This work develops adaptive time stepping algorithms for the approximation of a functional of a diffusion with jumps based on a jump augmented Monte Carlo Euler–Maruyama method, which achieve a prescribed precision. The main result is the derivation of new expansions for the time discretization error, with computable leading order term in a posteriori form, which are based on stochastic flows a...

This paper presents a sliding mode filter for removing noise. It effectively removes impulsive noise and high-frequency noise with producing smaller phase lag than linear filters. In addition, it is less prone to overshoot than previous sliding mode filters and it does not produce chattering. It is computationally inexpensive, and thus suitable for realtime applications. The proposed sliding mo...

In this paper, we propose an implicit gradient descent algorithm for the classic k-means problem. The implicit gradient step or backward Euler is solved via stochastic fixed-point iteration, in which we randomly sample a mini-batch gradient in every iteration. It is the average of the fixed-point trajectory that is carried over to the next gradient step. We draw connections between the proposed...

Delay integro-differential equations are very important in biology, medicine and many other fields. If we take random noise into account, we can obtain many stochastic delay integro-differential equations (SDIDEs). As a special case of stochastic functional differential equations (SFDEs), the fundamental theory of existence and uniqueness of the solution of SDIDEs can be regarded similarly to t...

We give a unified method to derive the strong convergence rate of backward Euler scheme for monotone SDEs in $L^p(\Omega)$-norm, with general $p \ge 4$. The results are applied SODEs polynomial growth coefficients. also generalize argument Galerkin-based SPDEs coefficients driven by multiplicative trace-class noise.

This paper studies the chaotic behavior of a series ferroresonant circuit. At first, we derive a state variable equation containing a nonlinear hysteretic circuit element represented by a Chua-type magnetization model. Second, the backward Euler method with automatic modification gives transient response of the state variable equation. The characteristic values of the state transition matrix ar...

A convergence analysis is presented for the implicit Euler and Lie splitting schemes when applied to nonlinear parabolic equations with delay. More precisely, we consider a vector field which is the sum of an unbounded dissipative operator and a delay term, where both point delays and distributed delays fit into the framework. Such equations are frequently encountered, e.g. in population dynami...