On the convergence of spectral deferred correction methods

In this work we analyze the convergence properties of the Spectral Deferred Correction (SDC) method originally proposed by Dutt et al. [BIT, 40 (2000), pp. 241--266]. The framework for this high-order ordinary differential equation (ODE) solver is typically described wherein a low-order approximation (such as forward or backward Euler) is lifted to higher order accuracy by applying the same low-order method to an error equation and then adding in the resulting defect to correct the solution. Our focus is not on solving the error equation to increase the order of accuracy, but on rewriting the solver as an iterative Picard integral equation solver. In doing so, our chief finding is that it is not the low-order solver that picks up the order of accuracy with each correction, but it is the underlying quadrature rule of the right hand side function that is solely responsible for picking up additional orders of accuracy. Our proofs point to a total of three sources of errors that SDC methods carry: the error at the current time point, the error from the previous iterate, and the numerical integration error that comes from the total number of quadrature nodes used for integration. The second of these two sources of errors is what separates SDC methods from Picard integral equation methods; our findings indicate that as long as difference between the current and previous iterate always gets multiplied by at least a constant multiple of the time step size, then high-order accuracy can be found even if the underlying "solver" is inconsistent the underlying ODE. From this vantage, we solidify the prospects of extending spectral deferred correction methods to a larger class of solvers to which we present some examples.Comment: 29 page

Convergence and stability analysis of stochastic optimization algorithms

This thesis is concerned with stochastic optimization methods. The pioneering work in the field is the article “A stochastic approximation algorithm” by Robbins and Monro [1], in which they proposed the stochastic gradient descent; a stochastic version of the classical gradient descent algorithm. Since then, many improvements and extensions of the theory have been published, as well as new versions of the original algorithm. Despite this, a problem that many stochastic algorithms still share, is the sensitivity to the choice of the step size/learning rate. One can view the stochastic gradient descent algorithm as a stochastic version of the explicit Euler scheme applied to the gradient flow equation. There are other schemes for solving differential equations numerically that allow for larger step sizes. In this thesis, we investigate the properties of some of these methods, and how they perform, when applied to stochastic optimization problems