Identification of the Source for Full Parabolic Equations

In this work, we consider the problem of identifying the time independent source for full parabolic equations in $\mathbb{R}^n$ from noisy data. This is an ill-posed problem in the sense of Hadamard. To compensate the factor that causes the instability, a family of parametric regularization operators is introduced, where the rule to select the value of the regularization parameter is included. This rule, known as regularization parameter choice rule, depends on the data noise level and the degree of smoothness that it is assumed for the source. The proof for the stability and convergence of the regularization criteria is presented and a H\"older type bound is obtained for the estimation error. Numerical examples are included to illustrate the effectiveness of this regularization approach.

This work aims to the determination, from noisy measurements taken at an arbitrary fixed time, of the realvalued function of n real variables, independent of time, in an evolutionary equation of transport in an unbounded domain. This is an ill-posed problem because the high frequency components of arbitrarily small data errors can lead to arbitrarily large errors in the solution [9,19].
Here, a family of regularization operators is designed to compensate the factor that causes the instability of the inverse operator. The parametric regularization operators lead to a family of well-posed problems that approximates the given ill-posed problem. The regularization operator family proposed here turns out to be an n-dimensional generalization of the modified regularization method considered in [29,30,46,40,48]. In these articles the authors estimate the source of the one-dimensional equation of heat from data measured in a fixed moment of time (t = 1) by adding a penalizing term and the parameter choice rule depends on the norm of the unknown function. In contrast, this paper analyzes the general n-dimensional parabolic equation while relaxing the conditions on the assumptions for the regularization process.
The stability and convergence of this regularization method are analyzed and a Hölder type bound is derived for the estimation error. In order to illustrate the regularization performance, some numerical examples for the 1D, 2D and 3D cases are included.

The ill-posed mathematical framework
We consider the problem of determining the source f for the following parabolic equation x ∈ R n , t > 0, u(x, 0) = 0, x ∈ R n , u(x, t 0 ) = y(x), x ∈ R n , t 0 > 0, where α 2 , ν > 0, β ∈ R n are given, ∆ denotes the Laplacian operator, ∇ denotes the Nabla operator and " · " is the usual inner product in R n . Note that this is a linear parabolic equations with constant coefficients. The existence and uniqueness of the solution to (1) is discussed in [49]. It is assumed that u(·, t), f (·) ∈ L 2 (R n ) are unknown functions and that y ∈ L 2 (R n ) can be measured with certain noise level δ, i.e., the data function where δ M ∈ R >0 represents the maximum level of noise. In practice , δ M may be estimated from the error committed by the measuring instruments. The analysis of the equation with boundary values and initial conditions in (1) is perform in the frequency space.
By using the above definition (3), the system (1) can be written in the frequency space as where Solving (4) in the frequency space, we obtain the solution Sinceû(ξ, t 0 ) =ŷ(ξ), an expression for the source in the frequency space is obtained by evaluating the equation where Denotingf Λ(ξ) increases without bound as ξ → ∞ amplifying the high frequency components of the observation error y(ξ) −ŷ δ (ξ). This fact might lead to a large estimation error f −f δ L 2 (R n ) even for small observation errors, hence one of the Hadamard conditions is not satisfied [13].

Regularization operators
In this section we propose a regularization operator taking into account the inestability factor in the inverse operator. We notice that the resulting operator is equivalent to the one that is obtained by using the quasi-reversibility method [20]. Basic theoretical issues related to regularization operators are included, more information can be found in [9,19].
Definition 2. Let T : Y −→ X, X and Y be Hilbert spaces and T be an unbounded operator. A regularization strategy for T is a family of linear and bounded operators Let us define the parametric family of linear operators R µ : where Λ(ξ) is given in (7) and µ is the regularization parameter. Note that the denominator in (10) was introduced for stabilization purposes. The properties for the operator family {R µ , µ > 0} are stated in the following theorem.
Theorem 1. Let us consider the problem of identifying f from noisy data y δ (x) measured at a given time instant t 0 > 0, where δ is the noise level defined in (2). Let the functions u and f satisfy the following differential equation with initial condition and let {R µ } be the family of operators defined in (10). Then, for every y(x) = u(x, t 0 ) there exists an a-priori parameter choice rule for µ > 0 such that the pair (R µ , µ) is a convergent regularization method for solving the identification problem (11).
Proof. The factor Λ(ξ) 1 + µ 2 ξ 2 is bounded for all ξ since it is continuous for all ξ ∈ R n and lim ξ →∞ Hence, for µ > 0, R µ is a linear continuous operator and we have that forŷ ∈ L 2 (R n ), then R µ is a regularization strategy for Λ. Therefore, by Proposition 3.4 in [9], for y(x) = u(x, t 0 ) there exists an a-priori parameter choice rule µ such that (R µ , µ) is a convergent regularization method for solving (6). The regularized solution to the inverse problem in the frequency space is given bŷ Therefore, an estimated function to f in (11) is given by the expression

Error analysis
In order to analyze the regularization performance, we first introduce some results that will be used later to obtain a bound for the error between the source f (x) and its estimate f δ,µ (x), that we will be referred to as the regularization error.
Proof. Euler's formula for a complex number ω = a + bi, and the parity for sine and cosine yields e −(a+bi) = e −a cos(b) − ie −a sin(b). Then, adding and subtracting 2e −a = 2e −Re(ω) , after algebraic operations one gets and the proof is completed Proof. First, let us consider the function f in (0, 1). Differentiating, in this case, we have f (x) = Then the function f is increasing in (0, 1) and On the other hand, for Therefore we have that f (x) < 1 1 − e −1 ∀x > 0 and since . Moreover, for α 2 , ν > 0 the following inequality Proof. Since a 2 + b 2 ≥ 2ab for all a, b ∈ R, we have that and k has only one critical point at ρ = 0. Consider three cases: • α 2 < νµ 2 : then the function k reaches its global maximum value ν at ρ = 0.
• α 2 > νµ 2 : since k is an even function and it is increasing for ρ >0 with lim and the proof is completed.
Definingf µ (ξ) := Λ(ξ) 1 + µ 2 ξ 2ŷ (ξ), by (6) it follows that and by the definition of the H p (R n )-norm given in (17), we have From the triangle inequality, then (21)- (22) and the definition of the regularized source (12) yield to From [45], sup ξ εR ≤ max µ p , µ 2 , this result together with Lemma 4 and the assumption ŷ −ŷ δ ≤ δ lead to , by Parseval's identity and the linearity of the Fourier transform, A-priori regularization parameter is usually chosen to be dependent on a-prior bound for the H p norm of the source and data noise. For the numerical examples the bound is generally assumed to be 1 [29,30,46,40,48] which can lead to erroneous estimates when f H p (R n ) > 1.
Here, we include a rule of choice for the regularization parameter that only depends on data noise. The following theorem is aimed to the estimate error for this case. Theorem 3. Consider the inverse problem of determining the source f (x) in (1)- (2). Let f δ,µ (x) be the regularization solution given in (13) and assume that f H p (R n ) is bounded in H p (R n ) for some 0 < p < ∞ (17). Then choosing there exists a constant K independent of δ such that Proof. From now on we denote · = · L 2 (R n ) . From the proof of Theorem 2 we have that , by Parseval's identity and the linearity of the Fourier transform, where Remark 1. Note p = ∞ is excluded since and the error bound in that case the error bound is K.
A particular case of mathematical interest is for p = 2, where we obtain Remark 2. If a bound δ M > 1 is allowed for noise in measurements, in order to keep 0 < µ < 1 one can take . (27) In that case, for

Numerical examples
In this section we consider few examples with a source f ∈ R n , n = 1, 2, 3 to illustrate the performance of the regularization operator. For each of them we have chosen different values for the parameters α 2 , β, ν, t 0 and a set of standard deviation values { 1 , ..., k } for the data noise. The space is uniformly discretized and a data set {y δ 1 , ..., y δ N } is obtained by evaluating the solution u(x, t) at a fixed time instant t 0 and adding noise, that is, where G is the uniform grid defined on R n and i , i = 1, ..., N are realizations of the normally distributed random variable η with mean 0 and standard deviation . By denoting y i = y(x i ), i = 1, .., N , since the noise level δ satisfies (2), the error y − y δ = (y 1 − y δ 1 , ..., y N − y δ N ) = ( 1 , ..., N ), is numerically integrated using the Simpson's method to obtained an approximated value for δ = δ( ). Then δ M in (2) is chosen to be an upper bound for δ. In practice, δ M can be estimated based on the measuring instruments used in data collection.
The results of the estimated sources (non-regularized and the regularized one) are plotted. A table of errors is included for each example which shows the absolute and relative errors.

Examples 1D
Two examples are considered for the one-dimensional inverse source problem defined in (1).
Example 1. For this example the source f is defined by 10 ≤ x ≤ 20, 0, in another case.
in another case.

Examples 2D
Two examples are considered for the two-dimensional inverse source problem defined in (1).
Example 3. For this example the source f is defined by in another case.

Examples 3D
One example are considered for the three-dimensional inverse source problem defined in (1).
Example 5. For this example the source f is defined by in another case.
With modeling parameter values α 2 = 0.4, β = (1, −0.5, −0.5), ν = 0.997, t 0 = 3, p = 3 and = 0.035. For this case, we consider either x = 0, y = 0 or z = 0 to plot the resulting estimated sources, the non-regularized and the regularized one.    This work focus on the problem of the inverse source for full parabolic equations in R n . A family of regularization operators is defined in order to deal with the ill-posedness the problem. It was designed to compensate the instability factor in the inverse operator. A rule of choice for the regularization parameters is also included which is based on the noise level assumed in the data set and the smoothness of the source to be identified. We demonstrate that for the parameter choice rule proposed here, the method is stable and a bound of Hölder type is obtained for the regularization error.
The numerical examples show good estimates for the different n-variables sources, n = 1, 2, 3, determined from data with different noise levels. Moreover, the sources used for the numerical examples belong to different Hilbert spaces and all of them show a good performance of the regularization approach adopted.