An improved Tikhonov regularization method combined with the exponential filter function

Abstract

Tikhonov regularization is one of the most popular methods for solving linear discrete ill-posed problems. This approach involves transforming the original problem into a penalized least-squares problem, yielding a solution that exhibits greater robustness against data inaccuracies and computational errors that may occur during the solving process. The choice of the regularization matrix significantly influences the accuracy of the resulting solution. In this paper, we propose a novel method for selecting the regularization matrix based on exponential filter functions, which have a unique connection with Tikhonov filter functions. Our proposed Tikhonov-exponential regularization method only requires one parameter, similar to the traditional Tikhonov regularization method. Computational examples demonstrate the advantages of our proposed method.

Keywords

ill-conditioned problems regularization filter function exponential filter regularization matrix

Introduction

This paper is focused on the calculation of an estimated solution for linear least-squares problems

\min_{x \in R^{n}} ‖ A x - b ‖

(1)

with a matrix

A \in R^{m \times n}

of undetermined rank. In other words, A has multiple singular values of varying magnitudes near zero. Problems involving minimization equation (1) with such matrices are generally known as discrete ill-conditioned problems. The aforementioned problems frequently arise from the discretization of linear ill-conditioned problems. The vector

b \in R^{m}

in equation (1) denotes measured data and is frequently subject to an unknown error

e \in R^{m}

. For the sake of notational simplicity, we assume that

m \geq n

; however, the techniques described also can be utilized when

m < n

. In this paper, the Euclidean vector norm is represented by

‖ \cdot ‖

Let $\hat{b} \in R^{m}$ represent the vector without errors that corresponds to $b$ , in other words,

b = \hat{b} + e .

(2)

where e denotes the error or noise. The consistent assumption is made for the infeasible system of linear equations with an error-free right-hand side, denoted as

A x = \hat{b}

(3)

The matrix $A^{+}$ represents the Moore-Penrose generalized inverse of matrix A, and it denotes the least-square solution with minimum norm for the consistent system (3). The generalized inverse solution to equation (1), as provided by

A^{+} b = A^{+} \hat{b} + A^{+} e = \hat{x} + A^{+} e

(4)

Typical approximation of $\hat{x}$ is often not meaningful due to the significant propagation of error $e$ , which is influenced by the large norm of $A^{+}$ . To address this challenge, a commonly employed approach involves substituting problem (1) with an adjacent problem that yields a solution less susceptible to the impact of error $e$ in measured data $b$ . Tikhonov regularization, one of the widely adopted techniques for achieving this objective, replaces equation (1) with the solution obtained from a least-squares problem

\min_{x \in R^{n}} {{‖ A x - b ‖}^{2} + λ^{2} {‖ L x ‖}^{2}},

(5)

where the scalar

λ \geq 0

is the regularization parameter and the matrix

L \in R^{p \times n}, 1 \leq p \leq n

is commonly known as the regularization matrix. For further insights on Tikhonov regularization, refer to Gerth.¹ Usually, the value of p, which represents the number of L’s rows, is either less than or equal to n. However, there are also cases where

p > n

is involved in the application of regularization matrices. The objective of regularization matrix L is to minimize the distributed error

A^{+} e

while preserving crucial characteristics of the exact solution

\hat{x}

. In many cases, the identity matrix I serves as a common choice for regularization matrices L. Recent studies have discussed various approaches for constructing regularization matrices.^2–8

In this paper, we will introduce an exponential regularization matrix, which can serve as a viable alternative to the identity matrix when there is limited knowledge about the exact solution $\hat{x}$ . Similar to using the identity matrix $I$ for regularization, the proposed matrix $L$ allows for easy computation of solution (4) when the singular value decomposition (SVD) of matrix A is readily accessible. Furthermore, the proposed Tikhonov-exponential regularization method only has one regularization parameter the same as Tikhonov regularization method. Numerical illustrations demonstrate that the regularization method proposed in this study consistently produces more precise approximations $x_{λ}$ of $\hat{x}$ compared to the alternative regularization methods examined in this paper.

This paper is organized as follows. We first provide a comprehensive discussion on regularization techniques, including Tikhonov regularization and two families of fractional Tikhonov methods. Additionally, our proposed regularization matrix is introduced. Then, illustrative instances of the four regularization techniques discussed. Finally, we show some numerical examples.

Regularization methods and filter functions

We initially present the SVD of matrix A, followed by an exploration of regularization techniques using Tikhonov methods. The SVD is a well-known format that can be utilized to represent the factorization of A

A = U Σ V^{T} = \sum_{j = 1}^{n} u_{j} σ_{j} v_{j}^{T},

(6)

where

U = [u_{1}, u_{2}, \dots, u_{m}] \in R^{m \times m}

and

V = [v_{1}, v_{2}, \dots, v_{n}] \in R^{n \times n}

are both orthogonal matrices, i.e.,

U^{T} U = V^{T} V = I_{n}

, and

Σ = diag [σ_{1}, σ_{2}, \dots, σ_{n}] \in R^{m \times n}

is a diagonal matrix, which may have a rectangular shape. Its diagonal entries consist of nonnegative singular values of

σ_{j}

. These singular values are arranged in a specific order

σ_{1} \geq σ_{2} \geq \dots \geq σ_{n} \geq 0

Tikhonov regularization and fractional Tikhonov regularization

By inserting equation (6), variables $\tilde{x} = V^{T} x$ , $\tilde{b} = U^{T} b$ , and $L = I$ into equation (5), we obtain the problem of least-squares with penalty term

\min_{\tilde{x} \in R^{n}} {{‖ Σ \tilde{x} - \tilde{b} ‖}^{2} + λ^{2} {‖ \tilde{x} ‖}^{2}}

with solution

{\tilde{x}}_{λ} = {(Σ^{T} Σ + λ^{2} I)}^{- 1} Σ^{T} \tilde{b}

(7)

for

λ > 0

. The equation (5) yields the solution as

x_{λ} = V {\tilde{x}}_{λ}

. It meets

(A^{T} A + λ^{2} I) x_{λ} = A^{T} b .

(8)

The regularization parameter $λ$ plays an important role in determining the sensitivity of $x_{λ}$ to error e in measured data b and the proximity of $x_{λ}$ to $\hat{x}$ . The appropriate value for $λ$ cannot be predetermined, but must be calculated during the solution process. Assuming that a bound

‖ e ‖ \leq ε

is available. The determination of the regularization parameter

λ

can be achieved by using the discrepancy principle (DP)

‖ A x_{λ} - b ‖ = η ε,

(9)

or, alternatively, so that

{‖ Σ {\tilde{x}}_{λ} - \tilde{b} ‖}^{2} = η^{2} ε^{2},

(10)

where the constant

η > 1

is unrelated to

ε

, is provided by the user and remains unchanged. To solve this nonlinear equation for

λ

, one possible approach is utilizing Newton’s method. Typically,

λ

exhibits a decrease in correlation with

ε

It is enlightening to examine the filtering mechanisms employed in the discussed methodology. Studies on regularization techniques utilizing filter functions can be referenced in Chen et al.⁵ The solution of equation (1) without regularization can be represented as

x = A^{+} b = \sum_{j = 1}^{n} \frac{u_{j}^{T} b}{σ_{j}} v_{j} .

It is advantageous to utilize filter functions $f_{λ, σ_{j}}$ in order to express approximate solutions $x_{λ}$ of equation (1), i.e.,

x_{λ} = \sum_{j = 1}^{n} f_{λ, σ_{j}} \frac{u_{j}^{T} b}{σ_{j}} v_{j}

(11)

The filter functions illustrate how the regularization technique affects the alteration of components. For instance, we can represent the Tikhonov solution $x_{λ} = V {\tilde{x}}_{λ}$ , which is defined by equation (7) with $L = I$ , in the following manner

x_{λ} = \sum_{j = 1}^{n} f_{λ, σ_{j}}^{(Tik)} \frac{u_{j}^{T} b}{σ_{j}} v_{j}

with the filter functions

f_{λ, σ_{j}}^{(Tik)} = \frac{σ_{j}^{2}}{σ_{j}^{2} + λ^{2}} .

(12)

Recently, a novel approach to regularization called fractional Tikhonov regularization^9,10 was introduced by Klann and Ramlau. They investigated a family of filter functions in their study

f_{γ, λ, σ_{j}}^{(KR)} = {(\frac{σ_{j}^{2}}{σ_{j}^{2} + λ^{2}})}^{γ},

(13)

where

0 \leq γ \leq 1

is an exponent be applied to the filter function

f_{λ, σ_{j}}^{(Tik)}

, leading to the generation of the fractional filter function. The counterpart of equation (8) related to the filter function (13) is provided in

{(A^{T} A + λ^{2} I)}^{γ} x_{λ, γ} = {(A^{T} A)}^{(γ - 1)} A^{T} b .

Hochstenbach and Reichel explored the assessment of residual error in Tikhonov regularization by employing a semi-norm that incorporates a fractional exponent of the Moore-Penrose generalized inverse of $A A^{T}$ as a weighting matrix.^9,11–13 The normal equations corresponding to the Tikhonov minimization problem (4) involving a symmetric positive semidefinite matrix can be expressed as follows

({(A^{T} A)}^{(γ + 1) / 2} + λ^{2} I) x_{λ, γ} = {(A^{T} A)}^{(γ - 1) / 2} A^{T} b .

with the filter functions

f_{γ, λ, σ_{j}}^{(HR)} = \frac{σ_{j}^{γ + 1}}{σ_{j}^{γ + 1} + λ^{2}}, γ > 0 .

(14)

For the two families of fractional Tikhonov regularization methods discussed above, standard Tikhonov regularization is recovered for $γ = 1$ .

Tikhonov-exponential regularization method

Introduce

L = E_{λ} V^{T}

(15)

with

E_{λ} = diag [\exp (\frac{- σ_{1}^{2}}{λ^{2}}), \exp (\frac{- σ_{2}^{2}}{λ^{2}}), \dots, \exp (\frac{- σ_{n}^{2}}{λ^{2}})] .

Analogously to equation (7), we obtain

{\tilde{x}}_{λ} = {(Σ^{T} Σ + λ^{2} E_{λ}^{2})}^{- 1} Σ^{T} \tilde{b}

(16)

and the corresponding approximate solution of equation (1) is provided and fulfills the given conditions

(A^{T} A + λ^{2} L^{T} L) x_{λ} = A^{T} b;

(17)

cf. equation (8). The value of

λ

utilized in equation (16) remains consistent with that employed in Section (2.1). Our regularization method, which incorporates Tikhonov-exponential technique with L defined by equation (15), can be formulated as

x_{λ} = \sum_{j = 1}^{n} f_{λ, σ_{j}}^{(Tik - \exp)} \frac{u_{j}^{T} b}{σ_{j}} v_{j}

with the filter functions

f_{λ, σ_{j}}^{(Tik - \exp)} = \frac{σ_{j}^{2}}{σ_{j}^{2} + λ^{2} \exp (\frac{- 2 σ_{j}^{2}}{λ^{2}})} .

(18)

Theorem 2.1. $f (λ, σ)$ defined by equation (18) has characters

(1) | f (λ, σ) | \leq 1 for all λ > 0 and 0 < σ \leq ‖ A ‖;

(2) | f (λ, σ) | < \frac{σ}{2 λ} for all λ > 0 and 0 < σ \leq ‖ A ‖;

(3) \lim_{λ \to 0} f (λ, σ) = 1 for every 0 < σ \leq ‖ A ‖ .

Proof. Obviously, (1) and (3) are valid, so only to prove (2).

Since $σ^{2} + λ^{2} \exp (\frac{- 2 σ^{2}}{λ^{2}}) \geq 2 σ λ \exp (\frac{- σ^{2}}{λ^{2}})$ and $\exp (\frac{- σ^{2}}{λ^{2}}) \geq 1$ , Property (2) follow from the elementary estimate

\frac{σ^{2}}{σ^{2} + λ^{2} \exp (\frac{- 2 σ^{2}}{λ^{2}})} \leq \frac{σ^{2}}{2 σ λ \exp (\frac{- σ^{2}}{λ^{2}})} \leq \frac{σ}{2 λ \exp (\frac{- σ^{2}}{λ^{2}})} < \frac{σ}{2 λ}

for all

λ, σ > 0

. Theorem 2.1 shows that the filter function

f (λ, σ)

defined by equation (18) is a regularizing filter, and it paves the way for the realization of regularization method.¹⁴

Theorem 2.2. The Tikhonov filter functions $\frac{σ_{j}^{2}}{σ_{j}^{2} + λ^{2}}$ can be considered approximations of the exponential filter functions $1 - \exp (\frac{- σ_{j}^{2}}{λ^{2}})$ .

Proof. The Taylor series expansion of $\exp (\frac{σ_{j}^{2}}{λ^{2}})$ is given by

\exp (\frac{σ_{j}^{2}}{λ^{2}}) = 1 + \frac{σ_{j}^{2}}{λ^{2}} + O (σ_{j}^{4})

By using the first two terms of this Taylor series expansion to approximate

\exp (\frac{σ_{j}^{2}}{λ^{2}})

, we obtain

1 - \exp (\frac{- σ_{j}^{2}}{λ^{2}}) \approx 1 - \frac{1}{1 + σ_{j}^{2} / λ^{2}} = 1 - \frac{λ^{2}}{σ_{j}^{2} + λ^{2}} = \frac{σ_{j}^{2}}{σ_{j}^{2} + λ^{2}} .

It can be inferred from equation (7) that the utilization of Tikhonov regularization with $L = I$ and $λ > 0$ results in a reduction of all elements in $Σ^{T} \tilde{b}$ , which corresponds to the solution components $v_{j}$ of $x_{λ}$ . This is why the standard Tikhonov regularization can sometimes result in an overly smoothed solution, even when the regularization parameter is well-chosen. However, it would be more reasonable to selectively attenuate only the solution components with high index of $v_{j}$ , while leaving the solution components with low index unchanged. As we have observed, it is the small singular values that lead to significant error e propagation in b and thus are filtered out by the low-pass filter $f_{λ, σ_{j}}^{(Tik)}$ defined in equation (12). As a result we consider the choice of $L = E_{λ} V^{T}$ be difined by equation (15), where the components $\exp (- σ_{j}^{2} / λ^{2})$ in $E_{λ}$ is a special designed high-pass filter according to Theorem 2.2.

Figure 1 illustrates the performance of the filter functions $f_{λ, σ_{j}}^{(Tik)}$ , $f_{0.8, λ, σ_{j}}^{(KR)}$ , $f_{0.6, λ, σ_{j}}^{(HR)}$ and $f_{λ, σ_{j}}^{(Tik - \exp)}$ for $λ = 10^{- 2}$ . The constituent parts of the solution (11) related to singular values with low index are observed to be less attenuated by $f_{λ, σ_{j}}^{(Tik - \exp)}$ compared to other filter functions, thereby potentially yielding computed approximate solutions of equation (1) that exhibit superior quality when compared to standard Tikhonov regularization or fractional Tikhonov regularization.

Figure 1.

Semilog plots of filter functions f corresponding to Tihonov regularization (black line), Klann-Ramlau approach (blue line), Hochstenbach-Reichel approach (green line) and Tikhonov-exponential regularization (red line) as function of singular values $σ$ . The regularization parameter has a value of $λ = 10^{- 2}$ .

The aim is to prevent the significant spread of error e in b into the calculated estimated solution $x_{λ}$ . Therefore, it is desirable for matrix $L^{T} L$ to possess a small norm, as this may assist us in determining a precise approximation of $\hat{x}$ .

Theorem 2.3. Consider the definition of L as given in equation (15), assuming that $λ$ and $σ_{j}$ are both greater than zero. Then ${‖ L^{T} L ‖}_{F}^{2} < {‖ I ‖}_{F}^{2}$ , where ${‖ ‖}_{F}$ represents the Frobenius norm.

Proof.

{‖ L^{T} L ‖}_{F}^{2} = {‖ E_{λ}^{2} ‖}_{F}^{2} = \sum_{j = 1}^{n} \exp (\frac{- 2 σ_{j}^{2}}{λ^{2}}) < n = {‖ I ‖}_{F}^{2} .

The determination of the regularization parameter $λ$ , using either the regularization matrix $L = I$ or (2.10), is based on the DP with respect to $L = I$ , as explained in Section 2.1. According to Theorem 2.3, the regularization matrix employed in equation (8) with $L = I$ exhibits a greater deviation from $A^{T} A$ compared to the regularization matrix (15) employed in equation (17). This implies that the solution of regularized normal equation (17) might provide a more precise estimation of the exact solution $\hat{x}$ than the solution of regularized normal equation (8). The numerical illustrations in Section 3 consistently support the observation mentioned.

Numerical examples

The numerical simulations were conducted using the Regularization Tools MATLAB package.¹⁵ The examples provided are derived from discretizing integral equations of the first kind

\int_{a}^{b} κ (x, y) f (x) d x = z (y), \begin{array}{l} c \leq y \leq \end{array} d

(19)

where

κ (x, y)

is a given smooth kernel function,

z (y)

is observable and

f (x)

is estimated via the integral equation. The discretizations

A \in R^{n \times n}

of the kernel

κ (x, y)

and discrete approximations

\hat{f} \in R^{n}

of the solution f in equation (19) are calculated using MATLAB functions.¹⁵ In all examples, we assume

n = 200

. To acquire vector b in equation (1),

e \in R^{n}

with zero mean and random entries following a normal distribution is added to

\hat{b} = A \hat{x}

, as described in equation (2). The scaling of vector

e

ensures a specified noise level

\frac{‖ e ‖}{‖ \hat{b} ‖}

. For our experiments, we examine error levels of 10%, 5%, 1%, and 0.1%. The regularization parameter

λ

is calculated using the DP with

ε = ‖ e ‖

, while

η = 1

is chosen according to equation (10) in the calculated examples.

Assume $x_{c}$ represent the solution obtained through the four regularization techniques discussed in Section 2. The choice of the parameter $γ$ in equation (13) and equation (14) are 0.8 and 0.6, respectively. We are care about the relative error (RE) $\frac{‖ x_{c} - \hat{x} ‖}{‖ \hat{x} ‖}$ , which is determined through the utilization of these regularization techniques. The occurrence of this error is dependent on the components present in the error vector e. To obtain a comprehensive understanding of the typical behavior exhibited by these regularization techniques, we present, in each example, the mean REs in $x_{c}$ over 2000 iterations for every level of noise.

Example 3.1. The problem of phillips is first examined in Hansen.¹⁵ Discretization of the famous first-kind Fredholm integral equation devised by D. L. Phillips. Let

ϕ (x) = {\begin{cases} 1 + \cos (\frac{π x}{3}), | x | < 3, \\ 0, | x | \geq 3 . \end{cases}

The smooth kernel function

κ (x, y)

, function on the right side

z (y)

, and solution to equation (19)

f (x)

are provided by

κ (x, y) = ϕ (y - x), f (x) = ϕ (x),

z (y) = (6 - | y |) (1 + \frac{1}{2} \cos (\frac{π y}{3})) + \frac{9}{2 π} \sin (\frac{π | y |}{3})

and

a = c = - 6, b = d = 6

. Table 1 presents the mean REs in the calculated solutions for every level of noise. We observe that Tikhonov-exponential regularization usually render solutions of higher quality at 1%, 5% and 10% noise levels, which indicates that Tikhonov-exponential method has good noise immunity and stability.

Table 1.

Example 3.1: mean REs for the phillips problem.

Noise level (%)	Tik	KR	HR	Tik-exp
10.0	$5.12 \cdot 10^{- 2}$	$4.40 \cdot 10^{- 2}$	$5.83 \cdot 10^{- 2}$	$2.42 \cdot 10^{- 2}$
5.0	$3.52 \cdot 10^{- 2}$	$3.10 \cdot 10^{- 2}$	$3.69 \cdot 10^{- 2}$	$2.31 \cdot 10^{- 2}$
1.0	$2.03 \cdot 10^{- 2}$	$1.83 \cdot 10^{- 2}$	$1.67 \cdot 10^{- 2}$	$1.62 \cdot 10^{- 2}$
0.1	$8.71 \cdot 10^{- 2}$	$7.63 \cdot 10^{- 2}$	$5.30 \cdot 10^{- 2}$	$5.64 \cdot 10^{- 2}$

Example 3.2. The test problem of heat is considered from Hansen.¹⁵ The employed inverse heat equation is a Volterra integral equation, integrated over the interval [0,1]. The smooth kernel function is $κ (x, y) = h (y - x)$ with

h (x) = \frac{x^{- 3 / 2}}{2 τ \sqrt{π}} \exp (- \frac{1}{4 τ^{2} x}),

where the parameter

τ

determines the degree of ill-conditioned in the kernel of this problem, and

τ = 5

results in a well-conditioned matrix, whereas

τ = 1

leads to an ill-conditioned matrix. We choose

τ = 5

in this experiment. The integral equation is discretized by means of simple collocation and the midpoint rule. Table 2 presents the mean REs for every level of noise. It can be observed that Tikhonov-exponential regularization consistently produces the lowest mean errors at 0.1%, 1% and 10% noise levels.

Table 2.

Example 3.2: mean REs for the heat problem.

Noise level (%)	Tik	KR	HR	Tik-exp
10.0	$7.96 \cdot 10^{- 2}$	$6.45 \cdot 10^{- 2}$	$6.17 \cdot 10^{- 2}$	$5.09 \cdot 10^{- 2}$
5.0	$4.48 \cdot 10^{- 2}$	$3.61 \cdot 10^{- 2}$	$3.41 \cdot 10^{- 2}$	$3.53 \cdot 10^{- 2}$
1.0	$1.12 \cdot 10^{- 2}$	$8.99 \cdot 10^{- 2}$	$8.20 \cdot 10^{- 2}$	$1.72 \cdot 10^{- 2}$
0.1	$1.50 \cdot 10^{- 2}$	$1.22 \cdot 10^{- 2}$	$9.58 \cdot 10^{- 2}$	$4.96 \cdot 10^{- 2}$

Example 3.3. This particular test example utilizes the gravity problem from Hansen¹⁵ on deconvolution and regularization with Toeplitz matrices, and models a 1-D gravity surveying problem. The smooth kernel function $κ (x, y)$ and solution to equation (19) $f (x)$ are provided by

κ (x, y) = d {({d^{2} + (y - x)}^{2})}^{- 3 / 2}, f (x) = \sin (π x) + 0.5 \sin (2 π x),

and

0 \leq x \leq 1, d = 0.25

. The constant d controls the decay of the singular values (the larger the d, the faster the decay). The integration interval is

[0, 1]

. Table 3 presents the mean REs in the calculated solutions across different levels of noise. Tikhonov-exponential regularization usually gives the smallest REs at 1%, 5% and 10% noise levels.

Table 3.

Example 3.3: mean REs for the gravity problem.

Noise level (%)	Tik	KR	HR	Tik-exp
10.0	$6.71 \cdot 10^{- 2}$	$5.95 \cdot 10^{- 2}$	$7.02 \cdot 10^{- 2}$	$4.58 \cdot 10^{- 2}$
5.0	$5.15 \cdot 10^{- 2}$	$4.60 \cdot 10^{- 2}$	$4.95 \cdot 10^{- 2}$	$3.66 \cdot 10^{- 2}$
1.0	$2.99 \cdot 10^{- 2}$	$2.70 \cdot 10^{- 2}$	$2.46 \cdot 10^{- 2}$	$2.27 \cdot 10^{- 2}$
0.1	$1.47 \cdot 10^{- 2}$	$1.34 \cdot 10^{- 2}$	$1.02 \cdot 10^{- 2}$	$1.14 \cdot 10^{- 2}$

Example 3.4. We examine the issue of i_laplace as presented in Hansen,¹⁵ which involves discretizing an inverse Laplace transformation by means of Gauss-Laguerre quadrature using

κ (x, y) = \exp (- x y), f (x) = x^{2} \exp (- x / 2), z (y) = \frac{2}{2 y + 1},

The integration interval is

[0, \infty)

. Table 4 presents the mean REs in the calculated solutions over 2000 runs for various levels of noise. Tikhonov-exponential regularization is generally associated with the lowest relative errors at noise levels of 1%, 5%, and 10%. This suggests that the Tikhonov-exponential method exhibits robust noise immunity and stability.

Table 4.

Example 3.4: mean REs for the i_laplace problem.

Noise level (%)	Tik	KR	HR	Tik-exp
10.0	$1.72 \cdot 10^{- 2}$	$1.55 \cdot 10^{- 2}$	$1.39 \cdot 10^{- 2}$	$1.36 \cdot 10^{- 2}$
5.0	$1.31 \cdot 10^{- 2}$	$1.17 \cdot 10^{- 2}$	$9.61 \cdot 10^{- 2}$	$8.60 \cdot 10^{- 2}$
1.0	$6.26 \cdot 10^{- 2}$	$5.60 \cdot 10^{- 2}$	$4.06 \cdot 10^{- 2}$	$4.60 \cdot 10^{- 2}$
0.1	$2.54 \cdot 10^{- 2}$	$2.23 \cdot 10^{- 2}$	$1.12 \cdot 10^{- 2}$	$1.44 \cdot 10^{- 2}$

Conclusions

The selection of the regularization matrix is a crucial and challenging task. We propose an innovative method to determine the regularization matrix by utilizing exponential filter functions. By incorporating this new regularization matrix, Tikhonov regularization only has one regularization parameter. Regularization matrix using the SVD of the matrix A has both advantage and disadvantage. The disadvantage is that the SVD of A is needed to calculate the least-squares problem (1.6). The advantage, however, is that, like the original Tikhonov regularization, the improved regularization method in our manuscript can be calculated without adding any additional parameters, except for the regularization parameter. There are many improved Tikhonov regularization methods that require additional parameters, such as fractional Tikhonov regularization, which requires the parameter $γ$ to be specified. Numerical illustrations demonstrate the effective performance of the regularization matrix determined using this approach. Additionally, for large-scale problems, the Tikhonov-exponential regularization method can be employed by initially projecting them onto Krylov spaces with lower dimensions.

Footnotes

Acknowledgements

The authors acknowledge the National Natural Science Foundation of China (Grant no. 41804136) for the support of the subject.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (Grant no. 41804136).

ORCID iD

Xiaoniu Zeng

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