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# Stability and Error Analysis of Reduced-Order Methods Based on POD with Finite Element Solutions for Nonlocal Diffusion Problems

1 School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710129, China

2 Xi’an Key Laboratory of Scientific Computation and Applied Statistics, Northwestern Polytechnical University, Xi’an, 710129, China

* Corresponding Author: Yufeng Nie. Email:

(This article belongs to the Special Issue: Advances in Methods of Computational Modeling in Engineering Sciences, a Special Issue in Memory of Professor Satya Atluri)

*Digital Engineering and Digital Twin* **2024**, *2*, 49-77. https://doi.org/10.32604/dedt.2023.044180

**Received** 23 July 2023; **Accepted** 05 December 2023; **Issue published** 31 January 2024

## Abstract

This paper mainly considers the formulation and theoretical analysis of the reduced-order numerical method constructed by proper orthogonal decomposition (POD) for nonlocal diffusion problems with a finite range of nonlocal interactions. We first set up the classical finite element discretization for nonlocal diffusion equations and briefly explain the difference between nonlocal and partial differential equations (PDEs). Nonlocal models have to handle double integrals when using finite element methods (FEMs), which causes the generation of algebraic systems to be more challenging and time-consuming, and discrete systems have less sparsity than those for PDEs. So we establish a reduced-order model (ROM) for nonlocal diffusion equations to alleviate the calculation load and expedite the solving process. The ROM is constructed using FE solutions in a small time interval as snapshots and has much fewer degrees of freedom than FEMs. We focus on discussing the existence, stability, and error estimates of the reduced-order solutions, which have not been considered in previous research for nonlocal models. Several numerical examples are presented to validate the theoretical conclusions and to show that the ROM is quite effective for solving nonlocal equations. Moreover, we systematically explore the effect of different parameters on the behavior of the POD algorithms. Both theoretical and experimental results offer valuable insights for developing more reliable and efficient algorithms.## Keywords

Nonlocal models based on long-range interactions have broad applications in various research fields [1–5]. Differing from the classical modeling method for PDEs, nonlocal models employ integral-type operators instead of differential operators in space to establish more generalized integro-differential equations, which rely on the interaction between points within a finite distance. Due to their characteristics, nonlocal models can provide a more precise description in the presence of anomalous behaviors, discontinuities, and singularities, which are challenging for partial differential equation models. In particular, the peridynamic (PD) model proposed by Silling [1] allows for discontinuities in the displacement field and has been successfully applied to fracture and damage for diverse materials and structures [4,6,7]. Moreover, nonlocal diffusion models can depict much more general stochastic jump processes relative to Brownian motion corresponding to normal diffusion [8,9], which allows for discontinuities and jump behaviors. Nonlocal diffusion models can also describe widespread analogous diffusion by selecting some special nonlocal operators, such as fractional derivative operators [10].

Although nonlocal models exhibit a better modeling capability for complex physical phenomena than classical local models, they need to deal with an additional layer of integration, which results in algebraic systems having higher computational and assembly costs. Besides, one has to face higher solving costs of discrete systems due to much lower sparsity than that for PDEs. Thus, it is urgent to develop efficient numerical methods for solving nonlocal problems. There have been great efforts on fast algorithms for solving various high-dimensional systems, and one of the most widely used methods is model reduction based on proper orthogonal decomposition (POD) and Galerkin projection [11,12]. The POD technique [13] can significantly decrease the degrees of freedom of classical numerical methods, and has been applied to various fields of PDE problems, such as computational fluid dynamics (CFD) [12,14], compressible fluid flow as well as incompressible fluid flow [15,16], phase field [17], miscible displacement [18], supersonic flow [19], hydraulic fracturing [20,21] and so on.

In recent years, there also have been some efforts to develop reduced-order methods for nonlocal models. Gunzburger et al. [22] applied the POD method to the parametrized time-dependent nonlocal diffusion problem in the 1D case and trained a system with only a few degrees of freedom using parameterized equations, achieving almost the same accuracy as finite element discrete models. A POD-based fast algorithm was developed for one-dimensional nonlocal parabolic equations and nonlocal wave equations, which vastly speeds up the process of solving algebraic systems while maintaining high accuracy [23]. Considering that Galerkin methods need to compute multiple integrals, a reduced-order fast reproducing kernel collocation method is proposed to solve 2D nonlocal diffusion equations and peridynamic equations [24]. In this method, to get rid of the high computational complexity in the projection of inhomogeneous volume constraints, a mixed reproducing kernel approximation with nodal interpolation property is introduced to make the projection explicit, thereby truly improving computational efficiency without compromising accuracy.

Nevertheless, to our knowledge, there are very few works on the relevant theoretical analysis of POD reduced-order algorithms for nonlocal problems with finite element methods or meshfree methods. Thus, by drawing inspiration from ideas similar to those used in classical numerical analysis methods and leveraging existing techniques on analysis of POD-based reduced-order methods for PDEs, we demonstrate the existence, stability, and convergence of the POD reduced-order solutions for nonlocal models through different methodologies, and supply several numerical experiments to validate the theoretical results.

The rest of the paper is organized as follows. In Section 2, we introduce the nonlocal diffusion model with Dirichlet volume constraints and a finite scope of nonlocal interactions. In Section 3, we briefly derive the weak form for the nonlocal diffusion equation, and then give its finite element discretization, including the handling of mesh partitioning for nonlocal domains as well as the quadrature on an Euclidean ball, and provide some significant theoretical conclusions. The reduced-order model constructed by POD for nonlocal diffusion equations is given in Section 4, and we present the existence and stability analysis as well as the error estimates for the reduced-order solutions. In Section 5, we provide several numerical tests to verify the accuracy and efficiency of the reduced-order method and explore the effect of various parameters on the performance of the algorithm. Conclusions and summaries are given in Section 6.

The mathematical definitions and corresponding notations that are used throughout the article will be first introduced. Let

where

We consider the following unsteady nonlocal diffusion problem with Dirichlet volume constraints:

where

and

Without loss of generality, we also assume in this paper that the kernel

3 Finite Element Method for Nonlocal Diffusion Problems

The weak formulation of the nonlocal problem (2) can be derived by the same procedure as the classical PDE setting. Applying Green’s first identity of the nonlocal vector calculus [26] with a test function

where

where

and

Based on Eq. (6), we can define the energy norm

Therefore, the weak formulation of the nonlocal problem (2) can be defined as follows:

Given

The well-posedness results for the problem (9) have been supplied in previous references [9,10,25,27].

3.2 Finite Element Grid for Nonlocal Domain

We assume that

Then we denote

• every vertex of

• all elements

• the vertices of the triangulations of

As a result, the triangulation of the entire domain

3.3 Finite Element Discretization

In this subsection, we consider the finite element discretization of the weak Eq. (9) using continuous piecewise-polynomial spaces, and restrict ourselves to Lagrange-type basis functions defined on a set of nodes. Let

Let

The finite element approximation

and from the properties of basis functions, we have

We substitute Eq. (11) to Eq. (10) and let

where

where the mass matrix is

3.4 Balls Approximation for Stiffness Matrix Computing

Differing from the local PDEs whose stiffness matrices only involve a single integral, the stiffness matrices of nonlocal equations contain double integrals as follows:

In Eq. (14), the inner integral is defined on Euclidean balls, which may cause much more difficulties for the computation and assembly process and may result in a loss of the convergence order, since classical quadrature rules, such as Gauss quadrature, cannot be performed well on that curved domain.

To overcome these difficulties, D’Elia et al. [29] used polyhedral approximate balls denoted by

The nocaps strategy is briefly outlined here. To begin with, we seek the simplices

Consequently, substituting

where

We define a uniform partition for the time interval

where

where

One can observe that, for the system (17), different values of

Lemma 3.1. [31] If the solution

where

4 Reduced-Order Model Based on POD for Nonlocal Diffusion Problems

In this section, we first present the procedure of model reduction for nonlocal diffusion problems using the POD method and Galerkin projection. Then, the existence, stability, and convergence of the reduced-order solutions will be deduced.

4.1 Establishment of the Reduced-Order Model

We first select the initial

After getting the set of snapshots

Let

We get the singular values

The POD bases are optimal in the sense of least squares. In other words, the POD method provides a way to seek the best m-dimensional approximate subspace for a given set of data [33,34], and the square error of the POD subspace relative to the set of snapshots is

The reduced-order solution of nonlocal diffusion problems can be expressed as a linear combination of POD bases and particular solutions

where

Substituting the solution (23) into the system (13) gives

Then, employing the Galerkin projection onto the subspace spanned by

Furthermore, we introduce the corresponding initial condition and get a low-dimensional ODEs

Discreting the system (26) by

where

The same as the previous finite element system, we only consider the Crank-Nicolson scheme for the reduced-order model (27).

Remark 1. It is worth noting that the reduced-order model only needs to solve an m-dimensional system at each time node, whereas the model (17) requires solving an algebraic system with

4.2 Stability and Error Analysis for Reduced-Order Solutions

Since the coefficient matrix

To discuss the stability and convergence of the reduced-order solutions, we redefine the corresponding coefficient vectors as

where m denotes the number of POD bases. Analogous to the formulation of FE solutions, the reduced-order solution

where

We now give the stability result of the reduced-order method as follows.

Theorem 4.1. The series of the reduced-order solutions

where

Proof. For

It follows that

According to the definition of particular solutions in Eq. (20), one can get

where

Furthermore, by the orthogonality of POD basis vectors, we can obtain

where

If

Setting

One can observe that

then substituting it into the Eq. (33) yields

Furthermore,

There exist

It follows that

For any

thus, we have

Choosing

Summating from

It implies that

Finally, according to the discrete Gronwall’s lemma (see Lemma 1.4.1 in [35] ) and the above inequality, we have

where

Combining (31) and (34), we can derive the result for Theorem 4.1.

In this section, we present the error analysis for the numerical solutions

Method I: Let

Let

and, clearly,

and

which further explains the meaning of the errors

By the definition of the errors, we observe that the error

Noting that

If the POD technique is further applied to model reduction, it will give rise to additional model errors

Combining Eq. (23), the equivalent high-dimensional form of the system (26) can be represented as

Denoting

Then, multiplying both sides of the above equation system by the projection matrix

where we have substituted

Let

where

By Gronwall’s lemma [36] and

According to the Cauchy-Schwarz inequality, we have

Further information can also be obtained, i.e.,

Finally, combining (35) and (40), we get

so that

where

where

Theorem 4.2. If the solution

where

Method II: We next present an alternative method to estimate the errors of the reduced-order solutions. Differing from the above method, we now use the following matrix approaches based on the fully discrete format.

For

By the properties of

It follows that

where

where

When

Similarly, the Crank-Nicolson scheme of the reduced-order model is denoted as

From Eqs. (44) and (45), we obtain

Summing over

which leads to

Because

It follows that

From the discrete Gronwall’s lemma, we have

so that

Consequently, we obtain that

By (43), (46), Lemma 3.1 and the triangle inequality, we have

where

Theorem 4.3. If the solution

where

Remark 2. The error factor

In this section, several numerical examples are used to verify the theoretical results and investigate the impact of different parameters on the performance of the POD reduced-order algorithm. For the 2D case, we set the computational domain

The error is computed by

where

Example 1. This example examines the performance of the reduced-order algorithm. We consider a manufactured exact solution

The errors and computational time of the FEM and ROM at

Example 2. We test the effect of different parameters on the reduced-order algorithm to verify the theoretical results of the previous section in this example. The manufactured analytical solution is

Theoretically, as time

Next, we test the influences of the sampling range on the ROM. Considering that the time step is fixed, the size of the sampling range is equivalent to the number of snapshots. Fig. 9a shows the relationship between the errors of the reduced-order solutions at

We now focus on the number of POD bases. Fig. 10a shows the relationship between the error of the reduced-order numerical solutions and the number of POD base vectors with a sampling interval

Example 3: In this example, we explore the impact of size parameters on the POD numerical results. Firstly, the “sampling time step” is studied, which refers to the time increment employed for calculating snapshots in the sampling interval, and can differ from the time step

For the sake of simplicity, we consider a one-dimensional problem as follows:

where

with the exact solution

The distribution of singular values and POD subspace errors of the above generated snapshot matrices are presented by Figs. 11a and 11b. Furthermore, Figs. 11c and11d respectively indicate the relationship between the j-th singular value

Below is the research about the mesh size

where

and the relationship between the modified singular values and subspace errors with different

Similar to the situation of the sampling time step

In this work, we have studied the fast reduced-order method for 2D nonlocal diffusion models based on the POD technique and Galerkin projection. The reduced-order formulation is established first, and the existence, stability as well as convergence of the reduced-order solutions are demonstrated for the nonlocal diffusion equation. In addition, we present the error estimates of the POD solutions through the utilization of two diverse methodologies. Three numerical experiments are supplied to verify the effectiveness of the algorithm and the soundness of the theoretical analysis for the nonlocal diffusion model. Importantly, we conduct a systematic analysis of how different parameters affect the performance of the reduced-order algorithm building upon the proposed theoretical and test results.

Although the reduced-order methods for nonlocal equations have been proposed, they lack a comprehensive theoretical framework. Therefore, the research undertaken in this paper is meaningful and can provide some valuable references for the development of more stable and efficient algorithms. The possibility of extending the theoretical framework to nonlinear problems and even more complex nonlocal problems, such as peridynamic equations, will be studied in future works.

Acknowledgement: The authors would like to acknowledge the support of National Natural Science Foundation of China and the Nation Key R&D Program of China.

Funding Statement: This research was supported by National Natural Science Foundation of China (No. 11971386) and the Nation Key R&D Program of China (No. 2020YFA0713603).

Author Contributions: The authors confirm contribution to the paper as follows: study conception and design: H.L. Zhang, Y.F. Nie; data collection: H.L. Zhang; analysis and interpretation of results: H.L. Zhang, Y.F. Nie; draft manuscript preparation: H.L. Zhang; paper revision: M.N. Yang, J. Wei. All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials: The data used to support the findings of this study are available in the manuscript.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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