Solution mapping approach

Consider a dynamic model that describes the time evolution of species concentrations. 

The responses t] can be species concentrations at given observation times (y), concentration peaks, peak locations, induction times, flame speeds, etc., i.e., the experimental targets we intend to match. We can refer to functions g in equation (4) as response functional relationships, regardless of whether they are given in an analytical, tabular, or numerical form. In most applications one only needs to know several of these response functional relationships. It is these functions g that enter into the objective function evaluations by equation (1) or (2). 

The essence of the SM technique is approximation of functions g by simple algebraic expressions s within a subset H of parameter space 0. The approximating functions for the responses are obtained using the methodology of the response surface technique [17,19,27], by means of a relatively small number of computer simulations, referred as computer experiments. They are performed at pre-selected combinations of the parameter values and the entire set of these combinations is called a design of computer experiments. The computer experiments are performed using the complete dynamic model (3) and the functions obtained in this manner are referred as surrogate models. 

The use of a simple polynomial form as a surrogate model decreases the computational cost of the objective function evaluation by orders of magnitude. Not only does it make the solution of the inverse problem possible for large-scale dynamic models, but it also allows one to use more elaborate numerical methods of optimization, enables a rigorous statistical analysis of confidence regions [30,31], and ties in closely with a more general approach to model analysis. Data Collaboration, discussed later in the text. 

We will continue with the discussion of some practical aspects of the SM approach, but first we need to introduce concepts of effect sparsity, active variables, and variable transformation. 

---

Scheuer A, Hirsch O, Hayes R, Vogel H, Votsmeier M (2011) Efficient simulation of an ammonia oxidation reactor using a solution mapping approach. Catal Today 175 (1) 141-146 Votsmeier M, Scheuer A, Droehner A, Vogel H, Gieshoff J (2010) Simulation of automotive NH3 oxidation catalysts based on pre-computed rate data from mechanistic surface kinetics. Catal Today 151 (3-4) 271-277... 

Because the mapping approach treats electronic and nuclear dynamics on the same dynamical footing, its classical limit can be employed to study the phase-space properties of a nonadiabatic system. With this end in mind, we adopt a onemode two-state spin-boson system (Model IVa), which is mapped on a classical system with two degrees of freedom (DoF). Studying various Poincare surfaces of section, a detailed phase-space analysis of the problem is given, showing that the model exhibits mixed classical dynamics [123]. Furthermore, a number of periodic orbits (i.e., solutions of the classical equation of motion that return to their initial conditions) of the nonadiabatic system are identified and discussed [125]. It is shown that these vibronic periodic orbits can be used to analyze the nonadiabatic quantum dynamics [126]. Finally, a three-mode model of nonadiabatic photoisomerization (Model III) is employed to demonstrate the applicability of the concept of vibronic periodic orbits to multidimensional dynamics [127]. 

In another paper from the same group (40), a new solution-based approach for linear epitope mapping based on ACE/MS is demonstrated using beta-endorphin as a model substance. The procedure can briefly be described... 

In this section we present some applications of the LAND-map approach for computing time correlation functions and time dependent quantum expectation values for realistic model condensed phase systems. These representative applications demonstrate how the methodology can be implemented in general and provide challenging tests of the approach. The first test application is the spin-boson model where exact results are known from numerical path integral calculations [59-62]. The second system we study is a fully atomistic model for excess electronic transport in metal - molten salt solutions. Here the potentials are sufficiently reliable that findings from our calculations can be compared with experimental results. 

The constants a, b, and m in eqn [3] depend on the solute and on the chromatographic system. b = (ka) " , where ka is the retention factor in a pure nonpolar solvent. Equation [2] or [3] can be used as the basis of optimization of the composition of two-component (binary) mobile phases in NPLC, using a common window diagram or overlapping resolution mapping approach, as illustrated in an example in Figure 3. 

Another way to approach this is to use the solution map to multiply the original volume by a ratio of pressures that will result in an increase in volume. 

Many of the Examples use a unique visual approach in the Strategize Step, where you ll be shown how to draw a solution map for a problem. 

Stack M. M., Rodling J., Mathew M. T., Jawan H., Huang W. et al. (2010b), Microabrasion-corrosion of a Co-Cr/UHMWPE couple in Ringer s solution An approach to construction of mechanism and synergism maps for application to bio-implants . Wear, 269, 376-82. (doi 10.1016/j.wear.2010.04.022)... 

A convenient and constructive approach to attain symplectic maps is given by the composition of symplectic maps, which yields again a symplectic map. For appropriate Hk, the splittings (6) and (7) are exactly of this form If the Hk are Hamiltonians with respect to the whole system, then the exp rLnk) define the phase flow generated by these Hk- Thus, the exp TL-Hk) are symplectic maps on the whole phase space and the compositions in (6) and (7) are symplectic maps, too. Moreover, in order to allow for a direct numerical realization, we have to find some Hk for which either exp(rL-Kfc) has an analytic solution or a given symplectic integrator. 

Problems of inclusions in solids are also treated by exact elasticity approaches such as Muskhelishvili s complex-variable-mapping techniques [3-9]. In addition, numerical solution techniques such as finite elements and finite differences have been used extensively. 

P - pixei this solution is identical to the one given by Wiener inverse-filter in Eq. (11). This shows that Wiener approach is a particular case in MAP framework. 

The lower a graph is more interesting. While initially the Poincar6 phase portrait looks the same as before (point E, inset 2c) an interval of hysteresis is observed. The saddle-node bifurcation of the pericxiic solutions occurs off the invariant circle, and a region of two distinct attractors ensues a stable, quasiperiodic one and a stable periodic one (Point F, inset 2d). The boundary of the basins of attraction of these two attractors is the one-dimensional (for the map) stable manifold of the saddle-type periodic solutions, SA and SB. One side of the unstable manifold will get attract to one attractor (SC to the invariant circle) while the other side will approach die other attractor (SD to die periodic solution). 

A typical configuration of a SECM system is shown in Fig. 36.6. In this case the solution contains oxidized (Ox) species (denoted mediators) that are reduced on the active part of the microelectrode yielding the reduced (Red) species. The figure also shows a possible reaction of the Red species with the electrode, with the reaction rate If is very large, the approach of the tip to the surface will result in an increase in the reduction reaction (current) on the tip because the regeneration of Ox on the tip will be more efficient in a smaller gap. On tfie otfier fiand, if k is close to zero, the only effect of the tip approach to the surface wifi be the depletion of the Ox species in the gap upon reduction, whose diffusion from the bulk of the solution is now hindered by the probe. These two mechanisms, which result in the positive and negative feedback operation modes, can be used to map the reaction rate k, on the surface. 

---