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Median filtering is a non-linear smoothing technique widely used in digital image processing to remove noise while retaining sharp edges. It is particularly well suited to removing outliers (impulse noise) or granular artifacts (speckle…
We introduce a modified algorithm to perform nonlinear filtering of a time series by locally linear phase space projections. Unlike previous implementations, the algorithm can be used not only for a posteriori processing but includes the…
As an alternative to PINNs, a Deep Ritz framework is proposed to solve fully nonlinear PDEs. A least-squares algorithm is advocated to decouple the nonlinearities from the variational features of several fully nonlinear PDEs. A splitting…
The binary-forking model is a parallel computation model, formally defined by Blelloch et al. very recently, in which a thread can fork a concurrent child thread, recursively and asynchronously. The model incurs a cost of $\Theta(\log n)$…
This article introduces a novel approach to learning monotone neural networks through a newly defined penalization loss. The proposed method is particularly effective in solving classes of variational problems, specifically monotone…
We consider nonlinear filters for diffusion processes when the observation and signal noises are small and of the same order. As the noise intensities approach zero, the nonlinear filter can be approximated by a certain variational problem…
This paper introduces a novel algorithmic framework for a deep neural network (DNN), which in a mathematically rigorous manner, allows us to incorporate history (or memory) into the network -- it ensures all layers are connected to one…
Although the linear method is one of the most robust algorithms for optimizing non-linearly parametrized wavefunctions in variational Monte Carlo, it suffers from a memory bottleneck due to the fact at each optimization step a generalized…
A string matching -- and more generally, sequence matching -- algorithm is presented that has a linear worst-case computing time bound, a low worst-case bound on the number of comparisons (2n), and sublinear average-case behavior that is…
We propose a nonlinear filter for noise removal based on the Laplacian for 1D and 2D data. The method utilizes the solution to a fourth-order nonlinear PDE involving the Laplacian for data reconstruction. Evolution equations are introduced…
In recent years, machine learning (ML) has been proposed to devise data-driven parametrisations of unresolved processes in dynamical numerical models. In most cases, the ML training leverages high-resolution simulations to provide a dense,…
The development of data-informed predictive models for dynamical systems is of widespread interest in many disciplines. We present a unifying framework for blending mechanistic and machine-learning approaches to identify dynamical systems…
Developing reduced-order models for turbulent flows, which contain dynamics over a wide range of scales, is an extremely challenging problem. In statistical mechanics, the Mori-Zwanzig (MZ) formalism provides a mathematically formal…
The aim of this paper is to provide a variational interpretation of the nonlinear filter in continuous time. A time-stepping procedure is introduced, consisting of successive minimization problems in the space of probability densities. The…
Machine unlearning is an emerging technique that removes the influence of a subset of training data (forget set) from a model without full retraining, with applications including privacy protection, content moderation, and model correction.…
Nonlinear filtering problems are encountered in many applications, and one solution approach is the extended Kalman filter, which is not always convergent. Therefore, it is crucial to identify conditions under which the extended Kalman…
A generalized dynamical robust nonlinear filtering framework is established for a class of Lipschitz differential algebraic systems, in which the nonlinearities appear both in the state and measured output equations. The system is assumed…
In this technical note, a recursive set-membership filtering algorithm for discrete-time nonlinear dynamical systems subject to unknown but bounded process and measurement noises is proposed. The nonlinear dynamics is represented in a…
In this paper we propose a new non-linear classifier based on a combination of locally linear classifiers. A well known optimization formulation is given as we cast the problem in a $\ell_1$ Multiple Kernel Learning (MKL) problem using many…
A class of algorithms for the solution of discrete material optimization problems in electromagnetic applications is discussed. The idea behind the algorithm is similar to that of the sequential programming. However, in each major iteration…