Related papers: On the efficient computation of high-order derivat…
The nature of so-called differential-algebraic operators and their approximations is constitutive for the direct treatment of higher-index differential-algebraic equations. We treat first-order differential-algebraic operators in detail and…
Several quantities important in condensed matter physics, quantum information, and quantum chemistry, as well as quantities required in meta-optimization of machine learning algorithms, can be expressed as gradients of implicitly defined…
In this paper, we consider the numerical pricing of financial derivatives using Radial Basis Function generated Finite Differences in space. Such discretization methods have the advantage of not requiring Cartesian grids. Instead, the nodes…
In this paper we propose an algorithm for the numerical solution of arbitrary differential equations of fractional order. The algorithm is obtained by using the following decomposition of the differential equation into a system of…
Accurate derivatives are important for efficiently locally traversing and converging in quantum optimization landscapes. By deriving analytically exact control derivatives (gradient and Hessian) for unitary control tasks, we show here that…
Over the last decade, it has been demonstrated that many systems in science and engineering can be modeled more accurately by fractional-order than integer-order derivatives, and many methods are developed to solve the problem of fractional…
We consider high-order splitting schemes for large-scale differential Riccati equations. Such equations arise in many different areas and are especially important within the field of optimal control. In the large-scale case, it is critical…
Gradient-based techniques are becoming increasingly critical in quantitative fields, notably in statistics and computer science. The utility of these techniques, however, ultimately depends on how efficiently we can evaluate the derivatives…
We introduce a high-order numerical scheme for fractional ordinary differential equations with the Caputo derivative. The method is developed by dividing the domain into a number of subintervals, and applying the quadratic interpolation on…
We address the task of higher-order derivative evaluation of computer programs that contain QR decompositions and real symmetric eigenvalue decompositions. The approach is a combination of univariate Taylor polynomial arithmetic and matrix…
A unified explicit form for difference formulas to approximate the fractional and classical derivatives is presented. The formula gives finite difference approximations for any classical derivatives with a desired order of accuracy at nodal…
A consequent approach is proposed to construct symplectic force-gradient algorithms of arbitrarily high orders in the time step for precise integration of motion in classical and quantum mechanics simulations. Within this approach the basic…
If several independent algorithms for a computer-calculated quantity exist, then one can expect their results (which differ because of numerical errors) to follow approximately Gaussian distribution. The mean of this distribution,…
Weighted finite-state machines are a fundamental building block of NLP systems. They have withstood the test of time -- from their early use in noisy channel models in the 1990s up to modern-day neurally parameterized conditional random…
The problem of recovering partial derivatives of high orders of bivariate functions with finite smoothness is studied. Based on the truncation method, a numerical differentiation algorithm was constructed, which is optimal by the order,…
Differentiation is a cornerstone of computing and data analysis in every discipline of science and engineering. Indeed, most fundamental physics laws are expressed as relationships between derivatives in space and time. However, derivatives…
This paper presents a high-order accurate numerical quadrature algorithm for evaluating integrals over curved surfaces and regions defined implicitly via a level set of a given function restricted to a hyperrectangle. The domain is divided…
Differentiable programming is revolutionizing computational science by enabling automatic differentiation (AD) of numerical simulations. While first-order gradients are well-established, second-order derivatives (Hessians) for implicit…
We present a method for the numerical calculation of derivatives of functions of general complex matrices. The method can be used in combination with any algorithm that evaluates or approximates the desired matrix function, in particular…
The Riemann-Liouville formula for fractional derivatives and integrals (differintegration) is used to derive formulae for matrix order derivatives and integrals. That is, the parameter for integration and differentiation is allowed to…