Related papers: Reduced basis approximation and a~posteriori error…
We derive the adjoint path-kernel method for computing parameter-gradients (linear responses) of SDEs. Its cost is almost independent of the number of parameters, and it works for non-hyperbolic systems with parameter-controlled…
Data assimilation leads naturally to a Bayesian formulation in which the posterior probability distribution of the system state, given the observations, plays a central conceptual role. The aim of this paper is to use this Bayesian…
The focus is on a model reduction framework for parameterized elliptic eigenvalue problems by a reduced basis method. In contrast to the standard single output case, one is interested in approximating several outputs simultaneously, namely…
Regressions created from experimental or simulated data enable the construction of metamodels, widely used in a variety of engineering applications. Many engineering problems involve multi-parametric physics whose corresponding…
This paper studies the numerical approximation of parametric time-dependent partial differential equations (PDEs) by proper orthogonal decomposition reduced order models (POD-ROMs). Although many papers in the literature consider reduced…
We propose and analyze an inexact gradient method based on incremental proper orthogonal decomposition (iPOD) to address the data storage difficulty in time-dependent PDE-constrained optimization, particularly for a data assimilation…
This paper addresses variational data assimilation from a learning point of view. Data assimilation aims to reconstruct the time evolution of some state given a series of observations, possibly noisy and irregularly-sampled. Using automatic…
Data assimilation (DA) in the geophysical sciences remains the cornerstone of robust forecasts from numerical models. Indeed, DA plays a crucial role in the quality of numerical weather prediction, and is a crucial building block that has…
Variational data assimilation is a technique for combining measured data with dynamical models. It is a key component of Earth system state estimation and is commonly used in weather and ocean forecasting. The approach involves a…
We derive a posteriori error estimators for an optimal control problem governed by a convection-reaction-diffusion equation; control constraints are also considered. We consider a family of low-order stabilized finite element methods to…
Data Assimilation is the process in which we improve the representation of the state of a physical system by combining information coming from a numerical model, real-world observations, and some prior modelling. It is widely used to model…
Modelling partial differential equations (PDEs) is of crucial importance in science and engineering, and it includes tasks ranging from forecasting to inverse problems, such as data assimilation. However, most previous numerical and machine…
In this paper, we propose a data-driven model reduction method to solve parabolic inverse source problems efficiently. Our method consists of offline and online stages. In the off-line stage, we explore the low-dimensional structures in the…
Recent work on Path-Dependent Partial Differential Equations (PPDEs) has shown that PPDE solutions can be approximated by a probabilistic representation, implemented in the literature by the estimation of conditional expectations using…
Partial differential equations can be used to model many problems in several fields of application including, e.g., fluid mechanics, heat and mass transfer, and electromagnetism. Accurate discretization methods (e.g., finite element or…
In this paper the authors study a non-linear elliptic-parabolic system, which is motivated by mathematical models for lithium-ion batteries. One state satisfies a parabolic reaction diffusion equation and the other one an elliptic equation.…
We consider mixed finite element approximation of a singularly perturbed fourth-order elliptic problem with two different boundary conditions, and present a new measure of the error, whose components are balanced with respect to the…
Reduced model spaces, such as reduced basis and polynomial chaos, are linear spaces $V_n$ of finite dimension $n$ which are designed for the efficient approximation of families parametrized PDEs in a Hilbert space $V$. The manifold…
We present a novel approach for solving steady-state stochastic partial differential equations (PDEs) with high-dimensional random parameter space. The proposed approach combines spatial domain decomposition with basis adaptation for each…
We consider parameter identification problems in parametrized partial differential equations (PDE). This leads to nonlinear ill-posed inverse problems. One way to solve them are iterative regularization methods, which typically require…