Article
Mathematics, Applied
P. Edel, Y. Maday
Summary: In this work, dual natural-norm is utilized to derive a posteriori error bounds with a stability constant of O(1) for parametrized linear equations. These error bounds are then translated into effective practical error estimators for reduced basis approximations using an offline/online strategy. The practical dual natural-norm error estimator is proven to outperform the classical inf-sup based error estimators in the self-adjoint case. Numerical results demonstrate its ability to accurately estimate the reduced basis approximation error, even for non-self-adjoint problems, surpassing the classical inf-sup based error estimator.
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Julia Novo, Samuele Rubino
Summary: This study examines the Proper Orthogonal Decomposition (POD) stabilized methods for the Navier-Stokes equations and analyzes two cases for approximating velocity and pressure. The numerical experiments demonstrate the accuracy and performance of the schemes in different scenarios.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2021)
Article
Mathematics, Applied
Dietmar Gallistl
Summary: Two a posteriori error estimates for the inf-sup constant are presented, providing upper and lower bounds for eigenvalue and eigenfunction errors, with a convergence of the reliability constant to 1 as mesh size decreases for guaranteed enclosures of the inf-sup constant on sufficiently fine meshes. Suboptimal efficiency estimate is noted for the second error estimate.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2021)
Article
Computer Science, Interdisciplinary Applications
Shantanu Shahane, Surya Pratap Vanka
Summary: We introduce an exponentially convergent semi-implicit meshless algorithm for solving Navier-Stokes equations in complex domains. The algorithm discretizes partial derivatives using radial basis functions (RBF) as interpolants. A collocation method is employed to derive interpolation coefficients. The algorithm shows exponential convergence with decreasing discretization errors. It has the potential to accurately and efficiently solve time-dependent and steady state fluid flows in complex domains.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics
P. Acevedo Tapia, C. Amrouche, C. Conca, A. Ghosh
Summary: This study proves the existence and uniqueness of weak and strong solutions in W-1,W-p(Ω) and W-2,W-p(Ω) with minimal regularity on the friction coefficient α. Additionally, uniform estimates are deduced for the solution with respect to α, allowing for analysis of the solution's behavior as α approaches infinity.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Correction
Mathematics, Applied
Javier de Frutos, Bosco Garcia-Archilla, Julia Novo
Summary: The proof of Lemma 5 in the study by de Frutos et al. (J Sci Comput 80: 1330-1368, 2019) is found to be incorrect, leading to an alternative statement and proof for Lemma 5. The new statement results in a halving of the convergence order of pressure in spatial mesh size, with corresponding changes in results relying on Lemma 5.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Computer Science, Software Engineering
Mario Stojanovic, Francesco Romano, Hendrik C. Kuhlmann
Summary: MaranStable is a software used for performing three-dimensional linear stability analyses of steady two-dimensional non-isothermal multiphase flows in canonical geometries. Different approximations to the Navier-Stokes equations can be selected, and the calculations are based on MATLAB with the use of parallelized operators. The software provides an easy-to-use graphical user interface for accessing MaranStable.
Article
Mathematics, Applied
Naveed Ahmed, Gunar Matthies
Summary: The paper utilizes higher-order discontinuous Galerkin methods for temporal discretizations of the transient Navier-Stokes equations, and stabilizes the spatial discretization using inf-sup stable pairs of finite element spaces with a one-level local projection stabilization method. Optimal error bounds for the velocity with viscosity parameter-independent constants are obtained for both the semidiscrete and fully discrete cases, and numerical results confirm the theoretical predictions.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2021)
Article
Mathematics, Applied
Saskia Dietze, Martin A. Grepl
Summary: Model Predictive Control (MPC) is a well-established approach for solving infinite horizon optimal control problems, but its application to large-scale systems is computationally expensive. In this study, the reduced basis method is employed as a low-dimensional surrogate model for the finite time optimal control problem, allowing for efficient offline-online computation. The proposed RB-MPC approach guarantees asymptotic stability of the closed-loop system and offers an adaptive strategy for choosing the prediction horizon. Numerical results illustrate the theoretical properties of the approach.
APPLIED MATHEMATICS AND COMPUTATION
(2023)
Article
Computer Science, Interdisciplinary Applications
Pao-Hsiung Chiu
Summary: The conventional DFIB method may suffer from spurious force oscillations (SFO) due to mass imbalance when integrating drag/lift forces around solid obstacles in incompressible flows. To solve this issue, we propose a novel method called cDFIB that effectively suppresses SFO for both small and large time step sizes.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Roman Frolov, Peter Minev, Aziz Takhirov
Summary: This article introduces a direction splitting method combined with a nonlinear iteration for compressible Navier-Stokes equations in spherical coordinates, suitable for simulations on the sphere or entire sphere. The method demonstrates good convergence and stability in the range of Mach numbers [10(-2), 10(-6)], with parameters affecting the solution shown in a geophysical test case. The algorithm is well-suited for massive parallel implementation, as demonstrated by excellent weak scalability results.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS
(2021)
Article
Mathematics, Applied
Xiaobing Feng, Hailong Qiu
Summary: This paper deals with fully discrete mixed finite element approximations of the time-dependent stochastic Stokes equations with multiplicative noise. Strong convergence rates are established for both velocity and pressure approximations in a time-averaged fashion. A stochastic inf-sup condition is used in a nonstandard way to obtain error estimates for pressure approximation.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Engineering, Multidisciplinary
Victor Bayona, Mario Sanchez-Sanz, Eduardo Fernandez-Tarrazo, Manuel Kindelan
Summary: This research focuses on developing a high-order meshfree method to model combustion inside complex geometries using radial basis functions-generated finite differences, aiming to identify different combustion regimes and improve conversion efficiency.
APPLIED MATHEMATICAL MODELLING
(2021)
Article
Chemistry, Physical
Christian Neiss, Steffen Fauser, Andreas Goerling
Summary: Recently, sigma-functionals have been introduced as new correlation functionals in Kohn-Sham (KS) methods. When used in a post-self-consistent field manner in a Gaussian basis set framework, sigma-functional methods are computationally efficient and highly accurate for main group chemistry. They can reach a chemical accuracy of 1 kcal/mol for reaction and transition state energies. Sigma-functional methods yield accurate geometries and vibrational frequencies for main group molecules superior to conventional KS methods and RPA methods.
JOURNAL OF CHEMICAL PHYSICS
(2023)
Article
Mathematics, Applied
N. Verma, S. Kumar
Summary: This paper discusses and analyzes virtual element approximations for the nonstationary Stokes problem on polygonal meshes. The proposed scheme is based on pressure-velocity formulations and constructs virtual element spaces for velocity and pressure that satisfy the discrete inf-sup (LBB) condition. Through suitable energy and L-2 projection operators, optimal error estimates are established for both semi and fully discrete schemes under minimal regularity assumptions on continuous solutions, validated through numerical experiments.
Article
Engineering, Multidisciplinary
Marco Tezzele, Lorenzo Fabris, Matteo Sidari, Mauro Sicchiero, Gianluigi Rozza
Summary: This study presents a structural optimization pipeline for modern passenger ship hulls, focusing on reducing metal raw materials used during manufacturing. Using advanced model order reduction techniques, the dimensionality of input parameters and outputs of interest is reduced. The method incorporates parameter space reduction through active subspaces into the proper orthogonal decomposition with interpolation method in a multi-fidelity setting. Comprehensive testing and error analysis demonstrate the effectiveness and usefulness of the method, especially during the preliminary design phase with high dimensional parameterizations.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
(2023)
Article
Mathematics, Applied
Monica Nonino, Francesco Ballarin, Gianluigi Rozza, Yvon Maday
Summary: This manuscript presents a reduced order model based on POD-Galerkin for unsteady fluid-structure interaction problems. The model utilizes a partitioned algorithm with semi-implicit treatment of coupling conditions. It extends existing works on reduced order models for fluid-structure interaction to unsteady problems.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Pierfrancesco Siena, Michele Girfoglio, Francesco Ballarin, Gianluigi Rozza
Summary: In this work, a machine learning-based Reduced Order Model (ROM) is proposed for the investigation of hemodynamics in a patient-specific configuration of Coronary Artery Bypass Graft (CABG). The ROM method extracts a reduced basis space using a Proper Orthogonal Decomposition (POD) algorithm and employs Artificial Neural Networks (ANNs) for computation. The Full Order Model (FOM) is represented by the Navier-Stokes equations discretized using a Finite Volume (FV) technique. The novelties of this study include the use of FV method in a patient-specific configuration, a data-driven ROM technique, and a mesh deformation strategy based on Free Form Deformation (FFD) technique. The performance of the ROM approach is analyzed in terms of error and speed-up achieved.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Engineering, Civil
Armin Sheidani, Sajad Salavatidezfouli, Giovanni Stabile, Gianluigi Rozza
Summary: In this paper, high-fidelity CFD simulations were used to examine the wake characteristics of an H-shaped vertical-axis wind turbine. Proper Orthogonal Decomposition (POD) was applied to analyze the computed flow field in the near wake of the rotor. The performance of different turbulence models was assessed, and the significant time and length scales of the predictions were highlighted using the extracted POD modes.
JOURNAL OF WIND ENGINEERING AND INDUSTRIAL AERODYNAMICS
(2023)
Article
Mathematics, Applied
Nicola R. Franco, Andrea Manzoni, Paolo Zunino
Summary: In this research, a constructive approach based on Deep Neural Networks is proposed for the approximation of the parameter-to-solution map in parameter dependent Partial Differential Equations (PDEs). The approach overcomes the limitations of existing algorithms and provides high fidelity approximation and small approximation errors. Numerical experiments demonstrate the superior performance of the proposed approach compared to traditional methods.
MATHEMATICS OF COMPUTATION
(2023)
Article
Mathematics, Applied
Martin W. W. Hess, Annalisa Quaini, Gianluigi Rozza
Summary: This work presents a new approach for reducing the models of time-dependent parametric partial differential equations based on data. By using a multi-step procedure, the proposed method can accurately recover field solutions from a limited number of large-scale simulations. Numerical experiments on the Rayleigh-Benard cavity problem demonstrate the effectiveness of this approach in both medium and high Grashof number regimes, especially in the latter regime approaching turbulent and chaotic behavior. One major advantage of this method is its ability to recover frequencies that are not present in the sampled data.
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2023)
Article
Mathematics, Interdisciplinary Applications
Dario Coscia, Laura Meneghetti, Nicola Demo, Giovanni Stabile, Gianluigi Rozza
Summary: Convolutional Neural Network (CNN) is a crucial architecture in deep learning, with trainable filters used for convolution on discrete input data. This paper introduces a continuous version of a trainable convolutional filter that can handle unstructured data. The new framework expands the usage of CNNs for more complex problems beyond discrete domains. Experimental results demonstrate that the continuous filter achieves comparable accuracy to state-of-the-art discrete filters and can be utilized as a building block in current deep learning architectures for solving problems in unstructured domains as well.
COMPUTATIONAL MECHANICS
(2023)
Article
Multidisciplinary Sciences
Michela C. Massi, Nicola R. Franco, Andrea Manzoni, Anna Maria Paganoni, Hanla A. Park, Michael Hoffmeister, Hermann Brenner, Jenny Chang-Claude, Francesca Ieva, Paolo Zunino
Summary: Within the precision medicine framework, the stratification of individual genetic susceptibility based on inherited DNA variation is crucial. Traditional Polygenic Risk Scores (PRS) approaches face challenges in modeling complex high-order non-linear SNP-SNP interactions and their effect on the phenotype. In this study, we propose a novel approach called High-order Interactions-aware Polygenic Risk Score (hiPRS), which incorporates high-order interactions and provides a simple and interpretable model. Through comprehensive simulations and real data analysis, hiPRS demonstrates superior performance in scoring and interpretability compared to state of the art methods.
Article
Chemistry, Analytical
Giorgio Gobat, Stefania Fresca, Andrea Manzoni, Attilio Frangi
Summary: Micro-electro-mechanical systems are complex structures used in various applications as sensors and actuators. This paper proposes a deep learning technique to generate accurate and real-time reduced order models for simulation and optimization of complex systems. Extensive testing on micromirrors, arches, gyroscopes, and other structures demonstrates the reliability and effectiveness of the deep learning approach in replicating complex dynamical evolutions.
Article
Nanoscience & Nanotechnology
Michele Girfoglio, Annalisa Quaini, Gianluigi Rozza
Summary: This paper introduces a pressure-based solver developed within OpenFOAM for the Euler equations. The solver uses conservative form equations with density, momentum, and total energy as variables. Two Large Eddy Simulation models are considered for stabilization and sub-grid process capturing. Numerical results demonstrate the accuracy of the approach.
Article
Engineering, Multidisciplinary
F. Mohammadizadeh, S. G. Georgiev, G. Rozza, E. Tohidi, S. Shateyi
Summary: This paper introduces the 0-Hilfer fractional Black-Scholes (0-HFBS) equation, and focuses on the existence of solutions and numerical methods. The equation is converted into a system of linear algebraic equations by collocating it with the boundary and initial conditions at Chebyshev-Gauss-Lobato points. The convergence of the method is proved, and some test problems are provided to demonstrate the effectiveness of the approach.
ALEXANDRIA ENGINEERING JOURNAL
(2023)
Article
Computer Science, Interdisciplinary Applications
Nicola Clinco, Michele Girfoglio, Annalisa Quaini, Gianluigi Rozza
Summary: In this paper, a filter stabilization technique for the mildly compressible Euler equations is presented, which relies on an indicator function to identify the regions where artificial viscosity is needed. By adopting the Evolve Filter-Relax (EFR) algorithm, the proposed technique shows superior stability and less dissipation compared to linear filter and Smagorinsky-like models, especially when using a function based on approximate deconvolution operators.
COMPUTERS & FLUIDS
(2023)
Article
Mathematics, Applied
Nicola Demo, Maria Strazzullo, Gianluigi Rozza
Summary: In this work, we propose applying physics informed supervised learning strategies to parametric partial differential equations. Our main goal is to simulate parametrized phenomena in a short amount of time by utilizing physics information for learning. This is achieved through the use of physics information in the loss function (standard physics informed neural networks), augmented input (extra feature employment), and guiding the construction of an effective neural network structure (physics informed architecture). The methodology has been tested for various equations and optimal control framework.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Review
Engineering, Multidisciplinary
Francesco Romor, Marco Tezzele, Markus Mrosek, Carsten Othmer, Gianluigi Rozza
Summary: Multi-fidelity models are important for combining information from different numerical simulations, surrogates, and sensors. This study focuses on approximating high-dimensional scalar functions with low intrinsic dimensionality. By introducing a low-dimensional bias, the curse of dimensionality can be overcome, especially for many-query applications. A gradient-based reduction of the parameter space through active subspaces or a nonlinear transformation of the input space is used to build a low-fidelity response surface, enabling nonlinear autoregressive multi-fidelity Gaussian process regression without the need for new simulations. This approach has great potential in engineering applications with limited data.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
(2023)
Article
Mechanics
Nicola Demo, Marco Tezzele, Gianluigi Rozza
Summary: In this work, a novel approach is introduced to enhance the precision of reduced order models by utilizing a multi-fidelity perspective and DeepONets. The integration of model reduction with machine learning residual learning allows the neural network to learn and infer the error introduced by the simplification operation. The exploitation of high-fidelity information for building the reduced order model and learning the residual is emphasized in the framework.
ADVANCED MODELING AND SIMULATION IN ENGINEERING SCIENCES
(2023)