arXiv subjects

Lorenzo Tamellini

Publications and source records attributed to Lorenzo Tamellini.

At least 19 recordsLinked to original sources

arXiv2026

Sparse-grids-like surrogate models enhanced with gradient information

This work concerns surrogate modeling for quantities of interest (QoI) arising from parametric non-linear partial differential equations (PDEs). More specifically, we consider extending the sparse-grids surrogate modeling approach to incorporate derivatives of the QoI with respect to the PDE parameters. We discuss why this operation is not straightforward and propose a hybrid approach in which a sparse-grid scheme provides the collocation points in the parameter domain and a suitable polynomial space, but the surrogate model is built with a least-squares approach. We showcase our approach on several numerical tests, and we discuss in particular how its performance crucially depends on the relative cost and accuracy of evaluating the derivatives of the QoI compared to evaluating the QoI itself.

math.NA↗

arXiv2026

Bi-fidelity sparse-grid interpolation driven by a local-error estimator

Sparse grids based on Lagrange polynomials have become one of the staple methods for approximating functions that are high-dimensional and expensive to evaluate, in the context e.g. of PDE-based parametric design exploration. They are however known to be inefficient for problems requiring local refinement, such as when the target function exhibits localized features or sharp gradients. While locally-refined sparse grids based e.g. on piecewise linear polynomials are a well-established alternative to circumvent this problem, in this work we present a strategy for improving the local efficiency of Lagrangian sparse grids. We do so by building the sparse grid approximation incrementally and evaluating the function only at collocation points at which a suitable (and crucially, zero-cost) error indicator suggest that incorporating the function evaluation would significantly change the landscape of the approximation. The remaining collocation points are instead assigned values predicted by the already available sparse grid, i.e., following a bifidelity approach that reduces costs while preserving accuracy. The effectiveness of this methodology is demonstrated on several benchmark analytical functions and an engineering application concerning flashback phenomena in hydrogen-fueled perforated burners.

cs.CE↗

arXiv2026

Towards uncertainty quantification of a model for cancer-on-chip experiments

This study is a first step towards using data-informed differential models to predict and control the dynamics of cancer-on-chip experiments. We consider a conceptualized one-dimensional device, containing a cancer and a population of white blood cells. The interaction between the cancer and the population of cells is modeled by a chemotaxis model inspired by Keller-Segel-type equations, which is solved by a Hybridized Discontinuous Galerkin method. Our goal is using (synthetic) data to tune the parameters of the governing equations and to assess the uncertainty on the predictions of the dynamics due to the residual uncertainty on the parameters remaining after the tuning procedure. To this end, we apply techniques from uncertainty quantification for parametric differential models. We first perform a global sensitivity analysis using both Sobol and Morris indices to assess how parameter uncertainty impacts model predictions, and fix the value of parameters with negligible impact. Subsequently, we conduct an inverse uncertainty quantification analysis by Bayesian techniques to compute a data-informed probability distribution of the remaining model parameters. Finally, we carry out a forward uncertainty quantification analysis to compute the impact of the updated (residual) parametric uncertainties on the quantities of interest of the model. The whole procedure is sped up by using surrogate models, based on sparse-grids, to approximate the mapping of the uncertain parameters to the quantities of interest.

cs.CE↗

arXiv2025

Noise-robust multi-fidelity surrogate modelling for parametric partial differential equations

We address the challenge of constructing noise-robust surrogate models for quantities of interest (QoIs) arising from parametric partial differential equations (PDEs), using multi-fidelity collocation techniques; specifically, the Multi-Index Stochastic Collocation (MISC). In practical scenarios, the PDE evaluations used to build a response surface are often corrupted by numerical noise, especially for the low-fidelity models. This noise, which may originate from loose solver tolerances, coarse discretisations, or transient effects, can lead to overfitting in MISC, degrading surrogate quality through nonphysical oscillations and loss of convergence, thereby limiting its utility in downstream tasks like uncertainty quantification, optimisation, and control. To correct this behaviour, we propose an improved version of MISC that can automatically detect the presence of solver noise during the surrogate model construction and then ignore the exhausted fidelities. Our approach monitors the spectral decay of the surrogate at each iteration, identifying stagnation in the coefficient spectrum that signals the onset of noise. Once detected, the algorithm selectively halts the use of noisy fidelities, focusing computational resources on those fidelities that still provide meaningful information. The effectiveness of this approach is numerically validated on two challenging test cases: a parabolic advection--diffusion PDE with uncertain coefficients, and a parametric turbulent incompressible Navier--Stokes problem. The results showcase the accuracy and robustness of the resulting multi-fidelity surrogate and its capability to extract relevant information, even from under-resolved meshes not suitable for reliable single-fidelity computations.

math.NA↗

arXiv2024

An efficient slope stability algorithm with physically consistent parametrisation of slip surfaces

This paper presents an optimised algorithm implementing the method of slices for analysing the stability of slopes. The algorithm adopts an improved physically based parameterisation of slip lines according to their geometrical characteristics at the endpoints, which facilitates the identification of all viable failure mechanisms while excluding unrealistic ones. The minimisation routine combines a preliminary discrete calculation of the factor of safety over a coarse grid covering the above parameter space with a subsequent continuous exploration of the most promising region via the simplex optimisation. This reduces computational time up to about 92% compared to conventional approaches that rely on the discrete calculation of the factor of safety over a fine grid covering the entire search space. Significant savings of computational time are observed with respect to recently published heuristic algorithms, which enable a continuous exploration of the entire parametric space. These efficiency gains are particularly advantageous for numerically demanding applications like, for example, the statistical assessment of slopes with uncertain mechanical, hydraulic and geometrical properties. The novel physically based parametrisation of the slip geometry and the adoption of a continuous local search allow exploration of parameter combinations that are necessarily neglected by standard grid-based approaches, leading to an average improvement in accuracy of about 5%.

cs.CE↗

arXiv2024

Democratizing Uncertainty Quantification

Uncertainty Quantification (UQ) is vital to safety-critical model-based analyses, but the widespread adoption of sophisticated UQ methods is limited by technical complexity. In this paper, we introduce UM-Bridge (the UQ and Modeling Bridge), a high-level abstraction and software protocol that facilitates universal interoperability of UQ software with simulation codes. It breaks down the technical complexity of advanced UQ applications and enables separation of concerns between experts. UM-Bridge democratizes UQ by allowing effective interdisciplinary collaboration, accelerating the development of advanced UQ methods, and making it easy to perform UQ analyses from prototype to High Performance Computing (HPC) scale. In addition, we present a library of ready-to-run UQ benchmark problems, all easily accessible through UM-Bridge. These benchmarks support UQ methodology research, enabling reproducible performance comparisons. We demonstrate UM-Bridge with several scientific applications, harnessing HPC resources even using UQ codes not designed with HPC support.

cs.MS↗

arXiv2024

Two Nitsche-based mixed finite element discretizations for the seepage problem in Richards' equation

This paper proposes two algorithms to impose seepage boundary conditions in the context of Richards' equation for groundwater flows in unsaturated media. Seepage conditions are non-linear boundary conditions, that can be formulated as a set of unilateral constraints on both the pressure head and the water flux at the ground surface, together with a complementarity condition: these conditions in practice require switching between Neumann and Dirichlet boundary conditions on unknown portions on the boundary. Upon realizing the similarities of these conditions with unilateral contact problems in mechanics, we take inspiration from that literature to propose two approaches: the first method relies on a strongly consistent penalization term, whereas the second one is obtained by an hybridization approach, in which the value of the pressure on the surface is treated as a separate set of unknowns. The flow problem is discretized in mixed form with div-conforming elements so that the water mass is preserved. Numerical experiments show the validity of the proposed strategy in handling the seepage boundary conditions on geometries with increasing complexity.

math.NA↗

arXiv2024

Data-informed uncertainty quantification for laser-based powder bed fusion additive manufacturing

We present an efficient approach to quantify the uncertainties associated with the numerical simulations of the laser-based powder bed fusion of metals processes. Our study focuses on a thermomechanical model of an Inconel 625 cantilever beam, based on the AMBench2018-01 benchmark proposed by the National Institute of Standards and Technology (NIST). The proposed approach consists of a forward uncertainty quantification analysis of the residual strains of the cantilever beam given the uncertainty in some of the parameters of the numerical simulation, namely the powder convection coefficient and the activation temperature. The uncertainty on such parameters is modelled by a data-informed probability density function obtained by a Bayesian inversion procedure, based on the displacement experimental data provided by NIST. To overcome the computational challenges of both the Bayesian inversion and the forward uncertainty quantification analysis we employ a multi-fidelity surrogate modelling technique, specifically the multi-index stochastic collocation method. The proposed approach allows us to achieve a 33\% reduction in the uncertainties on the prediction of residual strains compared with what we would get basing the forward UQ analysis on a-priori ranges for the uncertain parameters, and in particular the mode of the probability density function of such quantities (i.e., its ``most likely value'', roughly speaking) results to be in good agreement with the experimental data provided by NIST, even though only displacement data were used for the Bayesian inversion procedure.

cs.CE↗

arXiv2023

The Sparse Grids Matlab kit -- a Matlab implementation of sparse grids for high-dimensional function approximation and uncertainty quantification

The Sparse Grids Matlab Kit provides a Matlab implementation of sparse grids, and can be used for approximating high-dimensional functions and, in particular, for surrogate-model-based uncertainty quantification. It is lightweight, high-level and easy to use, good for quick prototyping and teaching; however, it is equipped with some features that allow its use also in realistic applications. The goal of this paper is to provide an overview of the data structure and of the mathematical aspects forming the basis of the software, as well as comparing the current release of our package to similar available software.

cs.MS↗

arXiv2023

Lowering the Entry Bar to HPC-Scale Uncertainty Quantification

Treating uncertainties in models is essential in many fields of science and engineering. Uncertainty quantification (UQ) on complex and computationally costly numerical models necessitates a combination of efficient model solvers, advanced UQ methods and HPC-scale resources. The resulting technical complexities as well as lack of separation of concerns between UQ and model experts is holding back many interesting UQ applications. The aim of this paper is to close the gap between advanced UQ methods and advanced models by removing the hurdle of complex software stack integration, which in turn will offer a straightforward way to scale even prototype-grade UQ applications to high-performance resources. We achieve this goal by introducing a parallel software architecture based on UM-Bridge, a universal interface for linking UQ and models. We present three realistic applications from different areas of science and engineering, scaling from single machines to large clusters on the Google Cloud Platform.

cs.DC↗

arXiv2023

Sparse-grids uncertainty quantification of part-scale additive manufacturing processes

The present paper aims at applying uncertainty quantification methodologies to process simulations of powder bed fusion of metal. In particular, for a part-scale thermomechanical model of an Inconel 625 super-alloy beam, we study the uncertainties of three process parameters, namely the activation temperature, the powder convection coefficient and the gas convection coefficient. First, we perform a variance-based global sensitivity analysis to study how each uncertain parameter contributes to the variability of the beam displacements. The results allow us to conclude that the gas convection coefficient has little impact and can therefore be fixed to a constant value for subsequent studies. Then, we conduct an inverse uncertainty quantification analysis, based on a Bayesian approach on synthetic displacements data, to quantify the uncertainties of the two remaining parameters, namely the activation temperature and the powder convection coefficient. Finally, we use the results of the inverse uncertainty quantification analysis to perform a data-informed forward uncertainty quantification analysis of the residual strains. Crucially, we make use of surrogate models based on sparse grids to keep to a minimum the computational burden of every step of the uncertainty quantification analysis. The proposed uncertainty quantification workflow allows us to substantially ease the typical trial-and-error approach used to calibrate power bed fusion part-scale models, and to greatly reduce uncertainties on the numerical prediction of the residual strains. In particular, we demonstrate the possibility of using displacement measurements to obtain a data-informed probability density function of the residual strains, a quantity much more complex to measure than displacements.

cs.CE↗

arXiv2023

Combining noisy well data and expert knowledge in a Bayesian calibration of a flow model under uncertainties: an application to solute transport in the Ticino basin

Groundwater flow modeling is commonly used to calculate groundwater heads, estimate groundwater flow paths and travel times, and provide insights into solute transport processes within an aquifer. However, the values of input parameters that drive groundwater flow models are often highly uncertain due to subsurface heterogeneity and geologic complexity in combination with lack of measurements/unreliable measurements. This uncertainty affects the accuracy and reliability of model outputs. Therefore, parameters' uncertainty must be quantified before adopting the model as an engineering tool. In this study, we model the uncertain parameters as random variables and use a Bayesian inversion approach to obtain a posterior,data-informed, probability density function (pdf) for them: in particular, the likelihood function we consider takes into account both well measurements and our prior knowledge about the extent of the springs in the domain under study. To keep the modelistic and computational complexities under control, we assume Gaussianity of the posterior pdf of the parameters. To corroborate this assumption, we run an identifiability analysis of the model: we apply the inversion procedure to several sets of synthetic data polluted by increasing levels of noise, and we determine at which levels of noise we can effectively recover the "true value" of the parameters. We then move to real well data (coming from the Ticino River basin, in northern Italy, and spanning a month in summer 2014), and use the posterior pdf of the parameters as a starting point to perform an Uncertainty Quantification analysis on groundwater travel-time distributions.

cs.CE↗

arXiv2022

Combining the Morris Method and Multiple Error Metrics to Assess Aquifer Characteristics and Recharge in the Lower Ticino Basin, in Italy

Groundwater flow model accuracy is often limited by the uncertainty in model parameters that characterize aquifer properties and aquifer recharge. Aquifer properties such as hydraulic conductivity can have an uncertainty spanning orders of magnitude. Meanwhile, parameters used to configure model boundary conditions can introduce additional uncertainty. In this study, the Morris Method sensitivity analysis is performed on multiple quantities of interest to assess the sensitivity of a steady-state groundwater flow model to uncertain input parameters. The Morris Method determines which of these parameters are less influential on model outputs. Uninfluential parameters can be set constant during subsequent parameter optimization to reduce computational expense. Combining multiple quantities of interest (e.g., RMSE, groundwater fluxes) when performing both the Morris Method and parameter optimization offers a more complete assessment of groundwater models, providing a more reliable and physically consistent estimate of uncertain parameters. The parameter optimization procedure also provides us an estimate of the residual uncertainty in the parameter values, resulting in a more complete estimate of the remaining uncertainty. By employing such techniques, the current study was able to estimate the aquifer hydraulic conductivity and recharge rate due to rice field irrigation in a groundwater basin in Northern Italy, revealing that a significant proportion of surficial aquifer recharge (approximately 81-94%) during the later summer is due to the flood irrigation practices applied to these fields.

cs.CE↗

arXiv2021

Comparing Multi-Index Stochastic Collocation and Multi-Fidelity Stochastic Radial Basis Functions for Forward Uncertainty Quantification of Ship Resistance

This paper presents a comparison of two multi-fidelity methods for the forward uncertainty quantification of a naval engineering problem. Specifically, we consider the problem of quantifying the uncertainty of the hydrodynamic resistance of a roll-on/roll-off passengers ferry advancing in calm water and subject to two operational uncertainties (ship speed and payload). The first four statistical moments (mean, variance, skewness, kurtosis), and the probability density function for such quantity of interest (QoI) are computed with two multi-fidelity methods, i.e., the Multi-Index Stochastic Collocation (MISC) method and an adaptive multi-fidelity Stochastic Radial Basis Functions (SRBF) algorithm. The QoI is evaluated via computational fluid dynamics simulations, which are performed with the in-house unsteady Reynolds-Averaged Navier-Stokes (RANS) multi-grid solver $χ$navis. The different fidelities employed by both methods are obtained by stopping the RANS solver at different grid levels of the multi-grid cycle. The performance of both methods are presented and discussed: in a nutshell, the findings suggest that, at least for the current implementations of both algorithms, MISC could be preferred whenever a limited computational budget is available, whereas for a larger computational budget SRBFs seem to be preferable, thanks to its robustness to the numerical noise in the evaluations of the QoI.

math.NA↗

arXiv2021

On the convergence of adaptive stochastic collocation for elliptic partial differential equations with affine diffusion

Convergence of an adaptive collocation method for the stationary parametric diffusion equation with finite-dimensional affine coefficient is shown. The adaptive algorithm relies on a recently introduced residual-based reliable a posteriori error estimator. For the convergence proof, a strategy recently used for a stochastic Galerkin method with an hierarchical error estimator is transferred to the collocation setting. Extensions to other variants of adaptive collocation methods (including the classical one proposed in the paper "Dimension-adaptive tensor-product quadratuture" Computing (2003) by T. Gerstner and M. Griebel) is explored.

math.NA↗

arXiv2020

A note on tools for prediction under uncertainty and identifiability of SIR-like dynamical systems for epidemiology

We provide an overview of the methods that can be used for prediction under uncertainty and data fitting of dynamical systems, and of the fundamental challenges that arise in this context. The focus is on SIR-like models, that are being commonly used when attempting to predict the trend of the COVID-19 pandemic. In particular, we raise a warning flag about identifiability of the parameters of SIR-like models; often, it might be hard to infer the correct values of the parameters from data, even for very simple models, making it non-trivial to use these models for meaningful predictions. Most of the points that we touch upon are actually generally valid for inverse problems in more general setups.

stat.ME↗

arXiv2020

Uncertainty Quantification of Ship Resistance via Multi-Index Stochastic Collocation and Radial Basis Function Surrogates: A Comparison

This paper presents a comparison of two methods for the forward uncertainty quantification (UQ) of complex industrial problems. Specifically, the performance of Multi-Index Stochastic Collocation (MISC) and adaptive multi-fidelity Stochastic Radial Basis Functions (SRBF) surrogates is assessed for the UQ of a roll-on/roll-off passengers ferry advancing in calm water and subject to two operational uncertainties, namely the ship speed and draught. The estimation of expected value, standard deviation, and probability density function of the (model-scale) resistance is presented and discussed; the required simulations are obtained by the in-house unsteady multi-grid Reynolds Averaged Navier-Stokes (RANS) solver $χ$navis. Both MISC and SRBF use as multi-fidelity levels the evaluations on the different grid levels intrinsically employed by the RANS solver for multi-grid acceleration; four grid levels are used here, obtained as isotropic coarsening of the initial finest mesh. The results suggest that MISC could be preferred when only limited data sets are available. For larger data sets both MISC and SRBF represent a valid option, with a slight preference for SRBF, due to its robustness to noise.

math.NA↗

arXiv2020

Compressive Isogeometric Analysis

This work is motivated by the difficulty in assembling the Galerkin matrix when solving Partial Differential Equations (PDEs) with Isogeometric Analysis (IGA) using B-splines of moderate-to-high polynomial degree. To mitigate this problem, we propose a novel methodology named CossIGA (COmpreSSive IsoGeometric Analysis), which combines the IGA principle with CORSING, a recently introduced sparse recovery approach for PDEs based on compressive sensing. CossIGA assembles only a small portion of a suitable IGA Petrov-Galerkin discretization and is effective whenever the PDE solution is sufficiently sparse or compressible, i.e., when most of its coefficients are zero or negligible. The sparsity of the solution is promoted by employing a multilevel dictionary of B-splines as opposed to a basis. Thanks to sparsity and the fact that only a fraction of the full discretization matrix is assembled, the proposed technique has the potential to lead to significant computational savings. We show the effectiveness of CossIGA for the solution of the 2D and 3D Poisson equation over nontrivial geometries by means of an extensive numerical investigation.

math.NA↗