Relaxation analysis in a data driven problem with a single outlier

Röger, Matthias; Schweizer, Ben

doi:10.1007/s00526-020-01773-x

Cited by 7 publications

(7 citation statements)

References 24 publications

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“…In a deterministic framework, solving the datadriven problem then entails minimizing an appropriate distance between the points z h = ( h , σ h ) and y h = ( h , σ h ). Appropriate notions of convergence of the data set D h → D ensuring convergence of solutions, as well as related notions of relaxation in the infinite-dimensional setting, have been set forth in [7,9,10]. We note that the paradigm is strictly data-driven and modelfree in the sense that solutions are obtained, or approximated, directly from the data set without recourse to any intervening modeling of the data.…”

Section: −→ Predictionmentioning

confidence: 99%

“…Data-Driven (DD) solvers seek to determine the material state y ∈ D that is closest to being admissible, in the sense of E, or, alternatively, the admissible state z ∈ E that is closest to being a possible state of the material, in the sense of D. Optimality is understood in the sense of a suitable norm, e. g., for the set-up in Example 2.2, we may choose i. e., we wish to determine the state z ∈ E of the system that is admissible and closest to the data set D, or, equivalently, the point y ∈ D in the material data set that is closest to being admissible. Evidently, if E is affine and D is compact, e. g., consisting of a finite collection of points, then the DD problem (10) has solutions by the Weierstrass extreme-value theorem. More generally, in [7, Cor.…”

Section: Materials Characterizationmentioning

confidence: 99%

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Model-free Data-Driven Inference

Conti¹,

Hoffmann²,

Ortiz³

2021

Preprint

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We present a model-free data-driven inference method that enables inferences on system outcomes to be derived directly from empirical data without the need for intervening modeling of any type, be it modeling of a material law or modeling of a prior distribution of material states. We specifically consider physical systems with states characterized by points in a phase space determined by the governing field equations. We assume that the system is characterized by two likelihood measures: one µD measuring the likelihood of observing a material state in phase space; and another µE measuring the likelihood of states satisfying the field equations, possibly under random actuation. We introduce a notion of intersection between measures which can be interpreted to quantify the likelihood of system outcomes. We provide conditions under which the intersection can be characterized as the athermal limit µ∞ of entropic regularizations µ β , or thermalizations, of the product measure µ = µD × µE as β → +∞. We also supply conditions under which µ∞ can be obtained as the athermal limit of carefully thermalized (µ h,β h ) sequences of empirical data sets (µ h ) approximating weakly an unknown likelihood function µ. In particular, we find that the cooling sequence β h → +∞ must be slow enough, corresponding to quenching, in order for the proper limit µ∞ to be delivered. Finally, we derive explicit analytic expressions for expectations E[f ] of outcomes f that are explicit in the data, thus demonstrating the feasibility of the model-free data-driven paradigm as regards making convergent inferences directly from the data without recourse to intermediate modeling steps.

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Section: −→ Predictionmentioning

confidence: 99%

Section: Materials Characterizationmentioning

confidence: 99%

Model-free Data-Driven Inference

Conti¹,

Hoffmann²,

Ortiz³

2021

Preprint

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“…In this case, the set of data converges in a weak sense to some distribution, see [CHO21]. See also [RS20] for the analysis of single outliers in measurements.…”

Section: Range Of Measurement Constant (Unbounded) Increasingmentioning

confidence: 99%

A data-driven approach to viscous fluid mechanics -- the stationary case

Lienstromberg¹,

Schiffer²,

Schubert³

2022

Preprint

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We introduce a data-driven approach to the modelling and analysis of viscous fluid mechanics. Instead of including constitutive laws for the fluid's viscosity in the mathematical model, we suggest to directly use experimental data. Only a set of differential constraints, derived from first principles, and boundary conditions are kept of the classical PDE model and are combined with a data set. The mathematical framework builds on the recently introduced data-driven approach to solid-mechanics [KO16, CMO18]. We construct optimal data-driven solutions that are material model free in the sense that no assumptions on the rheological behaviour of the fluid are made or extrapolated from the data. The differential constraints of fluid mechanics are recast in the language of constant rank differential operators. Adapting abstract results on lower-semicontinuity and -quasiconvexity, we show a Γconvergence result for the functionals arising in the data-driven fluid mechanical problem. The theory is extended to compact nonlinear perturbations, whence our results apply to both inertialess fluids and flows with finite Reynolds number. Data-driven solutions provide a new relaxed solution concept. We prove that the constructed data-driven solutions are consistent with solutions to the classical PDEs of fluid mechanics if the data sets have the form of a monotone constitutive relation.

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“…We first show that this property can be localized, in the sense that the limits can be assumed to be constant. A related question on localization has been raised in [RS19].…”

Section: Div-curl-closed Materials Data Setsmentioning

confidence: 99%

“…These difficulties notwithstanding, in [CMO18] conditions on the data set ensuring existence are set forth, and it is shown that classical solutions are recovered when the data set takes the form of a graph. The approach developed in [CMO18] has been used for the study of relaxation in a data-driven model with a single outlier, which was motivated by the study of porous media [RS19]. This latter connection shows that Data-Driven problems generalize classical problems and subsume them as special cases.…”

Section: Introductionmentioning

confidence: 99%

Data-Driven Finite Elasticity

Conti

Müller²,

Ortiz

2020

Arch Rational Mech Anal

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We extend to finite elasticity the Data-Driven formulation of geometrically linear elasticity presented in Conti, Müller, Ortiz, Arch. Ration. Mech. Anal. 229, 79-123, 2018. The main focus of this paper concerns the formulation of a suitable framework in which the Data-Driven problem of finite elasticity is well-posed in the sense of existence of solutions. We confine attention to deformation gradients F ∈ L p (Ω; R n×n ) and first Piola-Kirchhoff stresses P ∈ L q (Ω; R n×n ), with (p, q) ∈ (1, ∞) and 1/p+1/q = 1. We assume that the material behavior is described by means of a material data set containing all the states (F, P ) that can be attained by the material, and develop germane notions of coercivity and closedness of the material data set. Within this framework, we put forth conditions ensuring the existence of solutions. We exhibit specific examples of two-and three-dimensional material data sets that fit the present setting and are compatible with material frame indifference.

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Relaxation analysis in a data driven problem with a single outlier

Cited by 7 publications

References 24 publications

Model-free Data-Driven Inference

Model-free Data-Driven Inference

A data-driven approach to viscous fluid mechanics -- the stationary case

Data-Driven Finite Elasticity

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