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Ordinary differential equations (ODEs) are widely used to characterize the dynamics of complex systems in real applications. In this article, we propose a novel joint estimation approach for generalized sparse additive ODEs where…
Sampling logconcave functions arising in statistics and machine learning has been a subject of intensive study. Recent developments include analyses for Langevin dynamics and Hamiltonian Monte Carlo (HMC). While both approaches have…
We develop a numerical method for solving a system of nonlinear integral equations involving two integral terms: at the current time t, one integral is taken from 0 to t, and a different integral is taken from t to infinity. We prove the…
In this paper, we propose a unified framework, the Hessian discretisation method (HDM), which is based on four discrete elements (called altogether a Hessian discretisation) and a few intrinsic indicators of accuracy, independent of the…
In this work we apply the Deep Galerkin Method (DGM) described in Sirignano and Spiliopoulos (2018) to solve a number of partial differential equations that arise in quantitative finance applications including option pricing, optimal…
We present a spectrally accurate numerical method for finding non-trivial time-periodic solutions of non-linear partial differential equations. The method is based on minimizing a functional (of the initial condition and the period) that is…
We study a class of linear ordinary differential equations (ODE)s with distributional coefficients. These equations are defined using an {\it intrinsic} multiplicative product of Schwartz distributions which is an extension of the…
In this paper, a class of high-order methods to numerically solve Functional Differential Equations with Piecewise Continuous Arguments (FDEPCAs) is discussed. The framework stems from the expansion of the vector field associated with the…
Solving Algebra Problems with Geometry Diagrams (APGDs) is still a challenging problem because diagram processing is not studied as intensively as language processing. To work against this challenge, this paper proposes a hologram reasoning…
Generalized additive index models (GAIMs) offer a flexible semiparametric framework for capturing complex data relationships, balancing the interpretability of parametric models with the flexibility of nonparametric approaches. However,…
In this paper, we propose a novel numerical method for Path-Dependent Partial Differential Equations (PPDEs). These equations firstly appeared in the seminal work of Dupire [2009], where the functional It\^o calculus was developed to deal…
In this paper it is shown how to solve numerically eigenvalue problems associated to second order linear ordinary differential equations, containing also terms which depend on the variable. A didactic presentation of the Numerov Method is…
The purpose of this article is to propose ODE based approaches for the numerical evaluation of matrix functions $f(A)$, a question of major interest in the numerical linear algebra. To this end, we model $f(A)$ as the solution at a finite…
Nonlinear (systems of) ordinary differential equations (ODEs) are common tools in the analysis of complex one-dimensional dynamic systems. In this paper we propose a smoothing approach regularized by a quasilinearized ODE-based penalty in…
Solving systems of ordinary differential equations (ODEs) is essential when it comes to understanding the behavior of dynamical systems. Yet, automated solving remains challenging, in particular for nonlinear systems. Computer algebra…
Realizations of stochastic process are often observed temporal data or functional data. There are growing interests in classification of dynamic or functional data. The basic feature of functional data is that the functional data have…
We deal with the numerical solution of linear partial differential equations (PDEs) with focus on the goal-oriented error estimates including algebraic errors arising by an inaccurate solution of the corresponding algebraic systems. The…
We propose a numerical homogenization method for scalar linear partial differential equations with rough coefficients, that integrates classical coarse-scale solvers with quantum subroutines for fine-scale corrections. Inspired by the…
We describe a quantum algorithm for preparing states that encode solutions of non-homogeneous linear partial differential equations. The algorithm is a continuous-variable version of matrix inversion: it efficiently inverts differential…
Scientific studies often require the precise calculation of derivatives. In many cases an analytical calculation is not feasible and one resorts to evaluating derivatives numerically. These are error-prone, especially for higher-order…