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We provide a convenient framework for the study of the well-posedness of a variety of abstract (integro)differential equations in general Banach function spaces. It allows us to extend and complement the known theory on the maximal…
Regularization is a core component of modern inverse problems, as it helps establish the well-posedness of the solution of interest. Popular regularization approaches include variational regularization and iterative regularization. The…
We give a construction allowing to construct local renormalised solutions to general quasilinear stochastic PDEs within the theory of regularity structures, thus greatly generalising the recent results of [BDH16,FG16,OW16]. Loosely…
The theory of regularity structures sets up an abstract framework of modelled distributions generalising the usual H\"older functions and allowing one to give a meaning to several ill-posed stochastic PDEs. A key result in that theory is…
A pluri-Lagrangian structure is an attribute of integrability for lattice equations and for hierarchies of differential equations. It combines the notion of multi-dimensional consistency (in the discrete case) or commutativity of the flows…
We review a collection of models of random simplicial complexes together with some of the most exciting phenomena related to them. We do not attempt to cover all existing models, but try to focus on those for which many important results…
In this paper, we study the regularity of several notions of Lipschitz solutions to the minimal surface system with an emphasis on partial regularity results. These include stationary solutions, integral weak solutions, and viscosity…
We investigate the use of models from the theory of regularity structures as features in machine learning tasks. A model is a polynomial function of a space-time signal designed to well-approximate solutions to partial differential…
In the article random functions in modules over the octonion algebra and Cayley-Dickson algebras are investigated. For their study transition measures with values in the octonion algebra and Cayley-Dickson algebras are used. Stochastic…
The notion of periodic two-scale convergence and the method of periodic unfolding are prominent and useful tools in multiscale modeling and analysis of PDEs with rapidly oscillating periodic coefficients. In this paper we are interested in…
For a class of piecewise deterministic Markov processes we introduce a stochastic calculus which is a certain non-Gaussian counterpart to the classical Malliavin calculus. As an application we investigate the regularity of densities of…
In this work we prove convergence of renormalised models in the framework of regularity structures [Hai14] for a wide class of variable coefficient singular SPDEs in their full subcritical regimes. In particular, we provide for the first…
Training materials through periodic drive allows to endow materials and structures with complex elastic functions. As a result of the driving, the system explores the high dimensional space of structures, ultimately converging to a…
Singularities of a statistical model are the elements of the model's parameter space which make the corresponding Fisher information matrix degenerate. These are the points for which estimation techniques such as the maximum likelihood…
We prove a general equivalence statement between the notions of models and modelled distributions over a regularity structure, and paracontrolled systems indexed by the regularity structure. This takes in particular the form of a…
This paper presents a detailed theoretical analysis of the three stochastic approximation proximal gradient algorithms proposed in our companion paper [49] to set regularization parameters by marginal maximum likelihood estimation. We prove…
In this paper we study on smooth bounded domains the global regularity (up to the boundary) for weak solutions to systems having $p$-structure depending only on the symmetric part of the gradient.
We provide an overview of recent progress in statistical inverse problems with random experimental design, covering both linear and nonlinear inverse problems. Different regularization schemes have been studied to produce robust and stable…
We unify and extend the semigroup and the PDE approaches to stochastic maximal regularity of time-dependent semilinear parabolic problems with noise given by a cylindrical Brownian motion. We treat random coefficients that are only…
This paper discusses basic results and recent developments on variational regularization methods, as developed for inverse problems. In a typical setup we review basic properties needed to obtain a convergent regularization scheme and…