Related papers: Advanced Differential Equations: Asymptotics & Per…
The method of statistical differentials, which approximates the mean and variance of transformations of random variables is used in many areas of mathematics. This paper will discuss the conditions under which such an approximation will be…
Differential equation discovery, a machine learning subfield, is used to develop interpretable models, particularly in nature-related applications. By expertly incorporating the general parametric form of the equation of motion and…
Finite differences have been widely used in mathematical theory as well as in scientific and engineering computations. These concepts are constantly mentioned in calculus. Most frequently-used difference formulas provide excellent…
Differentiable physics provides a new approach for modeling and understanding the physical systems by pairing the new technology of differentiable programming with classical numerical methods for physical simulation. We survey the rapidly…
Techniques are proposed for solving integral equations of the first kind with an input known not precisely. The requirement that the solution sought for includes a given number of maxima and minima is imposed. It is shown that when the…
Analytic perturbation theory for matrices and operators is an immensely useful mathematical technique. Most elementary introductions to this method have their background in the physics literature, and quantum mechanics in particular. In…
Several different methods exist for efficient approximation of paths in multiscale stochastic chemical systems. Another approach is to use bursts of stochastic simulation to estimate the parameters of a stochastic differential equation…
We deliver a call to arms for probabilistic numerical methods: algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such…
Partial Differential Equations (PDE) are fundamental to model different phenomena in science and engineering mathematically. Solving them is a crucial step towards a precise knowledge of the behaviour of natural and engineered systems. In…
Conformance checking techniques let us find out to what degree a process model and real execution data correspond to each other. In recent years, alignments have proven extremely useful in calculating conformance statistics. Most techniques…
We perform a systematic comparison of various numerical schemes for the approximation of interface problems. We consider unfitted approaches in view of their application to possibly moving configurations. Particular attention is paid to the…
Dynamic graph algorithms have seen significant theoretical advancements, but practical evaluations often lag behind. This work bridges the gap between theory and practice by engineering and empirically evaluating recently developed…
Symmetries play an critical role in finding analytic solutions to nonlinear differential equations. A symmetry is a mapping of the solutions of the differential equation into the solutions and have been studied extensively for over a…
We provide a `user guide' to the literature of the past twenty years concerning the modeling and approximation of discontinuous solutions to nonlinear hyperbolic systems that admit small-scale dependent shock waves. We cover several classes…
Optimal prediction methods compensate for a lack of resolution in the numerical solution of time-dependent differential equations through the use of prior statistical information. We present a new derivation of the basic methodology, show…
In this paper, we will present advanced discretization methods for solving retarded potential integral equations. We employ a $C^{\infty}$-partition of unity method in time and a conventional boundary element method for the spatial…
A new method of approximation scheme with potential application to a general interacting quantum system is presented. The method is non-perturbative, self- consistent, systematically improvable and uniformly applicable for arbitrary…
Computational methods for fractional differential equations exhibit essential instability. Even a minor modification of the coefficients or other entry data may switch good results to the divergent. The goal of this paper is to suggest the…
Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and…
Designing and analyzing algorithms with provable performance guarantees enables efficient optimization problem solving in different application domains, e.g.\ communication networks, transportation, economics, and manufacturing. Despite the…