*2.2. Tikhonov Regularization Method*

In this section, we briefly present the idea of the Tikhonov regularization method [19], which is usually applied to stabilize ill-posed problems, such as our inverse problem. Normally, the considered inverse problem can be represented by the system of *m* linear equations with *n* unknowns, as

$$\mathbf{Ax} = \mathbf{b}^c,\tag{5}$$

where **b** is the vector in the right-hand side, which is perturbed by some noise , and **x** is the solution of the system (5) after perturbation. Tikhonov regularization replaces the inverse problem (5) by a minimization problem to obtain an efficiently approximate solution, which can be described as

$$\underset{\mathbf{x}\in\mathbb{R}^{n}}{\arg\min} \left\{ \left\| \left\| \mathbf{A}\mathbf{x} - \mathbf{b}^{\mathbf{c}} \right\|^{2} + \lambda \left\| \mathbf{x} \right\|^{2} \right\} \right\}\tag{6}$$

where *λ* > 0 is a regularization parameter balancing the weighting between the two terms of the function and · is the standard Euclidean norm. To reformulate the above minimization problem (6), we obtain

$$\underset{\mathbf{x}\in\mathbb{R}^n}{\arg\min} \left\{ \left\| \begin{bmatrix} \mathbf{A} \\ \sqrt{\lambda}\mathbf{I} \end{bmatrix} \mathbf{x} - \begin{bmatrix} \mathbf{b}^c \\ \mathbf{0} \end{bmatrix} \right\|^2 \right\}.$$

Clearly, this is a linear least-square problem in **x**. Then, the above problem turns out to be the normal equation of the form

> **A** <sup>√</sup>*λ***<sup>I</sup> A** <sup>√</sup>*λ***<sup>I</sup> x** = **A** <sup>√</sup>*λ***<sup>I</sup> b 0** .

To simplify the above equation, the solution **x** under the regularization parameter *λ* (denoted by **x***λ*) can be computed by

$$\mathbf{x}\_{\lambda} = (\mathbf{A}^{\top}\mathbf{A} + \lambda\mathbf{I})^{-1}\mathbf{A}^{\top}\mathbf{b}^{\epsilon}. \tag{7}$$

We can see that the accuracy of **x***<sup>λ</sup>* in (7) depends on the regularization parameter *λ*, which plays an important role in the calculation: A large regularization parameter may over-smoothen the solution, while a small regularization parameter may lose the ability to stabilize the solution. Therefore, a suitable choice of the regularization parameter *λ* is very significant for finding a stable approximate solution. There are many approaches for choosing a value of the parameter *λ*, such as the discrepancy principle criterion, the generalized cross-validation, the L-curve method, and so on. Nevertheless, the regularization parameter *λ* in this paper is chosen according to Morozov's discrepancy principle combined with Newton's method, which has been proposed in Reference [20]. We provide the procedure for calculating the optimal regularization parameter *λ* below, which can be carried out by the following steps:

**Step 1:** Set *n* = 0 and give an initial regularization parameter *λ*<sup>0</sup> > 0. **Step 2:** Compute **x***λ<sup>n</sup>* = (**AA** + *λn***I**)−1**Ab**. **Step 3:** Compute ∇**x***λ<sup>n</sup>* = −(**A<sup>A</sup>** + *<sup>λ</sup>n***I**)−1**x***λ<sup>n</sup>* . **Step 4:** Compute *<sup>G</sup>*(*λn*) = **Ax***λ<sup>n</sup>* <sup>−</sup> **<sup>b</sup>** <sup>2</sup> <sup>−</sup> 2. **Step 5:** Compute *G* (*λn*) = <sup>2</sup>*λ<sup>n</sup>* **<sup>A</sup>**∇**x***λ<sup>n</sup>* <sup>2</sup> + <sup>2</sup>*λ*<sup>2</sup> *<sup>n</sup>*∇**x***λ<sup>n</sup>* 2. **Step 6:** Compute *<sup>λ</sup>n*+<sup>1</sup> <sup>=</sup> *<sup>λ</sup><sup>n</sup>* <sup>−</sup> *<sup>G</sup>*(*λn*) *<sup>G</sup>*(*λn*). **Step 7:** If *λn*+<sup>1</sup> − *λ<sup>n</sup>* < *δ* for a tolerance *δ*, end. Else, set *n* = *n* + 1 and return to Step 2.

Therefore, we receive the optimal regularization parameter *λ*, which is the terminal value *λ<sup>n</sup>* obtained from the above procedure. When the regularization parameter *λ* is fixed as the mentioned optimal value, we can directly obtain the corresponding regularized solution by (7).

### **3. Numerical Algorithms for Direct and Inverse Problems of TVIDE**

In this section, we apply the FIM-SCP described in Section 2.1 to devise the numerical algorithms for solving both the direct and inverse TVIDE problems (1), in order to obtain accurate approximate results. Let *u* be an approximate solution of *v* in (1). Then, we have the following linear TVIDE over the domain Ω = (0, *L*) × (0, *T*]:

$$u\_t(\mathbf{x},t) + \mathcal{L}u(\mathbf{x},t) = \int\_0^t \kappa\_1(\mathbf{x},\eta)u(\mathbf{x},\eta)d\eta + \int\_0^\mathbf{x} \kappa\_2(\xi,t)u(\xi,t)d\xi + F(\mathbf{x},t), \tag{8}$$

subject to the initial condition

$$
\mu(\mathbf{x},0) = \phi(\mathbf{x}), \quad \mathbf{x} \in [0,L], \tag{9}
$$

and the boundary conditions

$$
\mu^{(r)}(b, t) = \psi\_r(t), \ t \in [0, T], \tag{10}
$$

for *b* ∈ {0, *L*} and *r* ∈ {0, 1, 2, ..., *n* − 1}, where *t* and *x* represent time and space variables, respectively. Additionally, *κ*1, *κ*2, *F*, *φ*, and *ψ<sup>r</sup>* are given continuous functions and L is the spatial linear differential operator of order *<sup>n</sup>* defined by <sup>L</sup> :<sup>=</sup> <sup>∑</sup>*<sup>n</sup> <sup>i</sup>*=<sup>0</sup> *pi*(*x*, *<sup>t</sup>*) *<sup>d</sup><sup>i</sup> dx<sup>i</sup>* , where *pi*(*x*, *<sup>t</sup>*) is given and sufficiently smooth.
