The general theory of compatibility conditions for the differentiability of solutions to initial-boundary value problems is well known. This paper introduces the application of that theory to numerical solutions of partial differential equations and its ramifications on the performance of high-order methods. Explicit application of boundary conditions (BCs) that are independent of the initial c...

We consider the Fast Diffusion Equation (FDE) ut = ∆u in a bounded smooth domain Ω ⊂ R with homogeneous Dirichlet conditions; the exponent range is ms = (d − 2)+/(d + 2) < m < 1. It is known that bounded positive solutions u(t, x) of such problem extinguish in a finite time T , and also that such solutions approach a separate variable solution u(t, x) ∼ (T − t)1/(1−m)f(x), as t → T−. We show th...

Abstract. Consider a Kirchhoff plate ∂2 t u+Δ 2u−∂2 t Δu = 0 in Ω× (0, T ), with boundary data u = Δu = 0 on ∂Ω×(0, T ) and unknown initial data u(·, 0) = u0 and ∂tu(·, 0) = u1 in Ω. We study an inverse problem of determining (u0, u1) from an interior observation u|ω×(0,T ). Here Ω is a bounded domain, ω a nonempty open subset of Ω, and T > 0 a suitable time duration. By means of an iterative t...

We present a new multiagent learning algorithm, RVσ(t), that builds on an earlier version, ReDVaLeR . ReDVaLeR could guarantee (a) convergence to best response against stationary opponents and either (b) constant bounded regret against arbitrary opponents, or (c) convergence to Nash equilibrium policies in self-play. But it makes two strong assumptions: (1) that it can distinguish between self-...

An important problem in the implementation of Markov Chain Monte Carlo algorithms is to determine the convergence time, or the number of iterations before the chain is close to stationarity. For many Markov chains used in practice this time is not known. Even in cases where the convergence time is known to be polynomial, the theoretical bounds are often too crude to be practical. Thus, practiti...

In this paper, we focus on the application of the Peaceman-Rachford splitting method (PRSM) to a convex minimization model with linear constraints and a separable objective function. Compared to the Douglas-Rachford splitting method (DRSM), another splitting method from which the alternating direction method of multipliers originates, PRSM requires more restrictive assumptions to ensure its con...

Gravitational search algorithm (GSA) has been successfully applied to many scientific and engineering applications in the past few years. In the original GSA and most of its variants, every agent learns from all the agents stored in the same elite group, namely K best . This type of learning strategy is in nature a fully-informed learning strategy, in which every agent has exactly the same glob...

This paper presents some theoretical convergence characteristics of Keep-Best Reproduction (KBR), a selection strategy for genetic algorithms (GAs). We have previously introduced KBR and reported encouraging results in the traveling salesman domain (Wie98a) where KBR was compared with the standard replacement strategy of replacing the two parents by their two children (STDS). Here we demonstrat...

The logistic loss is strictly convex and does not attain its infimum; consequently the solutions of logistic regression are in general off at infinity. This work provides a convergence analysis of gradient descent applied to logistic regression under no assumptions on the problem instance. Firstly, the risk is shown to converge at a rate O(ln(t)/t). Secondly, the parameter convergence is charac...