Learning nonlocal regularization operators

Holler, Gernot; Kunisch, Karl

doi:10.3934/mcrf.2021003

Cited by 15 publications

(9 citation statements)

References 28 publications

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“…Nonlocal operators: Nonlocal operators arise in various areas such as nonlocal and fractional diffusions [45,12,11,2,1,5,8,50,48], de-noising and regularization by nonlocal kernels [24,15,19], multi-agent systems with nonlocal interaction [37,36,26] and nonlocal networks [49,31]. The inverse problem for nonlocal diffusions has been studied in [22,28] from a single solution.…”

Section: Related Workmentioning

confidence: 99%

Nonparametric learning of kernels in nonlocal operators

Lu¹,

An²,

Yu³

2022

Preprint

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Nonlocal operators with integral kernels have become a popular tool for designing solution maps between function spaces, due to their efficiency in representing long-range dependence and the attractive feature of being resolution-invariant. In this work, we provide a rigorous identifiability analysis and convergence study for the learning of kernels in nonlocal operators. It is found that the kernel learning is an ill-posed or even ill-defined inverse problem, leading to divergent estimators in the presence of modeling errors or measurement noises. To resolve this issue, we propose a nonparametric regression algorithm with a novel data adaptive RKHS Tikhonov regularization method based on the function space of identifiability. The method yields a noisy-robust convergent estimator of the kernel as the data resolution refines, on both synthetic and real-world datasets. In particular, the method successfully learns a homogenized model for the stress wave propagation in a heterogeneous solid, revealing the unknown governing laws from real-world data at microscale. Our regularization method outperforms baseline methods in robustness, generalizability and accuracy.Preprint. Under review.

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Section: Related Workmentioning

confidence: 99%

Nonparametric learning of kernels in nonlocal operators

Lu¹,

An²,

Yu³

2022

Preprint

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“…The approach we take in this context, which we call the method of nonlocal regularization with quasi-reversibility perturbation (NRQRP), stems from earlier works (e.g., [202,203]), where the associated problems were considered in the form of abstract parabolic equations. Multiple versions of the nonlocal regularization and general regularization techniques have been proposed in the past [204][205][206], many in the context of image reconstructions and inverse scattering problems, and optimality issues of nonlocal regularization operators from certain classes have recently been studied in [207]. However, as seen in the previous sections, our approach is different as it exploits the nature of nonlocal conditions, where an assumption that the spectrum of Hamiltonian is contained in the horizontal strip of the complex plane plays an important role.…”

Section: Example 2 Let Us Consider a Version Of Problem (mentioning

confidence: 99%

Mathematical Models with Nonlocal Initial Conditions: An Exemplification from Quantum Mechanics

Sytnyk

Melnik

2021

MCA

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Nonlocal models are ubiquitous in all branches of science and engineering, with a rapidly expanding range of mathematical and computational applications due to the ability of such models to capture effects and phenomena that traditional models cannot. While spatial nonlocalities have received considerable attention in the research community, the same cannot be said about nonlocality in time, in particular when nonlocal initial conditions are present. This paper aims at filling this gap, providing an overview of the current status of nonlocal models and focusing on the mathematical treatment of such models when nonlocal initial conditions are at the heart of the problem. Specifically, our representative example is given for a nonlocal-in-time problem for the abstract Schrödinger equation. By exploiting the linear nature of nonlocal conditions, we derive an exact representation of the solution operator under assumptions that the spectrum of Hamiltonian is contained in the horizontal strip of the complex plane. The derived representation permits us to establish the necessary and sufficient conditions for the problem’s well-posedness and the existence of its solution under different regularities. Furthermore, we present new sufficient conditions for the existence of the solution that extend the existing results in this field to the case when some nonlocal parameters are unbounded. Two further examples demonstrate the developed methodology and highlight the importance of its computer algebra component in the reduction procedures and parameter estimations for nonlocal models. Finally, a connection of the considered models and developed analysis is discussed in the context of other reduction techniques, concentrating on the most promising from the viewpoint of data-driven modelling environments, and providing directions for further generalizations.

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“…Over the last years, the study of variational models with nonlocal features has attracted increased interest in the community, motivated by the desire to develop a solid understanding of global effects, long-range interactions, and singular behavior in physical phenomena and technical applications, which standard local modeling approaches cannot capture. To mention but a few selected examples from the recent literature, functionals with a nonlocal character appear in the theory of phase transitions [22,49], in peridynamics [13,36], in new models of hyperelasticity [11,12], in image processing [15,23] or in machine learning applications [4,30]. From the mathematical perspective, the presence of nonlocality in variational problems requires substantially different techniques from standard ones, which often rest on localization arguments and are therefore not applicable.…”

Section: Introductionmentioning

confidence: 99%

Cartesian convexity as the key notion in the variational existence theory for nonlocal supremal functionals

Kreisbeck¹,

Ritorto²,

Zappale³

2022

Preprint

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Motivated by the direct method in the calculus of variations in L ∞ , our main result identifies the notion of convexity characterizing the weakly * lower semicontinuity of nonlocal supremal functionals: Cartesian level convexity. This new concept coincides with separate level convexity in the one-dimensional setting and is strictly weaker for higher dimensions. We discuss relaxation in the vectorial case, showing that the relaxed functional will not generally maintain the supremal form. Apart from illustrating this fact with examples of multi-well type, we present precise criteria for structure-preservation. When the structure is preserved, a representation formula is given in terms of the Cartesian level convex envelope of the (diagonalized) original supremand. This work does not only complete the picture of the analysis initiated in [Kreisbeck & Zappale, Calc. Var. PDE, 2020], but also establishes a connection with double integrals. We relate the two classes of functionals via an L p -approximation in the sense of Γ-convergence for diverging integrability exponents. The proofs exploit recent results on nonlocal inclusions and their asymptotic behavior, and use tools from Young measure theory and convex analysis.

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Learning nonlocal regularization operators

Cited by 15 publications

References 28 publications

Nonparametric learning of kernels in nonlocal operators

Nonparametric learning of kernels in nonlocal operators

Mathematical Models with Nonlocal Initial Conditions: An Exemplification from Quantum Mechanics

Cartesian convexity as the key notion in the variational existence theory for nonlocal supremal functionals

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