Localized Solutions of Nonlinear Stationary Problems

Abstract Some classes of generalized Schrödinger stationary problems are studied. Under appropriated conditions is proved the existence of at least 1 + $\begin{array}{} \sum_{i=2}^{m} \end{array}$ dim Vλi pairs of nontrivial solutions if a parameter involved in the equation is large enough, where Vλi denotes the eigenspace associated to the i-th eigenvalue λi of laplacian operator with homogeneous Dirichlet boundary condition.

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Zur Lösung der BETHE-GOLDSTONE-Gleichung bei nicht-verschwindendem Gesamtimpuls I

Zeitschrift für Naturforschung A ◽

10.1515/zna-1963-0415 ◽

1963 ◽

Vol 18 (4) ◽

pp. 531-538

Author(s):

Dallas T. Hayes

Keyword(s):

Approximate Solution ◽

Analytical Solution ◽

Nuclear Matter ◽

Analytical Solutions ◽

Projection Operator ◽

Center Of Mass ◽

Separable Potential ◽

Localized Solutions ◽

Non Local ◽

Square Well

Localized solutions of the BETHE—GOLDSTONE equation for two nucleons in nuclear matter are examined as a function of the center-of-mass momentum (c. m. m.) of the two nucleons. The equation depends upon the c. m. m. as parameter due to the dependence upon the c. m. m. of the projection operator appearing in the equation. An analytical solution of the equation is obtained for a non-local but separable potential, whereby a numerical solution is also obtained. An approximate solution for small c. m. m. is calculated for a square-well potential. In the range of the approximation the two analytical solutions agree exactly.

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Meta-Descent for Online, Continual Prediction

Proceedings of the AAAI Conference on Artificial Intelligence ◽

10.1609/aaai.v33i01.33013943 ◽

2019 ◽

Vol 33 ◽

pp. 3943-3950

Author(s):

Andrew Jacobsen ◽

Matthew Schlegel ◽

Cameron Linke ◽

Thomas Degris ◽

Adam White ◽

...

Keyword(s):

Gradient Descent ◽

Large Scale ◽

Time Series Prediction ◽

Real Data ◽

Second Order ◽

Stochastic Gradient Descent ◽

Step Size ◽

Vector Approximation ◽

Prediction Problems ◽

Stationary Problems

This paper investigates different vector step-size adaptation approaches for non-stationary online, continual prediction problems. Vanilla stochastic gradient descent can be considerably improved by scaling the update with a vector of appropriately chosen step-sizes. Many methods, including AdaGrad, RMSProp, and AMSGrad, keep statistics about the learning process to approximate a second order update—a vector approximation of the inverse Hessian. Another family of approaches use meta-gradient descent to adapt the stepsize parameters to minimize prediction error. These metadescent strategies are promising for non-stationary problems, but have not been as extensively explored as quasi-second order methods. We first derive a general, incremental metadescent algorithm, called AdaGain, designed to be applicable to a much broader range of algorithms, including those with semi-gradient updates or even those with accelerations, such as RMSProp. We provide an empirical comparison of methods from both families. We conclude that methods from both families can perform well, but in non-stationary prediction problems the meta-descent methods exhibit advantages. Our method is particularly robust across several prediction problems, and is competitive with the state-of-the-art method on a large-scale, time-series prediction problem on real data from a mobile robot.

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