Harmonic inversion problem

The inverse problem would be well defined if we knew the temperature or the harmonic function... 

One of the important consequences of these results can be revealed when one regards the observed interference as an inverse problem. In fact, from (4.3), observing the relative phase corresponds to the broadband measurement of the excursion time T and thus the estimation of the individual harmonic phase [24]. Measuring attosecond excursion times offer a crucial basis... 

This demonstrates the equivalence between the harmonic inversion and the moment problem [2],... 

There is more to the presented strategy than the equivalence Eq. (308). To illuminate an additional advantage of the presented formalism, we shall analyze its application to the quantification problem (harmonic inversion) with the goal of determining the key spectral parameters uk, dk. To this end, we rewrite the matrix equation (302) as follows ... 

This type of problem, known as quantification, spectral analysis or harmonic inversion, can be solved by the FPT for any theoretically generated/ simulated time signals or experimentally measured data. The obtained solutions gain in their value by the manner in which the accompanying and unavoidable problem of noise is solved. This is done by unequivocal... 

The fast Fade transform successfully overcomes the problems of the FFT. As a nonlinear transform, the FFT suppresses noise from the analyzed time signals. Most importantly, the FFT completely obviates postprocessing via fitting or any other subjective adjustments. This is accomplished by a direct quantification of the time signal under study through exact spectral analysis, which provides the unique solutions for the inverse harmonic (HI) problem [1, 2]. The solution of the HI problem contains four real-valued spectral parameters (one complex frequency and one complex amplitude) for each resonance or peak in the associated frequency spectrum. From these spectral parameters, the metabolite concentrations are unequivocally extracted. This bypasses the ubiquitous uncertainties inherent to fitting, such... 

It is well known that the permeability of a one dimensional system, in which there is incompressible single phase flow, can be replaced by the harmonic mean. This ensures the same total flux for a given pressure drop. Now consider the inverse problem, where the flux and the pressure drop are given, and the problem is to determine the permeability distribution. In this case it is easy to see why the problem has a non-unique solution. All distributions with the same harmonic mean are candidates - all give complete agreement with observations and all leave the physics of the problem unchanged without error. 

We shall proceed as follows. We shall first diagonalize the Schrbdinger problem [Eq. (3.46)] with respect to the vibrational and rotational quantum numbers (Section 5.1). We arrive in this way at a Schrodinger equation in the variable p with an effective potential function for each vibration—rotation state. A least squares procedure that includes the numerical integration of the Schrodinger equation for this effective Hamiltonian will be used to determine the harmonic force field and the doubleminimum inversion potential function for ( NHa, NHs), ( ND3, NTa) and NH2D, ND2H (Section 5.2). 

We can see from Eqs. (3.5) [see also Appendix in Ref. ] that the force constants F , F etc., are generally mass dependent quantities. To arrive at the iso-topically invariant potential function we must therefore express these quantities in terms involving mass independent valence force constants and to fit these to experimental spectra [cf. ]. For ammonia, this would represent a really formidable numerical problem. Taking into account the proposed limits of our model, the fact that we are mainly interested in the inversion—rotation structure of the spectra, we have overcome the above mentioned difficulties in the following way [see for details] (i) all the enharmonic force constants in Eq. (5.5) were neglected (ii) the p-dependent contributions to the harmonic force constants F [see Eq. (4.7)] were neglected (iii) the least squares fit of the double-minimum potential function parameters and the p-independent harmonic force constants were performed for light isotopes ( NHs, NHs) and heavy isotopes ( ND3, NTs) separately. 

Laplace of Fourier transforms can be used to solve wave propagation problems. In certain special cases, for example, for harmonic excitations, the inversion integral can be evaluated directly or through the use of residues theory. However, in the general case an analytical evaluation in impracticable. 

The inversion double well potential in ammonia is a difficult problem that has been the object of numerous theoretical studies over the years [2-12], We chose to model this potential as the superposition of a harmonic part [modeling its overall shape to the zeroth Hamiltonian Ho<2Uq2), and two Gaussian terms (modeling the barrier), supplemented by three small corrections to the overall shape, of fourth, sixth, and eighth powers in (q2) ... 

In eonneetion with the implementation in the Rietveld codes, the Dollase March model and the spherical harmonics approach, for pole distributions determination, is developed in the next two parts. The problem of pole figure inversion is outside the scope of this chapter. 

Some time ago Schmidt and Korzeniewski [11] presented a simple onedimensional tight binding limit analysis of the problem of band states for ionic conduction. Although the potential energy function used was harmonic, delocalization was considered in the same manner as was used years ago in the Wall-Glockler [41] analysis of the inversion doubling of ammonia. Because of the simplicity of the calculation, and its ease of application, I summarize it here. The treatment is limited, at this... 

An alternative and very accurate way to solve the problem was given by Aquino [34,35], which is based on a procedure developed by Campoy and Palma [189,190] for free (unbounded) systems. This method has been successfully applied in the following contexts The spherically confined harmonic oscillator [13], computation of the Einstein coefficients of the ID asymmetrically confined harmonic oscillator [169], confined 2D hydrogen atom [185], and also in the study of free (unbounded) systems as the inversion frequencies of NH3, in which the inversion potential is modeled by a 20th-degree polynomial [191], and in the Mitra potential [192]. 

Therefore the following question arises in spite of the fact that the 3-dimension harmonic-oscillator potential is nonlocal and has specific properties, is it possible to derive potential forms which, put into the standard Schrodinger equation, will give about the same spectrum as the QZO model This is a standard question in inverse scattering problems [33]. Classical potentials giving approximately the same spectrum as the g-deformed, one-dimensional harmonic oscillator have been obtained either through the use of standard perturbation theory [34], or within the WKB approximation [35]. 

Finally, for this section we note that the valence interactions in Eq. [1] are either linear with respect to the force constants or can be made linear. For example, the harmonic approximation for the bond stretch, 0.5 (b - boV, is linear with respect to the force constant If a Morse function is chosen, then it is possible to linearize it by a Taylor expansion, etc. Even the dependence on the reference value bg can be transformed such that the force field has a linear term k, b - bo), where bo is predetermined and fixed, and is the parameter to be determined. The dependence of the energy function on the latter is linear. [After ko has been determined the bilinear form in b - bo) can be rearranged such that bo is modified and the term linear in b - bo) disappears.] Consequently, the fit of the force constants to the ab initio data can be transformed into a linear least-squares problem with respect to these parameters, and such a problem can be solved with one matrix inversion. This is to be distinguished from parameter optimizations with respect to experimental data such as frequencies that are, of course, complicated functions of the whole set of force constants and the molecular geometry. The linearity of the least-squares problem with respect to the ab initio data is a reflection of the point discussed in the previous section, which noted that the ab initio data are related to the functional form of empirical force fields more directly than the experimental data. A related advantage in this respect is that, when fitting the ab initio Hessian matrix and determining in this way the molecular normal modes and frequencies, one does not compare anharmonic and harmonic frequencies, as is usually done with respect to experimental results. 

Rather than dealing with k, nuclear science has focused in its inverse, the incompressibility coefficient K. Another minor change is that rather than dealing with a problem that is intrinsically 3-dimensional, the analysis is reduced to a one-dimensional problem of the stiffness with respect to harmonic vibrations of a uniform (albeit infinite) sphere of radius R. 

In this equation, Y is the spherical harmonic rotational wavefunction for rotational quantum number N and projection Mj, and p is the parity under inversion, equal to either +1 or —1 (see Problem 6.32). The rest of the... 

---