PDEs with Compressed Solutions

Sparsity plays a central role in recent developments in signal processing, linear algebra, statistics, optimization, and other fields. In these developments, sparsity is promoted through the addition of an $L^1$ norm (or related quantity) as a constraint or penalty in a variational principle. We apply this approach to partial differential equations that come from a variational quantity, either by minimization (to obtain an elliptic PDE) or by gradient flow (to obtain a parabolic PDE). Also, we show that some PDEs can be rewritten in an $L^1$ form, such as the divisible sandpile problem and signum-Gordon. Addition of an $L^1$ term in the variational principle leads to a modified PDE where a subgradient term appears. It is known that modified PDEs of this form will often have solutions with compact support, which corresponds to the discrete solution being sparse. We show that this is advantageous numerically through the use of efficient algorithms for solving $L^1$ based problems.Comment: 21 pages, 15 figure

One-log call iterative solution of the Colebrook equation for flow friction based on Pade polynomials

The 80 year-old empirical Colebrook function zeta, widely used as an informal standard for hydraulic resistance, relates implicitly the unknown flow friction factor lambda, with the known Reynolds number Re and the known relative roughness of a pipe inner surface epsilon* ; lambda = zeta(Re, epsilon* ,lambda). It is based on logarithmic law in the form that captures the unknown flow friction factor l in a way that it cannot be extracted analytically. As an alternative to the explicit approximations or to the iterative procedures that require at least a few evaluations of computationally expensive logarithmic function or non-integer powers, this paper offers an accurate and computationally cheap iterative algorithm based on Pade polynomials with only one log-call in total for the whole procedure (expensive log-calls are substituted with Pade polynomials in each iteration with the exception of the first). The proposed modification is computationally less demanding compared with the standard approaches of engineering practice, but does not influence the accuracy or the number of iterations required to reach the final balanced solution.Web of Science117art. no. 182

Iterative Decoupling Method for High-Precision Imaging of Complex Surfaces

Nonlinear systems and interaction forces are pervasive in many scientific fields, such as nanoscale metrology and materials science, but their accurate identification is challenging due to their complex behaviour and inaccessibility of measured domains. This problem intensifies for continuous systems undergoing distributed, coupled interactions, such as in the case of topography measurement systems, measuring narrow and deep grooves. Presented is a method to invert a set of nonlinear coupled equations, which can be functions of unknown distributed physical quantities. The method employs a successive approach to iteratively converge to the exact solution of the set of nonlinear equations. The latter utilizes an approximate yet invertible model providing an inexact solution, which is evaluated using the hard-to-invert exact model of the system. This method is applied to the problem of reconstructing the topography of surface contours using a thin and long vibrating fiber. In nanoscale metrology, measuring inaccessible deep and narrow grooves or steep walls becomes difficult and singular when attempting to extract distributed nonlinear interactions that depend on the topography. We verify our method numerically by simulating the Van der Waals (VdW) interaction forces between a nanofiber and a nanoscale deep groove, and experimentally by exploiting magnetic interactions between a magnetic topography and a vibrating, elastic beam. Our results validate the ability to accurately reconstruct the topography of normally inaccessible regions, making it a possible enhancement for traditional point based AFM measurements, as well as for other nonlinear inverse problems

Effective linear damping and stiffness coefficients of nonlinear systems for design spectrum based analysis

A stochastic approach for obtaining reliable estimates of the peak response of nonlinear systems to excitations specified via a design seismic spectrum is proposed. This is achieved in an efficient manner without resorting to numerical integration of the governing nonlinear equations of motion. First, a numerical scheme is utilized to derive a power spectrum which is compatible in a stochastic sense with a given design spectrum. This power spectrum is then treated as the excitation spectrum to determine effective damping and stiffness coefficients corresponding to an equivalent linear system (ELS) via a statistical linearization scheme. Further, the obtained coefficients are used in conjunction with the (linear) design spectrum to estimate the peak response of the original nonlinear systems. The cases of systems with piecewise linear stiffness nonlinearity, along with bilinear hysteretic systems are considered. The seismic severity is specified by the elastic design spectrum prescribed by the European aseismic code provisions (EC8). Monte Carlo simulations pertaining to an ensemble of nonstationary EC8 design spectrum compatible accelerograms are conducted to confirm that the average peak response of the nonlinear systems compare reasonably well with that of the ELS, within the known level of accuracy furnished by the statistical linearization method. In this manner, the proposed approach yields ELS which can replace the original nonlinear systems in carrying out computationally efficient analyses in the initial stages of the aseismic design of structures under severe seismic excitations specified in terms of a design spectrum

An Efficient Three Step Method For finding the Root Of Non-linear Equation with Accelerated convergence.

We have made an effort to design an accurate numerical strategy to be applied in the vast computing domain of numerical analysis. The purpose of this research is to develop a novel hybrid numerical method for solving a nonlinear equation, That is both quick and computationally cheap, given the demands of today's technological landscape. Sixth-order convergence is demonstrated by combining the classical Newton method, on which this method is largely based, with another two-step third-order iterative process. The effectiveness index for this novel approach is close to 1.4309, and it requires only five evaluations of the functions without a second derivative. The findings are compared to standard practice. The provided technique demonstrates higher performance in terms of computational efficiency, productivity, error estimation, and CPU times. Moreover, its accuracy and performance are tested using a variety of examples from the existing literature. Keywords: efficient scheme, nonlinear application, nonlinear functions, error estimation, computational cost