相关论文: Data Assimilation: A Mathematical Introduction

Following the processing of individual topics of elementary school mathematics as content of empirical theories the question is adressed wether the associated conception of mathematics finds itself under established concepts, and how it can…

Data-driven methods play an increasingly important role in discovering geometric, structural, and semantic relationships between 3D shapes in collections, and applying this analysis to support intelligent modeling, editing, and…

We give a constructive approach for the study of integral representations of classical solutions to Poisson equations under some integrability conditions on data functions.

Variational data assimilation technique applied to identification of optimal approximations of derivatives near boundary is discussed in frames of one-dimensional wave equation. Simplicity of the equation and of its numerical scheme allows…

This draft book offers a comprehensive and rigorous treatment of the mathematical principles underlying modern deep learning. The book spans core theoretical topics, from the approximation capabilities of deep neural networks, the theory…

Embedding is a common technique for analyzing multi-dimensional data. However, the embedding projection cannot always form significant and interpretable visual structures that foreshadow underlying data patterns. We propose an approach that…

In this note, we offer a palatable introduction to the field of arithmetic dynamics. That is, we study the patterns that arise when iterating a polynomial map. This note is accessible to those who have taken an introductory proof based…

Data mining is about obtaining new knowledge from existing datasets. However, the data in the existing datasets can be scattered, noisy, and even incomplete. Although lots of effort is spent on developing or fine-tuning data mining models…

We consider data assimilation for the heat equation using a finite element space semi-discretization. The approach is optimization based, but the design of regularization operators and parameters rely on techniques from the theory of…

The paper considers a universal approach that allows one to quite simply obtain nonlinear asymptotic estimates of various summation functions. It is shown the application of this approach to the asymptotic estimation of divergent Dirichlet…

Some issues in the numerical treatment of the conformal field equations are discussed. In particular, the problem of obtaining smooth initial data for the hyperboloidal initial value problem is described and solution methods are presented.

This note collects together useful unit conversions and numerical values from early universe cosmology. It is a quick reference that can be used to make easy order-of-magnitude estimates. Included are tables for unit conversions, the…

Continuous data assimilation (CDA) techniques, most notably the nudging approach proposed by Azouani, Olson, and Titi (AOT), have been shown to be very successful in deterministic frameworks for achieving long-time synchronization between…

Variational data assimilation optimizes for an initial state of a dynamical system such that its evolution fits observational data. The physical model can subsequently be evolved into the future to make predictions. This principle is a…

Motivated by the challenge of moment recovery in hydrodynamic approximation in kinetic theory, we propose a data-driven approach for the hydrodynamic models. Inspired by continuous data assimilation, our method introduces a relaxation-based…

In this article, we show that the continuous data assimilation algorithm is valid for the 3D primitive equations of the ocean. Namely, the $L^2$ norm of the assimilated solution converge to that of the reference solution at an exponential…

In this article, we prove that data assimilation by feedback nudging can be achieved for the three-dimensional quasi-geostrophic equation in a simplified scenario using only large spatial scale observables on the dynamical boundary. On this…

An introduction to applied mathematics written for students in engineering and science. Focus is on a rigorous presentation that also builds understanding by discussion, analogy, and examples. Discussion of concepts involved in modeling…

Data assimilation (DA) aims at forecasting the state of a dynamical system by combining a mathematical representation of the system with noisy observations taking into account their uncertainties. State of the art methods are based on the…

The global activity fields of a nuclear core can be reconstructed using data assimilation. Data assimilation allows to combine measurements from instruments, and information from a model, to evaluate the best possible activity within the…