On some extremal positive definite functions.
Received radar signal power maximizing problems in planetary exploration, analyzing abelian group real line and circle problems
Garsia, A.↗
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A multi-sheath model for highly nonlinear plasma wakefields
An improved description for nonlinear plasma wakefields with phase velocities near the speed of light is presented and compared against fully kinetic particle-in-cell simulations. These wakefields are excited by intense particle beams or lasers pushing plasma electrons radially outward, creating an ion bubble surrounded by a sheath of electrons characterized by the source term S≡−1enp(ρ−Jz/c), where ρ and Jz are the charge and axial current densities, respectively. Previously, the sheath source term was described phenomenologically with a positive-definite function, resulting in a positive definite wake potential. In reality, the wake potential is negative at the rear of the ion column which is important for self-injection and accurate beam loading models. To account for this, we introduce a multi-sheath model in which the source term, S, of the plasma wake can be negative in regions outside the ion bubble. Using this model, we obtain a new expression for the wake potential and a modified differential equation for the bubble radius. Numerical results obtained from these equations are validated against particle-in-cell simulations for unloaded and loaded wakes. The new model provides accurate predictions of the shape and duration of trailing bunch current profiles that flatten plasma wakefields. It is also used to design a trailing bunch for a desired longitudinally varying loaded wakefield. We present beam loading results for laser wakefields and discuss how the model can be improved for laser drivers in future work. Finally, we discuss differences between the predictions of the multi- and single-sheath models for beam loading.
Dalichaouch, T. N. (ORCID:0000000247350150)↗
Asymptotic form of the longitudinal correlation function for isotropic homogeneous turbulence
An asymptotic form is derived for the longitudinal correlation function for isotropic homogeneous turbulence in an incompressible fluid governed by the Navier-Stokes equation. The result is obtained from an analysis of the algebraic-differential structure of the two-point correlation tensor contained in the complex-valued Fourier transform of the probability measure over the turbulence ensemble, which reveals that the Hopf characteristic functional (the complex-valued Fourier transform of the probability measure) satisfies the Fourier interference inequality. Consequences of the expression obtained, in which the correlation function is a positive definite function of the inverse cube of the spatial coordinate as it approaches infinity, are shown to include the nonexistence of the Loitsianskii invariant, and the solution is shown to be consistent with empirical formulas.
Rosen, G.↗
Denoising and Extension of Response Functions in the Time Domain
Response functions of quantum systems, such as electron Green’s functions, magnetic, or charge susceptibilities, describe the response of a system to an external perturbation. They are the central objects of interest in field theories and quantum computing and measured directly in experiment. Further, response functions are intrinsically causal. In equilibrium and steady-state systems, they correspond to a positive spectral function in the frequency domain. Since response functions define an inner product on a Hilbert space and thereby induce a positive definite function, the properties of this function can be used to reduce noise in measured data and, in equilibrium and steady state, to construct positive definite extensions for data known on finite time intervals, which are then guaranteed to correspond to positive spectra.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Nonlinear Matrix Approximation with Radial Basis Function Components
We introduce and investigate matrix approximation by decomposition into a sum of radial basis function (RBF) components. An RBF component is a generalization of the outer product between a pair of vectors, where an RBF function replaces the scalar multiplication between individual vector elements. Even though the RBF functions are positive definite, the summation across components is not restricted to convex combinations and allows us to compute the decomposition for any real matrix that is not necessarily symmetric or positive definite. We formulate the problem of seeking such a decomposition as an optimization problem with a nonlinear and non-convex loss function. Several modern versions of the gradient descent method, including their scalable stochastic counterparts, are used to solve this problem. We provide extensive empirical evidence of the effectiveness of the RBF decomposition and that of the gradient-based fitting algorithm. While being conceptually motivated by singular value decomposition (SVD), our proposed nonlinear counterpart outperforms SVD by drastically reducing the memory required to approximate a data matrix with the same L2 error for a wide range of matrix types. For example, it leads to 2 to 6 times memory save for Gaussian noise, graph adjacency matrices, and kernel matrices. Moreover, this proximity-based decomposition can offer additional interpretability in applications that involve, e.g., capturing the inner low-dimensional structure of the data, retaining graph connectivity structure, and preserving the acutance of images.
Rebrova, Elizaveta↗
Compactly‐Supported Nonstationary Kernels for Computing Exact Gaussian Processes on Big Data
The Gaussian process (GP) is a widely used method for analyzing large-scale data sets, including spatio-temporal measurements of nonlinear processes that are now commonplace in the environmental sciences. Traditional implementations of GPs involve stationary kernels (also termed covariance functions) that limit their flexibility, and exact methods for inference that prevent application to data sets with more than about 10,000 points. Modern approaches to address stationarity assumptions generally fail to accommodate large data sets, while all attempts to address scalability focus on approximating the Gaussian likelihood, which can involve subjectivity and lead to inaccuracies. In this work, we explicitly derive an alternative kernel that can discover and encode both sparsity and nonstationarity. We embed the kernel within a fully Bayesian GP model and leverage high-performance computing resources to enable the analysis of massive data sets. We demonstrate the favorable performance of our novel kernel relative to existing exact and approximate GP methods across a variety of synthetic data examples. Furthermore, we conduct space–time prediction based on more than 1 million measurements of daily maximum temperature and verify that our results outperform state-of-the-art methods in the Earth sciences. More broadly, having access to exact GPs that use ultra-scalable, sparsity-discovering, nonstationary kernels allows GP methods to truly compete with a wide variety of machine learning methods.
Gaussian processes↗
Strictly positive real transfer functions revisited
The two most commonly used definitions of strictly positive real (SPR) transfer functions are reviewed. Contrary to what has been suggested in the literature, it is proven that the least restrictive (weak) definition of the two is clearly related to the Yacubovich-Kalman lemma. The relationship between time- and frequency-domain conditions pertaining to the weak definition of SPR systems is established.
Lozano-Leal, Rogelio↗
Davidon-Broyden rank-one minimization methods in Hilbert space with application to optimal control problems
The Davidon-Broyden class of rank one, quasi-Newton minimization methods is extended from Euclidean spaces to infinite-dimensional, real Hilbert spaces. For several techniques of choosing the step size, conditions are found which assure convergence of the associated iterates to the location of the minimum of a positive definite quadratic functional. For those techniques, convergence is achieved without the problem of the computation of a one-dimensional minimum at each iteration. The application of this class of minimization methods for the direct computation of the solution of an optimal control problem is outlined. The performance of various members of the class are compared by solving a sample optimal control problem. Finally, the sample problem is solved by other known gradient methods, and the results are compared with those obtained with the rank one quasi-Newton methods.
Straeter, T. A.↗
Lyapunov-based control designs for flexible-link manipulators
A feedback controller for the stabilization of closed-loop systems is proposed which is based on the Liapunov stability criterion. A feedback control law is first generated for the linear portion of the system equation using linear control theory. A feedback control is then designed for the nonlinear portion of the system equation by making negative the time derivative of a positive definite Liapunov function.
Juang, Jer-Nan↗
Density Estimation with Mercer Kernels
We present a new method for density estimation based on Mercer kernels. The density estimate can be understood as the density induced on a data manifold by a mixture of Gaussians fit in a feature space. As is usual, the feature space and data manifold are defined with any suitable positive-definite kernel function. We modify the standard EM algorithm for mixtures of Gaussians to infer the parameters of the density. One benefit of the approach is it's conceptual simplicity, and uniform applicability over many different types of data. Preliminary results are presented for a number of simple problems.
Macready, William G.↗
Generalized invariance principles and the theory of stability.
Description of some recent extensions of the invariance principle to more generalized dynamical systems where the state space is not locally compact and the flow is unique only in the forward direction of time. A sufficient condition for asymptotic stability of an invariant set is obtained which does not require that the Liapunov function be positive-definite. A recently developed generalized invariance principle is described which is applicable to functional differential equations, partial differential equations, and, in particular, to certain stability problems arising in thermoelasticity, viscoelasticity, and distributed nonlinear networks.
Lasalle, J. P.↗
Stability of a class of interconnected evolution systems
Stability conditions for a class of interconnected systems modeled by linear abstract evolution equations and a memoryless nonlinearity are derived. These conditions are stated in terms of the passivity of each of the subsystems and can be considered as a partial generalization of the hyperstability theorem. A Liapunov function approach is used in the proof without requiring the positive definiteness of the Liapunov function. Application to the robustness analysis of the infinite-dimensional linear quadratic regulator is also discussed.
Wen, John T.↗
Taking Measure of the Milky Way
We intend to use SIM to make definitive measurements of fundamental structural and dynamical parameters of the Milky Way. The important niche in dynamical parameter space afforded by SIM can be exploited to resolve, with unprecedented precision, a number of classical problems of Galactic astronomy. In addition, we have developed new tests of the Galactic mass distribution specifically designed for data with the special properties of SIM products. Our proposed suite of experiments will utilize the SIM Astrometric Grid as well as complementary observations of star clusters and other strategically-selected, distant "test particles" for a definitive characterization of the major components (bulge, disk, halo, satellite system) of the Milky Way. Specifically, our goals will be: 1) The determination of two fundamental parameters that play a central role in virtually every problem in Galactic astronomy, namely (a) the solar distance to the center of the Milky Way, R(sub 0); (b) the solar angular velocity around the Galactic: center, omega(sub 0). 2) The measurement of fundamental dynamical properties of the Milky Way, among them (a) the pattern speed of the central bar (b) the rotation field and velocity-dispersion tensor in the disk (c) the kinematics (mean rotational velocity and velocity dispersion tensor) of the halo as a function of position 3. The definition of the mass distribution of the Galaxy, which is dominated by the presence of dark matter. We intend to measure (a) the relative contribution of the disk and halo to the gravitational potential (b) the local volume and surface mass density of the disk (c) the shape, mass and extent of the dark halo of the Milky Way out to 250 kpc.
Majewski, Steven R.↗
A study of aircraft cruise
The long range aircraft cruise problem is analyzed using a model intermediate in complexity between energy model and point mass model. It is shown that this formulation imbeds the classical steady state cruise as the central member along with several other oscillatory extremals. The oscillatory cruise trajectories are shown to exist if the Hessian of the function QD/VT with respect to altitude and airspeed is positive definite. An expression for predicting the frequency of oscillation is developed. Qualitative effects of increasing the vehicle thrust and improving the L/D are discussed. Numerical results for two fighter aircraft and a transport aircraft are given. While oscillatory cruise mode exists for the two fighter aircraft, steady state cruise at full throttle is found to be optimal for the transport aircraft. A second variation analysis to bring out the reason for fuel savings is developed. It is shown that whenever the Hessian of the function QD/VT is positive definite, the second variation will be zero along the classical steady state cruise arc, indicating that a neighboring extremal is competitive. Comparisons with the previous point mass and energy modeled results are given.
Menon, P. K. A.↗
A variable-metric algorithm employing linear and quadratic penalties
A variable-metric algorithm is described that uses both linear and quadratic penalty terms for handling nonlinear constraints. Quadratic penalty coefficients are adjusted in a process which maintains a positive-definite matrix of second partial derivatives of the function without generating the large positive eigenvalues which cause zigzagging and slow convergence. The schemes suggested use inferred second-order properties not only in terms of the variable metric of the Davidson-Fletcher-Powell algorithm (or its relatives) but by estimating of second directional derivatives by fitting cubics to various functions along search directions.
Kelley, H. J.↗
Two-nucleon 𝑆-wave interactions at the SU(3) flavor-symmetric point with 𝑚 𝑢𝑑 ≈ 𝑚$^{phys}_𝑠$: A first lattice QCD calculation with the stochastic Laplacian Heaviside method
We report on the first application of the stochastic Laplacian Heaviside method for computing multiparticle interactions with lattice QCD to the two-nucleon system. Like the Laplacian Heaviside method, this method allows for the construction of interpolating operators which can be used to construct a set of positive-definite two-nucleon correlation functions, unlike nearly all other applications of lattice QCD to two nucleons in the literature. It also allows for a variational analysis in which optimal linear combinations of the interpolating operators are formed that couple predominantly to the eigenstates of the system. Utilizing such methods has become of paramount importance to help resolve the discrepancy in the literature on whether two nucleons in either isospin channel form a bound state at pion masses heavier than physical, with the discrepancy persisting even in the SU(3)-flavor-symmetric point with all quark masses near the physical strange quark mass. This is the first in a series of papers aimed at resolving this discrepancy. In the present work, we employ the stochastic Laplacian Heaviside method without a hexaquark operator in the basis at a lattice spacing of 𝑎 ≈0.086 fm, lattice volume of 𝐿 = 48𝑎 ≈ 4.1 fm and pion mass 𝑚 𝜋 ≈ 714 MeV. With this setup, the observed spectrum of two-nucleon energy levels strongly disfavors the presence of a bound state in either the deuteron or dineutron channel.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Layout optimization with algebraic multigrid methods
Finding the optimal position for the individual cells (also called functional modules) on the chip surface is an important and difficult step in the design of integrated circuits. This paper deals with the problem of relative placement, that is the minimization of a quadratic functional with a large, sparse, positive definite system matrix. The basic optimization problem must be augmented by constraints to inhibit solutions where cells overlap. Besides classical iterative methods, based on conjugate gradients (CG), we show that algebraic multigrid methods (AMG) provide an interesting alternative. For moderately sized examples with about 10000 cells, AMG is already competitive with CG and is expected to be superior for larger problems. Besides the classical 'multiplicative' AMG algorithm where the levels are visited sequentially, we propose an 'additive' variant of AMG where levels may be treated in parallel and that is suitable as a preconditioner in the CG algorithm.
Regler, Hans↗
Positive definiteness of gravitational field energy.
Total gravitation field energy functional with one extremum
Brill, D.↗