Large-scale linear equation set distributed solving method and system based on clustering network

Abstract

The invention discloses a distributed solving method and system for a large-scale linear equation set based on a clustering network, and belongs to the technical field of electric digital data processing, and the method comprises the steps: S1, constructing the large-scale linear equation set; s2, decomposing the augmented matrix, firstly dividing the augmented matrix into 1 different parts according to columns, and then subdividing each part into li sub-parts according to rows; s3, constructing a clustering multi-agent network, wherein each agent has a corresponding selection matrix and serves as a calculation unit to solve a small-scale linear equation; forming a cluster i by a group of intelligent agents; each agent ij solves a small-scale linear equation set through iterative calculation by using own information and information obtained by direct communication with adjacent agents in all clusters in the solving process; and S4, performing distributed solution on the large-scale linear equation through the clustering multi-agent network. According to the method, the large-scale linear equation set is rapidly solved through the clustering network, and the efficiency of large-scale data processing is improved.

Description

Technical Field

[0001] The invention belongs to the technical field of electronic digital data processing, and in particular relates to a distributed solution method and system for large-scale linear equations based on clustering networks. Background Art

[0002] With the rapid development of artificial intelligence, big data analysis, digitalization and other fields, science and technology has quietly entered a new era of network big data. This era is characterized by unprecedented amounts of big data, which are the key factors driving the change of the times. They have simultaneously opened up such a new revolution in all fields, corners, and all aspects of society. In this era of big data covered by the Internet, the research on distributed computing and distributed optimization algorithms based on multi-agent networks has received great attention due to its wide application in the fields of natural science and engineering technology.

[0003] In particular, the distributed solution algorithm of linear equations based on multi-agent networks has become the focus of research and practice due to its unique advantages in computing and processing capabilities. In the past few decades, with the development of multi-agent systems and distributed control, distributed computing has had a significant impact in many intelligent technologies and engineering. These fields include power system flow calculation, material balance calculation in the chemical industry, multi-robot coordination, and distributed optimization, etc.

[0004] As an important part of distributed computing, distributed algorithms for solving linear equations have been widely studied. However, as the scale of linear equations continues to grow, the distributed solution problem of large-scale linear equations becomes increasingly complex. Therefore, it is necessary to design a method that can further reduce the computational burden of intelligent agents, which is of great significance.

[0005] In the power system, power flow calculation is the key link to ensure the stable operation of the power grid, and distributed computing can effectively process large-scale power grid data and improve computing efficiency. In the chemical industry, material balance calculation is an important means to optimize the production process. Distributed computing can monitor and adjust the production process in real time and improve resource utilization. In a multi-robot system, coordinating tasks and path planning between multiple robots is the key to achieving efficient collaboration. Distributed computing can realize real-time communication and decision-making between robots and improve overall efficiency.

[0006] However, as the scale of problems continues to expand, traditional distributed algorithms face the problem of excessive computational burden. Therefore, researchers are committed to developing new algorithms to reduce the computational burden of intelligent agents and improve the efficiency and scalability of distributed computing. These studies are not only of great significance to theoretical research, but also have broad application prospects in practical applications. Through continuous optimization and innovation, distributed computing and optimization algorithms will play a more important role in the future and promote technological progress and innovation in various fields of society. Summary of the invention

[0007] In order to solve the technical problems existing in the known technology, the present invention provides a distributed solution method and system for large-scale linear equations based on clustering networks, which quickly solves large-scale linear equations through clustering networks, thereby improving the efficiency of large-scale data processing.

[0008] The first object of the present invention is to provide a large-scale distributed solution system for linear equations based on a clustering network, comprising:

[0009] S1, construct a large-scale linear system of equations:

[0010] Ax=b (1.1)

[0011] in, is the known coefficient matrix, is a known vector, is the unknown vector that needs to be solved;

[0012] S2, decomposition augmented matrix [Ab], see Figure 2 , the augmented matrix is ​​first divided into l different parts by column, and then each part is further divided into l i sub-parts;

[0013] S3, build a clustering multi-agent network:

[0014] Each agent i j With the corresponding selection matrix [A ij b ij ] and used as a computational unit to solve the small-scale linear equation A ij x ij =b ij ; A group of intelligent agents Constitute a cluster i, i = 1, 2…, l; each agent i j In the process of solving the problem, it uses its own information and the information obtained by direct communication with neighboring agents in all clusters to solve small-scale linear equations through iterative calculations.

[0015] S4, performing distributed solving of large-scale linear equations through the clustered multi-agent network.

[0016] Preferably, S2 comprises:

[0017] S201, divide the augmented matrix [Ab] into the following l parts:

[0018]

[0019] S202, each of the divided parts [A i b i ] is decomposed into the following l i Sub-sections:

[0020]

[0021] in,

[0022] Preferably, in S3: each agent i j The neighborhood relationship of is represented by an undirected graph: in, is a set of vertices; is a collection of edges; an agent can communicate with multiple agents in another cluster, and the neighbor relationship between agents is represented by an undirected graph to describe.

[0023] Preferably, in S4, agent i j ,i=1,2,…,l,j=1,2,…,l i The update rules of are collectively expressed as:

[0024]

[0025] in, is the intelligent agent i j The state vector of is the intelligent agent i j Auxiliary vector of ; η is the step size; is satisfied and The extraction matrix of Represents agent i j The set of neighbors in cluster i; Represents agent i j In the network The set of neighbors in ;

[0026] Each agent i j Knowing its corresponding selection matrix [A ij b ij ], using the selection matrix [A ij b ij] directly calculates the sub-part corresponding to the extracted auxiliary vector and extracts it in the agent i j The state vector With auxiliary vector Dimension transformation between It means that the agent is an agent that can exchange information with agents in other clusters in the cluster; the discrete-time distributed algorithm based on clustering network (1.4) is concisely expressed as:

[0027]

[0028] in, is the state vector of cluster i, which is composed of the column set of the state vectors of the agents in the cluster, that is, Representing an Agent Auxiliary vector in cluster i In the formula, Indicates that it contains l i An undirected connected graph network of agent nodes, The Laplace matrix of in represents an undirected connected graph network containing l agent nodes, The Laplace matrix of

[0029] make

[0030]

[0031] Preferably, S1 includes: constructing a large-scale linear equation group based on the power system power flow calculation problem, and the elements in x represent the voltage and phase angle in the power flow calculation.

[0032] Preferably, S1 includes: constructing a large-scale linear equation system based on the calculation problem of drug dosage in biology, and x represents the calculated value of drug dosage in biology.

[0033] Preferably, S1 includes: constructing a large-scale linear equation group based on the atmospheric state estimation calculation problem of weather forecast, and x represents the atmospheric state parameter in the weather forecast.

[0034] A second object of the present invention is to provide a large-scale distributed solution system for linear equations based on a clustering network, comprising:

[0035] Large-scale linear equations building module, build large-scale linear equations:

[0036] Ax=b (1.1)

[0037] in, is the known coefficient matrix, is a known vector, is the unknown vector that needs to be solved;

[0038] Decomposition module, decomposes the augmented matrix [Ab], first divides the augmented matrix [Ab] into l different parts by column, and then each part is subdivided into l by row i sub-parts;

[0039] Clustering multi-agent network building module, building a clustering multi-agent network:

[0040] Each agent i j With the corresponding selection matrix [A ij b ij ] and used as a computational unit to solve the small-scale linear equation A ij x ij =b ij ; A group of intelligent agents Constitute a cluster i, i = 1, 2…, l; each agent i j In the process of solving the problem, it uses its own information and the information obtained by direct communication with neighboring agents in all clusters to solve small-scale linear equations through iterative calculations.

[0041] The distributed solution module performs distributed solution to large-scale linear equations through the clustered multi-agent network.

[0042] The third object of the present invention is to provide a computer-readable storage medium storing a computer program, which, when executed by a processor, implements the above-mentioned distributed solution method for large-scale linear equations based on clustering networks.

[0043] The fourth object of the present invention is to provide a computer program product, comprising a computer program, which, when executed by a processor, implements the above-mentioned distributed solution method for large-scale linear equations based on clustering networks.

[0044] The advantages and positive effects of the present invention are:

[0045] The core idea of ​​the present invention is to first construct a large-scale linear equation system corresponding to a specific practical problem. This process involves in-depth analysis and understanding of the practical problem in order to accurately convert the problem into a mathematical model. Then, the present invention adopts an innovative method, that is, the augmented matrix of the constructed large-scale linear equation system is decomposed one by one. This decomposition process is one of the key steps of the present invention, which can effectively simplify the complexity of the large-scale linear equation system, thereby laying a solid foundation for the subsequent solution steps.

[0046] After completing the successive decomposition of the augmented matrix, the present invention further constructs a clustered multi-agent network. This network is composed of multiple agents, each of which can independently perform calculations and information processing. By assigning the task of solving large-scale linear equations to these agents, the present invention realizes distributed computing. This distributed solution method not only significantly reduces the computational difficulty of large-scale linear equations, but also greatly improves the efficiency of data processing.

[0047] In order to verify the effectiveness of the present invention, the present application has conducted a number of experiments. The experimental results show that the present invention has demonstrated excellent performance in multiple fields. Specifically, the present invention can well meet the needs of power system flow calculation problems. In the power system, flow calculation is a key link to ensure the stable operation of the power grid. The present invention can quickly and accurately calculate the voltage, current, and phase angle distribution of each node in the power grid through an efficient distributed solution method. In addition, the present invention has also shown great potential in the field of biology, especially in the calculation of drug dosage. Through precise mathematical models and efficient calculation methods, the present invention can help researchers quickly determine the optimal dosage of drugs, thereby improving the therapeutic effect and reducing side effects. Finally, the present invention has also demonstrated its application value in the field of weather forecasting, especially in the problem of atmospheric state estimation. Through the distributed computing method of the present invention, meteorologists can process large amounts of meteorological data more efficiently, thereby improving the accuracy and timeliness of weather forecasts.

[0048] In summary, the present invention is not only innovative in theory, but also shows significant advantages in practical applications. By constructing a large-scale linear equation system and using a clustered multi-agent network for distributed solution, the present invention effectively reduces the difficulty of calculation, improves data processing efficiency, and achieves satisfactory results in multiple fields such as power system flow calculation, biological drug dosage calculation, and weather forecast. BRIEF DESCRIPTION OF THE DRAWINGS

[0049] Figure 1 A flowchart of a preferred embodiment of the present invention;

[0050] Figure 2 This is a flow chart of S2 in a preferred embodiment of the present invention;

[0051] Figure 3 A schematic diagram of a clustered multi-agent network in a preferred embodiment of the present invention;

[0052] Figure 4 It is a schematic diagram of simulation results in a preferred embodiment of the present invention;

[0053] Figure 5 This is a network diagram of a power system with five branches and five nodes in a preferred embodiment of the present invention. DETAILED DESCRIPTION

[0054] In order to further understand the content, features and effects of the present invention, the following embodiments are listed and described in detail with reference to the accompanying drawings:

[0055] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the technical solutions in the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0056] First embodiment:

[0057] See also Figure 1 , a distributed solution method for large-scale linear equations based on clustering networks, including:

[0058] S1, construct a large-scale linear system of equations:

[0059] Ax=b (1.1)

[0060] in, is the known coefficient matrix, is a known vector, is the unknown vector that needs to be solved;

[0061] For example, in the process of power system flow calculation, it is often necessary to construct a mathematical model of the power system flow, and these mathematical models often contain some large-scale linear equations; in this case, the elements in x can represent the voltage and phase angle of the power system flow;

[0062] A large-scale linear equation system is used to solve the power flow calculation problem of a five-branch five-node power system, such as Figure 5 As shown, G1 is the balancing node, G2 is the rated power operation node; T1 and T2 are transformers; L1, L2, and L3 are branches; and S3 is the load.

[0063] Based on the Newton-Raphson iterative method, the following modified equation can be obtained:

[0064]

[0065] Among them, Δδ and ΔU are unknown vectors to be determined; ΔP is the active power change vector; ΔQ is the reactive power change vector; H, N, J, L are sensitivity matrices, whose elements can be obtained by the following formula:

[0066]

[0067] Where, i, j = 1, 2, ..., 5, i ≠ j; P i is the active power at node i; U i is the voltage at node i; G ij is the conductance between node i and node j; δ ij is the phase angle difference between node i and node j; B ij is the susceptance between node i and node j; Q i is the reactive power at node i. Based on the above analysis, solving the power flow calculation problem of a 5-branch 5-node power grid can be transformed into solving the linear equation system Ax = b, where

[0068]

[0069] In biological research, x can be the calculated value of drug dosage, and linear equations can be used to describe the growth model of biological populations, gene expression patterns in genetics, etc. In the medical field, linear equations can be used to calculate drug dosage and analyze biomarkers.

[0070] In weather forecasting, x can represent an estimate of the atmospheric state: temperature forecast, wind direction and speed forecast, precipitation forecast;

[0071] In numerical weather forecasting, the solution of a system of linear equations often arises in the process of data assimilation. Data assimilation is the process of combining observational data with a forecast model, which requires solving a large system of linear equations to optimally determine the initial state of the atmosphere. This process often involves statistical methods such as Kalman filters, which can essentially be viewed as the problem of solving a system of linear equations.

[0072] For example, suppose there is a set of meteorological observations, including temperature, humidity, wind speed and direction, which can be represented as a vector. At the same time, there is a numerical weather forecast model that provides an approximate representation of the state of the atmosphere, which can be represented as a vector. The goal of data assimilation is to find an optimal estimate of the state of the atmosphere that best fits all the observations.

[0073] In this process, a linear system of equations may be constructed that contains the relationship between the observed data and the model approximation. By solving this linear system of equations, an adjusted estimate of the atmospheric state can be obtained, which is as close as possible to the model prediction while satisfying all the observed data constraints. This adjusted estimate is the solution to the linear system of equations, which has practical physical meaning because it represents a best estimate of the atmospheric state.

[0074] In practical applications, this solution can be used to initialize the numerical weather forecast model, thereby improving the accuracy of the forecast. This process is a key step in numerical weather forecasting because it directly affects the starting point of the forecast model, which in turn affects the reliability and accuracy of subsequent forecasts.

[0075] The explanation for large scale is as follows: When the dimension of the linear equation system is very large, such as in the state estimation problem of large-scale power systems, centralized methods may be difficult to handle because a large amount of computing resources and storage space are required, and the demand for communication resources is also high. In this case, distributed algorithms provide an effective solution. Distributed algorithms can decompose large-scale linear equation systems into multiple small-scale sub-equation systems that can be solved in parallel on different computing nodes. This approach takes advantage of the advantages of multi-agent systems to disperse the computational burden, reduce communication requirements, and improve data security and privacy protection.

[0076] Specifically, there is no clear dimensionality limit to determine when a distributed algorithm must be used. This decision depends on a variety of factors, including the sparsity of the system of equations, the structure of the coefficient matrix, the available computing resources, and the requirements for computational efficiency and communication costs. In practical applications, such as power system analysis, when the size of the system of equations grows to tens of millions of dimensions or even higher, distributed methods become very useful.

[0077] In general, distributed algorithms are appropriate for solving linear systems of equations when the size of the system grows beyond the ability of a single machine to handle it efficiently, or when the structure of the problem allows for parallelization. This typically involves large-scale problems such as power flow calculations in power systems, where the size of the system of equations may be related to the number of nodes and branches in the system. In these cases, distributed methods can effectively exploit the advantages of multi-agent systems to achieve efficient and reliable solutions.

[0078] Without loss of generality, assume that the linear system of equations (1.1) has at least one solution.

[0079] When the scale of the linear equation system is large, the complexity of the traditional centralized algorithm will increase, thus placing higher requirements on computing power. The distributed algorithm can decompose a large-scale linear equation system into multiple small-scale linear equation systems for distributed solution, which is simple and efficient. Therefore, the distributed algorithm provides a feasible and promising algorithm for solving large-scale linear equation systems. In order to solve linear equations in a distributed manner, it is necessary to divide the augmented matrix [A b] associated with the linear equation system (1.1) into several matrices [A ij b ij], each matrix corresponds to a small-scale linear equation. Then a clustered multi-agent network is constructed to describe the communication between agents solving these small-scale linear equations.

[0080] S2, decompose the augmented matrix [Ab], the augmented matrix is ​​first divided into l different parts by column, and each part is further divided into l by row i Specifically, the augmented matrix [Ab] is divided into the following l parts:

[0081]

[0082] Then, each part [A i b i ] is decomposed into the following l i Sub-sections:

[0083]

[0084] in,

[0085] S3, build a clustering multi-agent network:

[0086] Each agent i j It is known that the corresponding augmented matrix [A ij b ij ] and used as a computational unit to solve the small-scale linear equation A ij x ij =b ij . A group of intelligent agents Constitute a cluster i, i = 1, 2…, l. Each agent i j In the process of solving the problem, the agent uses its own information and the information obtained through direct communication with neighboring agents in all clusters to solve the small-scale linear equations through iterative calculation. This communication allows agents to exchange data directly related to them, which enhances the accuracy and speed of local solutions. Each agent i j The neighborhood relationship can be represented by an undirected graph: in is a set of vertices; is a set of edges. An agent can communicate with multiple agents in another cluster, and the neighbor relationship between these agents is represented by an undirected graph to describe.

[0087] To better understand the relationship between the decomposition of a system of linear equations and the construction of a clustered multi-agent network, this section provides the following example: Consider the following system of linear equations;

[0088]

[0089] Different columns are used to represent different clusters, and the partitioning of the agents can also be inferred based on the partitioning of the rows. According to the above partitioning, the linear equation system can be divided into the following smaller-scale linear equation systems;

[0090]

[0091] Then, according to the partition of the above augmented matrix, the following can be constructed Figure 3 The clustered multi-agent network shown in Figure 1 is used to solve the above linear equations; each agent corresponds to a small-scale linear equation, and each agent can communicate directly with its neighbors, regardless of whether its neighbors are in the same cluster as it or not. The decomposition of the linear equations is consistent with the construction of the clustered multi-agent network, providing an important foundation for the design of distributed algorithms.

[0092] S4, performing distributed solving of large-scale linear equations through the clustered multi-agent network.

[0093] Algorithm Preparation Lemma

[0094] First, a key lemma is introduced to promote the development of the main results of this chapter. This lemma plays a necessary bridge role in the process of solving problems and establishes an important theoretical foundation for us. Based on this, a distributed solution algorithm for solving linear equations is proposed by constructing a clustering multi-agent network.

[0095] Lemma 1: For the following matrix Φ:

[0096]

[0097] Matrix Φ i is a real number, where i = 2, 3, 4, 5, Φ 3 and Φ 5 is a positive semidefinite matrix and Then, all eigenvalues ​​of the matrix Φ are negative real numbers or 0. Furthermore, if 0 is an eigenvalue of the matrix Φ, then the algebraic multiplicity of the eigenvalue of the matrix 0 is equal to its geometric multiplicity.

[0098] Lemma 2: Consider the following non-zero matrix Φ:

[0099]

[0100] Matrix Φ i is a real number, where i = 2, 3, 4, 5, Φ 3 and Φ 5 is a positive semidefinite matrix and Then, for the matrix Ψ=I+ηΦ, where The following conditions must be met:

[0101] 1) The eigenvalues ​​of a matrix Ψ are real numbers and lie in the interval (-1,1]. Furthermore, if 1 is an eigenvalue of the matrix Ψ, then the algebraic multiplicity of the eigenvalue of 1 is equal to its geometric multiplicity.

[0102] 2) There exists a non-singular matrix T, the matrix and matrix The following conditions must be met:

[0103]

[0104] in, The eigenvalues ​​of are real numbers and are in the interval (-1,1). The eigenvalues ​​of are real numbers and are in the interval (-∞,0), and σ is the algebraic multiplicity at the eigenvalue 1 of the matrix Ψ.

[0105] Proof: 1) First, prove statement 1). According to Lemma 5.1, the eigenvalues ​​of the matrix Φ are real numbers or less than 0. Specifically, when the eigenvalue of the matrix Φ is 0, its algebraic multiplicity is equal to its geometric multiplicity. This means that if every eigenvalue of the matrix Φ is 0. Applying Jordan decomposition to the matrix Φ yields Φ=0, which conflicts with the statement that Φ is a non-zero matrix. Therefore, the matrix Φ must have a non-zero eigenvalue and satisfy λ min (Φ)<0. This means The definition is reasonable and good. Therefore, the eigenvalue of ηΦ is a real number and lies in the range of (-2,0]. Furthermore, it can be directly proved that the eigenvalue of Ψ=I+ηΦ is also a real number and lies in the range of (-1,1]. Based on the fact that the geometric multiplicity at the eigenvalue 0 of the matrix Φ is equal to its algebraic multiplicity, by comparing and observing, it can be concluded that the algebraic multiplicity at the eigenvalue 1 of the matrix Ψ=I+ηΦ is equal to its geometric multiplicity. Therefore, the statement 1) of Lemma 2 is proved.

[0106] 2) For the matrix Ψ, the algebraic multiplicity of its eigenvalue 1 is equal to its geometric multiplicity. According to the Jordan decomposition and Ψ = I + ηΦ, it can be concluded that there exists a non-singular matrix T such that equation (1.13) holds, where the multiplicity of the eigenvalue 1 of the matrix Ψ and the multiplicity of the eigenvalue 0 of the matrix Φ is σ. Since non-singular transformations of a matrix do not change the eigenvalues ​​of the matrix, then the matrix and Ψ have the same eigenvalues, the matrix and Φ also have the same eigenvalues. This means that the matrix All eigenvalues ​​of are in the interval (-1,1). A similar algorithm can verify that the matrix All eigenvalues ​​of are real and in the interval (-∞,0). Therefore, Lemma 2 is proved.

[0107] Next, we will introduce the distributed algorithm for linear discrete-time systems based on clustered multi-agent networks, and prove and analyze the convergence and correctness of the algorithm.

[0108] Based on the above matrix decomposition, the problem of solving a system of linear equations can be transformed into the following two sub-problems:

[0109] The first problem is to ensure that the solutions of all agents in the same cluster are the same, that is:

[0110]

[0111] Where k is the number of iterations, is the intelligent agent i j The result obtained at time step k;

[0112] The second problem is to ensure that the solutions of the l clusters are consistent, that is,

[0113]

[0114] Where k represents the number of algorithm iterations.

[0115] is the total number of l in the i-th part at time step k i The set of sequences of solutions of agents, namely

[0116] In view of the above two sub-problems, this section proposes a distributed solution algorithm for linear equations based on clustering networks, where the intelligent agent i j ,i=1,2,…,l,j=1,2,…,l i The update rule of can be collectively expressed as

[0117]

[0118] in is the intelligent agent i j The state vector of is the intelligent agent i j Auxiliary vector of ; η is the step size; is satisfied and The extraction matrix. Represents agent i j The set of neighbors in cluster i. Represents agent i j In the network The set of neighbors in .

[0119] In the above algorithm, each agent i jKnowing its corresponding selection matrix, the selection matrix can be directly used to calculate the sub-part corresponding to the extracted auxiliary vector and j The state vector With auxiliary vector Transform dimensions between . It means that the agent can exchange information with agents in other clusters in the cluster. In addition, the discrete-time cluster network-based distributed algorithm (1.4) can be concisely expressed as:

[0120]

[0121] in is the state vector of cluster i, which is composed of the column set of the state vectors of the agents in the cluster, that is, Representing an Agent Auxiliary vector in cluster i; In the formula Indicates that it contains l i The undirected connected graph network of agent nodes has a Laplacian matrix of in Represents an undirected connected graph network containing l agent nodes, and its Laplacian matrix is

[0122] make

[0123]

[0124] Next, we will conduct an in-depth analysis of the convergence of a distributed algorithm for solving linear discrete-time systems based on clustering networks for solving linear equations. The analysis aims to determine whether the algorithm can effectively reach the predetermined solution under various conditions and identify the key factors that affect the convergence speed and stability.

[0125] Theorem 1: Assume that the linear system (1.1) is solvable and the graph and i=1,2,…,l is a connected graph. If Then the distributed solution algorithm for linear equations based on clustering networks can ensure that the state vector exponentials of the agents in the same cluster converge to the same vector, and the column set of all identical vectors in different clusters is one of the solutions to the linear equation system (1.1).

[0126] Proof: Based on the representation relationship in equation (1.5), this section will prove Theorem 5.1 from two aspects. First, we will explore the relationship between the equilibrium point of the distributed solution algorithm for linear equations based on clustering networks and the solution of the linear equations. Second, we will discuss whether the distributed solution algorithm for linear equations based on clustering networks can converge to the equilibrium point.

[0127] Step 1: In this step, we will prove the equilibrium point of the distributed solution algorithm for linear equations based on clustering networks. Existence, among which is a solution to the linear system of equations.

[0128] According to Ax ★ =b and (1.2), we can get the following relationship:

[0129]

[0130] On this basis, according to equation (1.3), the above equation is equivalent to:

[0131]

[0132] Given that agent i j The extraction matrix E ij The definition of , (1.17) can be expressed in the following compact form:

[0133]

[0134] Equation (1.18) can be further expressed as:

[0135]

[0136] Because of the figure It is connected. According to the properties of the connected graph, So there must be a vector exists, so that the following equation holds:

[0137]

[0138] based on In fact, multiplying both sides of equation (1.20) by E yields:

[0139]

[0140] from It can be inferred that This also means:

[0141]

[0142] In the above derivation, the establishment of equations (1.20) and (1.22) means that It is the equilibrium point of the distributed solution algorithm (1.5) for linear equations based on clustering networks. This also means that the solution of the linear equations is part of the equilibrium point of the distributed solution algorithm (1.5) for linear equations based on clustering networks.

[0143] On the other hand, assuming is the equilibrium point of the distributed solution algorithm for linear equations based on clustering networks (1.5), where Then equation (1.20) and equation (1.22) hold. That is, Right now according to There must be a vector And satisfy This also means Therefore, the following equation can be obtained through analysis:

[0144]

[0145] Multiply both sides of equation (1.20) by You can get:

[0146]

[0147] based on It can be deduced that:

[0148]

[0149] That is:

[0150]

[0151] The above equation can be restated as follows:

[0152]

[0153] based on available

[0154]

[0155] Equation (1.28) can be simplified to:

[0156]

[0157] definition This definition means that Au = b. Based on the above derivation, it can be concluded that a part of the equilibrium point of the distributed solution algorithm (1.5) for linear equations based on clustering networks is a solution to the linear equations.

[0158] Step 2: Assumptions It is an equilibrium point of the distributed solution algorithm of linear equations based on clustering network. Definition According to equation (1.5):

[0159] e(k+1)=Ψ 1 e(k)(1.30)

[0160] It can be seen that the matrix Ψ 1 and Φ 1 The structure of Ψ matches the structure of Ψ and Φ in Lemma 5.2. The matrix Ψ 1 and Φ 1 If Lemma 2 is satisfied, then Ψ 1 The eigenvalues ​​of are in the interval (-1,1], and the matrix Ψ 1 The algebraic multiplicity of an eigenvalue 1 is equal to its geometric multiplicity.

[0161] According to Lemma 2, the matrix T exists and satisfies (1.13). Definition Where T = [T 1 T 2 ], then the error system (1.30) can be written as follows:

[0162]

[0163] From Lemma 5.2, we can see that the matrix The eigenvalues ​​of are real numbers and are in the interval (-1,1). This means:

[0164]

[0165] therefore

[0166]

[0167] Since the matrix T exists and satisfies (1.13), then:

[0168]

[0169] From the above equation, we can see that Φ 1 T 2 =0, that is, T 2 The image subspace belongs to Φ 1 According to the nuclear space. It can be inferred that e(k) exponentially converges to the matrix Φ 1 of nuclear space.

[0170] Therefore, the discrete-time distributed algorithm (1.5) based on clustering networks converges exponentially to in belongs to the matrix Φ 1 Any vector in the kernel space. Based on this, we can conclude that Also because Satisfies equations (1.20) and (1.22), so It is also an equilibrium point of the distributed solution algorithm (1.5) for linear equations based on clustering networks. Therefore, there exists a vector Make

[0171]

[0172] It is established. From the above analysis, we can see that the distributed solution algorithm (1.5) of linear equations based on clustering network converges exponentially to the equilibrium point. The proof is complete.

[0173] In the above cluster network-based distributed solution algorithm for linear equations, each agent only knows a sub-part of the original linear equations, and the dimensions of the state variables of each agent can be different. The agents in each cluster reach a consensus on a vector through inter-cluster communication, and the purpose of inter-cluster communication is to ensure that the set of these vectors is the solution to the linear equations.

[0174] Second embodiment:

[0175] A distributed solution system for large-scale linear equations based on clustering networks, used to implement the above-mentioned distributed solution method for large-scale linear equations based on clustering networks, the system comprising:

[0176] Large-scale linear equations building module, build large-scale linear equations:

[0177] Ax=b (1.1)

[0178] in, is the known coefficient matrix, is a known vector, is the unknown vector that needs to be solved;

[0179] Decomposition module, decomposes the augmented matrix [Ab], first divides the augmented matrix [Ab] into l different parts by column, and then each part is subdivided into l by row i sub-parts;

[0180] Clustering multi-agent network building module, building a clustering multi-agent network:

[0181] Each agent i j With the corresponding selection matrix [A ij b ij ] and used as a computational unit to solve the small-scale linear equation A ij x ij =bij ; A group of intelligent agents Constitute a cluster i, i = 1, 2…, l; each agent i j In the process of solving the problem, it uses its own information and the information obtained by direct communication with neighboring agents in all clusters to solve small-scale linear equations through iterative calculations.

[0182] The distributed solution module performs distributed solution to large-scale linear equations through the clustered multi-agent network.

[0183] The third embodiment: A computer-readable storage medium stores a computer program, which, when executed by a processor, implements the above-mentioned distributed solution method for large-scale linear equations based on clustering networks.

[0184] Fourth embodiment: A computer program product, comprising a computer program, which, when executed by a processor, implements the above-mentioned distributed solution method for large-scale linear equations based on clustering networks.

[0185] Simulation Verification

[0186] Consider solving the following linear equation system Ax=b, where A and b are specifically expressed as:

[0187]

[0188] Based on the above specific parameters of A and b in the linear equation system Ax=b, it can be calculated that the linear equation system has a unique solution, and its solution is Therefore, according to the distributed solution algorithm of linear equations based on clustering network, the following Figure 3 The clustered multi-agent network shown in is used to solve the above linear equations.

[0189] According to the division of A and b, the construction of the clustering network can derive λ min Φ 1 =-215.6877. According to Theorem 1, the step size can be selected as η = 0.004,

[0190]

[0191] To describe the error between the results obtained by the discrete-time distributed algorithm based on clustering networks and the solution of the linear equations. When the initial conditions of the distributed solution algorithm based on clustering networks are randomly selected and the value range is [-5,5], the simulation results are as follows Figure 4 As shown in Figure 1, it is not difficult to find that the state vector of each cluster is the only solution that converges exponentially to the linear equations. This verifies the validity of Theorem 1.

[0192] The present invention aims at the problem of distributed solution of large-scale linear equations, with the aim of reducing the computational burden of intelligent agents and improving the efficiency and effectiveness of the distributed solution algorithm of linear equations by simplifying the computational process. The present invention also proves that the proposed distributed algorithm reaches consistency for a part of a solution to a linear equation in the same class, and all column sets of the same vectors reach consistency in different classes, and all column sets of the same vectors are a solution to the linear equations.

[0193] In the above embodiments, it can be implemented in whole or in part by software, hardware, firmware or any combination thereof. When the use is implemented in whole or in part in the form of a computer program product, the computer program product includes one or more computer instructions. When the computer program instructions are loaded or executed on a computer, the process or function described in the embodiment of the present invention is generated in whole or in part. The computer may be a general-purpose computer, a special-purpose computer, a computer network, or other programmable device. The computer instructions may be stored in a computer-readable storage medium, or transmitted from one computer-readable storage medium to another computer-readable storage medium. For example, the computer instructions may be transmitted from one website site, computer, server or data center by wired (e.g., coaxial cable, optical fiber, digital subscriber line (DSL) or wireless (e.g., infrared, wireless, microwave, etc.) mode) to another website site, computer, server or data center. The computer-readable storage medium may be any available medium that a computer can access or a data storage device such as a server or data center that includes one or more available media integrated. The available medium may be a magnetic medium (e.g., a floppy disk, a hard disk, a tape), an optical medium (e.g., a DVD), or a semiconductor medium (e.g., a solid-state hard disk SolidState Disk (SSD)), etc.

[0194] The above description is only a preferred embodiment of the present invention and does not limit the present invention in any form. Any simple modification, equivalent changes and modifications made to the above embodiments based on the technical essence of the present invention are within the scope of the technical solution of the present invention.

Claims

1. A distributed solution method for large-scale linear equations based on clustering networks, characterized in that: include: S1, construct a large-scale linear system of equations: Ax = b(1.1) where is the known coefficient matrix, is a known vector, is the unknown vector that needs to be solved; S2, decompose the augmented matrix [Ab], refer to Figure 2, first divide the augmented matrix [Ab] into l different parts by column, and then subdivide each part into l by row i sub-parts; S3, build a clustering multi-agent network: Each agent i j With the corresponding selection matrix [A ij b ij ] and used as a computational unit to solve the small-scale linear equation A ij x ij =b ij ; A group of intelligent agents Constitute a cluster i, i = 1, 2…, l; each agent i j In the process of solving the problem, it uses its own information and the information obtained by direct communication with neighboring agents in all clusters to solve small-scale linear equations through iterative calculations. S4, performing distributed solving of large-scale linear equations through the clustered multi-agent network.

2. The distributed solution method for large-scale linear equations based on clustering network according to claim 1, characterized in that S2 include: S201, divide the augmented matrix [Ab] into the following l parts: S202, each of the divided parts [A i b i ] is decomposed into the following l i Sub-sections: in, 3. The distributed solution method for large-scale linear equations based on clustering network according to claim 2, characterized in that: In S3: Each agent i j The neighborhood relationship of is represented by an undirected graph: in, is a set of vertices; is a collection of edges; an agent can communicate with multiple agents in another cluster, and the neighbor relationship between agents is represented by an undirected graph to describe.

4. The distributed solution method for large-scale linear equations based on clustering network according to claim 3, characterized in that: In S4, agent i j ,i=1,2,…,l,j=1,2,…,l i The update rules of are collectively expressed as: in, is the intelligent agent i j The state vector of is the intelligent agent i j Auxiliary vector of ; η is the step size; is satisfied and The extraction matrix of Represents agent i j The set of neighbors in cluster i; Represents agent i j In the network The set of neighbors in ; Each agent i j Knowing its corresponding selection matrix [A ij b ij ], using the selection matrix [A ij b ij ] directly calculates the sub-part corresponding to the extracted auxiliary vector and extracts it in the agent i j The state vector With auxiliary vector Dimension transformation between It means that the agent can exchange information with agents in other clusters in the cluster; the discrete-time distributed algorithm based on cluster network (1.4) is concisely expressed as: in, is the state vector of cluster i, which is composed of the column set of the state vectors of the agents in the cluster, that is, Representing an Agent Auxiliary vector in cluster i; In the formula, Indicates that it contains l i An undirected connected graph network of agent nodes, The Laplace matrix of in represents an undirected connected graph network containing l agent nodes, The Laplace matrix of make 5. The distributed solution method for large-scale linear equations based on clustering network according to any one of claims 1 to 4, characterized in that: S1 includes: constructing a large-scale linear equation group based on the power system power flow calculation problem, and the elements in x represent the voltage and phase angle in the power flow calculation.

6. The distributed solution method for large-scale linear equations based on clustering network according to any one of claims 1 to 4, characterized in that: S1 includes: A large-scale linear equation system is constructed based on the calculation problem of drug dosage in biology, and x represents the calculated value of drug dosage in biology.

7. The distributed solution method for large-scale linear equations based on clustering network according to any one of claims 1 to 4, characterized in that: S1 includes: A large-scale linear equation system is constructed based on the atmospheric state estimation calculation problem for weather forecast, where x represents the atmospheric state parameters in weather forecast.

8. A distributed solution system for large-scale linear equations based on clustering networks, characterized in that: include: Large-scale linear equations building module, build large-scale linear equations: Ax=b (1.1) in, is the known coefficient matrix, is a known vector, is the unknown vector that needs to be solved; Decomposition module, decomposes the augmented matrix [Ab], first divides the augmented matrix [Ab] into l different parts by column, and then each part is subdivided into l by row i sub-parts; Clustering multi-agent network building module, building a clustering multi-agent network: Each agent i j With the corresponding selection matrix [A ij b ij ] and used as a computational unit to solve the small-scale linear equation A ij x ij =b ij ; A group of intelligent agents Constitute a cluster i, i = 1, 2…, l; each agent i j In the process of solving the problem, it uses its own information and the information obtained by direct communication with neighboring agents in all clusters to solve small-scale linear equations through iterative calculations. The distributed solution module performs distributed solution to large-scale linear equations through the clustered multi-agent network.

9. A computer-readable storage medium storing a computer program, which, when executed by a processor, implements the distributed solution method for large-scale linear equations based on clustering networks as described in any one of claims 1 to 7.

10. A computer program product, comprising a computer program, characterized in that When the computer program is executed by a processor, the distributed solution method for large-scale linear equations based on clustering networks described in any one of claims 1 to 7 is implemented.

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