Applying Radial Basis Functions and Partition of Unity for Solving Heating Equations Optimal Control Issues

In numerous situations, normal differential equations are not anymore sufficient for describing the processes to be improved; instead, partial differential equations need to be implemented. An essential function is fulfilled by the ideal partial differential equations governing the control issue (OCI) in numerous scientific and engineering disciplines. Numerous physical occurrences, including heat transfer, dispersion, electromagnetic radiation, liquid moves, and freeze techniques, can be simulated using equations based on partial differentials. Many scholars in various domains have shown interest in optimization issues with PDE restrictions due to their widespread usage in different disciplines and sectors, such as the energy sector [1-4]. As a result, numerous attempts have been made to find effective and practical remedies to these issues. When pursuing the goal role of this class of optimization issues the expressions turn into sparse optimization issues:

$\frac{\partial y}{\partial t} - \Delta y = w + g~in~\Omega \times (0,T)$    (1)

when the $L_1$ norm is present.

Currently, as we are aware, Stedler investigated them for the first time in 2007 and suggested Newton-typed techniques as a solution. Since then, other approaches have been put forth, several of which are presented below. For their solution, Porcelli et al. [5] employed a preconditioner and an overall semi-smooth Newton's method. For the best sparse management of semi-linear parabolic equations, Langer et al. [6] used the unorganized time-space finite element approach. Phan and Gillis [7], considering non-smooth and not convex issues related to optimization, a novel inertial block minimizing approach is introduced. An adaptable finite element technique of sparse optimum control of fractional diffusing has been recently provided by Otárola [8]. Utilizing the inside technique, Pearson et al. [9] offered a successful approach.

The heat equation control issue will be examined in this work in the following manner:

${{\min }_{y,w}}\frac{1}{2}\|y-\overset{\scriptscriptstyle\smile}{y}\|_{{{L}_{2}}(\Omega )}^{2}+\frac{\beta }{2}\|w\|_{{{L}_{2}}(\Omega )}^{2}+\gamma \|w{{\|}_{{{L}_{1}}(\Omega )}}$    (2)

subject to:

$\frac{\partial y}{\partial t}-\Delta y=w+g\text{ in }\Omega \times (0,T)$    (3)

$y(x,0)={{y}_{0}}(x)\text{ in }\Omega $    (4)

$w\in {{W}_{\xi \lambda }}=\{h(x)\mid \xi \le w\le \lambda ,a.e.\text{ on }\Omega \}$    (5)

The system's state variable is called $y$. Within the framework of the heat formula, the temperature distributions across the geographic domain $\Omega$ at a specific time are represented by $y$ and $\check{y}$ represents the desired or goal state. It's the temperature distribution we want to attain with the best possible control. The controlling function that we are attempting to optimize is represented by $w$ with the regularization coefficient $\beta>0$. The expression $\gamma\|w\|_{L_1(\Omega)}$ is regarded as an example of regularization applied to the issue to limit the possible solutions and keep them from becoming chaotic or excessive. This kind of regularization can enhance the model's responsiveness to slight variations in information in addition to its predictability.

The collection of all allowable functions for controlling is represented by $W_{\xi \lambda}$ also, the control function's elements are represented by $h(x)$ at every point $x$ in the region $\Omega$.

The state formula, the adjoint formula, and the gradient formula for the control are among the optimality criteria:

$\frac{\partial y}{\partial t}-\Delta y=w+g,y(x,0)={{y}_{0}}(x),y=0\text{ on }\partial \Omega $    (6)

$-\frac{\partial p}{\partial t}-\Delta p=y-\overset{\scriptscriptstyle\smile}{y},P(x,T)=0,p=0\text{ on }\partial \Omega$    (7)

$-p+\beta w+\mu =0$    (8)

$\begin{matrix}w-\max (0,w+c(\mu -\gamma ))-\min (0,w+c(\mu +\gamma ))=0 \\\max (0,(w-b)+c(\mu -\gamma ))+\min (0,(w-a)+c(\mu +\gamma ))=0 \\\end{matrix}$    (9)

The state variables (temperature) in this case are represented by $y$, the control variable by $w$, the adjoint state by $p$, and the Lagrange multiplier related to the control requirements by $\mu$ and $c>0$.

Due to their numerous applications in environmental science, engineering, and other domains, optimal control issues (OCIs) of the equation of heat represent an important subject of study.

Talib and Al Dulaimi [10] presented an optimized design for conical cavity receivers incorporating helically baffled paths to enhance thermal performance. Also, Different optimization algorithms were used to find the size of a stand-alone hybrid power system (HES) that would supply power to the isolated residential loads. A stand-alone hybrid power system consists of three components: photovoltaic (PV) panels and diesel generators (DG). Several optimization algorithms were evaluated in this research to determine the most efficient size of a stand-alone hybrid power system in order to minimize the PV arrays, the number of DGs, and the total system cost, thus minimizing the cost of energy (COE) [11]. The approach enhances the stability and accuracy of the numerical solutions, especially in the presence of sharp gradients. Basic techniques for interpolating dispersed data, particularly in higher dimensional fields, are Radial Basis Function (RBF) algorithms. Kansa [12] originally presented this approach for resolving partial differential equations (PDEs) in 1990. An RBF technique was utilized by Rad et al. [13] to determine the optimal controls for parabolic scattered parameter systems utilizing a quadratic price function. Pearson used RBF clustering methods to address the Poisson controlling issue [14].

In the past 20 years, Radial Basis Functions have become a very common technique to solve partial differential equations. Although RBF procedures are often straightforward to solve many problems, global solutions unfortunately have a drawback in that the larger the problem, the more complex the solution becomes. The cost of computing to solve dense linear systems goes up. To overcome these shortcomings, good efforts were taken to localize RBF collocation methods. Several articles have employed the partition unity method, which is a helpful tool in this area. The Partition of Unity (PU) technique has been used for interpolation since about 1960 [15]; lately, the PU method has been used with RBFs [16-19]. To get a global approximation of the outcome of the two-dimensional Klein-Gordon equation on a specified bounded domain, Ahmadi Darani proposed the use of a localized radial basis function approach [20]. Cavoretto and De Rossi [21] employed a collocation strategy derived from the basis function for radials partition with unity (RBF-PU) in an adaptable refinements method for solving Poisson issues. Garmanjani et al. [22] employed the finite difference (FD) scheme-based RBF Partition of Unity technique (RBF-PUM) for the initial-boundary values issue. The RBF-PU method for spatial discretization of partially integro-differential formulas was created by Fereshtian et al. [23]. The elliptic interface problems were solved by Gholampour et al. [24] via using the RBF partition of the unity local method. Recently, Mirzaee [25] developed an RBF Partition of Unity technique for PDEs using a direct discretization technique. Also, Garmanjani et al. [26] provide a fresh approach based on the RBF-PU approach and the domain deconstruction scheme, in which the physical domain must be divided into two subdomains, each of which defines a forward or backward subproblem, each of which is handled using a radial basis function. The meshless technique uses a finite difference algorithm for the time derivatives and a Partition of Unity for the spatial dimension.

1.1 Comparative analysis and justification of RBF-PU superiority

Computational Efficiency: Traditional methods like FEM and global RBF methods suffer from high computational costs, especially in high-dimensional problems. The RBF-PU method addresses this by partitioning the domain into subdomains and performing local RBF approximations, resulting in smaller, sparse matrices. This localization reduces the complexity from $O\left(N^3\right)$ for global methods to $O\left( N.n^{2} \right)$, making RBF-PU more scalable.

Handling Complex Geometries: Unlike FEM, which requires a structured mesh, the meshless RBF-PU method easily handles irregular geometries. Nodes can be freely distributed, and subdomains overlap for smooth transitions, allowing for adaptive refinement in regions of interest.

Accuracy and Sparsity Control: The RBF-PU method naturally promotes sparsity by focusing on local subdomains, which is beneficial for sparse optimal control problems. It includes rigorous error analysis, showing reduced approximation errors with finer subdomain and time step refinements.

Boundary Conditions: RBF-PU handles boundary conditions more efficiently than traditional methods by adjusting local interpolants near the domain boundaries. This local handling enhances numerical stability and accuracy.

Theoretical Justification: By using compactly supported RBFs and partitioning the domain, RBF-PU results in sparse matrices that are easier to solve. This combination of local approximation with global continuity offers superior numerical stability and efficiency compared to traditional methods.

The proposed Radial Basis Function-Partition of Unity (RBF-PU) method offers a more flexible and computationally efficient approach to solving optimal control problems governed by the heat equation. Unlike traditional global RBF methods, the RBF-PU technique utilizes local approximations within overlapping subdomains, significantly reducing computational complexity and enhancing scalability for high-dimensional problems. This approach effectively captures the spatio-temporal sparsity of control functions, a key aspect in practical applications where control is needed only in specific regions.

To expand with recent developments in optimal control using Radial Basis Functions (RBF), and to discuss the challenges and limitations that previous works have faced, the following points should be considered:

1. Recent developments in RBF for optimal control problems

Local RBF Methods: While global RBF methods were traditionally used for solving partial differential equations, their computational inefficiency for large-scale problems led researchers to explore local methods. For example, Garmanjani et al. [22] employed a local RBF Partition of Unity (RBF-PUM) method based on finite difference schemes to tackle initial-boundary value problems, enhancing computational efficiency and accuracy.

RBF for Fractional Optimal Control: Otárola [8] introduced an adaptive finite element method for the sparse optimal control of fractional diffusion, using RBF for spatial discretization to handle fractional differential equations effectively. This advancement was crucial for applications involving complex dynamics like diffusion processes.

Hybrid Methods: Mirzaei [25] proposed the direct RBF-PU method, which integrates RBF Partition of Unity with other numerical techniques like finite difference schemes to achieve better results in solving partial differential equations (PDEs). This approach paved the way for dealing with problems involving intricate geometries and varying control needs.

2. Challenges and limitations of previous works

Computational Complexity: Early RBF methods, particularly global ones, were computationally expensive due to the need to solve dense linear systems, especially as problem size increased. Kansa [12] initially introduced RBF for PDEs, but its global nature required solving increasingly complex systems as the domain expanded, limiting its scalability.

Accuracy vs. Efficiency: Several methods attempted to balance the accuracy of control solutions with computational efficiency. For instance, Pearson [14] employed RBF clustering methods to address the Poisson control problem. However, clustering led to compromises in control accuracy due to the coarse spatial discretization. This created a need for localized, fine-tuned methods.

Lack of Sparsity Control: Many traditional optimization techniques failed to incorporate sparsity control in their objective functions effectively. Techniques introduced by Porcelli et al. [5] and Langer et al. [6] for sparse control optimization offered some improvements, but their reliance on global methods limited their applicability in problems with intricate spatio-temporal dynamics.

Mesh Dependence: Earlier finite element methods, relied heavily on structured meshes, making them less suitable for domains with complex geometries or dynamically changing boundaries. In contrast, the mesh-free nature of RBF-PU methods, as proposed in this work, offers more flexibility.

How the current RBF-PU approach overcomes these challenges:

Local RBF-PU Method: The current approach leverages the RBF-PU technique to split the domain into overlapping subdomains, allowing for localized RBF approximations. This addresses the computational complexity challenge by reducing the size of matrices involved, enabling the handling of larger problems more efficiently.

Enhanced Sparsity: By incorporating an L₂ norm in the objective function, the proposed RBF-PU approach directly promotes sparsity in control functions. Unlike previous works, which struggled to optimize sparse controls in high-dimensional spaces, this method offers a more refined solution without compromising accuracy.

Meshless Flexibility: The RBF-PU method used in this study is inherently mesh-free, eliminating the need for predefined grids. This is particularly advantageous when dealing with irregular domains or problems requiring dynamic adaptability, which traditional finite element methods often found challenging.

Error Analysis and Validation: Unlike many earlier works, this study provides a detailed error analysis, showing that the approximation error decreases with reduced subdomain size and time step. This validation offers confidence in the method's accuracy and robustness, addressing a common limitation in prior methods.

The structure of this paper is as follows: Section 1 is an introduction and Section 2 includes a quick overview of RBF techniques. The RBF-PU approach can resolve limited heat formula optimization issues, according to its explanation in Section 3. In addition, Section 4 presents some numerical data that illustrates the accuracy and efficacy of the proposed technique. Subsequently, in Section 5, we provide the conclusions.

The Radial Basis Function Partitioning of the Unity Procedure (RBF-PU) is explained and applied to the heat formula in this section. The RBF-PU approach divides the area $\Omega$ into $N$ subdomains or adjustments, $\Omega_1, \Omega_2, \ldots, \ldots, \Omega_N$. This strategy maintains accuracy while needing less computational effort.

3.1 Division of local RBF interpolants and unity

Assume that $\left\{\Omega_i\right\}_{i=1}^N$, that is, $\Omega \subseteq \cup_{i=1}^N \Omega_i$. Establish a new division of unity $\left\{\omega_i\right\}_{i=1}^N$, subordinate to the covering $\left\{\Omega_i\right\}_{i=1}^N$, so as follows:

$\sum\limits_{i=1}^{N}{{{\omega }_{i}}}(x)=1,x\in \Omega$    (22)

where, the weight function is found nonnegative, compactly accepted, and continuous, $\omega_i: \Omega_i \rightarrow R$ has $\operatorname{supp}\left(\omega_i\right) \subseteq \Omega_i$.

For each subdomain, create a local RBF interpolant of the form $\psi_u^i: \Omega_i \rightarrow R$.

$\psi _{u}^{i}=\sum\limits_{i=1}^{{{N}_{i}}}{\zeta _{i}^{k}}\phi \left( {{\left\| x-x_{i}^{k} \right\|}_{2}} \right)$    (23)

$N_i$ represents the number of localization points in $\Omega_i$. A global RBF-PUM interpolant for the whole domain $\Omega$ is stated as:

$\begin{matrix} {{\psi }_{u}}(x)=\sum\limits_{i=1}^{N}{{{\omega }_{i}}}(x)\psi _{u}^{i}= \sum\limits_{i=1}^{N}{{{\omega }_{i}}}(x)\sum\limits_{i=1}^{{{N}_{i}}}{\zeta _{i}^{k}}\phi \left( {{\left\| x-x_{i}^{k} \right\|}_{2}} \right),x\in \Omega  \\\end{matrix}$    (24)

3.2 Compactly asserted weight functions

It is found the weighted function $\omega_i(x)$ is supported compactly on $\Omega_i$. The improved compactly provided function that we employ is as follows:

$\Psi (\eta )=\left\{ \begin{array}{*{35}{l}} {{(1-\eta )}^{5}},0\le \eta \le 1  \\(5\eta +1),\eta >1  \\\end{array} \right.$  (25) 

The open cover portions will be selected following the heat equation's characteristics. The scaling-modified Wendland functions will therefore produce:

${{\Psi }_{i}}(x)={{\Psi }^{2}}\left( \frac{{{\left\| x-{{\varsigma }_{i}} \right\|}_{2}}}{{{n}_{i}}} \right),i=1,\ldots ,N$    (26)

where, $\eta_i$ and $\zeta_i$ are the centers and efficient temperature diffusion radii of regions $\Omega_i, i=1, \ldots, N$.

3.3 Approximation of the global interpolant

The formula that follows is used for approximating the global interpolant:

${{y}_{N}}(x)=\sum\limits_{i=1}^{N}{{{\omega }_{i}}}(x)\psi _{u}^{i}$    (27)

as well as

$\psi _{u}^{i}={{A}_{i}}{{\text{u}}^{i}}$    (28)

where, $A_i$ represents the RBF matrix locally.

3.4 Approximations of derivatives

The following methods can be used to approximate both first and second derivatives, respectively:

$\frac{\partial \tilde{y}}{\partial x}=\sum\limits_{i=1}^{N}{{{S}_{i}}}\left[ {{\left( {{U}_{i}} \right)}_{x}}{{A}_{i}}+{{U}_{i}}{{\left( {{A}_{i}} \right)}_{x}} \right]{{u}^{i}}$    (29)

and

$\frac{{{\partial }^{2}}\tilde{y}}{\partial {{x}^{2}}}=\sum\limits_{i=1}^{N}{{{S}_{i}}}\left[ {{\left( {{U}_{i}} \right)}_{xx}}{{A}_{i}}+2{{\left( {{U}_{i}} \right)}_{x}}{{\left( {{A}_{i}} \right)}_{x}}+{{U}_{i}}{{\left( {{A}_{i}} \right)}_{xx}} \right]{{u}^{i}}$    (30)

where, $U_i$ is a diagonal matrix with entries $\omega_i\left(x_i\right)$, and $A_i$ is the local RBF matrix.

3.5 Utilizing the heat equation

Use this technique to solve the heat equation:

$\frac{\partial y}{\partial t}-\beta \Delta y=w(x,t)+g(x,t),\text{in }\Omega \times (0,T)$    (31)

where, the temperature distribution is represented by $y(x, t)$, the control variable is represented by $w(x, t)$, and the outside heat source is represented by $g(x, t)$. The thermal dispersion parameter is represented here by $\beta$.

3.6 Error analysis

The precise answer is $y(x, t)$, and the calculation is $y_h(x, t)$. The definition of the error $e(t)$ is:

$e(t)=y(t)-{{y}_{N}}(t)$    (32)

For analysis of errors, the $L_2$ norm is employed.

$\|e(t){{\|}_{{{L}_{2}}(\Omega )}}={{\left\| y(t)-{{y}_{N}}(t) \right\|}_{{{L}_{2}}}}$    (33)

By applying Poisson's inequality, we arrive at:

$\|e(t){{\|}_{{{L}_{2}}(\Omega )}}\le K\left( {{h}^{\varsigma }}+{{\tau }^{\rho }} \right)$    (34)

To demonstrate that the error reduces as $h$ and $\tau$ decrease, we demonstrate that:

${{\left\| {{Y}_{N+1}}-{{Y}_{N}} \right\|}_{{{L}_{2}}}}\approx {{\left\| y\left( {{t}_{N+1}} \right)-{{y}_{h}}\left( {{t}_{N+1}} \right) \right\|}_{{{L}_{2}}}}\le K\left| \left( {{h}^{\varsigma }}+{{\tau }^{\rho }} \right) \right|$    (35)

where, $K$ represents a constant.

To prove the above, first employ the $L_2$ norms for analyzing errors. By Eq. (33):

$\|e(t){{\|}_{{{L}_{2}}(\Omega )}}={{\left\| y(t)-{{y}_{N}}(t) \right\|}_{{{L}_{2}}}}$    (36)

Applying the inequality of triangles, we arrive at:

$\|e(t){{\|}_{\left. {{L}_{2}}(s) \right)}}\le \|y(t)-\overset{\scriptscriptstyle\smile}{y}(t){{\|}_{{{L}_{2}}\left( {{s}_{1}} \right)}}+\|y(t)-\overset{\scriptscriptstyle\smile}{y}(t){{\|}_{{{L}_{2}}\left( {{s}_{2}} \right)}}$    (37)

where, the following reduces the spatial error:

$\|y(t)-\overset{\scriptscriptstyle\smile}{y}(t){{\|}_{{{L}_{2}}\left( {{\Omega }_{1}} \right)}}\le {{K}_{1}}{{h}^{\varsigma }}$    (38)

The following reduces the temporal error:

$\|y(t)-\overset{\scriptscriptstyle\smile}{y}(t){{\|}_{{{L}_{2}}\left( {{\Omega }_{2}} \right)}}\le {{K}_{2}}{{\tau }^{\rho }}$   (39)

Thus, the total error is:

${{\left\| y(t)-{{y}_{N}}(t) \right\|}_{{{L}_{2}}}}\le {{K}_{1}}{{h}^{\varsigma }}+{{K}_{2}}{{\tau }^{\rho }}$    (40)

Therefore, the total error at every time step $t_{N+1}$ is provided by:

${{\left\| {{Y}_{N+1}}-{{Y}_{N}} \right\|}_{{{L}_{2}}}}\approx {{\left\| y\left( {{t}_{N+1}} \right)-{{y}_{N}}\left( {{t}_{N+1}} \right) \right\|}_{{{L}_{2}}}}\le K\left| \left( {{h}^{\varsigma }}+{{\tau }^{\rho }} \right) \right|$    (41)

where, $K=K_1+K_2$.

This study demonstrates the effectiveness and viability of the Radial Basis Function-Partition of Unity (RBF-PU) method for solving optimal control problems governed by the heat equation. By localizing the approximation into subdomains and using compactly supported RBFs, the RBF-PU method achieves a high level of accuracy while maintaining computational efficiency. This makes it particularly suitable for large-scale, high-dimensional problems that are computationally intensive with traditional methods.

The findings highlight the RBF-PU method's flexibility in handling complex geometries and its natural promotion of sparsity, which is essential in scenarios with resource constraints. Additionally, the method's local refinement capabilities provide targeted accuracy improvements, making it an ideal choice for applications requiring precise control, such as environmental modeling (e.g., pollutant dispersion), biomedical engineering (e.g., targeted drug delivery), and financial optimization (e.g., portfolio management).

Significance: The results confirm that the RBF-PU method can be a powerful tool in optimal control problems, outperforming traditional methods in terms of scalability and adaptability. Its ability to handle complex boundary conditions locally further enhances its applicability across various fields.

Potential Extensions: Future work could focus on adaptive shape parameter selection to optimize accuracy and stability further. Additionally, exploring its application to other types of PDEs and control problems could broaden its utility.

In summary, the RBF-PU method offers a viable, efficient, and flexible approach for solving optimal control problems, paving the way for advancements in both research and practical applications.

Further work on our RBF-PU technique to address optimal control issues guided by the heat equation would entail a detailed convergent and error analysis. This will provide additional validation and enhancement of the technique's accuracy and resilience.