arXiv research

Paper quantifies neural operators' efficiency for solving nonlinear parabolic PDEs.

We introduce a data-based approach to estimating key quantities which arise in the study of nonlinear control systems and random nonlinear dynamical systems. Our approach hinges on the observation that much of the existing linear theory may be readily extended to nonlinear systems - with a reasonable expectation of suc…

Study approximates nonlinear functionals using deep ReLU networks.

Kernel approximation via nonlinear random feature maps is widely used in speeding up kernel machines. There are two main challenges for the conventional kernel approximation methods. First, before performing kernel approximation, a good kernel has to be chosen. Picking a good kernel is a very challenging problem in its…

Novel filter uses deep BSDE for nonlinear density approximation.

Novel autoencoder method approximates Koopman operator in low dimensions.

Factor graphs have recently gained increasing attention as a unified framework for representing and constructing algorithms for signal processing, estimation, and control. One capability that does not seem to be well explored within the factor graph tool kit is the ability to handle deterministic nonlinear transformati…

A novel model uses ODE-based random features to model nonlinear dynamical systems.

The purpose of this paper is to analyze and compute the early exercise boundary for a class of nonlinear Black--Scholes equations with a nonlinear volatility which can be a function of the second derivative of the option price itself. A motivation for studying the nonlinear Black--Scholes equation with a nonlinear vola…

Method solves complex optimization problems with high probability bounds.

Extends importance sampling to nonlinear models using adjoint operators.

Leveled Homomorphic Encryption (LHE) offers a potential solution that could allow sectors with sensitive data to utilize the cloud and securely deploy their models for remote inference with Deep Neural Networks (DNN). However, this application faces several obstacles due to the limitations of LHE. One of the main probl…

New method solves robust matrix completion using nonlinear equations.

Temporal Difference Learning analysis under non-i.i.d. data and nonlinear approximation.

Enhances RSCNs with hybrid regularization for nonlinear dynamics.

Richberg technique adapted for nonlinear subequations.

While there are convergence guarantees for temporal difference (TD) learning when using linear function approximators, the situation for nonlinear models is far less understood, and divergent examples are known. Here we take a first step towards extending theoretical convergence guarantees to TD learning with nonlinear…

In many compressive sensing problems today, the relationship between the measurements and the unknowns could be nonlinear. Traditional treatment of such nonlinear relationships have been to approximate the nonlinearity via a linear model and the subsequent un-modeled dynamics as noise. The ability to more accurately ch…

Kernel-based Bayesian filter for nonlinear systems using infinite-dimensional operators.

Study on statistical inference for nonlinear stochastic approximation with Markovian data.

This paper presents an efficient Bayesian framework for solving nonlinear, high-dimensional model calibration problems. It is based on a Variational Bayesian formulation that aims at approximating the exact posterior by means of solving an optimization problem over an appropriately selected family of distributions. The…

Paper uses Koopman operator and Nyström method for efficient nonlinear control.

We investigate finite-time decoupled convergence in nonlinear two-time-scale stochastic approximation.

Recently, some works have suggested methods to combine variational probabilistic inference with Monte Carlo sampling. One promising approach is via local optimal transport. In this approach, a gradient steepest descent method based on local optimal transport principles is formulated to transform deterministically point…

Paper presents a new backward deep BSDE method for solving nonlinear FBSDE problems.

The purpose of this paper is to construct the early exercise boundary for a class of nonlinear Black--Scholes equations with a nonlinear volatility depending on the option price. We review a method how to transform the problem into a solution of a time depending nonlinear parabolic equation defined on a fixed domain. R…

A deep learning method solves nonlinear filtering problems efficiently.

This paper is a follow up to the previous author's paper on convex optimization. In that paper we began the process of adjusting greedy-type algorithms from nonlinear approximation for finding sparse solutions of convex optimization problems. We modified there three the most popular in nonlinear approximation in Banach…

High-dimensional partial differential equations (PDE) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment (CVA) models, or portfolio optimization models. The PDEs in such applications are high-dimensional as the dimension corresponds to the number …

The purpose of this survey chapter is to present a transformation technique that can be used in analysis and numerical computation of the early exercise boundary for an American style of vanilla options that can be modelled by class of generalized Black-Scholes equations. We analyze qualitatively and quantitatively the…

Paper analyzes and proves convergence of a new method for solving complex PDEs.

Families of explicit solutions are found to a nonlinear Black-Scholes equation which incorporates the feedback-effect of a large trader in case of market illiquidity. The typical solution of these families will have a payoff which approximates a strangle. These solutions were used to test numerical schemes for solving …

Improved bounds for function approximation in nonlinear sets.

Develops a framework for learning nonlinear operators using Mercer kernels.

FNOs learn solution operators of dissipative equations efficiently via spectral methods.

We are interested in approximation of a multivariate function $f(x_1,\dots,x_d)$ by linear combinations of products $u^1(x_1)\cdots u^d(x_d)$ of univariate functions $u^i(x_i)$, $i=1,\dots,d$. In the case $d=2$ it is a classical problem of bilinear approximation. In the case of approximation in the $L_2$ space the bili…

DeepRSCN models nonlinear systems using stochastic configurations.

A new method improves Bayesian filtering in nonlinear systems.

A new numerical scheme approximates nonlinear filtering densities for noisy and partial measurements.

We study sparse approximate solutions to convex optimization problems. It is known that in many engineering applications researchers are interested in an approximate solution of an optimization problem as a linear combination of elements from a given system of elements. There is an increasing interest in building such …

We analyze SA with Markovian data and nonlinear updates, overcoming prior limitations.

Zap Q-learning is a recent class of reinforcement learning algorithms, motivated primarily as a means to accelerate convergence. Stability theory has been absent outside of two restrictive classes: the tabular setting, and optimal stopping. This paper introduces a new framework for analysis of a more general class of r…

Market illiquidity, feedback effects, presence of transaction costs, risk from unprotected portfolio and other nonlinear effects in PDE based option pricing models can be described by solutions to the generalized Black-Scholes parabolic equation with a diffusion term nonlinearly depending on the option price itself. Di…

Bayesian methods solve complex nonlinear PDEs efficiently.

This paper develops a fast algorithm for solving nonlinear PDEs using sparse Cholesky factorization.

A new model approximates complex functions in parameter space.

In this paper we propose a new robust algorithm to find the optimal static replicating portfolios for general nonlinear payoff functions and give the estimate of the rate of convergence that is absent in the literature. We choose the static replication by minimizing the error bound between the nonlinear payoff function…

Proposes an INLA-based method for state and parameter estimation in nonlinear systems.