Blood Flow Velocity Field Estimation Via Spatial Regression With PDE Penalization

Azzimonti, Laura; Sangalli, Laura M.; Secchi, Piercesare; Domanin, Maurizio; Nobile, Fabio

doi:10.1080/01621459.2014.946036

Cited by 33 publications

(90 citation statements)

References 22 publications

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“…This is useful in many applications of interest, and it is, for instance, the case of the estimation of the blood velocity field from echo-Doppler data presented in [1]; the echoDoppler data represent, in fact, the mean velocity of the blood cells on a subdomain within an artery and cannot be approximated with pointwise observations. Let D i ⊂ Ω, for i = 1, .…”

Section: Surface Estimator For Areal Datamentioning

confidence: 99%

“…Moreover, the smoothing parameter λ may be selected via standard methods as the AIC, the BIC, the UPRE, and the GCV criterion; see, e.g., [17,22,29] and references therein. In particular the GCV, which is available in a closed form for the proposed models, is detailed in [1].…”

Section: Thanks To the Fact That A H Is Continuous And Coercive Inmentioning

confidence: 99%

“…In [1], exploiting the linearity of the estimatorf h in the observations, some classical inferential tools such as approximate pointwise confidence bands and prediction intervals are derived, providing measures for uncertainty quantification within such models. Moreover, the smoothing parameter λ may be selected via standard methods as the AIC, the BIC, the UPRE, and the GCV criterion; see, e.g., [17,22,29] and references therein.…”

Section: Thanks To the Fact That A H Is Continuous And Coercive Inmentioning

confidence: 99%

“…It is thus important to choose correct diffusion, transport, reaction, and boundary terms, which translate the prior knowledge of the phenomenon. An application of this method to a real problem can be found in [1], where the penalized differential operator describes the physiological knowledge of a blood velocity profile. Notice also that by penalizing the misfit of the PDE Lf 0 − u, where u ∈ L 2 (Ω) is the known part of the forcing term, we are actually minimizing the contribution of the unknown forcing term g 0 .…”

Section: Introductionmentioning

confidence: 99%

“…We consider, in particular, phenomena where the field can be described by a partial differential equation (PDE) and has to satisfy some known boundary conditions. Spatial regression with PDE penalization (SR-PDE) has been developed in [1] for the estimation of the blood velocity field on an artery section, from echo-Doppler data. This technique has very broad applicability since PDEs are commonly used to describe phenomena behavior in many fields of physics, mechanics, biology, and engineering.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Mixed Finite Elements for Spatial Regression with PDE Penalization

Azzimonti¹,

Nobile²,

Sangalli³

et al. 2014

SIAM/ASA J. Uncertainty Quantification

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Abstract. We study a class of models at the interface between statistics and numerical analysis. Specifically, we consider nonparametric regression models for the estimation of spatial fields from pointwise and noisy observations, which account for problem-specific prior information, described in terms of a partial differential equation governing the phenomenon under study. The prior information is incorporated in the model via a roughness term using a penalized regression framework. We prove the well-posedness of the estimation problem, and we resort to a mixed equal order finite element method for its discretization. Moreover, we prove the well-posedness and the optimal convergence rate of the proposed discretization method. Finally the smoothing technique is extended to the case of areal data, particularly interesting in many applications.

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Section: Surface Estimator For Areal Datamentioning

confidence: 99%

Section: Thanks To the Fact That A H Is Continuous And Coercive Inmentioning

confidence: 99%

Section: Thanks To the Fact That A H Is Continuous And Coercive Inmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Mixed Finite Elements for Spatial Regression with PDE Penalization

Azzimonti¹,

Nobile²,

Sangalli³

et al. 2014

SIAM/ASA J. Uncertainty Quantification

Self Cite

View full text Add to dashboard Cite

show abstract

Modeling Spatially Dependent Functional Data by Spatial Regression with Differential Regularization

Bernardi¹,

Sangalli²

2021

Geostatistical Functional Data Analysis

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Modeling Anisotropy and Non‐Stationarity Through Physics‐Informed Spatial Regression

Tomasetto,

Arnone,

Sangalli

2024

Environmetrics

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Many spatially dependent phenomena that are of interest in environmental problems are characterized by strong anisotropy and non‐stationarity. Moreover, the data are often observed over regions with complex conformations, such as water bodies with complicated shorelines or regions with complex orography. Furthermore, the distribution of the data locations may be strongly inhomogeneous over space. These issues may challenge popular approaches to spatial data analysis. In this work, we show how we can accurately address these issues by spatial regression with differential regularization. We model the spatial variation by a Partial Differential Equation (PDE), defined upon the considered spatial domain. This PDE may depend upon some unknown parameters that we estimate from the data through an appropriate profiling estimation approach. The PDE may encode some available problem‐specific information on the considered phenomenon, and permit a rich modeling of anisotropy and non‐stationarity. The performances of the proposed approach are compared to competing methods through simulation studies and real data applications. In particular, we analyze rainfall data over Switzerland, characterized by strong anisotropy, and oceanographic data in the Gulf of Mexico, characterized by non‐stationarity due to the Gulf Stream.

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Blood Flow Velocity Field Estimation Via Spatial Regression With PDE Penalization

Cited by 33 publications

References 22 publications

Mixed Finite Elements for Spatial Regression with PDE Penalization

Mixed Finite Elements for Spatial Regression with PDE Penalization

Modeling Spatially Dependent Functional Data by Spatial Regression with Differential Regularization

Modeling Anisotropy and Non‐Stationarity Through Physics‐Informed Spatial Regression

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