Conditional Optimization and One Inverse Boundary Value Problem

Here we construct different approximate solutions of the plane inverse boundary value problem of aerohydrodynamics. In order to do this we solve some conditional optimization problems in the norms ‖ ⋅ ‖ 2 , ‖ ⋅ ‖ ∞ , and ‖ ⋅ ‖ 1 and some of their generalizations. We present the example clarifying the mathematical constructions and show that the supremum norm generalization seems to be the optimal one of all the functionals considered in the paper.


Introduction
The formulation and the first attempt in solution of the inverse boundary value problem of aerohydrodynamics are due to Tumashev and Mangler [1,2].
This problem is one of numerous boundary value problems.The example of the boundary value problem is the Hilbert one: given the real value of the analytic function on the known contour  it should be possible to reconstruct this function in the inner domain with respect to the contour  [3].
The example of the inverse boundary value problem is the following: Assume that we have two real functions (), V(),  ∈ [0, ], such that (0) = () and V(0) = V().It should be possible to reconstruct both the contour  of length  and the analytic in the domain bounded by  function whose real and imaginary parts at the contour  point with the arc parameter  coincide with the given functions (), V() [2].
The inverse boundary value problem solution methods can be applied for problems of aerohydrodynamics [2,4].For example, all the wing construction methods I know of reduce to two basic problems [5].The first of them is the plane wing section reconstruction, and the second is mutual positioning of these profiles.We consider the former of these problems.
Our problem is as follows: we have the air or fluid flow particles velocity distribution along the length of the unknown contour , that is, the function V(),  ∈ [0, ].So we know only the contour  length  and not the form of this profile.We also assume that the air or fluid flow is potential.
The inverse boundary value problem then is to find the shape of the contour  depending only on the information on the particles velocity.Since the velocity is given only in the finite set of points it seems natural to apply a spline (not necessarily linear) in order to define this function for all the contour points.This approach was widely used by the authors of [4].
This problem requires certain resolution conditions which appear only after we find the contour parametrization.So these conditions cannot be satisfied initially.Here we try to find a minimal in some sense initial data deformation so that this deformed data meets these conditions.Note that almost all of the equation systems in the article can be solved only numerically due to their transcendent form.
Let us recall the solution procedure from [2].
(1) Inverse Boundary Value Problem Solution.Let the velocity distribution V() along the unknown contour  be given; here  is the curve length parameter.We must find the contour  form.Assume that the contour length equals , the flow bifurcation happens at the point with the parameter value   ∈ (0, ), and the flow velocity vanishes at  = 0,  = .Let V() < 0 for  ∈ [0,   ) and V() > 0 for  ∈ (  , ].
We consider the constants Mathematical Problems in Engineering and the function Here the number  ℎ −   is the flow circulation.If it is greater than 0 then we have a positive lifting force for the wing profile.Now we have the standard external inverse boundary value problem for the complex potential  =  +  in the unknown contour  exterior.This function  has a simple pole and a logarithmic singularity at infinity and meets the boundary conditions We introduce the auxiliary analytic function.First note that the complex potential of the flow over the unit disc with the circulation equal to  ℎ −   has the following form: Here we have the relations Since the velocity is 0 in the critical points (where flow separates into two parts and where it vanishes) we have the following equations on the unknown constants  0 ,  * and two auxiliary angles relative to the flow critical points  1 and  2 : This system is uniquely resolvable [2].Now we equate the complex flow potential real parts of the contour  in the complex plane of coordinate  =  +  to the complex flow potential of the unit circle in the plane of coordinate  =   ,  ≥ 0: Then we obtain the relation  = () on the parameters of the contours.Finally we reconstruct the function  = (), which maps the unit disc exterior onto the flow domain in the plane  solving the Schwartz problem for the function ln   () at the unit disc exterior.The main object for our consideration is now the function (2) Resolvability Conditions.The profile possesses the mechanical sense in case the function V() of the previous section meets certain conditions.The first condition on the function log |V()| naturally appears when we reconstruct our contour and equate the residue of log |V()| at infinity to 0. Then the profile is a closed curve.
So the contour  closeness condition is as follows: the function log |V()| must meet the complex equality Here  and  are known constants.Thus the first condition defines the first Fourier coefficients with cos  and sin  for the function log |V()|.
We arrive at the second condition on log |V()| due to purely mechanical reasons.Since the flow velocity is fixed at infinity (this is a flight speed) we equate the value of log |V()| at infinity to some constant and obtain the following condition on log |V()|: Note that the constants , , and  values under conditions ( 8), ( 9) depend only on the form of Zhukovskii-Mitchell modified function [2,4].
So in the general case, that is, when the conditions do not hold, the problem does not have a solution and becomes an ill-posed problem.
(3) The Mathematical Problem.Ivanov [6] proposed application of quasisolutions to this problem.The notion of "quasisolution" was introduced earlier by Elizarov and Fokin [7,8] and in our case can be described as follows.
Definition 1.Let the inverse boundary problem with the given velocity V ∞ resolution procedure result in the function log |V()| such that its first Fourier coefficients do not equal the desired ones.Let one denote the set of functions with appropriate first Fourier coefficients by .Then one says that quasisolution of the problem is the aerodynamic contour for whose construction we apply some function  0 () ∈  ⊂  instead of log |V()|.Here one chooses  0 () in the normed space  so that Thus we need to modify the function log |V()| so that this modified function meets conditions (8), (9).Naturally this modification affects the initial data V().So we must change the initial velocity distribution in some way.
This modification is as follows.
We consider the function log |V()()| instead of the constructed log |V()|.Assume that log |V()()| meets conditions ( 8), (9).Clearly we need to make the log |V()| modification as small as possible; that is, we search for the function log |()| with the minimal possible norm.
In [6] one can find the quasisolution which in our notation minimizes the function log The solution procedure of [6] gave us log | f()| in the form of the Fourier polynomial which allowed log V() to meet the desired conditions.
It seems necessary to note that the articles [3,9] contain function minimization similar to one presented here with respect to the norm ‖ ⋅ ‖ ∞ .At the same time both the posed problems and the solution technique differ from that given in this paper.
We have purely mathematical problem.
We search for the function () minimal in some norm under the following restrictions on its Fourier coefficients: We preserve the former of these relations and we rewrite the two latter relations as follows: Without loss of generality we assume that  = 0.In the other case we simply consider the shifted variable  =  +  instead of the initially given .
Finally we have the following problem.We search for the function () minimal in some norm under the following three conditions:

Approximate Solution Which
Minimizes ‖‖ ∞ The results of the section were proved in [10].
Clearly the possible solution space choice yields the following result: Let  be the set of functions integrable on [0, 2] meeting conditions (13).Given these conditions it should be possible to find in  the extreme function  1 () such that Clearly the function F () ≡   /2 is the solution of the problem for  = 0, and  0 () is also the solution for the case of   = 0. Thus it feels natural to construct the solution similarly to [4] and consider the sum F ()+ 0 ().Nevertheless the solution minimizing the norm ‖‖ ∞ under condition of nonzero   and  differs from the given sum.
Let us find now the constants  > 0 and  ∈ [0, /2] so that  ∫ This nonlinear system has a unique solution.The constant  value is the solution of equation cos()/ = /  .This relation possesses a unique solution in the interval (0, /2).After we obtain the value of  we naturally determine the one and only value of  > 0.
Thus existence of  and  makes it possible to construct the desired function It is easy to verify that the function  1 () meets the last condition of (13).20) is the problem solution for   > 0.

Quasisolution Which Minimizes
Function class choice makes us search for the function which meets the equations   () ± () ≡ const in subsets of the interval [0, 2].Solutions of these equations are the functions  1 −  2  ±(1/) .The constants  1 and  2 must be chosen so that the solution is the continuous 2-periodic function on R.

Problem Formulation.
Let  be a subclass of 2periodic functions which meet conditions (13).Find the function  0 () ∈  ⊂ Let us construct the piecewise smooth function glued from the functions of the form ±(1 −  ±2 ).
For the case of we consider the function The unknown variables  and  are the solutions of the system In particular  is the solution of the equation Under condition (24) this equation has a unique solution in the interval (0, /2).

Proposition 3. The function given by relation (25) is the solution of the problem under condition (24).
Let us now consider the case of false relation (24).Then the solution is the function (28) Here the system on  and  is as follows: This system is uniquely resolvable for  ≥ /2.Clearly for  → 0 we have  → +∞ and () ≡ .

Proposition 4. The function given by relation (28) solves the problem for the case of
Note that the constant  describes the relative impact of the auxiliary function derivative on the approximate solution.The greater this constant is the closer the resulting minimizing function is to a piecewise-linear curve being the solution of the similar problem for the functional ‖  ‖ ∞ (cf. Figure 1).

Approximate Solution Minimizing ‖𝐹‖ 𝐿 1
The results of the section were given without proof in [11].
Here we solve the problem for the integral norm ‖‖ 1 = ∫  − |()|.We obtain the solution for the norm opposite to the norm ‖ ⋅ ‖ ∞ on the spaces   scale.
Since the solution exists only in the space of distributions (generalized functions) [12] it cannot be immediately applied to mechanical problems.Indeed let us first find the function with minimal possible norm ‖‖  1 meeting the condition Here   () is a Dirac -function [12].At the same time   () must be considered as a limit of a function sequence   () such that supp(  ) →  and ∫ 2 0   () = .

Problem Formulation.
Let us find in the set of all integrable on [0, 2] distributions meeting conditions (13), one with the minimal possible norm ‖‖  1 .

Problem Solution.
The first two solutions presented here are of purely mathematical interest since due to their noncontinuity they cannot be applied for any actual profile reconstruction.
Proposition 5.The function solves the posed problem.
The proof of this statement is similar to the one of [11].
Proof.Assume that there exists a function () such that ∫ The following also holds true:  Figure 2 shows velocity distributions for the norms ‖‖ 2 , ‖‖ ∞ + 1/2‖  ‖ ∞ and the functional (1, 3).The integral norm solutions are better in majority of points but in certain points the norm ‖‖ ∞ + 1/2‖  ‖ ∞ is the best possible one.This is true for all of the examples constructed by the author.Since the velocity distribution describes the contour these differences result in rather palpable differences in contours presented in Figure 3. Also direct calculations tell us that the boundary layer thickness is minimal in the case of the norm ‖‖ ∞ + 1/2‖  ‖ ∞ and maximal in the case of the functional  (1,3).Thus the drag (that is wind or fluid resistance) is also the smallest for the norm ‖‖ ∞ + 1/2‖  ‖ ∞ .

Norm in the Space 𝐿 𝑝
Let us find the function with minimal norm ‖‖   in the space of all functions integrable on [0, 2] and meeting conditions (13).
The problem then can be resolved by methods of calculus of variations.
Construction.Let us first solve the auxiliary problem.Given the function  : [−, ] → R it is possible to construct the class   of functions  : [−, ] → R which meet the condition ∫  −  ()  ()  =  > 0.