Spurious solutions

Instead of a two-component equation as in the non-relativistic case, for fully relativistic calculations one has to solve a four-component equation. Conceptually, fully relativistic calculations are no more complicated than non-relativistic calculations, hut they are computationally demanding, in particular, for correlated molecular relativistic calculations. Unless taken care of at the outset, spurious solutions can occur in variational four-component relativistic calculations. In practice, this problem is handled by employing kinetically balanced basis sets. The kinetic balance relation is... 

The remarkable fact, first demonstrated by Nakatsuji [18], is that for each p >2, CSE(p) is equivalent (in a necessary and sufficient sense) to the original Hilbert-space eigenvalue equation, Eq. (2), provided that CSE(p) is solved subject to boundary conditions (A -representability conditions) appropriate for the (p + 2)-RDM. CSE(p), in other words, is a closed equation for the (p+ 2)-RDM (which determines the (p + 1)- and p-RDMs by partial trace) and has a unique A -representable solution Dp+2 for each electronic state, including excited states. Without A -representability constraints, however, this equation has many spurious solutions [48, 49]. CSE(2) is the most tractable reduced equation that is still equivalent to the original Hilbert-space equation, and ultimately it is CSE(2) that we wish to solve. Importantly, we do not wish to solve CSE(2) for... 

At this point, we note that there is no mechanism presently built into the relaxation methods to prevent undesirable high-frequency noise from growing with each iteration. Any spurious solution 6(x) satisfies Eq. (1) (see also Chapter 1, Sections V.A and V.B) for co beyond the band limit. If we know that the object 6 is truly band limited, with frequency cutoff co = 2, we can band-limit both data i and first object estimate d(1). The relaxation methods cannot then propagate noise having frequencies greater than Q into an estimate o(k). (One possible exception involves computer roundoff error. Sufficient precision is usually available to avoid this problem.)... 

Fig. 15.9 Steady-state solutions for the benzene mole fraction from the simulation of benzene oxidation near a turning point in a perfectly stirred reactor. Depending on the starting estimates, a number of spurious (nonphysical) solultions may be encountered. The true solution is indicated by the filled circles, while the shaded diamonds indicate (sometimes spurious) solutions that are computed through various continuation sequences.

Variational principle in the Dirac theory spurious solutions, unexpected extrema and other traps 175... 

In addition, for highly nonlinear cases, the mesh size needed to avoid spurious solutions may be so small that this approach is not feasible, (ii) The CFD approach also uses averaged models (e.g., k-s model for turbulent flows) with closure schemes that are not always justified and contain adjustable constants. 

The first approach is the discretization of the convection and the diffusion operators of the PDEs, which gives rise to a large (or very large) system of effective low-dimensional models. The order of these low-dimensional models depend on the minimum mesh size (or discretization interval) required to avoid spurious solutions. For example, the minimum number of mesh points (Nxyz) necessary to perform a direct numerical simulation (DNS) of convective-diffusion equation for non-reacting turbulent flow is given by (Baldyga and Bourne, 1999)... 

For very fast reactions i.e. Da //> 1, criterion equation (6) gives the minimum number of mesh points required (to avoid spurious solutions) for a three-dimensional scalar CDR equation as... 

In many cases one finds that considerably more off-resonance experiments are performed than are strictly necessary. This is clearly a considerable waste of spectrometer time when weak samples are involved. In fact two separate coherent experiments will always suffice (195) provided that the decoupler power is known with precision. It is then necessary to solve a quartic equation. Since this will probably be done by iteration it may be helpful to perform one or two additional off-resonance experiments so that spurious solutions can be rejected. 

Table V shows examples of the gains obtained using the new computational scheme. The typical real-time/computer-time ratio was increased from 36/1 to 180/1. Perhaps, more significant is the fact that the Fade method has allowed us to obtain acceptable solutions in situations where Runge-Kutta either failed to converge or produced spurious solutions. One such instance is the integration of full differential equations for the free-radical species. The Fade method successfully computed solutions without algebraic substitution of stationary-state assumptions whereas Runge-Kutta failed to produce any solution.

It is easy to show that (141) preserves charge conjugation symmetry in the free particle problem. But perhaps what is just as important is that spinor basis sets satisfying (141) listed below generate no spurious solutions for JC > 0 of the sort reported by [39,41,42,91,92]. 

The possible severity of the problem has been shown by M. Stanke and J. Karwowski, Variationalprinciples in the Dirac theory Spurious solutions, unexpected extrema and other traps in A eir Trends in Quantum Systems in Chemistry and Physics, vol. I, pp. 175 190, eds. J. Maniani et al., Kluwer Academic Publishers, Dordrecht (2001). Sometimes an eigenfunction corresponds to a quite different eigenvalue. Nothing of that sort appears in non-relativistic calculations. 

The calculation described here consists of two nested loops, an inner loop that refines the mole fractions of the vapor, and an outer loop that refines the estimate for pressure. Care must be exercised to identify trivial solutions. A trivial solution is one for which the liquid and vapor phases have the same composition and same compressibility factor. In this case the fugadty coefficients of a component are the same in the two phases (i.e., Ki = i for all components) and all equations are satisfied. Such solution is physically acceptable only at the critical point, but spurious solutions of this type may appear at other pressures and temperatures. 

Thus the concept of perfect matching handles everything within the proposed framework guaranteeing completeness. Further, no spurious solutions are generated and such a framework is useful for large-scale applications. Now a number of fault (disturbance) and failure scenarios are discussed with respect to perfect control and imperfect control. 

In the recent work by HMBB, the wavelet transform (defined in the next section), W ia. b). is used to discriminate between the true and spurious solutions, denoted by and , respectively. Let a ir) correspond to the global maximum extremal point, in the scale variable, for the wavelet transform, W ia. r). at a fixed turning point location, r. 

On the other hand, for a spurious solution, high frequency noise (i.e. approximation errors in the MRF representation) conspire to make it satisfy the TPQ conditions. These solutions (i.e. ) must have the support of their wavelet transform lie outside of the quantization scale range. That is, Ow (flQ) oo), or Ow < aq. 

The above criteria, aq < for physical solutions, and < aq, for unphysical, spurious, solutions, serve to discriminate between the physical and unphysical TPQ-MRF generated solutions (HMBB). 

As argued previously, we then have ag(T<) < Ou,(rf). This condition was used within the TPQ-MRF representation to discriminate between the physical and spurious solutions generated (HMBB (2000)). The effective quantization scale in that case corresponded to the smallest scale up to which the MRF generated solutions satisfied the scalet equation. 

Before defining the basic function space and operator, we would like to exclude the possibility of spurious solutions with negative components which may interfere with the existence proof. The following artificial example shows that spurious solutions may indeed result from a non-... 

Spurious solution or point 15,63,64 Stability of a steady state 25, 33 Steady state 32, 46 Steady state approximation 5, 42 Stirred-tank reactor 28, 38 Stoichiometric coefficient 2 Stoichiometric matrix 2 Stoichiometry 1... 

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