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We establish that stabilization of a class of linear, hyperbolic partial differential equations (PDEs) with a large (nevertheless finite) number of components, can be achieved via employment of a backstepping-based control law, which is…
Physics-informed machine learning (PIML) has emerged as a promising alternative to conventional numerical methods for solving partial differential equations (PDEs). PIML models are increasingly built via deep neural networks (NNs) whose…
This paper presents a numerical method for variable coefficient elliptic PDEs with mostly smooth solutions on two dimensional domains. The PDE is discretized via a multi-domain spectral collocation method of high local order (order 30 and…
We consider a general path-dependent version of the hedging problem with price impact of Bouchard et al. (2019), in which a dual formulation for the super-hedging price is obtained by means of PDE arguments, in a Markovian setting and under…
In this paper we introduce a variant of the Syndrome Decoding Problem (SDP), that we call Restricted SDP (R-SDP), in which the entries of the searched vector are defined over a subset of the underlying finite field. We prove the…
In this paper we combine the theory of reproducing kernel Hilbert spaces with the field of collocation methods to solve boundary value problems with special emphasis on reproducing property of kernels. From the reproducing property of…
The kernel herding algorithm is used to construct quadrature rules in a reproducing kernel Hilbert space (RKHS). While the computational efficiency of the algorithm and stability of the output quadrature formulas are advantages of this…
We investigate an optimization problem that arises when working within the paradigm of Data-Driven Computational Mechanics. In the context of the diffusion-reaction problem, such an optimization problem seeks for the continuous primal…
The numerical solution of partial differential equations (PDEs) is difficult, having led to a century of research so far. Recently, there have been pushes to build neural--numerical hybrid solvers, which piggy-backs the modern trend towards…
We develop a convergence theory for non-monotone approximation schemes for fully nonlinear parabolic partial differential equations. Modern computational methods such as kernel-based collocation, spectral methods, physics-informed neural…
PDEs are central to scientific and engineering modeling, yet designing accurate numerical solvers typically requires substantial mathematical expertise and manual tuning. Recent neural network-based approaches improve flexibility but often…
We develop several deep learning algorithms for approximating families of parametric PDE solutions. The proposed algorithms approximate solutions together with their gradients, which in the context of mathematical finance means that the…
High-dimensional partial-differential equations (PDEs) arise in a number of fields of science and engineering, where they are used to describe the evolution of joint probability functions. Their examples include the Boltzmann and…
With the increases in computational power and advances in machine learning, data-driven learning-based methods have gained significant attention in solving PDEs. Physics-informed neural networks (PINNs) have recently emerged and succeeded…
Built upon previous work of the authors in (Deckelnick, Herbert, and Hinze, ESAIM: COCV 28 (2022)), we present a general shape optimisation framework based on the method of mappings in the $W^{1,\infty}$ topology together with a suitable…
In this paper, we introduce and analyze a class of numerical schemes that demonstrate remarkable superiority in terms of efficiency, the preservation of positivity, energy stability, and high-order precision to solve the time-dependent…
Matrix congruence extends naturally to the setting of tensors. We apply methods from tensor decomposition, algebraic geometry and numerical optimization to this group action. Given a tensor in the orbit of another tensor, we compute a…
We present a general shape optimisation framework based on the method of mappings in the $W^{1,\infty}$ topology. We propose steepest descent and Newton-like minimisation algorithms for the numerical solution of the respective shape…
In this paper, we study existence of solutions to a conformally invariant integral equation involving Poisson-type kernels. Such integral equation has a stronger non-local feature and is not the dual of any PDE. We obtain the existence of…
Developing efficient numerical algorithms for the solution of high dimensional random Partial Differential Equations (PDEs) has been a challenging task due to the well-known curse of dimensionality. We present a new solution framework for…