Related papers: Discrete Differential Principle for Continuous Smo…
This paper considers approximate smoothing for discretely observed non-linear stochastic differential equations. The problem is tackled by developing methods for linearising stochastic differential equations with respect to an arbitrary…
The accuracy of the numerical solution of a fractional differential equation depends on the differentiability class of the solution. The derivatives of the solutions of fractional differential equations often have a singularity at the…
In this article we discuss an important students' misconception about derivatives, that the expression of the derivative of the function contains the information as to whether the function is differentiable or not where the expression is…
In some real world applications, such as spectrometry, functional models achieve better predictive performances if they work on the derivatives of order m of their inputs rather than on the original functions. As a consequence, the use of…
In this paper, we introduce a fourth-order accurate finite element method for incompressible variable density flow. The method is implicit in time and constructed with the Taylor series technique, and uses standard high-order Lagrange basis…
In view of training increasingly complex learning architectures, we establish a nonsmooth implicit function theorem with an operational calculus. Our result applies to most practical problems (i.e., definable problems) provided that a…
Taylor series methods show a newfound promise for the solution of non-stiff ordinary differential equations (ODEs) given the rise of new compiler-enhanced techniques for calculating high order derivatives. In this paper we detail a new…
We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations $$ \partial_t u-\mathfrak{L}^{\sigma,\mu}[\varphi(u)]=f \quad\quad\text{in}\quad\quad…
Compatible Discrete Operator schemes preserve basic properties of the continuous model at the discrete level. They combine discrete differential operators that discretize exactly topological laws and discrete Hodge operators that…
A common machine learning task is to discriminate between normal and anomalous data points. In practice, it is not always sufficient to reach high accuracy at this task, one also would like to understand why a given data point has been…
We use the displacement operator to derive an infinite series of integer order derivatives for the Gr\"{u}nwald-Letnikov fractional derivative and show its correspondence to the Riemann-Liouville and Caputo fractional derivatives. We…
Current techniques for explainable AI have been applied with some success to image processing. The recent rise of research in video processing has called for similar work n deconstructing and explaining spatio-temporal models. While many…
Interatomic potentials are essential to go beyond ab initio size limitations, but simulation results depend sensitively on potential parameters. Forward propagation of parameter variation is key for uncertainty quantification, whilst…
The paper is concerned with non-linear Gaussian filtering and smoothing in continuous-discrete state-space models, where the dynamic model is formulated as an It\^{o} stochastic differential equation (SDE), and the measurements are obtained…
Function approximation is a generic process in a variety of computational problems, from data interpolation to the solution of differential equations and inverse problems. In this work, a unified approach for such techniques is…
Singular value decomposition is widely used in modal analysis, such as proper orthogonal decomposition and resolvent analysis, to extract key features from complex problems. SVD derivatives need to be computed efficiently to enable the…
We describe and study geometric properties of discrete circular and spherical means of directional derivatives of functions, as well as discrete approximations of higher order differential operators. For an arbitrary dimension we present a…
In numerous applications, surrogate models are used as a replacement for accurate parameter-to-observable mappings when solving large-scale inverse problems governed by partial differential equations (PDEs). The surrogate model may be a…
In this paper, we propose a new adaptation of the D-iteration algorithm to numerically solve the differential equations. This problem can be reinterpreted in 2D or 3D (or higher dimensions) as a limit of a diffusion process where the…
The principle of smooth fit is probably the most used tool to find solutions to optimal stopping problems of one-dimensional diffusions. It is important, e.g., in financial mathematical applications to understand in which kind of models and…