scholarly journals The average method is much better than average

2021 ◽  
Vol 16 (1) ◽  
pp. 37-56
Author(s):  
Lívia Boda ◽  
Istvan Faragó ◽  
Tamás Kalmár-Nagy
Keyword(s):  
Average Method ◽  
Original Problem ◽  
Operator Splitting ◽  
Powerful Method ◽  
Splitting Scheme ◽  
Third Order ◽  
Order Accuracy ◽  
Computational Performance ◽  
Commuting Matrices ◽  
Better Than

Operator splitting is a powerful method for the numerical investigation of complex time-dependent models, where the stationary (elliptic) part consists of a sum of several structurally simpler sub-operators. As an alternative to the classical splitting methods, a new splitting scheme is proposed here, the Average Method with sequential splitting. In this method, a decomposition of the original problem is sought in terms of commuting matrices. Wedemonstrate that third-order accuracy can be achieved with the Average Method. The computational performance of the method is investigated, yielding run times 1-2 orders of magnitude faster than traditional methods.

scholarly journals Quantitative Ultrafast Spectroscopy and Microscopy of Traditional and Soft Condensed Matter

2018 ◽  
Vol 8 (8) ◽  
pp. 1317
Author(s):  
Adam Card ◽  
Mohammad Mokim ◽  
Feruz Ganikhanov
Keyword(s):  
Condensed Matter ◽  
Potassium Titanyl Phosphate ◽  
Soft Condensed Matter ◽  
Phonon Modes ◽  
Third Order ◽  
Unit Cells ◽  
Coherent Optical Spectroscopy ◽  
Series Of Experiments ◽  
Tio6 Octahedron ◽  
Better Than

We demonstrate and analyze a series of experiments in traditional and soft condensed matter using coherent optical spectroscopy and microscopy with ultrafast time resolution. We show the capabilities of resolving both real and imaginary parts of the third-order nonlinearity in the vicinity of Raman resonances from a medium probed within microscopic volumes with an equivalent spectral resolution of better than 0.1 cm−1. We can differentiate between vibrations of various types within unit cells of crystals, as well as perform targeted probes of areas within biological tissue. Vibrations within the TiO6 octahedron and the ones for the Ti-O-P intergroup were studied in potassium titanyl phosphate crystal to reveal a multiline structure within targeted phonon modes with closely spaced vibrations having distinctly different damping rates (~0.5 ps−1 versus ~1.1 ps−1). We also detected a 1.7–2.6 ps−1 decay of C-C stretching vibrations in fat tissue and compared that with the corresponding vibration in oil.

Calibration of pneumotachographs using a calibrated syringe

2003 ◽  
Vol 95 (2) ◽  
pp. 571-576 ◽  
Author(s):  
Yongquan Tang ◽  
Martin J. Turner ◽  
Johnny S. Yem ◽  
A. Barry Baker
Keyword(s):  
Calibration Method ◽  
Linear Range ◽  
Data Sets ◽  
Constant Flow ◽  
Calibration Constant ◽  
Third Order ◽  
Polynomial Coefficients ◽  
Calibration Curves ◽  
Order Polynomial ◽  
Better Than

Pneumotachograph require frequent calibration. Constant-flow methods allow polynomial calibration curves to be derived but are time consuming. The iterative syringe stroke technique is moderately efficient but results in discontinuous conductance arrays. This study investigated the derivation of first-, second-, and third-order polynomial calibration curves from 6 to 50 strokes of a calibration syringe. We used multiple linear regression to derive first-, second-, and third-order polynomial coefficients from two sets of 6–50 syringe strokes. In part A, peak flows did not exceed the specified linear range of the pneumotachograph, whereas flows in part B peaked at 160% of the maximum linear range. Conductance arrays were derived from the same data sets by using a published algorithm. Volume errors of the calibration strokes and of separate sets of 70 validation strokes ( part A) and 140 validation strokes ( part B) were calculated by using the polynomials and conductance arrays. Second- and third-order polynomials derived from 10 calibration strokes achieved volume variability equal to or better than conductance arrays derived from 50 strokes. We found that evaluation of conductance arrays using the calibration syringe strokes yields falsely low volume variances. We conclude that accurate polynomial curves can be derived from as few as 10 syringe strokes, and the new polynomial calibration method is substantially more time efficient than previously published conductance methods.

scholarly journals Operator splitting around Euler-Maruyama scheme and high order discretization of heat kernels

2020 ◽  
Author(s):  
Toshihiro Yamada ◽  
Yuga Iguchi
Keyword(s):  
Commutative Algebra ◽  
Computational Algorithm ◽  
Diffusion Processes ◽  
Operator Splitting ◽  
Heat Kernels ◽  
Heat Equations ◽  
Discretization Method ◽  
Splitting Scheme ◽  
Order Operator ◽  
Numerical Examples

This paper proposes a general higher order operator splitting scheme for diffusion semigroups using the Baker-Campbell-Hausdorff type commutator expansion of non-commutative algebra and the Malliavin calculus. An accurate discretization method for the fundamental solution of heat equations or the heat kernel is introduced with a new computational algorithm which will be useful for the inference for diffusion processes. The approximation is regarded as the splitting around the Euler-Maruyama scheme for the density. Numerical examples for diffusion processes are shown to validate the proposed scheme.

Numerical Method with the Third-Order ENO Reconstruction Satisfying Two Conversation Laws for Linear Advection Equation

2011 ◽  
Vol 130-134 ◽  
pp. 2969-2972
Author(s):  
Rong San Chen ◽  
An Ping Liu
Keyword(s):  
Numerical Solutions ◽  
Advection Equation ◽  
Finite Volume Schemes ◽  
Third Order ◽  
The Third ◽  
Linear Advection Equation ◽  
Long Time ◽  
Super Convergence ◽  
Better Than ◽  
Eno Scheme

In recent years, Mao and his co-workers developed a class of finite-volume schemes for evolution partial differential equations, see [1-5].The schemes show a super-convergence quality and have good structure-preserving property in long-time numerical simulations. In [6], Chen and Ma developed a scheme which combine the idea of paper [5] and that of the the second-order ENO scheme [7]. In this paper, we propose a scheme which extend the result of [6] and obtain the scheme using the third-order ENO reconstruction. Numerical experiments show that our scheme is robust in long-time behaviors. Numerical solutions are far better than those of [6].

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