Nonlinear Heat Diffusion Problem Solution with Spatio-Temporal Constraints Based on Regularized Gauss–Newton and Preconditioned Krylov Subspaces

Abstract

In this work, we proposed a dynamic inverse solution with spatio-temporal constraints of the nonlinear heat diffusion problem in 1D and 2D based on a regularized Gauss–Newton and Krylov subspace with a preconditioner. The preconditioner is computed by approximating the Jacobian of the nonlinear system at each Gauss–Newton iteration. The proposed approach is used for estimation of the initial value from measurements of the last value by considering spatial and spatio-temporal constraints. The system is compared to a dynamic Tikhonov inverse solution and generalized minimal residual method (GMRES) with and without a preconditioner. The system is evaluated under noise conditions in order to verify the robustness of the proposed approach. It can be seen that the proposed spatio-temporal regularized Gauss–Newton method with GMRES and a preconditioner shows better estimation results than the other methods for both spatial and spatio-temporal constraints.

1. Introduction

Nonlinear inverse problems arise frequently in engineering and applied sciences, particularly in systems governed by nonlinear partial differential equations. These problems are often ill-posed, meaning that small perturbations in the data can lead to large deviations in the solution, making regularization techniques essential for obtaining stable and meaningful results. In [1], several methods are considered for the solution of the linear and nonlinear inverse problems in practical applications, such as image deblurring, X-ray tomography, backward parabolic problems, heat transfer or diffusion, and electrical impedance tomography. In [2], the integration of model-based and data-driven techniques to enhance stability and generalizability in inverse reconstructions is presented. For example, in [3], a Jacobian-Free Newton–Krylov (JFNK) method is introduced to address tumour growth modeling, a highly nonlinear and ill-posed setting. The authors develop efficient preconditioning strategies that improve the convergence of the Krylov inverse solver without the need to compute or store Jacobians explicitly, making the approach particularly suitable for large-scale biomedical simulations and also generalizable to other applications.

Other adaptive and scalable strategies have also been introduced to improve the performance and robustness of nonlinear inverse problem solvers. A classical and widely studied approach is the iteratively regularized Gauss–Newton method, introduced by [4], which incorporates regularization at each iteration to stabilize the solution process. This method has proven effective for nonlinear ill-posed problems and is supported by rigorous convergence theory. More recently, in [5] an efficient implementation of the Gauss–Newton method using generalized Krylov subspaces is developed, significantly improving computational performance for large-scale problems. Their approach also integrates Broyden-type secant updates to approximate the Jacobian, reducing the computational cost in nonlinear settings. In parallel, Rajan and Jose [6] addressed the discretization challenges associated with nonlinear inverse problems, proposing a scheme that enhances stability and convergence by maintaining the structural fidelity of the continuous problem. These advances highlight the ongoing need for methods that are not only accurate and robust but also scalable and adaptable to complex, time-dependent systems. In [7], a learning-based regularization framework is proposed using deep neural networks to model complex solution manifolds in inverse problems, enabling data-driven priors in ill-posed nonlinear settings. Moreover, Zhao et al. [8] introduces a regularized Newton–GMRES method that combines regularization with iterative solvers, enhancing convergence and stability for nonlinear systems. Despite these advances, most methods still assume static or homogeneous priors and do not explicitly model spatio-temporal dependencies in the solution space.

The dynamic inverse problem of recovering the initial condition in nonlinear heat diffusion systems presents significant challenges due to its ill-posedness, nonlinearity, and sensitivity to noise. In [9,10,11], the inverse heat conduction problem and the traditional computational methods used for its solution are presented with applications in the areas of manufacturing, aerospace, biomedical, and electronics cooling. In these reviews, two main factors are considered for almost all the applications: a detailed knowledge of the heat equation (forward problem) and the use of a regularized cost function for the solution of the ill-posed ill-conditioned problem. However, traditional approaches often treat spatial and temporal regularization independently or rely on static formulations that fail to take advantage of the underlying dynamics of the system by considering only the solution of an optimization problem [12].

In [13], the authors proposed the numerical solution by Tikhonov regularization of the surface heat flux history and temperature distribution in an inverse heat conduction problem with a nonlinear source term. Similarly, Shidfar et al. [14] presents an inverse heat conduction problem with a nonlinear source term, where the surface heat flux history of a heated conducting body is identified by approximating the unknown function using linear polynomial pieces determined from an optimization problem. Moreover, Liu et al. [15] propose a multigrid-based framework integrated with constraint data assimilation to solve inverse problems governed by nonlinear diffusion equations. Their method improves both accuracy and computational speed by exploiting multi-resolution grid structures and incorporating measurement constraints at different scales. For instance, in [16], the authors propose a solution to the inverse heat conduction problem in a one-dimensional domain with a moving boundary and temperature-dependent material properties, emphasizing the need for robust methods capable of handling dynamic, nonlinear systems. In addition, in [17], a fast, robust, and automatic selection of the regularization parameter in inverse heat conduction problems is proposed for linear and nonlinear systems, which facilitates the tuning of the inverse problem solution. Also, Ciofalo [18] explores the solution of an inverse heat conduction problem to reconstruct the steady-state distribution of the convection heat transfer coefficient on one wall of a slab from the temperature distribution on a plane embedded in the slab, assuming that the thermal boundary conditions on the other walls are known. However, while these methods emphasize solver efficiency or domain adaptation, they do not address the integration of spatio-temporal constraints directly into the regularization process.

In this work, we propose a spatio-temporal regularized Gauss–Newton method combined with GMRES and an efficient preconditioning strategy to solve the nonlinear heat equation in both 1D and 2D domains. The preconditioner is constructed by approximating the Jacobian of the nonlinear system at each Gauss–Newton iteration, enhancing convergence and numerical stability. The proposed approach is evaluated in the context of an initial value reconstruction problem, where the goal is to estimate the initial state from final-time measurements, incorporating both spatial and spatio-temporal regularization constraints. By embedding these constraints into a dynamic regularization framework, our method aims to bridge the gap between physical realism, computational feasibility, and generalizability across diverse nonlinear inverse problems. We compare our method against state-of-the-art alternatives, including a dynamic Tikhonov inverse solution and the GMRES method with and without preconditioning.

The main contributions of this work are as follows:

•

Spatio-Temporal Regularized Inverse Framework: We propose a dynamic inverse solution that incorporates spatio-temporal regularization constraints directly into the cost functional. This allows the reconstruction to respect both spatial smoothness and temporal consistency.

•

Integration with Krylov Subspace Solvers: The nonlinear inverse problem is solved using a regularized Gauss–Newton method, where each linearized step is efficiently computed using the GMRES algorithm. This Krylov-based approach enables scalability to large problem sizes.

•

Jacobian-Based Preconditioning: A lightweight preconditioner is introduced by approximating the Jacobian at each iteration, significantly enhancing convergence and numerical stability of the GMRES solver in both 1D and 2D.

The remainder of this paper is organized as follows: Section 2 introduces the nonlinear diffusion problem. Section 3 details the proposed Spatio-Temporal Regularized Gauss–Newton method, including its integration with GMRES and the preconditioner design. Section 4 and Section 5 present the numerical results, performance evaluation, and concluding remarks based on the comparative analysis with existing estimation techniques.

2. Nonlinear Diffusion Problem

2.1. Nonlinear 1D Diffusion

We consider the general nonlinear diffusion equation, also known as the porous medium equation, as described in [19], as follows:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial u(x,t)}{\partial t}}={\displaystyle \frac{\partial }{\partial x}}\left(\alpha \left(u(x,t)\right){\displaystyle \frac{\partial u(x,t)}{\partial x}}\right),\end{array}$$

(1)

where $\alpha \left(u\right)$ is the diffusion coefficient defined by

$$\begin{array}{c}\hfill \alpha \left(u\right)=u^m,\end{array}$$

(2)

where m is the order of the nonlinearity. The system is subject to homogeneous Neumann boundary conditions:

$${\left.{\displaystyle \frac{\partial u}{\partial x}}\right|}_{x=0}=0,{\left.{\displaystyle \frac{\partial u}{\partial x}}\right|}_{x=L},\forall t,$$

which physically represent no-flux conditions at the boundaries. We complete the initial boundary value problem by specifying the initial condition:

$$u(x,0)=u_0\left(x\right),x\in [0,L],$$

where $u_0\left(x\right)$ is a sufficiently regular function. The well-posedness of the nonlinear diffusion Equation (1) with Neumann boundary conditions and a smooth, strictly positive initial condition has been studied in [19], guaranteeing existence, uniqueness, and continuous dependence of the solution on the data for $m>0$.

The approximation of (1) can be formulated by using the explicit finite difference discretization of the nonlinear diffusion equation. Let $u_i^n$ denote the numerical approximation of $u(x_i,t_n)$ and ${\alpha }_i=\alpha \left(u_i^n\right)$ the diffusion coefficient evaluated at $u_i^n$. The explicit scheme is given by:

$$\begin{array}{c}\hfill u_i^{n+1}=u_i^n+{\displaystyle \frac{\Delta t}{\Delta x^2}}\left[{\alpha }_{i+1}(u_{i+1}^n-u_i^n)-{\alpha }_i(u_i^n-u_{i-1}^n)\right],\end{array}$$

(3)

for $i=2,\dots ,N_x-1$.

The Neumann boundary conditions are discretized as:

$$u_0^n=u_1^n,u_{N_x}^n=u_{N_x-1}^n,\forall n,$$

which means that there is no flow of the diffusing quantity across the boundaries.

The explicit scheme can be summarized as

$$\begin{array}{c}\hfill u_{j+l}={\mathcal{F}}_1\left(u_j\right),\end{array}$$

(4)

where ${\mathcal{F}}_l(·)$ is the nonlinear discretization of the nonlinear diffusion equation l steps ahead.

The explicit scheme is first-order accurate in time and second-order accurate in space, assuming sufficient smoothness of the solution $u(x,t)$ [20]. That is, the local truncation error satisfies:

$$\tau =\mathcal{O}(\Delta t)+\mathcal{O}(\Delta x^2).$$

Stability of the explicit scheme is governed by a nonlinear generalization of the Courant–Friedrichs–Lewy (CFL) condition [21]:

$$\begin{array}{c}\hfill \Delta t\le {\displaystyle \frac{\Delta x^2}{2{max}_i{\alpha }_i}}={\displaystyle \frac{\Delta x^2}{2{max}_i{\left(u_i^n\right)}^m}},\end{array}$$

(5)

to prevent instabilities and ensure boundedness of the numerical solution. This condition becomes more restrictive for large values of m or sharply peaked initial profiles. The convergence of the scheme follows from consistency and stability (under the CFL condition), and can be justified using a nonlinear generalization of Lax’s equivalence theorem [22], provided the solution remains smooth and the step sizes are properly chosen.

2.2. Nonlinear Diffusion Problem in 2D

We now consider the extension of the nonlinear diffusion equation (also known as the porous medium equation) to two spatial dimensions, as described in [19]:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial u(x,y,t)}{\partial t}}=\nabla ·\left(\alpha \left(u\right(x,y,t\left)\right)\nabla u(x,y,t)\right),(x,y)\in \Omega \subset {\mathbb{R}}^2,\end{array}$$

(6)

where $\alpha \left(u\right)=u^m$ is the nonlinear diffusion coefficient, and $m>0$ is the nonlinearity parameter. We consider Neumann boundary conditions on the boundary $\partial \Omega$:

$${\left.{\displaystyle \frac{\partial u}{\partial \mathbf{n}}}\right|}_{\partial \Omega }=0,\mathrm{for}\mathrm{all}t,$$

where $\mathbf{n}$ denotes the outward normal to the boundary, indicating that there is no flux across the domain boundaries. We complete the initial boundary value problem by prescribing the initial condition:

$$u(x,y,0)=u_0(x,y),(x,y)\in \Omega ,$$

where $u_0(x,y)$ is a regular function, and $u_0(x,y)\ge 0$. For $m>1$, this condition ensures the well-posedness of (6), including existence, uniqueness, and continuous dependence on the initial data, as discussed in [19].

To numerically approximate (6), we consider a uniform grid over a rectangular domain $\Omega =[0,L_x]\times [0,L_y]$, with grid spacings $\Delta x$ and $\Delta y$, and time step $\Delta t$. Let $u_{i,j}^n$ denote the approximation to $u(x_i,y_j,t_n)$, and ${\alpha }_{i,j}^n=\alpha \left(u_{i,j}^n\right)$. The explicit finite difference scheme can be written as:

$$\begin{array}{cc}\hfill u_{i,j}^{n+1}=u_{i,j}^n& +{\displaystyle \frac{\Delta t}{\Delta x^2}}\left[{\alpha }_{i+1,j}^n(u_{i+1,j}^n-u_{i,j}^n)-{\alpha }_{i,j}^n(u_{i,j}^n-u_{i-1,j}^n)\right]\hfill \\ \hfill & +{\displaystyle \frac{\Delta t}{\Delta y^2}}\left[{\alpha }_{i,j+1}^n(u_{i,j+1}^n-u_{i,j}^n)-{\alpha }_{i,j}^n(u_{i,j}^n-u_{i,j-1}^n)\right],\hfill \end{array}$$

(7)

for $i=2,\dots ,N_x-1$, $j=2,\dots ,N_y-1$. The Neumann boundary conditions are enforced using ghost points or mirrored values. A simple first-order approximation gives:

$$u_{0,j}^n=u_{1,j}^n,u_{N_x,j}^n=u_{N_x-1,j}^n,u_{i,0}^n=u_{i,1}^n,u_{i,N_y}^n=u_{i,N_y-1}^n,\mathrm{for}\mathrm{all}i,j,n.$$

This reflects the assumption of zero gradient (i.e., no flux) at the boundary. The explicit scheme can also be summarized by a nonlinear operator ${\mathcal{F}}_l(·)$, which propagates the solution l steps forward in time:

$$\begin{array}{c}\hfill u_{j+l}={\mathcal{F}}_l\left(u_j\right),\end{array}$$

(8)

where ${\mathcal{F}}_l$ depends nonlinearly on the values of u through the coefficient $\alpha \left(u\right)=u^m$.

The explicit scheme is first-order accurate in time and second-order accurate in space, assuming that $u(x,y,t)$ is smooth enough [20]. That is, the local truncation error is given by:

$$\tau =\mathcal{O}(\Delta t)+\mathcal{O}(\Delta x^2+\Delta y^2).$$

For stability, the time step must satisfy a two-dimensional generalization of the CFL condition [21]:

$$\begin{array}{c}\hfill \Delta t\le {\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{1}{\Delta x^2}}+{\displaystyle \frac{1}{\Delta y^2}}\right)}^{-1}·{\displaystyle \frac{1}{{max}_{i,j}{\alpha }_{i,j}^n}}={\displaystyle \frac{1}{2{max}_{i,j}{\left(u_{i,j}^n\right)}^m}}{\left({\displaystyle \frac{1}{\Delta x^2}}+{\displaystyle \frac{1}{\Delta y^2}}\right)}^{-1}.\end{array}$$

(9)

This condition ensures numerical stability and must be respected, especially when $m\gg 1$ or when u develops sharp gradients. The convergence of the scheme, under this condition, can be justified for smooth solutions via nonlinear extensions of Lax’s equivalence theorem [22].

This formulation serves as the basis for the forward simulation in inverse problems or data assimilation frameworks involving nonlinear parabolic PDEs in two dimensions. In the context of the inverse problem, this formulation is often vectorized by reshaping the matrix-valued solution $u\in {\mathbb{R}}^{N_x\times N_y}$ into a column vector $\mathbf{u}\in {\mathbb{R}}^{N_x{N}_y}$. This allows the forward operator ${\mathcal{F}}_l$ to be implemented as a nonlinear mapping in vector space, facilitating the application of optimization-based reconstruction methods.

3. Dynamic Inverse Solution by Gauss–Newton

The inverse problem is as follows: given a noisy observation $u_T^{\mathrm{obs}}$ at final time T, recover the initial condition $u_0$. This inverse problem is analyzed in [23,24] by considering a linear diffusion coefficient. It is worth noting that if T is too large, the solution of the heat equation tends to a spatially homogeneous (stationary) state due to the strong smoothing effect of diffusion. As a result, the inverse problem becomes severely ill-posed, since high-frequency components of the initial condition are exponentially damped and become unrecoverable from the final data.

This ill-posedness manifests in the decay of the singular values of the forward operator, which becomes more pronounced as T increases. For instance, in the linear 1D heat equation with homogeneous Neumann or Dirichlet boundary conditions, the Fourier or eigenmode components decay like $exp(-{\lambda }_{k}T)$, where ${\lambda }_k$ are the eigenvalues of the Laplacian. This implies that larger values of T result in more significant loss of information, especially in higher modes. Therefore, there exists a practical upper limit for the observation time T, beyond which the data $u_T^{\mathrm{obs}}$ no longer contain sufficient information to accurately reconstruct the initial condition. The precise restriction depends on the spatial resolution, the noise level, and the regularity of the initial condition. In practice, moderately small values of T are preferred to balance physical diffusion and numerical stability.

To address the instability for large T, it is necessary to incorporate regularization algorithms (e.g., Tikhonov, TV, or iterative regularization methods) that stabilize the inversion and suppress amplification of noise during reconstruction. The Gauss–Newton method is a widely used iterative approach for solving nonlinear inverse problems, particularly when the forward model can be linearized in the neighborhood of the current estimate. A rigorous formulation and convergence analysis of this method is provided in [5], where the authors propose an efficient implementation based on projections into generalized Krylov subspaces. Their framework allows the Gauss–Newton method to be applied effectively to large-scale problems by iteratively building a sequence of low-dimensional subspaces that approximate the solution space. Importantly, the convergence of the proposed algorithm is theoretically established under standard assumptions, and its numerical performance is demonstrated across several examples. This formulation provides a solid theoretical and computational foundation upon which we build in this work, extending the approach with spatio-temporal regularization and preconditioning strategies to address dynamic inverse problems governed by nonlinear diffusion equations. In this case, we proposed a regularized dynamic inverse solution with spatio-temporal constraints based on the Gauss–Newton method with preconditioned Krylov subspaces (by means of the GMRES algorithm). The proposed approach is described in Section 3.1 and Section 3.2.

3.1. Regularized Inverse Solution with Spatio-Temporal Constraints

A generalized cost function with spatio-temporal constraints can be defined as

$$\begin{array}{c}\hfill \underset{u_0}{min}{∥{\mathcal{F}}_{N_t}\left(u_0\right)-u_T^{\mathrm{obs}}∥}^2+\lambda {\parallel L{u}_0\parallel }^2+\gamma \sum _{n=0}^{N_t-2}{∥u_{n+1}-u_n∥}_2^2,\end{array}$$

(10)

where

•

${\mathcal{F}}_{N_t}\left(u_0\right)$ denotes the forward nonlinear model evaluated at final time T at sample ($N_t$), with initial condition $u_0$;

•

$\lambda >0$ is the regularization parameter;

•

L is the spatial constraint;

•

$\gamma >0$ is the temporal regularization parameter;

•

$u_n$ represents the approximate solution at time $t_n$, with $n=0,1,\dots ,N_t$.

3.2. Gauss–Newton Iterative Scheme for the Spatio-Temporal Regularized System with GMRES

The Gauss–Newton method iteratively linearizes the forward model and solves a regularized linear system [25]. At iteration k, we perform the following steps:

• Simulate the forward model at the current guess of the initial condition $u_0^{\left(k\right)}$ at $\left(k\right)$ iteration:

  $$u_T^{\left(k\right)}={\mathcal{F}}_{N_t}\left(u_0^{\left(k\right)}\right)$$
• Compute the residual:

  $$r^{\left(k\right)}=u_T^{\left(k\right)}-u_T^{\mathrm{obs}}$$
• Compute the Jacobian matrix of the forward model with respect to the initial condition:

  $$J^{\left(k\right)}\approx {\displaystyle \frac{\partial {\mathcal{F}}_{N_t}}{\partial u_0}}{|}_{u_0^{\left(k\right)}}$$
• Define the left-hand-side operator for the linearized system:

  $$A^{\left(k\right)}=J^{\left(k\right)\top }{J}^{\left(k\right)}+\lambda L^{\top }L+\gamma R_t$$

  with $R_t$ computed as:

  $$R_t=\sum _{n=0}^{N_t-2}{\left(J_{n+1}^{\left(k\right)}-J_n^{\left(k\right)}\right)}^T\left(J_{n+1}^{\left(k\right)}-J_n^{\left(k\right)}\right)$$
• Define the right-hand side:

  $$b^{\left(k\right)}=-J^{\left(k\right)\top }{r}^{\left(k\right)}-\lambda L^{\top }L{u}_0^{\left(k\right)}-\gamma g_t$$

  with $g_t$ computed as

  $$g_t=\sum _{n=0}^{N_t-2}{\left(J_{n+1}^{\left(k\right)}\right)}^T\left(u_{n+1}-u_n\right)$$
• Solve the regularized linear system:

  $$A^{\left(k\right)}\delta u_0^{\left(k\right)}=b^{\left(k\right)}$$

  using the GMRES algorithm, where a preconditioner $M^{\left(k\right)}$ is used to accelerate convergence, resulting en the system

  $${\left(M^{\left(k\right)}\right)}^{-1}{A}^{\left(k\right)}\delta u_0^{\left(k\right)}={\left(M^{\left(k\right)}\right)}^{-1}{b}^{\left(k\right)}$$
• Update the estimate of the initial condition:

  $$u_0^{(k+1)}=u_0^{\left(k\right)}+\delta u_0^{\left(k\right)}$$
• Check convergence:

  $$\parallel \delta u_0^{\left(k\right)}\parallel <ϵ\mathrm{or}\parallel r^{\left(k\right)}\parallel <ϵ$$

This process is repeated until convergence is achieved. Unlike a single-step linearized Tikhonov method or GMRES method, Gauss–Newton refines both the solution and the local linearization at each iteration, making it more robust for strongly nonlinear models such as the case where $\alpha \left(u\right)=u^m$.

3.3. Regularized Inverse Solution with Spatial Constraints

Since we are considering a nonlinear dynamic system, the following regularized nonlinear problem with spatial constraints can be defined:

$$\begin{array}{c}\hfill \underset{u_0}{min}{∥{\mathcal{F}}_{N_t}\left(u_0\right)-u_T^{\mathrm{obs}}∥}^2+\lambda {\parallel L{u}_0\parallel }^2,\end{array}$$

(11)

which is the cost function for spatio-temporal constraints but considering the temporal regularization parameter $\gamma =0$.

3.4. Gauss–Newton Iterative Scheme for the Spatially Regularized System with GMRES

The Gauss–Newton method considered in the last section is applied iteratively, replacing only the steps 4 and 5 as follows:

4.

Define the left-hand-side operator for the linearized system:

$$A^{\left(k\right)}=J^{\left(k\right)\top }{J}^{\left(k\right)}+\lambda L^{\top }L$$

5.

Define the right-hand side:

$$b^{\left(k\right)}=-J^{\left(k\right)\top }{r}^{\left(k\right)}-\lambda L^{\top }L{u}_0^{\left(k\right)}$$

This process is repeated until convergence is achieved.

3.5. Jacobian Approximation

We approximate the Jacobian of the nonlinear mapping ${\mathcal{F}}_{N_t}$ at the k-th Gauss–Newton iteration, by using central differences [26], as follows:

$$J^{\left(k\right)}\approx {\displaystyle \frac{\partial {\mathcal{F}}_{N_t}}{\partial u_0}}{|}_{u_0^{\left(k\right)}}\approx {\left[\begin{array}{c}{\displaystyle \frac{{\mathcal{F}}_{N_t}(u_0^{\left(k\right)}+\epsilon e_1)-{\mathcal{F}}_{N_t}(u_0^{\left(k\right)}-\epsilon e_1)}{2\epsilon }}\\ ⋮\\ {\displaystyle \frac{{\mathcal{F}}_{N_t}(u_0^{\left(k\right)}+\epsilon e_{N_x})-{\mathcal{F}}_{N_t}(u_0^{\left(k\right)}-\epsilon e_{N_x})}{2\epsilon }}\end{array}\right]}^{\top },$$

where $\epsilon >0$ is a small perturbation step size, and $e_i\in {\mathbb{R}}^{N_x}$ is the unit vector in the i-th direction. This gives a second-order accurate approximation to the true Jacobian at iteration k.

3.6. Preconditioner Selection

The preconditioner $M^{\left(k\right)}$ is computed at each iteration by considering

$$\begin{array}{c}\hfill M^{\left(k\right)}\approx \mathrm{diag}\left\{J^{\left(k\right)}\right\}.\end{array}$$

(12)

Therefore,

$$\begin{array}{c}\hfill {\left(M^{\left(k\right)}\right)}^{-1}{J}^{\left(k\right)}\approx I,\end{array}$$

(13)

is equivalent to solving a system that is closer to identity, and thus the solver converges much faster.

4. Results

In order to evaluate the proposed approach for the dynamic inverse solution by the proposed Gauss–Newton with GMRES and a preconditioner, a forward problem with a nonlinear system diffusion equation for the 1D and 2D cases is analyzed, and the estimation results are compared with the Gauss–Newton method using Tikhonov and GMRES without a preconditioner.

4.1. One-Dimensional Estimation Results

Initially, the forward problem with a nonlinear system diffusion equation for the 1D case is considered with $m=2$ and with initial condition $u_0$ defined by

$$\begin{array}{c}\hfill u_0=e^{-{\displaystyle \frac{{\left(x-\frac{L}{2}\right)}^2}{0.1}}},\end{array}$$

(14)

with $L=5$, $N_x=101$, $T=0.1$, $N_t=100$. To ensure the stability of the explicit numerical scheme, the time step $\Delta t$ must satisfy the nonlinear CFL condition given by (5). Since the maximum of the initial condition is ${max}_i{u}_i^0\approx 1$, and replacing the values in (5), the following calculation is obtained:

$$\begin{array}{c}\hfill \Delta t\le {\displaystyle \frac{{\left(0.05\right)}^2}{2·1^2}}={\displaystyle \frac{0.0025}{2}}=0.00125.\end{array}$$

Since the selected time step is $\Delta t=0.001<0.00125$, the CFL condition is satisfied.

The space-time dynamic behavior of the nonlinear diffusion equation without noise is shown Figure 1.

The observed $u_T$ with an additive noise of $0.01$ is shown in Figure 2.

The performance of the proposed approach is evaluated and compared with spatial and spatio-temporal constraints and considering several methods for the solution of the resulting regularized system in the Gauss–Newton method. The tolerance value $ϵ$ to check the convergence of the Gauss–Newton algorithm is $ϵ=0.01$. In addition, the algorithm is restricted to a maximum of 20 iterations.

4.1.1. Dynamic Inverse Solution with Spatial Constraints

The $u_0$ estimated solution is computed by using the proposed approach computing the spatially regularized Gauss–Newton system solution by the GMRES algorithm with and without a preconditioner and the exact Tikhonov solution assuming $\gamma =0$, where the selection of $\lambda$ is performed by considering the trade-off between the norm of a regularized solution and its fit to the given data (L-curve method) [24].

The estimated $u_0$ from observed $u_T$ by using the Tikhonov regularized solution, with $\lambda =0.05$, is shown in Figure 3.

The estimated $u_0$ from observed $u_T$ by using the GMRES without the preconditioner for the regularized solution with $\lambda =0.05$ is shown in Figure 4.

The estimated $u_0$ from observed $u_T$ by using the GMRES with a preconditioner of the regularized solution with $\lambda =0.01$ is shown in Figure 5.

It can be seen from Figure 3, Figure 4 and Figure 5 that the GMRES spatial regularization with a preconditioner is the solution that best approximates the true $u_0$.

4.1.2. Dynamic Inverse Solution with Spatio-Temporal Constraints

The $u_0$ estimated solution is computed by using the proposed approach computing the spatio-temporal regularized Gauss–Newton system solution by the GMRES algorithm with and without a preconditioner and the exact Tikhonov, where the selection of $\lambda$ and $\gamma$ are performed by considering the trade-off between the norm of the regularized spatial solution, the temporal solution, and its fit to the given data (L-surface method or L-curve generalization for multiple constraints).

The estimated $u_0$ from the observed $u_T$ by using the Tikhonov spatio-temporal regularized solution, with $\lambda =0.01$ and $\gamma =0.01$, is shown in Figure 6.

The estimated $u_0$ from the observed $u_T$ by using the GMRES without the preconditioner for the regularized spatio-temporal solution with $\lambda =0.04$ and $\gamma =0.01$ is shown in Figure 7.

The estimated $u_0$ from observed $u_T$ by using the GMRES with a preconditioner of the regularized spatio-temporal solution with $\lambda =0.004$ and $\gamma =0.01$ is shown in Figure 8.

It can be seen from Figure 6, Figure 7 and Figure 8, that the GMRES spatio-temporal regularization with a preconditioner is the solution that best approximates the true $u_0$ even if the spatio-temporal regularization is compared with the spatial regularization.

Figure 9 shows a comparison of the 3D simulation (noiseless) of the nonlinear diffusion equation shown in Figure 1, and the resulting constrained estimation surface for all the time interval $0\le t\le T$ by considering the proposed GMRES spatio-temporal regularization with a preconditioner is shown.

It is worth noting from Figure 9 that the constrained estimation surface of the nonlinear diffusion equation is similar to the simulated behavior. However, the solution of the inverse problem is the estimation of $u_0$ by considering the noisy u at a final time T. It is worth mentioning that Figure 9 shows soft transitions over time, which is enforced by the temporal regularization.

In order to quantitatively evaluate the proposed approach, the estimation error is computed as follows:

$$\begin{array}{c}\hfill e={(u_{est}-u_0)}^{\top }(u_{est}-u_0)\end{array}$$

(15)

As a result,

Table 1. Estimation error !D.

Gauss–Newton Method Solution	Estimation Error	Iterations	Computational Time per Iteration
Spatial GMRES	0.0953	7	$0.1094\pm 0.0165$ s
with a preconditioner
Spatial GMRES	0.3616	9	$0.1075\pm 0.0147$ s
without a preconditioner
Spatial Tikhonov	0.3355	6	$0.0901\pm 0.0077$ s
Spatio-temporal GMRES	0.0377	10	$0.1003\pm 0.0142$ s
with a preconditioner
Spatio-temporal GMRES	0.2898	14	$0.0842\pm 0.0149$ s
without a preconditioner
Spatio-temporal Tikhonov	0.0388	7	$0.0911\pm 0.0052$ s

summarizes the performance of each method in terms of the estimation error for the 1D case.

It can be seen in Table 1 that the lowest estimation error is achieved by the proposed approach, namely the spatio-temporal regularized Gauss–Newton method with GMRES and a preconditioner. In addition, it can be seen that the use of Krylov subspaces with a preconditioner reduces the number of iterations required for the convergence of the method as well as the estimation error in comparison with the method without a preconditioner. There are no significant differences in terms of computational time between the methods with and without a preconditioner; however, the estimation error with a preconditioner is lower for all the analyzed cases.

The proposed approach is also evaluated by considering a nonlinearity diffusion equation in 1D with $m=3$. This analysis is only performed for the proposed approach. As a result, the estimated $u_0$ from observed $u_T$ by using the GMRES with a preconditioner of the regularized spatio-temporal solution with $\lambda =0.01$ and $\gamma =0.01$ is shown in Figure 10.

Figure 11 shows a comparison of the 3D simulation (noiseless) of the nonlinear diffusion equation with $m=3$, and the resulting constrained estimation surface for all the time interval $0\le t\le T$ by considering the proposed GMRES spatio-temporal regularization with a preconditioner is shown.

It can be seen from Figure 11 that the temporal constraint enforces that the estimation surface has soft transitions between the temporal samples.

An additional simulation is performed for $T=1$ s and $N_t=1000$. The simulation, the measurements, and the estimation are shown in Figure 12.

It is worth noting that since the value of T is increased, the value of $N_t$ must also be increased in order to fulfil the CFL conditions:

$$\begin{array}{c}\hfill \Delta t\le {\displaystyle \frac{{\left(0.05\right)}^2}{2·1^2}}={\displaystyle \frac{0.0025}{2}}=0.00125.\end{array}$$

(16)

Since the selected time step is $\Delta t=0.001<0.00125$, the CFL condition is satisfied. In addition, it is worth noting that the estimation error for this case is 0.5562, and the number of iterations is 20. In comparison to the simulation of $T=0.1$, it can be seen that for this case it is harder to estimate the initial condition since the inverse problem becomes even more ill-conditioned.

4.2. Two-Dimensional Estimation Results

The forward problem is extended to the two-dimensional nonlinear diffusion equation with nonlinearity parameter $m=2$. The initial condition $u_0$ is defined as a 2D Gaussian function centered in the middle of the spatial domain:

$$\begin{array}{c}\hfill u_0(x,y)=exp\left(-{\displaystyle \frac{{(x-\frac{L_x}{2})}^2+{(y-\frac{L_y}{2})}^2}{0.1}}\right),\end{array}$$

(17)

where the spatial domain is defined by $x\in [0,L_x]$ and $y\in [0,L_y]$ with $L_x=L_y=5$. The spatial discretization is performed using $N_x=N_y=41$ grid points in each direction, which leads to step sizes $\Delta x={\displaystyle \frac{L_x}{N_x-1}}$ and $\Delta y={\displaystyle \frac{L_y}{N_y-1}}$. The total simulation time is $T=0.1$, with $N_t=50$ uniform time steps of size $\Delta t={\displaystyle \frac{T}{N_t}}$. We now compute the CFL limit according to (9), by considering that ${max}_{i,j}{u}_{i,j}^0\approx 1$:

$$\begin{array}{cc}\hfill \Delta t& \le {\displaystyle \frac{1}{2·1^2}}{\left({\displaystyle \frac{1}{{\left(0.125\right)}^2}}+{\displaystyle \frac{1}{{\left(0.125\right)}^2}}\right)}^{-1}\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{2}{0.015625}}\right)}^{-1}={\displaystyle \frac{1}{2}}·{\displaystyle \frac{0.015625}{2}}={\displaystyle \frac{0.015625}{4}}=0.00390625\hfill \end{array}$$

Since the selected time step is $\Delta t=0.002<0.00390625$, the CFL condition is satisfied.

The observed $u_T$ with an additive noise of $0.01$ is shown in Figure 13.

The estimated $u_0$ from observed $u_T$ by using the GMRES with a preconditioner of the regularized spatio-temporal solution with $\lambda =0.01$ and $\gamma =0.0001$ is shown in Figure 14.

A comparison with the Tikhonov method with spatial constraints and regularization parameter of $\lambda =0.01$ is performed. As a result,

Table 2. Estimation error in 2D.

Gauss–Newton Method Solution	Estimation Error	Iterations	Computational Time per Iteration
Spatial Tikhonov	0.4618	7	$17.9258\pm 0.5737$ s
Spatio-temporal GMRES with a preconditioner	0.4202	4	$23.3710\pm 0.5539$ s

summarizes the performance of each method in terms of the estimation error for the 2D simulation.

It can be seen from Table 2, the lowest estimation error is also achieved by the proposed approach, namely the spatio-temporal regularized Gauss–Newton method with GMRES and a preconditioner.

5. Conclusions

In this work, we proposed a dynamic inverse solution framework with spatio-temporal constraints for the nonlinear heat diffusion problem in both 1D and 2D, based on a regularized Gauss–Newton method integrated with Krylov subspaces and a preconditioner. The proposed approach was evaluated through numerical experiments and compared against classical methods, including Tikhonov regularization, and GMRES solvers with and without preconditioning, under both purely spatial and combined spatio-temporal regularization settings. The results consistently demonstrate that incorporating spatio-temporal constraints into the regularization significantly improves the accuracy of the estimated initial condition. Furthermore, the use of the Gauss–Newton method within a Krylov subspace framework, combined with an efficient Jacobian-based preconditioner, accelerates convergence and enhances numerical stability, particularly in the 2D case where problem size and complexity increase. Compared to the evaluated alternatives, the proposed method achieves lower estimation errors, especially under challenging conditions such as limited observations or high noise levels. These findings support the effectiveness of the proposed framework as a generalized and scalable solution strategy for nonlinear dynamic inverse problems governed by time-dependent partial differential equations.