In the process of rational development and utilisation of nuclear energy, people often face nuclear accidents such as lost and stolen radioactive sources; so, the means of searching for these sources quickly in highly radioactive environments is an important security challenge. In the past, these jobs were limited to workers specialising in nuclear technology. They used gammaray detection equipment to search for radioactive sources, but the search efficiency was low. The main purpose of this article is to design a metaheuristic algorithm based on imitating professional technicians to locate radioactive sources in a computeraided manner. At the same time, due to the complexity that may characterise the actual search, the search strategy must be optimised. The article established an intelligent random search model with human thinking. Finally, it was proved based on the mathematical theory that the complexity of the model search algorithm is linear, and the simulation experiment results show that the optimisation algorithm has good efficiency and fault tolerance.
Keywords
 nuclear radioactive source
 random search algorithm
 complexity
Every year, many commercial radioactive sources are used in medicine and industry in the world, which presents a potential hazard. In the case of losing radioactive sources, the means to quickly search and remove contaminated materials in the separate areas in the highlevel environment of radioactive sources is an important security issue in the application of nuclear technology. Many researchers have contributed to the search for radioactive sources.
Some studies in the past have improved the sensitivity of nuclear detection instruments to improve the efficiency of searching for radioactive sources. Various types of
Similarly, some scholars have applied mathematical methods to the study of nuclear technologyrelated issues. In 2000, Alpay and Shor [8] studied the use of point sensor measurements to locate point sources in a class of distributed systems described by partial differential equations (PDE). In 2007, Ristic et al. [9] published an article, in which the detection/estimation part of the radioactive material of unknown intensity of radiation in the Bayesian framework was solved using a particle filter. In 2010, Hamideen et al. [10] used the mathematical nonlinear equation to derive and use the mathematical nonlinear program to locate radioactive sources in one, two and threedimensional space. Using mathematical methods ensures that it becomes possible to break the professional barriers and allows more researchers to study the problem, but the requirements of these methods are harsh and far removed from the actual situation.
However, there are also many scholars whose work is dedicated to the use of robots in radioactive search. In this way, experts in the mechanical field also join in the research. Redus et al. [4] developed an integrated imaging sensor system, which combined a gammaray image of the distribution of radioactivity with a video image of the area, allowing a rapid and intuitive determination of the source location. The CEA and CYBERIA [5] have worked together to develop the RICA robot, which can locate and measure the activity of radioactive sources. In 2017, Chen et al. [6] set up a computational fluid dynamics model of leakage for an indoor timevarying contaminant source, together with establishing a search strategy using multirobot active olfaction. In 2019, Ardiny et al. [7] adapted and implemented behaviourbased and multicriteria decisionmaking (MCDM) approaches on an autonomous robot. The core technology of robot search is algorithm design; so, a continuous optimisation algorithm is a key to improve robot search.
The search problem is a relatively conventional problem in algorithm design and has a good research foundation. In 1995, Wacholder et al. [11] established an optimisation artificial neural networks model, which was developed for solving the illposed inverse transport problem associated with localising radioactive sources in a medium with known properties and dimensions. In 2005, Presler et al. [12] and others developed a simple and practical algorithm to use two detectors to measure the emission of gamma rays from a huge sample, so that the ratio of the detector's counting rate can be used to locate the position of the point source of the bulk volume. In 2009, Vilim and Klann [13] published a paper describing advanced algorithms developed and implemented for realtime source detection, location and tracking. In 2015, Shixiang [14] designed the structure of the radiation source manipulator and proposed a source search algorithm based on Gaussian process regression. In 2015, Zhang [15] designed a heuristic search algorithm for line scanning, which can quickly search unknown sources in a short time. Liu [16] investigated a double Qlearningbased anomalous source searching algorithm to navigate the detector searching for sources. Gong et al. [17] developed the source localisation algorithm based on the inversesquare law and statistical methods in the region where the source is present, and the unmanned aircraft is used to locate the sources. Computationassisted search has low cost and algorithm design has a good research foundation; so, this has become a relatively good research direction. Here a small amount of grid data and algorithms are used to locate the radioactive source.
In this article, the authors use a small amount of raster data and a simulated artificial search thinking design algorithm to locate radioactive sources, and theoretically prove that this method is linear in search time; so, this method provides a new way to efficiently search for radioactive sources.
The rectangular area containing the radiation source is divided into a grid and the corresponding grid node data is collected. Consider horizontal square area with point source Ω = {(
Then
Radiation dosage at the grid nodes
…  

…  
…  
⋮  ⋮  ⋮  ⋮  ⋮  
… 
When people search for radiation sources, they hold the detector to detect the direction of the maximum dosage horizontally and move towards the direction of the maximum dosage. Imitating the idea of human search results in a random search model.
From the grid in the rectangular area of the radiation source, we randomly generated the position coordinate of a single radiation source and selected a point as the starting point for searching. We compare the four dosages around the point, and then move to the point of maximum dosage; subsequently, we analyse the radiation dosage at the next grid node, and move to the location with the most significant amount until no further movement is warranted. Finally, the stop point is the radiation source.
Randomly generated source locations can be placed on grid nodes or within the grid. For the radiation source located within a grid, we determine the grid node whose radiation dosage is the closest to the radiation source according to the principle of adjacent radiation dosages comparison. Then we divide the small area centred on the node into smaller grids, and search for the maximum radiation dosage points on the smaller mesh. Then, we divide the small area centred on the maximum radiation dosage points into smaller spaces and search for the point with the maximum radiation dosage on the grid, and repeat the operation until the radiation source point is on the partitioned grid node (as shown in Figure 2) and until the radiation source is ascertained. We combine the hierarchical search with the random search algorithm to optimise the random search. In random search experiments, to simplify the problem and highlight the essence of random search, we can consider placing radioactive sources on the grid nodes. Next, we introduce the related algorithms needed in the random search of singlepoint radiation sources.
Before the establishment of the model, it is necessary to mesh the area where the radiation source is located. A coarse grid is used for searching in a large area, but when approaching the target, higher accuracy is required. It is necessary to search on a finer grid to ensure accuracy. Consider the random search of a singlepoint radiation source. First, we divide the coarse grids, and then subdivide the networks in a small range. This operation can effectively improve the speed of the algorithm and save storage space. The means to effectively establish the grid is the most basic problem of search; the details of the grid algorithm are given below (Algorithm 1).
Note: Mesh the area
When searching for radioactive sources randomly, select the searching path to find the sources on the grid. At the grid node, all dosages of surrounding points compare to the initial point, and the random initial point will move towards the location where the radiation dosage is highest in the surrounding four points. This cycle's steps continue until the point no longer moves and then we find out the maximum point to be the radioactive source. It is necessary to design a neighbour set search algorithm. See the neighbour set finding algorithm (Algorithm 2) for details.
If
If
If
If
If
If
If
If
When searching for radiation sources, we usually compare the radiation dosage, and then determine the direction of walking, and search for radiation sources through continuous analysing. Therefore, we design a random search model by imitating the human search experience. The main idea of the random search algorithm is to select the maximum radiation dosage point around the node using the neighbour set comparison method. Compare the dosage of the four points around a random point with the point and move to the maximum point. Repeat until they are no longer more significant than other points. See Random Search Algorithms 3 for details.
The following are some explanations for the random search algorithm. In step 3 of the random search algorithm, if the random point is the maximum point in the set of adjacent points, then output the random point. If the point (
When searching for radiation sources, some problems often arise, such as misreading of grid data, large low concentration area in the radiation area and moving in small radiation dosage area for a long time by random searching, which result in low searching efficiency and even the problem of not searching for radiation sources. Therefore, to solve the issues that may arise in actual searching, it is necessary to optimise the random searching algorithm and increase the robustness of the random search model.
In the actual search and collection of grid node data, the collected data may be misread, resulting in deviation of search results. For enhancing the adaptability of the random search algorithm, it is necessary to increase the ability of the algorithm to resist data misreading interference. Generally, there are only two cases when data misreading occurs: The first is that of the searched maximum point being misinterpreted and enlarged. The second is that the maximum amount of the searched data is misread and reduced. If the dosage of the maximum point searched randomly is misread and reduced, it will not affect the fact that the node is the maximum point or even the radiation source point. However, if the dosage of the searched maximum point is misread and enlarged, then the point can still be the exact maximum point, or may not be the maximum point. To determine whether the point is the maximum point of the real field, we need some judgement rules with which to distinguish. It is necessary to optimise the Random Search Algorithm.
Let
If
If
Here are some explanations for the radiation dosage maximum correction algorithm. The size of
In the real search, the situation in the search area is complex. Near a certain maximum value, data misreading results in high and low fluctuation of data, which may lead to a dead cycle when the radiation dosage correction algorithm is called. Dead cycles may occur when calling the Radiation Dosage Maximum Correction Algorithm. In order to avoid a dead cycle, it is necessary to limit each point to a maximum of one search. If the node is found again, it is necessary to forcibly output the starting point of the first radiation dosage correction. This considers implementation of the maximum correction algorithm in a large field grid area
When there is only one grid distance between the misread point data and the radiation source, the dosage correction algorithm may fail. However, it can make up for the defect of correction. By further subdividing the mesh by hierarchical search, it becomes possible to judge the real radiation source with maximum characteristics by the fine mesh nodes in a small area, so that the accuracy and efficiency of the whole algorithm are not affected.
When searching the maximum point, we locate the random search point in a flat area far from the radiation source, where the radiation dosage changes little or even does not change. Then, a random search in this low radiation dosage area will cause unnecessary waste and might even stop the search. It is also possible that the small fluctuation of data in the flat area will search for a local maximum point, which will interfere with random search. Therefore, for data in a low radiation dosage area, it is necessary to jump out quickly and reduce unnecessary random search time. Thus, we design an algorithm for quick jump out of regions with similar radiation dosage. The details are shown in Algorithm 5 for a quick escape from the low radiation area.
Here are some explanations for a fast escape algorithm in a low radiation dosage area. In step 4 of the algorithm, the radiation dosage threshold of the fourth step of the algorithm can be the average of the measured dosage of the whole area.
Sometimes the low radiation dosage area is too large. When using the escape strategy of Algorithm 5, it is possible to enter the small radiation dosage area again when randomly selecting a new search starting point outside the escape circle, which makes the escape effect unsatisfactory. At this time, the method of increasing the random moving step can result in quickly escaping the low radiation dosage area. The specific idea of quick escape pseudoarea is as if random. The starting point of the search is in the little radiation dosage area, and we choose the horizontal direction of the left (right) to calculate the radiation dosage differences one by one. If the radiation dosage difference exceeds a certain threshold, the random point will move to that point so that the cycle can quickly escape the pseudoarea. See local fast escape (Algorithm 6).
From
Following are some explanations for the fast escape algorithm in local low radiation region.
In Algorithm 6,  stands for an integral function in step 5. A quick escape pseudozone algorithm breaks the general random search rule. Instead of using the strategy of moving a grid step by comparing radiation dosages at one time, it first compares radiation dosages until the radiation dosages of nodes exceed a threshold standard at the starting point, and then moves a giant step to achieve fast search. By analogy, it can quickly find the maximum location.
For the threshold selection of a significant step, we determine that the criteria can be according to the specific situation of the actual radiation dosage field. If a giant step is required, the threshold criteria can be adjusted appropriately. Otherwise, the threshold criteria can be lower.
Sometimes the biggest point found in this area is not a true source of radiation. For example, data misreading causes a certain amount of data to be particularly large. Therefore, it is necessary to identify and distinguish between the real and erroneous sources of radiation. Therefore, after searching for the maximum point, it is necessary to determine whether the point is a radioactive source.
Assuming that the activity of point source is A (Bq), the formula for calculating the specific kinetic energy release rate in the air at R (m) is as follows [18]:
How to judge whether the point is a radioactive source? What is the basis? Nuclear physics tells us that the radiation dosage at any point in a radioactive particle field is inversely proportional to the square of the distance of the radioactive source [18]. Therefore, the radiation field of the radioactive source is characterised by a sharp concave function. The concavity and convexity of the radiation dosage curve near the radiation source are judged by the trend of radiation dosage variation near the radiation source. See Algorithm 7.
If {
The design of random search algorithm for singlepoint radioactive source is mainly configured for improving the running speed, and to reduce the storage space and enhance the adaptability of the random search algorithm; its function is mainly to improve the running speed and reduce the storage space. Moreover, the random search algorithm is closer to the blind search in the actual hunt, and it is a useful, simple and easytooperate random search algorithm. We add the algorithm of maximum point correction and low radiation dosage escape to the random search algorithm, and obtain the optimised random search algorithm. See random search optimisation algorithm (Algorithm 8) for details.
If
For output point (
If
If
Here are some explanations of the random search optimisation algorithm. When the search area is large, the hierarchical search algorithm and the random search algorithm can be used together to improve the search efficiency. See Algorithm 8 for details.
We have improved the search algorithm. If data misreading occurs, the maximum point correction algorithm should be called to help identify whether the maximum point of the search is a radioactive source. See maximum dosage correction algorithm (Algorithm 7) for details.
For preventing the search from wasting the search time in the flat area of the radiation dosage, it is also possible to call Algorithm 6 to escape the low radiation dosage low area and start a new random search in a flat area where the radiation dosage is small. The above explanations all indicate that random search has strong adaptability and is of an intelligent type. Different algorithms can make the random search algorithm more effective in solving problems encountered in the actual search.
When the number of radioactive sources in the search area is small, or even only one, performing a global random search in the search area may waste the search time. In such cases, it becomes necessary to pursue a strategy of finding a radioactive source to remove one. To reduce unnecessary search, improve the speed and efficiency of the random search model and design a search algorithm with less time and space complexity, we have developed the random search strategy and adopted a hierarchical search strategy. First, we divide the coarse mesh in the searching area. According to the random search principle of singlepoint radioactive sources, we search several nodes with large radiation dosage and then take the four small grids of small area
Assume that a random search area
Secondlevel Random Search is used to determine the maximum point (
Here are some explanations for the hierarchical random search. Under the fine mesh, sometimes, the dosage changes a little or might even remain without change. It may cause the neighbour set comparison algorithm to fail, and since the distance between adjacent grid nodes is large and the radiation dosage data of the grid node changes significantly, the random search of the neighbouring radiation dosage comparison is facilitated. The hierarchical random search uses fewer grid nodes; so, the time complexity and space complexity are small.
There are two kinds of algorithm complexity, time complexity and space complexity. Time complexity refers to the computational effort required to execute an algorithm. Space complexity refers to the memory space needed to implement the algorithm. (The complexity of the algorithm is reflected in the number of resources required by the computer when running the algorithm. The most important among computer resources are the time and space [i.e. storage] resources; so, the complexity is divided into time and space complexity).
The random search algorithm is simple and feasible, and it has strong operability and conforms to the actual search situation. Now we analyse the complexity, running time and storage space of the algorithm. We randomly generate the position coordinate of the search starting point (
First, we calculte the summation in the horizontal direction, as follows:
Similarly, we calculate the summation in the vertical direction, as follows:
Therefore, the complexity of running time average
If
The following analysis analyses the storage space required to search for humans when randomly searching for radioactive sources. The spatial complexity of a program is the amount of memory needed to run a program. In addition to the instructions, constants, variables and input data used by the storage space and the storage itself, a program needs some work units that operate on the data and an auxiliary space that stores some information required for realworld calculations. The storage space required for program execution includes the following two parts.
Fixed space: The size of this part of space is independent of the number of data input/output and the dosage. It mainly includes the space occupied by code space and data space. This part is a static space.
Variable space: This part of the space mainly includes the space allocated dynamically, and the space required for the recursive stack.
In the process of imitating the real artificial random search, it is not necessary to collect the radiation dosage of all the mesh nodes in the search area. At the randomly generated search starting point, we only compare the four points around the point and the dosage. As the subsequent nodes continue to be compared, we only compare the dosage of three points around the point. The step is continued iteratively until we are able to ascertain the location of the radiation dosage maximum point. Simulated random search requires the realtime dynamic call of radiation dosage at three points around the point; so, the simulated random search process does not require static storage space, only dynamic storage space. The average space for a single source is three times the time complexity; so, the space complexity
Let
If there are two point sources in the search area, the radiation field will become slightly more complex. If the two radiation sources are far away, whether the fields of the two radiation sources interfere with each other minimally or not at all, the two different radiation sources can be searched by searching one radiation source and clearing one radiation source by random searching twice. If two radiation sources are relatively close, or there are obstacles between them, the mapping results in the overlapping area of radiation measurement of two radiation sources. The overlapping regions may produce local maximum (as shown in Figure 5), but the local maximum are not radiation sources. Therefore, it is necessary to discuss the overlapping regions and how they are judged in the search of two radiation sources.
Since the radiation fields of the two radiation sources will overlap with each other, the radiation measurement value in the overlapping area will increase correspondingly, resulting in the formation of a new local maximum in the area. However, the real radiation source will not appear in the overlapping area; so, it is necessary to avoid entering the overlapping area or eliminating the random searched maximum radiation metering points after entering the field. Therefore, we define the overlapping space of the radiation source as a pseudoarea, and then give the judgement method of eliminating the maximum radiation metering points in the overlapping area.
Radiometric values near the radiation source will increase dramatically and rapidly, but in the overlapping area, the increase of radiation measurement values is relatively flat and slow, i.e. the trend of the change rate of radiation measurement is characterised by monotonous decrease. We use this feature to define the pseudozone of the overlapping area of the radiation measurement field.
If there are two radiation sources in the searched rectangular region
Let
Since the search only knows the radiation dosage of grid nodes, the radiation dosage function in the search area is unknown. In practice, it is difficult to get the analytic expression of radiation dosage function; so, it is impossible to use Definition 1 to judge the pseudoregion directly. In this paper, we construct the differential expression to approximate the pseudoregion by using the radiation dosage of the collected discrete grid nodes. The we proceed to give the definition of pseudoregion.
Assume that the set of points consisting of grid nodes in overlapping area
If there are multiple radiation sources in the area, the radiation dosage fields of numerous radiation sources will be superimposed to form multiple pseudoregions. The means to avoid entering the pseudoregions or to jump out of the pseudoregions after entering the pseudoregions must be considered in a random search. To enable random search to adopt multiple radiation sources, we design the pseudopoint discrimination algorithm, as follows. We form all the pseudopoints in the pseudoarea corresponding to the radiation dosage field.
We completely cover the searching region
Then
For the search of multiple sources, we first study the search of two points and then extend the search method to the search of numerous sources after clarifying the search method of two sources. For the case that there are two radiation sources in the area, we use a singlepoint random search strategy to search for two different radiation sources through discovery of two points.
First, in the search area of two radiation sources, we discuss the probability of success of searching one radiation source at a time. For the problem of searching two radioactive sources, since we generate the starting point randomly, any point of the starting point in the searching area is equally possible. According to the geometric probability model, the probability that a random search can find a radioactive source at a time is
We discuss the relationship between the size of pseudoregion and the probability of search success as follows (Table 2).
The influence of the size of pseudoregion in the whole search area on the probability of search success
 
 
 
 


>=0.5  >=0.8  >=0.95  1 
From the above analysis, we infer that as the proportion of pseudoregions in the whole search area decreases, the probability of successful search for radioactive sources will gradually increase. In the actual radiation field, although there is a case of radiation field superposition, in general 
We discuss two search strategies for radioactive sources. When there are two radiation sources in the radiation field, the random search will become more complex, and the pseudoregion generated by the interference superposition of radiation fields will make the search difficult. To search two radiation sources effectively, we need to design several kinds of search algorithms, among which multiple independent random searches is a simple and effective search method.
For the random search of twopoint radiation sources, we adopt multiple random searches strategies, and each random search is independent of every other. Let
Consider the following two situations when Ktime searching for two radiation sources:
If one source is found, the source will be removed, and the remaining one will be searched for as a single radiation source; then the probability of searching for the second radiation source for the first time is 1; so, the first radiation source will be searched only after analysing the
Independent repeated
The idea of random searching for twopoint radioactive source is to search independently and randomly for
The advantage of adding a pseudopoint judgement algorithm to the singlepoint random search algorithm is that the interference of the first radiation source can be used to eliminate the second radiation source after removing the first radiation source.
In the actual search, we mainly consider the number of independent random searches to find two different sources. When the number of searches is unknown, it is necessary to analyse the probability of successful searches. For two radiation source search problems, the probability that one radiation source can be successfully searched by random search at a time is
If the independent random search is carried out
Let event
The probability of event
The probability results of event
As can be seen from the above table, if
In
For ∀
Probability relationship between two times of independent random search and search success
0.95  0.0475  2.375^{*}10^{−3}  <=3.71^{*}10^{−11} 
The number of random searches and the size of pseudoregions
 
 
 
 
 

>=0.5  >=0.8  >=0.95  1  
<=3  <=2.25  <=2.053  2 
From the above analysis, we infer that with the decrease of the proportion of pseudoarea to the whole search area, the probability of successful search increases gradually. In the actual radiation field, although there is a case of radiation field superposition, in general 
Time complexity analysis of multiple independent random search is performed as follows:
If ∀
Therefore, the time complexity
When two radioactive sources search randomly for many times independently, the radiation dosage of nodes in the area will change correspondingly after removal of one radioactive source and update the grid node data. In the independent repeated search, we only label the searched radiation source and continue the random search. See the marked random search algorithm (Algorithm 12) for details.
If (
If (
It shows that the marked random search can remove the radioactive source without searching for a radioactive source. Therefore, we add a link facilitating comparison of the newly searched radioactive source with the first searched radioactive source to make up for the obstacle of searching caused by not clearing the radioactive source. Of course, we carry out the second random search after finding the first source. We enable the second search to approach the second source quickly and avoid repeating the search for the first source or entering the pseudoarea. Further, we generate the random starting point far away from the first source, so that the possibility of the second search for the second source is increased. This is shown by the algorithm of marked random search optimisation (Algorithm 13).
Compared with the general marker random search algorithm, the marker random search optimisation algorithm enhances the design of jumping out of the search to get the first radiation region. To avoid the second search entering the range of the first source or the low dosage area, let Order d_{1}, d_{2}, d_{3}, d_{4} be the distance from the first radiation source to the four edges of the search area. We choose the shortest d_{1} as the radius to draw a circle, and intersect the other three lines at points A, B and C. We select the maximum radioactive dosage point of the three points as the starting point of the random search to continue searching for the second one.
In the random search experiment, we hope to find the probability of success of two different radioactive sources under the given search times. In the marker independent subrandom search, we discuss two cases separately: Finding the two various radioactive sources only after conducting the possible search several times; and finding the two different radioactive sources after the search times.
In this case, the search for the first source is once in the first
Similarly, we regard K searches as independent events to calculate the probability of successful searches.
If
The relationship between the increase in random search times and the probability of successful search
0.405  0.081  3.645 × 10^{−8} 
When
If we find one source in the
We analyse the complexity of markup independent random search. We find the two different sources only after searching; so, the search time complexity
In the actual search process, we are concerned about not only how many independent searches are needed to search for the radioactive source but also the probability of searching for two different radioactive sources in the case of
The probability of finding two different radiation sources by calculating
Next, we discuss the probability of finding two different sources by searching
When
The relationship between the increase in random search times and the probability of successful search
0.182  0.588  0.959 
It can be seen from the table that as long as the number of searches is increased, the probability of successful search will increase, thereby ensuring that all radioactive sources can be searched for probabilistic analysis. We perform the random search twice to ascertain the probability of two different sources.
Order 
We then find partial derivatives for
This means that there is no stagnation point in the area
From the above probability analysis, it can be seen that the probability of two random searches to get two different radioactive sources generally does not exceed 0.5. In the same way, the probability that independent random search K times can search k different radioactive sources is relatively small; so, in order to improve the probability of search success, we need to optimise the random search. The related optimisation strategies will be shown in future research.
The time complexity of finding two different radioactive sources by independent random search
There are multiple point sources of radiation coexisting in the search area. A search algorithm similar to that of two sources can be used to search for radioactive sources. At the same time, a clear search algorithm can be used to remove multiple point sources of radiation one by one until all the node radiation dosage is 0. When the clearing random search finds the radioactive source, the corresponding radiometric dosage of the radioactive source at each grid node needs to be subtracted when removing the source, and the computational complexity will increase.
The main idea of the random search optimisation algorithm for multiple radioactive sources is that the random search algorithm for a single point is used to find a radioactive source and clear one.
In order to avoid entering into the pseudoarea formed by the superposition of radiation fields of the remaining radiation sources, and quickly enter into the strong radiation field of the next radiation source, we take the previous radioactive source point as the centre, and select the shortest distance
If point (
If (
If the search result (
If the search result (
When the first radiation source is found, the remaining radiation field may be composed of multiple point radiation sources coexisting. Therefore, when a new radiation source is found, it is necessary to repeatedly compare with the already searched radiation source to confirm whether it is a new radiation source.
If a pseudoradiation source point is randomly found, it is necessary to jump out of this area and generate a new search starting point outside the pseudoarea. It is possible that the minimum distance
Next, a search starting point is generated, each starting point is tested for four times at most and the remaining
From the above discussion, it can be seen that the probability model method of random search is applicable to the search of singlepoint, doublepoint or even multiple point radioactive sources. In addition, the time and space complexity of the search for radioactive sources in all three cases are linear levels
Random search simulation experiment of a singlepoint radiation source uses data of grid 1000 × 1000. The radiation source is located at (500,500), and the radiation dosage is 5.3 × 10^{11} units (Figure 10).
In order to verify the simulation experiment effect of the random search algorithm, the algorithm is executed 100 times; the coordinate position of the searched radiation source is (500,500) in the networks, 100 starting points are randomly selected and every search result is the same. The following table gives the coordinates of 100 the random search starting points (
Random search experiment of singlepoint radiation source maximum correction uses 1000 × 1000 as grid. The radiation source is located at (500,500) and dosage is 5.3 × 10^{11} units. Figure 14 indicates the flow of the singlepoint random optimisation search.
In order to verify the maximum correction algorithm, we carry out the random search at any point, and add a circle of 0 data around the grid data. In order to verify the recognition function of the random search for data with larger misreading, especially near the peak point (20,20), the radiation dosage was increased to 5.5 × 10^{11} unit. The point is verified by the maximum correction algorithm as the maximum point with the characteristics of a radiation source. Therefore, the design of the maximum correction algorithm meets practical needs.
In order to verify that the complexity of random search time is linear, we designed an experiment involving random search times and verified the correctness of the conclusion by the number of steps of random search of the radioactive source.
The verification test uses 1000 × 1000 grid data. The source is located at location (500,500), and the radiation measurement of the source is 5.3 × 10^{11} units. A search starting point is randomly generated on the search area plane, and starting from the starting point, the random search is stopped until the movement is stopped, the number of steps of the search movement of the record is output, and the average number of random search moving steps in the experiment is obtained. The experiment was done in three groups. Each group randomly generated 1000, 10000, 100000 search starting points; the number of steps for each random search movement was recorded and from the results thus obtained, Table 7 was generated.
Random search to find the average search step for the source
1  1000  1000  494 
2  10000  10000  497 
3  100000  100000  499 
Based on the experiment, it can be ascertained that the number of moving steps of the average search success is stable at about 496 steps. According to the time complexity formula for random search based on theory,
The singlepoint source grading search experiment uses data grid 1051 × 1051, where the source is located at (529,529), and the radiation measurement of the source is A unit. In order to verify the effectiveness of the hierarchical search experiment, the grid is divided into three levels of 1051 × 1051 in the experiment:
The first level of coarse mesh is divided into grids in the radiation field area. A data line is collected for each interval of 30 cm for
In the secondlevel meshing, at the grid node of the radioactive source obtained by the firstlevel random search, the field character centred on the point is selected, and a data line is collected for each interval of 10 cm for
In the thirdlevel meshing, at the grid node of the radioactive source obtained by the secondlevel random search, the field character centred on the point is selected, and a data line is collected for each interval of 1 cm in the field character lattice to obtain a local part. The grid of the secondlevel search represents the coordinates of the radio source point, which are (529,529), and obtained by random search under the secondlevel coarse grid.
In this paper, we searched outofcontrol point radioactive sources, adopted a humanlike search experience, established random search models for singlepoint and multipoint radioactive sources and optimised the model for the problems that may be encountered in actual search. Theory proves that this random search algorithm has linear running speed and linear storage space. Simulation experiments prove that the algorithm is feasible.
This article discusses the search of a singlepoint radioactive source in detail and theoretically proves that the search algorithm is linear. Theoretical proof is not available for twopoint and multipoint at present; so, future research is expected to discuss in detail the role of multipoint radioactive sources as a theoretical part of the search. At the same time, the method will be based on a twodimensional plane search strategy. In the future, a random search strategy in threedimensional space can be considered, and the search strategy will be more accurate.
The relationship between the increase in random search times and the probability of successful search
0.182  0.588  0.959 
Radiation dosage at the grid nodes
…  

…  
…  
⋮  ⋮  ⋮  ⋮  ⋮  
… 
Probability relationship between two times of independent random search and search success
0.95  0.0475  2.375^{*}10^{−3}  <=3.71^{*}10^{−11} 
The number of random searches and the size of pseudoregions
 
 
 
 
 

>=0.5  >=0.8  >=0.95  1  
<=3  <=2.25  <=2.053  2 
The influence of the size of pseudoregion in the whole search area on the probability of search success
 
 
 
 


>=0.5  >=0.8  >=0.95  1 
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Law of interest rate changes in financial markets based on the differential equation model of liquidity Basalt fibre continuous reinforcement composite pavement reinforcement design based on finite element model Industrial transfer and regional economy coordination based on multiple regression model Satisfactory consistency judgement and inconsistency adjustment of linguistic judgement matrix Spatial–temporal graph neural network based on node attention A contrastive study on the production of double vowels in Mandarin Research of cascade averaging control in hydraulic equilibrium regulation of heating pipe network Mathematical analysis of civil litigation and empirical research of corporate governance Health monitoring of Bridges based on multifractal theory Health status diagnosis of the bridges based on multifractal detrend fluctuation analysis Performance evaluation of college laboratories based on fusion of decision tree and BP neural network Application and risk assessment of the 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Xinheyuan, China Response Model of Teachers’ Psychological Education in Colleges and Universities Based on Nonlinear Finite Element Equations Institutional investor company social responsibility report and company performance Mathematical analysis of China's birth rate and research on the urgency of deepening the reform of art education Precision Machining Technology of Jewelry on CNC Machine Tool Based on Mathematical Modeling Firstprinciples calculations of magnetic and mechanical properties of Febased nanocrystalline alloy Fe_{80}Si_{10}Nb_{6}B_{2}Cu_{2} Computer Vision Communication Technology in Mathematical Modeling The Effect of Children’s Innovative Education Courses Based on Fractional Differential Equations Fractional Differential Equations in the Standard Construction Model of the Educational Application of the Internet of Things Optimization in Mathematics Modeling and Processing of New Type Silicate Glass Ceramics Has the belt and road initiative boosted the resident 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Statistics Equations Data Forecasting of AirConditioning Load in Large Shopping Malls Based on Multiple Nonlinear Regression System dynamics model of output of ball mill Optimisation of Modelling of Finite Element Differential Equations with Modern Art Design Theory Mathematical function data model analysis and synthesis system based on shortterm human movement Sensitivity Analysis of the Waterproof Performance of Elastic Rubber Gasket in Shield Tunnel Human gait modelling and tracking based on motion functionalisation Analysis and synthesis of function data of human movement The Control Relationship Between the Enterprise's Electrical Equipment and Mechanical Equipment Based on Graph Theory Financial Accounting Measurement Model Based on Numerical Analysis of Rigid Normal Differential Equation and Rigid Functional Equation Mathematical Modeling and Forecasting of Economic Variables Based on Linear Regression Statistics Design of Morlet wavelet neural network to solve the nonlinear influenza disease system Nonlinear Differential Equations in Crossborder Ecommerce Controlling Return Rate Differential equation model of financial market stability based on Internet big data 3D Mathematical Modeling Technology in Visualized Aerobics Dance Rehearsal System Children’s cognitive function and mental health based on finite element nonlinear mathematical model Motion about equilibrium points in the JupiterEuropa system with oblateness Fractional Differential Equations in Electronic Information Models Badminton players’ trajectory under numerical calculation method BIM Engineering Management Oriented to Curve Equation Model Optimal preview repetitive control for impulsefree continuoustime descriptor systems Development of main functional modules for MVB and its application in rail transit Study on the impact of forest fire prevention policy on the health of forest resources Mathematical Method to Construct the Linear Programming of Football Training The Size of Children's 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