The system under study consists of 4 nodes, where at most one of the nodes is assumed to be Byzantine faulty. The model checking effort is focused on verifying correctness of the simplified model of the protocol in the presence of a permanent Byzantine fault as well as confirmation of claims of determinism and linear convergence with respect to the self-stabilization period. Although model checking results of the simplified model of the protocol confirm the theoretical predictions, these results do not necessarily confirm that the protocol solves the general case of this problem. Modeling challenges of the protocol and the system are addressed. A number of abstractions are utilized in order to reduce the state space.
Malekpour, Mahyar R.↗
Model Checking a Byzantine-Fault-Tolerant Self-Stabilizing Protocol for Distributed Clock Synchronization Systems
This report presents the mechanical verification of a simplified model of a rapid Byzantine-fault-tolerant self-stabilizing protocol for distributed clock synchronization systems. This protocol does not rely on any assumptions about the initial state of the system. This protocol tolerates bursts of transient failures, and deterministically converges within a time bound that is a linear function of the self-stabilization period. A simplified model of the protocol is verified using the Symbolic Model Verifier (SMV) [SMV]. The system under study consists of 4 nodes, where at most one of the nodes is assumed to be Byzantine faulty. The model checking effort is focused on verifying correctness of the simplified model of the protocol in the presence of a permanent Byzantine fault as well as confirmation of claims of determinism and linear convergence with respect to the self-stabilization period. Although model checking results of the simplified model of the protocol confirm the theoretical predictions, these results do not necessarily confirm that the protocol solves the general case of this problem. Modeling challenges of the protocol and the system are addressed. A number of abstractions are utilized in order to reduce the state space. Also, additional innovative state space reduction techniques are introduced that can be used in future verification efforts applied to this and other protocols.
Malekpour, Mahyar R.↗
Finite-field Calculations of Molecular Polarizabilities Using Field-Induced Polarization Functions: Second- and Third-order Perturbation Correlation Corrections to the Coupled Hartree-Fock Polarizability of H2O
Ordinary Rayleigh-Schroudinger perturbation theory with Moller-Plesset (RSMP) partitioning is used to calculate second- and third-order correlation corrections to the CHF polarizability and dipole moment of the water molecule by a finite-field procedure. Pade approximants are found to be useful in accelerating the convergence of the property perturbation expansions. Field-induced polarization functions suitable for polarizability calculations are determined. The average polarizability calculated, neglecting vibrational averaging, with Dunning's (9s5p/4s-4s2p/2s) contracted GTO basis set augmented by field-induced lslp2d/lp polarization functions is within 3 per cent of the experimental result. Correlation corrections to the dipole moment and polarizability of the water molecule calculated by the finite-field RSMP and single + double excitation CI(SDCI) methods for the same basis set are found to be in close agreement. The RSMP approach has the advantages of being size-consistent and of being capable of greater efficiency than the SCDI method. Comparative calculations carried out using Epstein-Nesbet partitioning show that through third order RSEN correlation perturbation expansions for the dipole moment and polarizability are less rapidly convergent than RSMP expansions. However, reasonable accord with RSMP results can be achieved by using Pade approximants to accelerate the convergence of RSEN energy perturbation expansions. The convergence of RSMP property correlation expansions based on the zeroth-order uncoupled-Hartree-Fock (UCHF) and coupled-Hartree-Fock (CHF) approximations are compared through third order. Whereas the CHF + RSMP expansions are for practical purposes fully converged, the UCHF + RSMP expansions are not adequately converged.
Langhoff, S. R.↗
A Self-Stabilizing Synchronization Protocol for Arbitrary Digraphs
This paper presents a self-stabilizing distributed clock synchronization protocol in the absence of faults in the system. It is focused on the distributed clock synchronization of an arbitrary, non-partitioned digraph ranging from fully connected to 1-connected networks of nodes while allowing for differences in the network elements. This protocol does not rely on assumptions about the initial state of the system, other than the presence of at least one node, and no central clock or a centrally generated signal, pulse, or message is used. Nodes are anonymous, i.e., they do not have unique identities. There is no theoretical limit on the maximum number of participating nodes. The only constraint on the behavior of the node is that the interactions with other nodes are restricted to defined links and interfaces. This protocol deterministically converges within a time bound that is a linear function of the self-stabilization period. We present an outline of a deductive proof of the correctness of the protocol. A bounded model of the protocol was mechanically verified for a variety of topologies. Results of the mechanical proof of the correctness of the protocol are provided. The model checking results have verified the correctness of the protocol as they apply to the networks with unidirectional and bidirectional links. In addition, the results confirm the claims of determinism and linear convergence. As a result, we conjecture that the protocol solves the general case of this problem. We also present several variations of the protocol and discuss that this synchronization protocol is indeed an emergent system.
Malekpour, Mahyar R.↗
CyRRL (Cyber Resilient Reinforcement Learning for grid voltage control) [SWR-24-115]
This codebase contains a multi-agent, actor-critic reinforcement learning implementation for cyber-resilient grid voltage control. It uses a 123-bus OpenDSS system as the environment, with three-phase power flow translating nodal power injections into solved nodal voltages. The reward function penalizes deviations from nominal voltage as well as reactive power dispatch, while encouraging agents to take actions that result in fast convergence to nominal conditions. The codebase models false data injection attacks and includes functionality for training, testing, hyper-parameter tuning, and visualization.
Murphy, Sinnott [National Renewable Energy Laborat↗
A parallel variable metric optimization algorithm
An algorithm, designed to exploit the parallel computing or vector streaming (pipeline) capabilities of computers is presented. When p is the degree of parallelism, then one cycle of the parallel variable metric algorithm is defined as follows: first, the function and its gradient are computed in parallel at p different values of the independent variable; then the metric is modified by p rank-one corrections; and finally, a single univariant minimization is carried out in the Newton-like direction. Several properties of this algorithm are established. The convergence of the iterates to the solution is proved for a quadratic functional on a real separable Hilbert space. For a finite-dimensional space the convergence is in one cycle when p equals the dimension of the space. Results of numerical experiments indicate that the new algorithm will exploit parallel or pipeline computing capabilities to effect faster convergence than serial techniques.
Straeter, T. A.↗
Legendre-tau approximation for functional differential equations. III - Eigenvalue approximations and uniform stability
The stability and convergence properties of the Legendre-tau approximation for hereditary differential systems are analyzed. A characteristic equation is derived for the eigenvalues of the resulting approximate system. As a result of this derivation the uniform exponential stability of the solution semigroup is preserved under approximation. It is the key to obtaining the convergence of approximate solutions of the algebraic Riccati equation in trace norm.
Ito, K.↗
Theoretical study of the nitrogen atom hyperfine coupling constant
The nitrogen-atom isotropic hyperfine coupling constant A(iso) is studied as a function of improvements in both the one-particle and n-particle basis sets. The study underscores the importance of diffuse basis functions. For example, the (9s 5p) primitive set of Huzinaga (1965) augmented with an even-tempered diffuse s function yields values for A(iso) that are virtually identical to an energy-optimized (23s 12p) even-tempered set. The A(iso) constant is found to converge relatively quickly with increasing l quantum numbers: d, f, and g functions are estimated to contribute 2.5 + or - 0.2, 0.4 + or - 0.1, and 0.05 + or - 0.05 MHz, respectively. Full CI calibration calculations indicate that very high levels of correlation treatment are required for quantitative results. In addition, a strong coupling is observed between the one-particle and n-particle requirements. The best result, 10.4 MHz, is in excellent agreement with the accurate experimental value of 10.4509 MHz.
Bauschlicher, Charles W., Jr.↗
Optimal model reduction and frequency-weighted extension
In this paper the quadratically optimal model reduction problem for single-input, single-output systems is considered. The reduced order model is determined by minimizing the integral of the magnitude-squared of the transfer function error. It is shown that the numerator coefficients of the optimal approximant satisfy a weighted least squares problem and, on this basis, a two-step iterative algorithm is developed combining a least squares solver with a gradient minimizer. The existence of globally optimal stable solutions to the optimization problem is established, and convergence of the algorithm to stationary values of the cost function is proved. The formulation is extended to handle the frequency-weighted optimal model reduction problem. Three examples demonstrate the optimization algorithm.
Spanos, J. T.↗