Solving AKNS equations via Jacobi inversion problem

1999 ◽  
Vol 15 (4) ◽  
pp. 337-344 ◽  
Author(s):  
Zeng Yunbo
Keyword(s):  
Inversion Problem ◽  
Jacobi Inversion Problem ◽  
Jacobi Inversion

The Tax Inversion Problem: A Hard Nut to Crack

2017 ◽  
Author(s):  
Phillip Fuller ◽  
Henry Thomas
Keyword(s):  
Inversion Problem

scholarly journals On Galerkin Approximations of the Surface Active Quasigeostrophic Equations

2016 ◽  
Vol 46 (1) ◽  
pp. 125-139 ◽  
Author(s):  
Cesar B. Rocha ◽  
William R. Young ◽  
Ian Grooms
Keyword(s):  
Galerkin Approximation ◽  
Three Dimensional ◽  
Active Surface ◽  
Inversion Problem ◽  
Conservation Of Energy ◽  
Galerkin Approximations ◽  
Representation Of Solutions ◽  
Surface Active ◽  
Surface Buoyancy ◽  
Quasigeostrophic Equations

AbstractThis study investigates the representation of solutions of the three-dimensional quasigeostrophic (QG) equations using Galerkin series with standard vertical modes, with particular attention to the incorporation of active surface buoyancy dynamics. This study extends two existing Galerkin approaches (A and B) and develops a new Galerkin approximation (C). Approximation A, due to Flierl, represents the streamfunction as a truncated Galerkin series and defines the potential vorticity (PV) that satisfies the inversion problem exactly. Approximation B, due to Tulloch and Smith, represents the PV as a truncated Galerkin series and calculates the streamfunction that satisfies the inversion problem exactly. Approximation C, the true Galerkin approximation for the QG equations, represents both streamfunction and PV as truncated Galerkin series but does not satisfy the inversion equation exactly. The three approximations are fundamentally different unless the boundaries are isopycnal surfaces. The authors discuss the advantages and limitations of approximations A, B, and C in terms of mathematical rigor and conservation laws and illustrate their relative efficiency by solving linear stability problems with nonzero surface buoyancy. With moderate number of modes, B and C have superior accuracy than A at high wavenumbers. Because B lacks the conservation of energy, this study recommends approximation C for constructing solutions to the surface active QG equations using the Galerkin series with standard vertical modes.

scholarly journals A 4D-Var inversion system based on the icosahedral grid model (NICAM-TM 4D-Var v1.0) – Part 2: Optimization scheme and identical twin experiment of atmospheric CO<sub>2</sub> inversion

2017 ◽  
Vol 10 (6) ◽  
pp. 2201-2219 ◽  
Author(s):  
Yosuke Niwa ◽  
Yosuke Fujii ◽  
Yousuke Sawa ◽  
Yosuke Iida ◽  
Akihiko Ito ◽  
...  
Keyword(s):  
Transport Model ◽  
Surface Fluxes ◽  
Identical Twin ◽  
Inversion Problem ◽  
Model Errors ◽  
Optimization Scheme ◽  
Error Covariance ◽  
Optimal Setting ◽  
Source Sink ◽  
Quasi Newton

Abstract. A four-dimensional variational method (4D-Var) is a popular technique for source/sink inversions of atmospheric constituents, but it is not without problems. Using an icosahedral grid transport model and the 4D-Var method, a new atmospheric greenhouse gas (GHG) inversion system has been developed. The system combines offline forward and adjoint models with a quasi-Newton optimization scheme. The new approach is then used to conduct identical twin experiments to investigate optimal system settings for an atmospheric CO2 inversion problem, and to demonstrate the validity of the new inversion system. In this paper, the inversion problem is simplified by assuming the prior flux errors to be reasonably well known and by designing the prior error correlations with a simple function as a first step. It is found that a system of forward and adjoint models with smaller model errors but with nonlinearity has comparable optimization performance to that of another system that conserves linearity with an exact adjoint relationship. Furthermore, the effectiveness of the prior error correlations is demonstrated, as the global error is reduced by about 15 % by adding prior error correlations that are simply designed when 65 weekly flask sampling observations at ground-based stations are used. With the optimal setting, the new inversion system successfully reproduces the spatiotemporal variations of the surface fluxes, from regional (such as biomass burning) to global scales. The optimization algorithm introduced in the new system does not require decomposition of a matrix that establishes the correlation among the prior flux errors. This enables us to design the prior error covariance matrix more freely.

Obtaining stable solutions to the diffusion battery data inversion problem in the presence of noise

1997 ◽  
Vol 28 (2) ◽  
pp. 332
Author(s):  
J.J. Lloyd ◽  
C.J. Taylor ◽  
R.S. Lawson ◽  
R.A. Shields
Keyword(s):  
Inversion Problem ◽  
Data Inversion ◽  
Stable Solutions ◽  
Diffusion Battery
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