Search results for " numerical analysis."
showing 10 items of 103 documents
Guaranteed and computable error bounds for approximations constructed by an iterative decoupling of the Biot problem
2021
The paper is concerned with guaranteed a posteriori error estimates for a class of evolutionary problems related to poroelastic media governed by the quasi-static linear Biot equations. The system is decoupled by employing the fixed-stress split scheme, which leads to an iteratively solved semi-discrete system. The error bounds are derived by combining a posteriori estimates for contractive mappings with functional type error control for elliptic partial differential equations. The estimates are applicable to any approximation in the admissible functional space and are independent of the discretization method. They are fully computable, do not contain mesh-dependent constants, and provide r…
ADI schemes for valuing European options under the Bates model
2018
Abstract This paper is concerned with the adaptation of alternating direction implicit (ADI) time discretization schemes for the numerical solution of partial integro-differential equations (PIDEs) with application to the Bates model in finance. Three different adaptations are formulated and their (von Neumann) stability is analyzed. Ample numerical experiments are provided for the Bates PIDE, illustrating the actual stability and convergence behaviour of the three adaptations.
Neural Networks, Inside Out: Solving for Inputs Given Parameters (A Preliminary Investigation)
2021
Artificial neural network (ANN) is a supervised learning algorithm, where parameters are learned by several back-and-forth iterations of passing the inputs through the network, comparing the output with the expected labels, and correcting the parameters. Inspired by a recent work of Boer and Kramer (2020), we investigate a different problem: Suppose an observer can view how the ANN parameters evolve over many iterations, but the dataset is oblivious to him. For instance, this can be an adversary eavesdropping on a multi-party computation of an ANN parameters (where intermediate parameters are leaked). Can he form a system of equations, and solve it to recover the dataset?
Deep neural networks to recover unknown physical parameters from oscillating time series.
2022
PLOS ONE 17(5), e0268439 (2022). doi:10.1371/journal.pone.0268439
Fast MATLAB assembly of FEM matrices in 2D and 3D: Edge elements
2014
We propose an effective and flexible way to assemble finite element stiffness and mass matrices in MATLAB. We apply this for problems discretized by edge finite elements. Typical edge finite elements are Raviart-Thomas elements used in discretizations of H(div) spaces and Nedelec elements in discretizations of H(curl) spaces. We explain vectorization ideas and comment on a freely available MATLAB code which is fast and scalable with respect to time.
Randomized Block Frank–Wolfe for Convergent Large-Scale Learning
2017
Owing to their low-complexity iterations, Frank-Wolfe (FW) solvers are well suited for various large-scale learning tasks. When block-separable constraints are present, randomized block FW (RB-FW) has been shown to further reduce complexity by updating only a fraction of coordinate blocks per iteration. To circumvent the limitations of existing methods, the present work develops step sizes for RB-FW that enable a flexible selection of the number of blocks to update per iteration while ensuring convergence and feasibility of the iterates. To this end, convergence rates of RB-FW are established through computational bounds on a primal sub-optimality measure and on the duality gap. The novel b…
Multispectral image denoising with optimized vector non-local mean filter
2016
Nowadays, many applications rely on images of high quality to ensure good performance in conducting their tasks. However, noise goes against this objective as it is an unavoidable issue in most applications. Therefore, it is essential to develop techniques to attenuate the impact of noise, while maintaining the integrity of relevant information in images. We propose in this work to extend the application of the Non-Local Means filter (NLM) to the vector case and apply it for denoising multispectral images. The objective is to benefit from the additional information brought by multispectral imaging systems. The NLM filter exploits the redundancy of information in an image to remove noise. A …
Perfectly matched layers for the stationary Schrodinger equation in a periodic structure
2008
We construct a perfectly matched absorbing layer for stationary Schrodinger equation with analytic slowly decaying potential in a periodic structure. We prove the unique solvability of the problem with perfectly matched layer of finite length and show that solution to this problem approximates a solution to the original problem with an error that exponentially tends to zero as the length of perfectly matched layer tends to infinity.
Analytic and directional wavelet packets in the space of periodic signals
2019
The paper presents a versatile library of analytic and quasi-analytic complex-valued wavelet packets (WPs) which originate from discrete splines of arbitrary orders. The real parts of the quasi-analytic WPs are the regular spline-based orthonormal WPs designed in [2]. The imaginary parts are the so-called complementary orthonormal WPs, which, unlike the symmetric regular WPs, they are antisymmetric. Tensor products of 1D quasi-analytic WPs provide a diversity of 2D WPs oriented in multiple directions. For example, a set of the fourth-level WPs comprises 62 different directions. The designed computational scheme in the paper enables us to get fast and easy implementation of the WP transforms…
Beyond the mesh handling Maxwell's curl equations with an unconditionally leapfrog stable scheme
2013
Numerical solution of equations governing time domain simulations in computational electromagnetics, is usually based on grid methods in space and on explicit schemes for the time evolution. A predefined grid in the problem domain and a stability step size restriction must be accepted. Evidence is given that efforts need for overcoming these heavy constraints. Recently, the authors developed a meshless method to avoid the connective laws among the points scattered in the problem domain. Despite the good spatial properties, the numerical explicit integration used in the original formulation of the method provides,also in a meshless context, spatial and time discretization strictly interleave…