arXiv subjects

Nicholas Mueller

Publications and source records attributed to Nicholas Mueller.

6 recordsLinked to original sources

arXiv2026

Opal.jl: a comprehensive, composable framework for data assimilation in Julia

Data assimilation (DA) and inverse modelling are indispensable tools for combining physics-based models with observations, yet the software that implements them is often fragmented: sequential (Kalman and particle) filters and variational (3D/4D-Var) estimators are typically developed as separate codebases, each tied to a specific class of forward model - for example, simple time-marching schemes for ordinary differential equations (ODEs), or full finite element (FE) discretisations of partial differential equations (PDEs). In this work, we present Opal.jl, a Julia package that overcomes this fragmentation by providing a unified environment for a wide range of DA methods and forward model backends, all accessible behind a single high-level interface. Starting from a handful of basic methods, complex inference tools can be built by composing them with advanced capabilities, such as covariance localisation/inflation, online noise covariance estimation, bias-aware correction, and kriging-based calibration of reduced-order surrogates. The package integrates natively with the SciML ecosystem for ODE-governed systems, and with Gridap/GridapROMs for both full-order and reduced-order discretisations of PDEs. Notably, the API is designed to allow a reduced-order surrogate to replace a full-order solver with no change to the code, even at the driver level. We demonstrate the library on a Lorenz-63 benchmark, on a Van der Pol oscillator problem with biased observations, on a turbulent Navier-Stokes flow past a square cavity, where the unknown Reynolds number is inferred from sparse velocity and pressure measurements, and finally on a heat equation using a reduced-order surrogate instead of a more standard full-order model.

cs.MS↗

arXiv2025

Reduced basis solvers for unfitted methods on parameterized domains

In this paper, we present a unified framework for reduced basis approximations of parametrized partial differential equations defined on parameter-dependent domains. Our approach combines unfitted finite element methods with both classical and tensor-based reduced basis techniques -- particularly the tensor-train reduced basis method -- to enable efficient and accurate model reduction on general geometries. To address the challenge of reconciling geometric variability with fixed-dimensional snapshot representations, we adopt a deformation-based strategy that maps a reference configuration to each parameterized domain. Furthermore, we introduce a localization procedure to construct dictionaries of reduced subspaces and hyper-reduction approximations, which are obtained via matrix discrete empirical interpolation in our work. We extend the proposed framework to saddle-point problems by adapting the supremizer enrichment strategy to unfitted methods and deformed configurations, demonstrating that the supremizer operator can be defined on the reference configuration without loss of stability. Numerical experiments on two- and three-dimensional problems -- including Poisson, linear elasticity, incompressible Stokes and Navier-Stokes equations -- demonstrate the flexibility, accuracy and efficiency of the proposed methodology.

math.NA↗

arXiv2025

GridapROMs.jl: Efficient reduced order modelling in the Julia programming language

In this paper, we introduce GridapROMs, a Julia-based library for the numerical approximation of parameterized partial differential equations (PDEs) using a comprehensive suite of linear reduced order models (ROMs). The library is designed to be extendable and productive, leveraging an expressive high-level API built on the Gridap PDE solver backend, while achieving high performance through Julia's just-in-time compiler and advanced lazy evaluation techniques. GridapROMs is PDE-agnostic, enabling its application to a wide range of problems, including linear, nonlinear, single-field, multi-field, steady, and unsteady equations. This work details the library's key innovations, implementation principles, and core components, providing usage examples and demonstrating its capabilities by solving a fluid dynamics problem modeled by the Navier-Stokes equations in a 3D geometry.

math.NA↗

arXiv2024

A tensor-train reduced basis solver for parameterized partial differential equations on Cartesian grids

In this manuscript, we introduce the tensor-train reduced basis method, a novel projection-based reduced-order model designed for the efficient solution of parameterized partial differential equations. While reduced-order models are widely used for their computational efficiency compared to full-order models, they often involve significant offline computational costs. Our proposed approach mitigates this limitation by leveraging the tensor train format to efficiently represent high-dimensional finite element quantities. This method offers several advantages, including a reduced number of operations for constructing the reduced subspaces, a cost-effective hyper-reduction strategy for assembling the PDE residual and Jacobian, and a lower dimensionality of the projection subspaces for a given accuracy. We provide a posteriori error estimates to validate the accuracy of the method and evaluate its computational performance on benchmark problems, including the Poisson equation, heat equation, and transient linear elasticity in two- and three-dimensional domains. Although the current framework is restricted to problems defined on Cartesian grids, we anticipate that it can be extended to arbitrary shapes by integrating the tensor-train reduced basis method with unfitted finite element techniques.

math.NA↗

arXiv2023

Model order reduction with novel discrete empirical interpolation methods in space-time

This work proposes novel techniques for the efficient numerical simulation of parameterized, unsteady partial differential equations. Projection-based reduced order models (ROMs) such as the reduced basis method employ a (Petrov-)Galerkin projection onto a linear low-dimensional subspace. In unsteady applications, space-time reduced basis (ST-RB) methods have been developed to achieve a dimension reduction both in space and time, eliminating the computational burden of time marching schemes. However, nonaffine parameterizations dilute any computational speedup achievable by traditional ROMs. Computational efficiency can be recovered by linearizing the nonaffine operators via hyper-reduction, such as the empirical interpolation method in matrix form. In this work, we implement new hyper-reduction techniques explicitly tailored to deal with unsteady problems and embed them in a ST-RB framework. For each of the proposed methods, we develop a posteriori error bounds. We run numerical tests to compare the performance of the proposed ROMs against high-fidelity simulations, in which we combine the finite element method for space discretization on 3D geometries and the Backward Euler time integrator. In particular, we consider a heat equation and an unsteady Stokes equation. The numerical experiments demonstrate the accuracy and computational efficiency our methods retain with respect to the high-fidelity simulations.

math.NA↗

arXiv2022

Space-time reduced basis methods for parametrized unsteady Stokes equations

In this work, we analyse space-time reduced basis methods for the efficient numerical simulation of hemodynamics in arteries. The classical formulation of the reduced basis (RB) method features dimensionality reduction in space, while finite differences schemes are employed for the time integration of the resulting ordinary differential equation (ODE). Space-time reduced basis (ST-RB) methods extend the dimensionality reduction paradigm to the temporal dimension, projecting the full-order problem onto a low-dimensional spatio-temporal subspace. Our goal is to investigate the application of ST-RB methods to the unsteady incompressible Stokes equations, with a particular focus on stability. High-fidelity simulations are performed using the Finite Element (FE) method and BDF2 as time marching scheme. We consider two different ST-RB methods. In the first one - called ST-GRB - space-time model order reduction is achieved by means of a Galerkin projection; a spatio-temporal velocity basis enrichment procedure is introduced to guarantee stability. The second method - called ST-PGRB - is characterized by a Petrov--Galerkin projection, stemming from a suitable minimization of the FOM residual, that allows to automatically attain stability. The classical RB method - denoted as SRB-TFO - serves as a baseline for the theoretical development. Numerical tests have been conducted on an idealized symmetric bifurcation geometry and on the patient-specific one of a femoropopliteal bypass. The results show that both ST-RB methods provide accurate approximations of the high-fidelity solutions, while considerably reducing the computational cost. In particular, the ST-PGRB method exhibits the best performance, as it features a better computational efficiency while retaining accuracies in accordance with theoretical expectations.

math.NA↗