Numerical Solution for an Inverse Variational Problem

In the present work, firstly, we use a minimax equality to prove the existence of a solution of certain system of varitional equations and we provide a numerical approximation of such a solution. Then, we propose a numerical method to solve a collage-type inverse problem associated with the corresponding system, and illustrate the behaviour of the method with a numerical example.


Introduction
In this paper, we propose a numerical method to solve an inverse problem associated with a certain system of variational equations, which is based upon a generalization of the classical collage theorem. To this end, we first characterize, in terms of the existence of a scalar, the 1 arXiv:2105.08314v1 [math.NA] 18 May 2021 solvability of the following system x * 2 (y 2 − x 0 ) ≤ a 2 (y 2 , y 2 − x 0 ) . . . . . . . . .
, with x * i being continuous and linear functionals in a real reflexive Banach space and a i continuous bilinear forms in the same space.
Since Stampacchia's results in the 1960s, the study of variational inequalities and systems of variational equations has raised great interest, in part due to the fact that a wide range of optimization problems can be reformulated as variational problems. The concept of variational systems encompasses different types of problems, for example, in [30], the Nash equilibrium problem, the spatial equilibrium problem and the general equilibrium programming problem are modeled as a system of variational inequalities. In [14], the variational system includes certain mixed variational formulations associated with some elliptic problems. Also, we can find them associated with abstract economy, [1].
Another specific type problem of variational systems, and that is related to our work, is the so-called common solutions to variational inequalities problem, which consists of finding common solutions to a system of variational inequalities. There are different approaches to this kind of inequality systems: in [38] the definition domains of the functions of the system are closed and convex sets of a Hausdorff topological vector space. In [17], the system problem is dealt with for only two inequalities, a treatment that is generalized in [9].
In the study of the solution of variational equations or systems of variational equations, a wide range of techniques is used, including those of minimax type, [5,10,31], or those that use fixed point results and their associated iterative methods, such as those we detail next. In [1], the authors prove the existence of solution of certain variational systems by using a multivalued fixed point theorem. Also, one can find a proof of the existence of solution to a variational system with the Brouwer fixed point theorem in [38] as well as the construction of an iterative algorithm for approximating the unique solution of the system and a discussion of the convergence analysis of the algorithm.
In the present paper, we use a minimax technique to prove the existence of solution for the system of variational inequalities, Theorem 2.3, that, unlike the different results that appear in the articles mentioned above, characterizes the existence of solutions in closed subsets (not necessarily convex).
Once the conditions that ensure the existence of a solution for the system of variational equations have been established, we will deal with the inverse problem, i.e., assuming that the model which depends on different parameters has been established, and some empirical solutions have been obtained, we will try to approximate the parameters for which the empirical solutions obtained are an approximation of the solution of the theoretical model.
From the different approaches proposed in the literature to solve inverse problems, we rely on the approach of the so-called Collage theorem, which starts by considering the forward problem as a solution to a fixed point problem and deduces its analysis from Banach's fixed point theorem. We follow the line of different proposed generalizations of the Collage theorem, which are supported by different versions of the Lax-Milgram theorem, established, for example, to solve inverse problems associated with different families of integral or ordinary differential equations, ([7, 20, 21, 22]), or of partial differential equations ( [3,14,15,23,24,25,28]).
The paper is organized as follows. The first section begins with the presentation of our minimax tool and the variational system. Theorem 2.3 is the central point of this section, and provide us with a characterization of the solvability of the variational system. Moreover, from this theorem we derive a result which implies Stampacchia's theorem. The following section begins with a collage-type result that will be used in the numerical treatment of the inverse problem of a concrete example. To this end, we first propose a numerical approximation of the solution of the forward problem that we show in different tables and graphics. Finally, we finish our work with the conclusions.

The forward variational problem
In this section we deal with a result, Theorem 2.3, that generalizes the classic Stampacchia theorem. Indeed, it allows us to characterize the existence of a solution to a system of variational equations as that of a certain scalar. We should mention that minimax inequalities are a widely used technique in variational analysis: [2] is a good example. In [35], we see the equivalence between minimax results and the Hahn-Banach Theorem, and how these results are used as functional analytic tools.
The fundamental tool to establish this direct result is given by the following minimax inequality [18], which includes a not very restrictive convexity condition, which allows to characterize the validity of the minimax identity [32,33]. This concept of weak convexity is called infsup-convexity, and it appears with a nomenclature for first time as affine weakly convexlikeness in [37]. The infsup-convexity arises, in a natural way, when we deal with equilibrium and minimax problems. Definition 2.1 If X and Y are nonempty sets, a function g : Clearly, the infsup-convexity extends the concept of convex function, but also another types of weak convexity, such as convexlikeness ( [11]). Let us recall that a function f : The concept of infsup-convexity is used in the following minimax result, [18].

Theorem 2.2
Assume that X is a nonempty, convex and compact subset of a real topological vector space, Y is a nonempty set and g : X × Y −→ R is continuous and concave on X. Then, This minimax inequality is of the Hahn-Banach type, in the sense that it is equivalent to this central result of the functional analysis. In fact, the Hahn-Banach theorem and some of its generalizations have also been used to prove some variational results [34,36], even for some systems of variational equations [14,15] that include, as a particular case, those corresponding to the mixed variational formulation of the classical Babuška-Brezzi theory [6,16]. Now, we introduce in a precise way the forward problem involving a system of variational equations. Let E be a real and reflexive Banach space, let Y 1 , . . . , Y N be closed and nonempty In order to study this system, we note by x * the linear and continuous functional defined in Y as and let a : E N × E N → R be the continuous and bilinear form First, let us verify that the problem (2.1) is equivalent to finding an x 0 ∈ Y fulfilling the following condition: for each y ∈ Y , where x 0 denotes the vector (x 0 , . . . , x 0 ), The fact that (2.1) implies (2.2) follows from the sum of the inequalities and from the definition of x * and a. For the opposite implication, it is enough to take (y 1 , x 0 , . . . , x 0 ) as an element of Y in (2.2) to obtain the first inequality of (2.1). We derive the other inequalities with the same reasoning.
Then we present the characterization mentioned at the beginning of this section, that extends the case of an equation previously established in [29]: Assume that E is a real and reflexive Banach space, Y 1 , . . . , Y N are nonempty and closed subsets of E, x * 1 : E −→ R, . . . , x * N : E −→ R are continuous and linear functionals and define Then, we have that there exists , if, and only if, for some α ≥ 0, Y ∩ αB E = ∅, and the next inequality holds: Proof. We have that (2.1) ⇒ (2.3) just by taking α := x 0 .
If µ = −∞ there is nothing to prove. Otherwise, let f : X × Y → R be the function defined as and as we have by Theorem 2.2 the maximun is reached, and as a consequence, there exists which is equivalent to the inequality system (2.1). 2 Before stating our next result, we recall a technical lemma. Although it is proven [8, Assume that a : E × E −→ R is a bilinear continuous form and x * : E −→ R is a continuous and linear functional. Then, the next problem: find y ∈ Y such that is equivalent to the problem of finding y ∈ Y satisfying If we add certain more restrictive hypotheses to Theorem 2.3, we can equivalently express the condition (2.3) in a simpler way, extending the system version of Stampacchia theorem. x * (y) := x * 1 (y 1 ) + · · · + x * N (y N ), y ∈ E N , and a(x, y) := a 1 (x 1 , y 1 ) + · · · + a N (x N , y N ), and suppose that x ∈ E ⇒ a(x, x) ≥ 0.
if, and only if, there exists α ≥ 0 fulfilling Y ∩ αB E = ∅ and a(y, x)). On the one hand, if we take m = 1 in (2.3) then we have (2.5). On the other hand, we suppose that (2.5) is valid and let m ≥ 1, t ∈ ∆ m and y 1 , . . . , y m ∈ Y . Then where we have used the convexity of Y and that of the quadratic form associated with the bilinear form a and (2.5). 2 We conclude this section by proving that the version of systems of the classical Stampacchia theorem is a consequence of Corollary 2.5. Indeed, assume that E is a real Hilbert space, Y 1 , . . . , Y N are nonempty closed and convex subsets of E, x * 1 : E −→ R, . . . , x * N : E −→ R are continuous and linear functionals and a 1 : E × E −→ R, . . . , a N : E × E −→ R are bilinear, continuous and coercive forms. With the notations above, let x * be the continuous and linear functional defined as and let a : E N × E N → R be the continuous and bilinear form Let us note that, given a vector x ∈ αB E with α ≥ 0, x = α, we can select, without loss of generality, the norm of E N appropriately so that x = α.
In addition, if ρ 1 , . . . , ρ N are the coercivity constants of a 1 , . . . , a N and x ∈ E, then we Taking α > β, it follows that Y ∩ αB E = ∅ and and we arrive at (2.5).

The inverse varational problem
To solve the inverse problem associated with the system of variational inequalities (2.1), we will make use of the following collage-type result, which can be proved as a consequence of Stampacchia's theorem for a system of inequalities. In order to avoid expository complications, we previously introduce the following notation: for a real Banach space, E * is its topological dual space. Moreover, if J is a nonempty set and for each j ∈ J and i ∈ {1, . . . , N } x 1 j * , . . . , x N j * ∈ E * and a 1 j , . . . , a N j : E × E −→ R are continuos biliear forms, we denote by (x 1 j * , . . . , x N j * ) and (a 1 j , . . . , a N j ) the continuous and linear functional and the continuous bilinear form respectively.
Proof. Given y ∈ Y and j ∈ J and taking into account Corollary 2.5, we have In [3,20,23,25,26] we can observe the idea that we used for the application of this result in the resolution of the inverse problem. This reasoning have been previously used with the Banach fixed point theorem in a similar way in [27].
We finish our work with the following example, which consists of two clearly defined parts.
The first deals with solving the forward problem using Galerkin's method. To this end, we will work with a certain Schauder basis, a very versatile tool, since we can observe its use in differential and integral problems ( [3,4,12,13,14,15]). The second part of this example, and also its main objective, is the numerical treatment of the inverse problem, where we obtain the target functions thanks to the Galerkin method previously described.
We have made use of Galerkin's method to solve the m-dimensional variational problem in the subspaces generated by {g 3 , · · · , g m+2 }. In the following table we show the behavior of the approximation in terms of the errors made in the corresponding spaces, where (u m , v m ) is the solution obtained for the m-dimensional problem. Also, we present some graphics that illustrate the exact solutions, their approximations and the differences between these functions. Now we finally present the treatment of the inverse problem under the notation of the Theorem 3.1. We now turn to the resolution of the inverse problem. The method we follow to      m solve it is as follows: we obtain a solution of the forward problem for each j 0 ∈ J. We write this solution as x j . In the inverse problem, we try to find, whenever possible, a j 0 ∈ J such that Let us take into account that if Y is a closed affine subset of E, and y ∈ Y is a target element, we can solve our problem if, under the condition that inf j (ρ 1 j , . . . , ρ N j ) > 0, we are able to solve To show a concrete example, we take the variational inequality discussed above, with f (x) := sup ω ∈ H 1 0 (0, 1) ω = 1 |a j ((u m , ω), (v m , ω)) − x * (ω, ω)|.
Below we present a table where we can observe the different approximations of (λ 1 , λ 2 ) that we have obtained by taking n = 7 in the above expression and considering different targets (u m , v m ).

Conclusions
In this paper we have presented a numerical method to solve the inverse problem associated with a system of variational inequalities, which has been illustred in Example 3.2. To do this, firstly, we have used a minimax equality, Theorem 2.1, to prove a result that allows us to characterize the existence of a solution to the system of inequalities, Theorem 2.2. To solve the inverse problem, we have applied a consequence of Stampacchia's theorem, which is a collage-type result.