Exit Time Analysis for Approximations of Gradient Descent Trajectories Around Saddle Points

Tropical cyclone (TC) track predictions from the operational regional nonhydrostatic TC forecast system of the Taiwanese Central Weather Bureau (CWB) are examined for their sensitivities to initial and lateral boundary conditions. Five experiments are designed and discussed, each using a combination of different initial and lateral boundary conditions coming either from the CWB or the National Centers for Environmental Prediction (NCEP) global forecast system. Eight typhoons in the western Pacific Ocean with 51 cases in 2004 and 2005 are tested with the five designed experiments for the 3-day forecast. The average track forecasts are the best when both the initial and lateral boundary conditions are from the NCEP global forecast system. This reflects the generally superior performance of the NCEP global forecast system relative to that of the CWB. Using different lateral boundary conditions has a greater impact on the track than using different initial conditions. Diagnostics using piecewise inversion of potential vorticity perturbations are carried out to identify synoptic features surrounding the featured typhoon that impact the track the most in each experiment. For the two cases demonstrated with the largest track improvement using NCEP global fields, the diagnostics indicate that the prediction of the strength and extent of the subtropical high in the western Pacific plays the major role in affecting these storm tracks. Using the analysis and predictions of the CWB global forecast system as the initial and lateral boundary conditions produces an overintensified subtropical ridge in the regional TC forecast model. Because most of the typhoons studied are located in the southwestern peripheral of the western Pacific subtropical high, the stronger steering from the more intense and extended high system is the main cause of the poleward bias in the predicted typhoon tracks in the operational run, which uses the CWB global forecast fields. The study suggests that, when efforts are made to improve a regional TC forecast model, it is also critically important to improve the global forecast system that provides the lateral boundary and initial conditions to the regional system.

Influences of initial conditions and boundary conditions on regional and urban scale Eulerian air quality transport model simulations

Impact of initial and lateral open boundary conditions in a Regional Indian Ocean Model on Bay of Bengal circulation

The dynamic programming formulation of the forward principle of optimality in the solution of optimal control problems results in a partial differential equation with initial boundary condition whose solution is independent of terminal cost and terminal constraints. Based on this property, two computational algorithms are described. The first-order algorithm with minimum computer storage requirements uses only integration of a system of differential equations with specified initial conditions and numerical minimization in finite-dimensional space. The second-order algorithm is based on the differential dynamic programming approach. Either of the two algorithms may be used for problems with nondifferentiable terminal cost or terminal constraints, and the solution of problems with complicated terminal conditions (e.g., with free terminal time) is greatly simplified.

Impact of chemical initial and lateral boundary conditions on air quality prediction

Mesoscale data assimilation studies in the Middle Adriatic Sea

Satisfaction of the strict saddle property has become a standard assumption in non-convex optimization, and it ensures that many first-order optimization algorithms will almost always escape saddle points. However, functions exist in machine learning that do not satisfy this property, such as the loss function of a neural network with at least two hidden layers. First-order methods such as gradient descent may converge to non-strict saddle points of such functions, and there do not currently exist any first-order methods that reliably escape non-strict saddle points. To address this need, we demonstrate that regularizing a function with a linear term enforces the strict saddle property, and we provide justification for only regularizing locally, i.e., when the norm of the gradient falls below a certain threshold. We analyze bifurcations that may result from this form of regularization, and then we provide a selection rule for regularizers that depends only on the gradient of an objective function. This rule is shown to guarantee that gradient descent will escape the neighborhoods around a broad class of non-strict saddle points, and this behavior is demonstrated on numerical examples of non-strict saddle points common in the optimization literature.

§. 1 Introduction. In the present paper, it is proved that the maximum amplitudes of a dispersive wave, the individual waves of which have maximum and minimum group velocities, are propagated with those group velocities. According to the late Professor K. Sezawa and Dr. K. Kanai(1), the maximum amplitude should always be propagated with the group velocity of the individua_wave of infinite wave length, being inconsistent with our conclusion. It is revealed that the examples treated by them are not suitable to examine the present problem, because such dispersion laws as discussed by them have no maximum or minimum group velocities except those corresponding to infinite wave length.§. 2 Fundamental Equations.Notations. V: Amplitude, C: Phase velocity, xπ/f: Wave length, t: Time, x: Space coordinate measured in the direction of propagation.The solution being responsible for the initial conditions, V=V0(x)and _??_ when t=0The relation between c and f is assumen to be given as where U is the group velocity. (Fig. 1). Then (2.4) may be reduced to and V0, V1 and V2 are certain constants.From physical point of view, the valucs of V2(±) when t is large will be most interesting; thus we will confine our attention only to the evaluation of V2(±).§. 3 Evaluation of V for a moderate valus of t under a Special Assumption of the Initial Conditions. (Omitted)§. 4 Evaluation of V for a Large Value of t under General Initial Conditions.To evaluate the right side of (2.10) for a large value of t, we must at first consider the integral under the condition _??_ To apply the method of steepest descent, let us suppose that f is, in general, a complex variable and putIt may be concluded, after some detailed examinations, that the positions of saddle points and the steepest paths in the (X, Y) plane are given as shown in Fig. 5. (In this figure, _??_, _??_ and _??_ indicate the terminal point of integral, the sadde point and triple point _??_espectively. A saddle point coincides with a terminal point at _??_.)Fig. 6 shows the relation between P/mg and x.The integrals near the terminal points (f=a and _??_) and the saddle points are carried out. Finally, comparing these values of integrals with each other, we may come to the conclusion that the most important terms are the integrals V2(+) älong the paths through the triple points, where P/mg=U1/β-x/βt=±1, and they are given by. where V1=A+(U0-U1)α+β/mcos(_??_) and A is an integral constant.The above values are O(g-1/3), Whereas the integrals through the other saddle points or the terminal points are O(g-1/2) or O(g-1) respectively and the largest values of V2(-) obtained are O(g-1/2) Thus, for a large value of t (viz_??_ -g_??_1), the maximum amplitudes in the wave train are found to appear at the points x=(U1±β)t, in which U1+β and U1-β mean the maximum and minimum group velocities respectively.§. 5 Some Notes on the Results obtained. (4.19) and (4.20) show that in general cases the amplitude of the wave group when U has its maximum value are greater than that when U has its minimum value. It agrees with the observational fact for Rayleigh waves of higer degrees along a surface of stratified ground as already discussed by Dr. K. Kanai(1) or the present author(2).If the extremum value of U corresponds to f=∞, V2(+) will not have such large magnitude of order as (4.19) or (4.20), because in this case _??_(f) and _??_(f) may be small quantities of O(f-1), as will be easily shown by Riemann-Lebesque's Lemma.

Recent publications have presented successful implementations of simple direct adaptive control techniques in various applications. However, they also expose the fact that the convergence of the adaptive gains has remained uncertain. The gains may not converge to the ideal constant control gains predicted by the underlying linear time-invariant system considerations. As those prior conditions that were also needed for stability may not hold, this conclusion may raise doubts about the robustness of the adaptive system. This paper intends to show that the adaptive control performs perfect tracking even when the linear time-invariant solution does not exist. It is shown that the adaptation performs a ‘steepest descent’ minimization of the errors, ultimately ending with the appropriate set of control gains that fit the particular input command and initial conditions. The adaptive gains do asymptotically reach an appropriate set of bounded constant ideal gain values that solve the problem at task. Copyright © 2004 John Wiley & Sons, Ltd.

Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts

Hamiltonian systems are analyzed with a double homoclinic orbit connecting a saddle to itself. Competing centers exist. A small dissipative perturbation causes the stable and unstable manifolds of the saddle point to break apart. The stable manifolds of the saddle point are the boundaries of the basin of attraction for the competing attractors. With small dissipation, the boundaries of the basins of attraction are known to be tightly wound and spiral-like. Small changes in the initial condition can alter the equilibrium to which the solution is attracted. Near the unperturbed homoclinic orbit, the boundary of the basin of attraction consists of a large sequence of nearly homoclinic orbits surrounded by close approaches to the saddle point. The slow passage through an unperturbed homoclinic orbit (separatrix) is determined by the change in the value of the Hamiltonian from one saddle approach to the next. The probability of capture can be asymptotically approximated using this change in the Hamiltonian. The well-known leading-order change of the Hamiltonian from one saddle approach to the next is due to the effect of the perturbation on the homoclinic orbit. A logarithmic correction to this change of the Hamiltonian is shown to be due to the effect of the perturbation on the saddle point itself. It is shown that the probability of capture can be significantly altered from the well-known leading-order probability for Hamiltonian systems with double homoclinic orbits of the twisted type, an example of which is the Hamiltonian system corresponding to primary resonance. Numerical integration of the perturbed Hamiltonian system is used to verify the accuracy of the analytic formulas for the change in the Hamiltonian from one saddle approach to the next. (c) 1995 American Institute of Physics.

This study assesses the one-dimensional (1D) consolidation of unsaturated soil subjected to time-dependent loading based on the 1D consolidation theories of unsaturated soil. The differential quadrature method (DQM) is used to produce a general solution that considers various boundary conditions, various initial pore-water and pore-air distributions, and complex time-dependent loading. A special case in which the analytical solution is available in the literature is used for verification and accuracy analysis. It is found that the DQM solution can deliver more accurate results compared with the finite-difference method with a small number of sampling points. The general solution can avoid cumbersome computations in solving eigenequations encountered with the analytical solution. In addition, the proposed solution is more suitable for practical engineering because of its generality in complex initial, boundary, and loading conditions. Finally, the characteristics of the 1D consolidation of unsaturated soils under various initial pore pressure distributions, boundary conditions, and complex time-dependent loading conditions are investigated, and it is found that the initial and boundary conditions have a significant influence on the consolidation of unsaturated soils.

Adomian decomposition method for solving the Volterra integral form of the Lane–Emden equations with initial values and boundary conditions

Application of a modified donnell technique for the determination of critical loads of imperfect plates

A regional ocean model was used to simulate the annual cycle of the upper-ocean dynamics and its influence on ocean properties in the tropical western Indian Ocean. Surface winds and heat fluxes from the National Centers for Environmental Prediction (NCEP) reanalysis forced the model (Model_NCEP) with initial and lateral boundary conditions derived from the Simple Ocean Data Assimilation (SODA). The model findings were in good agreement with previous research, satellite and observational data as well as another model configuration forced by Comprehensive Ocean and Atmosphere Data Sets (COADS). The initial and lateral boundary conditions for Model_COADS were extracted from World Ocean Atlas 2001. Anticyclonic wind stress curl occurred to the north of Madagascar, and extended towards the Tanzanian coast throughout the year, leading to Ekman convergence and downwelling in that region. The lowest sea-surface height values during the year occurred between 5° and 12° S with an elongated and contracted shape. The East African Coastal Current (EACC) was in phase with the westward North-East Madagascar Current (NEMC) throughout the year with volume transports peaking in June through July in the model forced by NCEP reanalysis. The variability of the volume transport, ocean currents, temperature and salinity to the north of Madagascar on the path of the NEMC mirrored those in the middle Tanzanian shelf on the path of the EACC throughout the year. The NEMC seemed to influence the water masses on the Tanzanian shelf, with cooler and lower-salinity water in the South-West Monsoon, and warmer and saltier water during the North-East Monsoon.