Related papers: Discrete Differential Principle for Continuous Smo…
This paper concerns developing a numerical method of the Newton type to solve systems of nonlinear equations described by nonsmooth continuous functions. We propose and justify a new generalized Newton algorithm based on graphical…
We propose a novel method for drift estimation of multiscale diffusion processes when a sequence of discrete observations is given. For the Langevin dynamics in a two-scale potential, our approach relies on the eigenvalues and the…
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…
By using the $q$-analogue of van der Corput's method we study the divisor function in an arithmetic progression to modulus $q$. We show that the expected asymptotic formula holds for a larger range of $q$ than was previously known, provided…
We consider a generic diffusion on the 1D torus and give a simple representation formula for the large deviation rate functional of its invariant probability measure, in the limit of vanishing noise. Previously, this rate functional had…
We propose to combine smoothing, simulations and sieve approximations to solve for either the integrated or expected value function in a general class of dynamic discrete choice (DDC) models. We use importance sampling to approximate the…
Mathematical descriptions of flow phenomena usually come in the form of partial differential equations. The differential operators used in these equations may have properties such as symmetry, skew-symmetry, positive or negative…
In this paper we discuss a closed-form approximation of the likelihood functions of an arbitrary diffusion process. The approximation is based on an exponential ansatz of the transition probability for a finite time step $\Delta t$, and a…
Linear discriminant analysis improves class separability but struggles with non-linearly separable data. To overcome this, we introduce Deep Discriminant Analysis (DDA), which directly optimizes the Fisher criterion utilizing deep networks.…
For discretisations of hyperbolic conservation laws, mimicking properties of operators or solutions at the continuous (differential equation) level discretely has resulted in several successful methods. While well-posedness for nonlinear…
The usual approach to developing and analyzing first-order methods for smooth convex optimization assumes that the gradient of the objective function is uniformly smooth with some Lipschitz constant $L$. However, in many settings the…
Study of scattering process in the nonlocal interaction framework leads to an integro-differential equation. The purpose of the present work is to develop an efficient approach to solve this integro-differential equation with high degree of…
This paper is devoted to the question of constructing a higher order Faber spline basis for the sampling discretization of functions with higher regularity than Lipschitz. The basis constructed in this paper has similar properties as the…
We develop theory for nonlinear dimensionality reduction (NLDR). A number of NLDR methods have been developed, but there is limited understanding of how these methods work and the relationships between them. There is limited basis for using…
This study debuts a new spline dimensional decomposition (SDD) for uncertainty quantification analysis of high-dimensional functions, including those endowed with high nonlinearity and nonsmoothness, if they exist, in a proficient manner.…
In this paper, we derive a variant of the Taylor theorem to obtain a new minimized remainder. For a given function $f$ defined on the interval $[a,b]$, this formula is derived by introducing a linear combination of $f'$ computed at $n+1$…
Collision detection appears as a canonical operation in a large range of robotics applications from robot control to simulation, including motion planning and estimation. While the seminal works on the topic date back to the 80s, it is only…
We introduce and analyze an algorithm for the minimization of convex functions that are the sum of differentiable terms and proximable terms composed with linear operators. The method builds upon the recently developed smoothed gap…
This article develops a novel approach to the representation of singular integral operators of Calder\'on-Zygmund type in terms of continuous model operators, in both the classical and the bi-parametric setting. The representation is…
Optimizing neural networks with loss that contain high-dimensional and high-order differential operators is expensive to evaluate with back-propagation due to $\mathcal{O}(d^{k})$ scaling of the derivative tensor size and the…