Related papers: Discretisation of regularity structures
A stochastic PDE, describing mesoscopic fluctuations in systems of weakly interacting inertial particles of finite volume, is proposed and analysed in any finite dimension $d\in\mathbb{N}$. It is a regularised and inertial version of the…
PDE-constrained optimal control problems require regularisation to ensure well-posedness, introducing small perturbations that make the solutions challenging to approximate accurately. We propose a finite element approach that couples both…
We study the question of testing structured properties (classes) of discrete distributions. Specifically, given sample access to an arbitrary distribution $D$ over $[n]$ and a property $\mathcal{P}$, the goal is to distinguish between…
This article studies the effect of discretisation error on the stationary distribution of stochastic partial differential equations (SPDEs). We restrict the analysis to the effect of space discretisation, performed by finite element…
An exact discretization method is being developed for solving linear systems of ordinary fractional-derivative differential equations with constant matrix coefficients (LSOFDDECMC). It is shown that the obtained linear discrete system in…
Typically when designing distributed controllers it is assumed that the state-space model of the plant consists of sparse matrices. However, in the discrete-time setting, if one begins with a continuous-time model, the discretization…
Some prominent discretisation methods such as finite elements provide a way to approximate a function of $d$ variables from $n$ values it takes on the nodes $x_i$ of the corresponding mesh. The accuracy is $n^{-s_a/d}$ in $L^2$-norm, where…
The Old Quantum Mechanics actions discretization rules for periodic motions on the atomic scale (Bohr-Sommerfeld) have been suitably modified in order to account the gravitational field instead of the electrostatic one. The new rules are…
In this paper, we discuss the solution of certain matrix-valued partial differential equations. Such PDEs arise, for example, when constructing a Riemannian contraction metric for a dynamical system given by an autonomous ODE. We develop…
This paper shows how numerical methods on a regular grid in a box can be used to generate numerical schemes for problems in general smooth domains contained in the box with no need for a domain specific discretization. The focus is mainly…
The problem of computing differential constraints for a family of evolution PDEs is discussed from a constructive point of view. A new method, based on the existence of generalized characteristics for evolution vector fields, is proposed in…
We consider a class of parameter-dependent optimal control problems of elliptic PDEs with constraints of general type on the control variable. Applying the concept of variational discretization, [4], together with techniques from the…
Systems of partial differential equations lie at the heart of physics. Despite this, the general theory of these systems has remained rather obscure in comparison to numerical approaches such as finite element models and various other…
Mixed-dimensional partial differential equations arise in several physical applications, wherein parts of the domain have extreme aspect ratios. In this case, it is often appealing to model these features as lower-dimensional manifolds…
We consider the ill-posed inverse problem of identifying a nonlinearity in a time-dependent PDE model. The nonlinearity is approximated by a neural network, and needs to be determined alongside other unknown physical parameters and the…
We study the stochastic Allen-Cahn equation driven by a noise term with intensity $\sqrt{\varepsilon}$ and correlation length $\delta$ in two and three spatial dimensions. We study diagonal limits $\delta, \varepsilon \to 0$ and describe…
Regularization is used in many different areas of optimization when solutions are sought which not only minimize a given function, but also possess a certain degree of regularity. Popular applications are image denoising, sparse regression…
We derive a new discretisation method for first order PDEs of arbitrary spatial dimension, which is based upon a meshfree spatial approximation. This spatial approximation is similar to the SPH (smoothed particle hydrodynamics) technique…
We study the inverse conductivity problem with discontinuous conductivities. We consider, simultaneously, a regularisation and a discretisation for a variational approach to solve the inverse problem. We show that, under suitable choices of…
We consider the inverse conductivity problem with discontinuous conductivities. We show in a rigorous way, by a convergence analysis, that one can construct a completely discrete minimization problem whose solution is a good approximation…