Related papers: Adaptive integration of nonlinear evolution equati…
In this paper, we introduce a new adaptive data analysis method to study trend and instantaneous frequency of nonlinear and non-stationary data. This method is inspired by the Empirical Mode Decomposition method (EMD) and the recently…
We propose a novel direct transcription and solution method for solving nonlinear, continuous-time dynamic optimization problems. Instead of forcing the dynamic constraints to be satisfied only at a selected number of points as in direct…
We study the best low-rank Tucker decomposition of symmetric tensors. The motivating application is decomposing higher-order multivariate moments. Moment tensors have special structure and are important to various data science problems. We…
Formulated is a new systematic method for obtaining higher order corrections in numerical simulation of stochastic differential equations (SDEs), i.e., Langevin equations. Random walk step algorithms within a given order of finite $\Delta…
In stochastic optimization, a common tool to deal sequentially with large sample is to consider the well-known stochastic gradient algorithm. Nevertheless, since the stepsequence is the same for each direction, this can lead to bad results…
Near-optimal computational complexity of an adaptive stochastic Galerkin method with independently refined spatial meshes for elliptic partial differential equations is shown. The method takes advantage of multilevel structure in expansions…
In this paper, we investigate adaptive nonlinear regression and introduce tree based piecewise linear regression algorithms that are highly efficient and provide significantly improved performance with guaranteed upper bounds in an…
High-order semi-Lagrangian methods for kinetic equations have been under rapid development in the past few decades. In this work, we propose a semi-Lagrangian adaptive rank (SLAR) integrator in the finite difference framework for linear…
We present an energy-preserving mechanic formulation for dynamic quasi-brittle fracture in an Eulerian-Lagrangian formulation, where a second-order phase-field equation controls the damage evolution. The numerical formulation adapts in…
In this work, we propose a numerical approach for simulations of large deformations of interfaces in a level set framework. To obtain a fast and viable numerical solution in both time and space, temporal discretization is based on the…
A space-time adaptive method is presented for the reactive Euler equations describing chemically reacting gas flow where a two species model is used for the chemistry. The governing equations are discretized with a finite volume method and…
The theory of nonlinear balanced truncation provides a system-theoretic framework for model reduction that preserves important properties such as stability, controllability, and observability. We present a scalable algorithm for computing…
The computation of correspondences between shapes is a principal task in shape analysis. To this end, methods based on partial differential equations (PDEs) have been established, encompassing e.g. the classic heat kernel signature as well…
Motivated by reduction of computational complexity, this work develops sign-error adaptive filtering algorithms for estimating time-varying system parameters. Different from the previous work on sign-error algorithms, the parameters are…
In this manuscript, we propose matrix- and tensor-oriented methods for the numerical solution of the multidimensional evolutionary space-fractional complex Ginzburg--Landau equation. After a suitable spatial semidiscretization, the…
Consistency models have recently been introduced to accelerate sampling from diffusion models by directly predicting the solution (i.e., data) of the probability flow ODE (PF ODE) from initial noise. However, the training of consistency…
High-order DG methods have become a popular technique in computational fluid dynamics because their accuracy increases spectrally in smooth solutions with the order of the approximation. However, their main drawback is that increasing the…
This work proposes a low complexity nonlinearity model and develops adaptive algorithms over it. The model is based on the decomposable---or rank-one, in tensor language---Volterra kernels. It may also be described as a product of FIR…
Modern technological advances have enabled an unprecedented amount of structured data with complex temporal dependence, urging the need for new methods to efficiently model and forecast high-dimensional tensor-valued time series. This paper…
In this paper, we study numerical approximations for stochastic differential equations (SDEs) that use adaptive step sizes. In particular, we consider a general setting where decisions to reduce step sizes are allowed to depend on the…