Conditionally well-posed problems

Suppose we know a priori that the exact solution belongs to a set, C, of the solutions with the property that the inverse operator A, defined on the image AC, is continuous. 

Definition 4 The problem (2.14) is conditionally well-posed (Tikhonov s vtell-posed) if the follotmng conditions are met (i) we know a priori that a solution of (2.14) exists and belongs to a specified set C C M, (li) the operator. A is a one-to-one mapping of C onto AC C D, (iii) the operator is continuous on AC C D. 

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. 

Tikhonov and Arsenin (1977) introduced the mathematical principles for selecting the correctness set C. For exani))le, if the models are described by a finite number of bounded parameters, they form a correctness set C in the Euclidean space of the model parameters. This result can be generalized for any metric space. 

Definition 5 The subset K of a metric. space M is called compact if any sequence mi K of elements in K contains a convergent subsequence m/ K, which converges to an element m in K. 

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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. 

It is well known that, for the Navier-Stokes equations, the prescription of the velocity field or of the traction on the boundary leads to a well-posed problem. On the other hand, viscoeleistic fluids have memory the flow inside the domain depends on the deformations that the fluid has experienced before it entered the domain, and one needs to specify conditions at the inflow boundary. For integral models, an infinite number of such conditions are required. For differential models only a finite number of conditions are necessary (more and more as the number of relaxation times increases,. ..). The number... 

Like all other models, LES requires the specification of proper boundary and initial conditions in order to fully determine the system and obtain a mathematically well-posed problem. However, this concept deviates from the more familiar average models in that the boundary conditions apparently rep>-resent the whole fluid domain beyond the computational domain. Therefore, to specify the solution completely, these conditions must apply to all of the space-time modes it comprises. 

The condition (5-224) provides a third boundary condition through matching with the core solution for the radial velocity. Although (5-224)-(5-226) is a well-posed problem, we shall not solve it here. The solution for F0 was numerical, and thus the present problem must also be solved numerically. The most efficient approach is to solve the problems for F0 and F together as this avoids storing the solutions for F0 and/or interpolating to evaluate the coefficients in (5-225). 

Together with the governing equation and boundary condition, (9-129), the matching condition (9-157) yields a well-posed problem that could be solved, in principle, to obtain... 

To determine the additional conditions that must be imposed to obtain a well-posed problem for the velocity fields in the inner and outer domains, we must examine the matching conditions in the region of overlap between them. These matching conditions can be written in general form as... 

The mathematical formulation of a well-posed problem can be well- or ill-conditioned. 

Equation 6.13 shows that a solution can only be obtained if spectral data are linearly independent otherwise, the matrix (AT AO cannot be inverted. Inversion problans can also arise if the matrix is ill conditioned. In such cases, special numerical techniques should be used to provide the inverse of (A A). For this reason, MLR is a method indicated for well-posed problems. This method may fail if process outputs respond nonhnearly to spectral changes or if the characteristic wavelengths in X are not selected properly [46,49]. 

Bubble collapse is a well-posed problem that is, the initial condition is rest state, for analysis with viscoelastic constitutive equations (Pearson and Middleman, 1977 Papanastasiou et al., 1984). Such work would be valuable to test other extensional methods for solutions. Bubble growth and collapse measurements also have important applications for processing of foamed polymer. 

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 ... 

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... 

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. 

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. 

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. 

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.)... 

To analyze a physical problem analytically, we must obtain the governing equations that model the phenomenon adequately. Additionally, if the auxiliary equations pertaining to initial and boundary conditions are prescribed those are also well-posed, then conceptually getting the solution of the problem is straightforward. Mathematicians are justifiably always concerned with the existence and uniqueness of the solution. Yet not every solution of the equation of motion, even if it is exact, is observable in nature. This is at the core of many physical phenomena where ohservahility of solution is of fundamental importance. If the solutions are not observable, then the corresponding basic flow is not stable. Here, the implication of stability is in the context of the solution with respect to infinitesimally small perturbations. 

The problem consists in finding the appropriate conditions to prescribe for the extrastress tensor at the inflow boundary x = 0), so that the steady problem is well-posed. The method of analysis is a variant of the algorithm leading to Theorem 3.1. 

Note that, for Maxwell models with —1 < a < 1, relation (15) is satisfied locally in time provided it is satisfied at time < = 0. For a = 1, relation (15) is equivalent to relation (5), which insures that the initial value problem is well-posed this is a natural condition to impose on the stress. But, for a 1, condition (15) reveals that the model is not always of evolution type, which means that Hadamard instabilities can occur. (See Section 2.1)... 

In a recent work [42], Renardy characterizes a set of inflow boundary conditions which leads to a locally well-posed initial boundary value problem for the two-dimensional flow of an upper-convected Maxwell fluid transverse to a domain bounded by parallel planes. 

Definition 2 The problem (2.14) is correctly (or well) posed if the following conditions are satisfied (i) the solution m of equation (2.14) exists, (ii) the solution m of equation (2.14) is unique, (Hi) the solution m depends continuously on the left-hand side of equation (2.14) d. 

The condition number (CN) is used in the numerical examples to control well-posedness of the SLAE. In the case of high CN, the methods for ill-posed problems should be used [8],... 

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