Well-posed problems

On the concept of well-posedness for a difference problem. There is another matter which is one of some interest. In conformity with statements of problems of mathematical physics, it is fairly common to call a problem well-posed if the following conditions are satisfied ... 

We call set C the correctness set. In contrast to the standard well-posed problem, a conditionally well-posed problem docs not require solvability over the entire space. Also the requirement of the continuity of over the entire space M is substituted by the requirement of continuity over the image of C in M. Thus, introducing a correctness set makes even an ill-posed problem well-posed. 

In setting up and carrying out any engineering computation, we have three fundamental issues to resolve before a calculation should be attempted (1) Is the problem well-posed (2) How should the problem be formulated mathematically (3) What computational techniques can be applied These are not three separate issues rather, they are coupled. For example, the way a problem is posed influences problem formulation, and problem formulation influences computational technique. 

Is the problem well-posed This issue concerns whether we have enough information to compute the required unknowns. In phase and reaction-equilibrium computations, this issue is resolved by a proper application of the generalized phase rule it might not be properly resolved by a routine application of the Gibbs phase rule. In particular, we have discussed two kinds of subtleties that are often overlooked. 

We can call the problem well-posed . Nonetheless, under unfavoiuable circumstances thus when the initial estimates (in particular mass fractions and temperatures) are too far from it can also happen that by the procedure, we arrive at m3 = 0 and m, = /Mj = 0, along with some incidental values of 5 and Such possibility can never be avoided the measured values are then discarded. So in practice, an adjustment procwiure can fail either because the problem is not well-posed (it then most likely fails always), or because the initial estimates (measured values) are too bad . 

According to Hadamard, what makes a mathematical problem well posed are the existence and uniqueness of the solution and its stability with respect to the data. 

The above formulation is a well posed problem in optimal control theory and its solution can be obtained by the application of Pontryagin s Minimum Principle (Sage and White (1977)). 

We say that a difference problem (scheme) is well-posed if for all sufficiently small < /ly... 

Our next goal is to establish direct links between stability, approximation on an element u G B l and convergence to u for scheme (21). If scheme (21) is correct, then problem (30) for Zj is well-posed. Because of this fact, its solution obeys the estimate... 

Problem (11) is said to be well-posed if there exists a unique solution of equation (11) for any f E H and this solution continuously depends on the right-hand side /, so that... 

Darrieus and Landau established that a planar laminar premixed flame is intrinsically unstable, and many studies have been devoted to this phenomenon, theoretically, numerically, and experimentally. The question is then whether a turbulent flame is the final state, saturated but continuously fluctuating, of an unstable laminar flame, similar to a turbulent inert flow, which is the product of loss of stability of a laminar flow. Indeed, should it exist, this kind of flame does constitute a clearly and simply well-posed problem, eventually free from any boundary conditions when the flame has been initiated in one point far from the walls. 

As was pointed out earlier, within these ranges the degree of matching with the measured useful signal is the same for any set of parameters. Certainly, knowledge of variation of these parameters is the most important step in solving the inverse problem, because this table allows us to separate the stable from unstable parameters, and correspondingly, perform a transformation from the ill-posed to a well-posed problem. 

Generally speaking, for condition numbers less than 10 the parameter estimation problem is well-posed. For condition numbers greater than 1010 the problem is relatively ill-conditioned whereas for condition numbers 10 ° or greater the problem is very ill-conditioned and we may encounter computer overflow problems. 

When the parameters differ by more than one order of magnitude, matrix A may appear to be ill-conditioned even if the parameter estimation problem is well-posed. The best way to overcome this problem is by introducing the reduced sensitivity coefficients, defined as... 

When the parameters differ by several orders of magnitude between them, the joint confidence region will have a long and narrow shape even if the parameter estimation problem is well-posed. To avoid unnecessary use of the shape criterion, instead of investigating the properties of matrix A given by Equation 12.2, it is better to use the normalized form of matrix A given below (Kalogerakis and Luus, 1984) as AR. 

The knowledge required to implement Bayes formula is daunting in that a priori as well as class conditional probabilities must be known. Some reduction in requirements can be accomplished by using joint probability distributions in place of the a priori and class conditional probabilities. Even with this simplification, few interpretation problems are so well posed that the information needed is available. It is possible to employ the Bayesian approach by estimating the unknown probabilities and probability density functions from exemplar patterns that are believed to be representative of the problem under investigation. This approach, however, implies supervised learning where the correct class label for each exemplar is known. The ability to perform data interpretation is determined by the quality of the estimates of the underlying probability distributions. 

If we have a second order system, we can derive an analytical relation for the controller. If we have a proportional controller with a second order process as in Example 5.2, the solution is unique. However, if we have, for example, a PI controller (2 parameters) and a first order process, there are no unique answers since we only have one design equation. We must specify one more design constraint in order to have a well-posed problem. 

Vol. 1543 A. L. Dontchev, T. Zolezzi, Well-Posed Optimization Problems. XII, 421 pages. 1993. 

Figure 4.12 corresponds to objective functions in well-posed optimization problems. In Table 4.2, cases 1 and 2 correspond to contours of /(x) that are concentric circles, but such functions rarely occur in practice. Elliptical contours such as correspond to cases 3 and 4 are most likely for well-behaved functions. Cases 5 to 10 correspond to degenerate problems, those in which no finite maximum or minimum or perhaps nonunique optima appear. 

For well-posed quadratic objective functions the contours always form a convex region for more general nonlinear functions, they do not (see tlje next section for an example). It is helpful to construct contour plots to assist in analyzing the performance of multivariable optimization techniques when applied to problems of two or three dimensions. Most computer libraries have contour plotting routines to generate the desired figures. 

Today, there an established software tool set does exist for the primary task, the calculation of modes and the description of field propagation. Approaches based on the finite element method (FEM) and finite differences (FD) are popular since long and can be applied to complex problems . The wave matching method, Green functions approaches, and many more schemes are used. But, as a matter of fact, the more dominant numerical methods are, the more the user has to scrutinize the results from the physical point of view. Recent mathematical methods, which can control accuracy absolutely - at least if the problem is well posed, help the design engineer with this. ... 

One of the basic elements of the computational algorithm is the determination of dependent variables at the inlet/outlet boundaries of a computational domain representing a finite length combustor. The essence of the problem lies in the fact that the nonstationary flow field has to be considered throughout a whole (unbounded) physical space, and only in this case the problem is mathematically well-posed. When solving a specific problem numerically, one has to consider a computational domain of a finite size, in which boundary conditions at artificial boundaries are to be imposed. 

So how accurate are DFT calculations It is extremely important to recognize that despite the apparent simplicity of this question, it is not well posed. The notion of accuracy includes multiple ideas that need to be considered separately. In particular, it is useful to distinguish between physical accuracy and numerical accuracy. When discussing physical accuracy, we aim to understand how precise the predictions of a DFT calculation for a specific physical property are relative to the true value of that property as it would be measured in a (often hypothetical) perfect experimental measurement. In contrast, numerical accuracy assesses whether a calculation provides a well-converged numerical solution to the mathematical problem defined by the Kohn-Sham (KS) equations. If you perform DFT calculations, much of your day-to-day... 

Another reason for searching for analytical solutions is that we can only solve numerically a problem that is well posed mathematically. We must program a valid mathematical expression of the problem on the computer or the answers may be nonsense. The need for proper descriptions of the equations, initial and bomdary conditions, and stoichiometric relations among the variables is the same whether one is interested in an analytical or a numerical solution. 

The physical interpretation of this well-posed problem is that it represents the motion of a particle in a non-uniform medium whose instantaneous response is modulated by g (R(t)) at each time t. It was also shown that the Hamiltonian of Eq. (6), with g taken such that... 

Equality (1.20) is of primary importance because of the following reason. It is customary in most ionic transport theories to use the local electroneutrality (LEN) approximation, that is, to set formally e = 0 in (1.9c). This reduces the order of the system (1.9), (l.lld) and makes overdetermined the boundary value problems (b.v.p.s) which were well posed for (1.9). In particular, in terms of LEN approximation, the continuity of Ci and ip is not preserved at the interfaces of discontinuity of N, such as those at the ion-exchange membrane/solution contact or at the contact of two ion-exchange membranes or ion-exchangers, etc. Physically this amounts to replacing the thin internal (boundary) layers, associated with N discontinuities, by jumps. On the other hand, according to (1-20) at local equilibrium the electrochemical potential of a species remains continuous across the interface. (Discontinuity of Cj, ip follows from continuity of p2 and preservation of the LEN condition (1.13) on both sides of the interface.)... 

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