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As wireless sensor networks (WSNs) are increasingly being deployed in some important applications, it becomes imperative that we consider application requirements in in-network processes. We intend to use a WSN to aid information querying and navigation within a dynamic and real-time environment. We propose a novel method that relies on the heat diffusion equation to finish the navigation process conveniently and easily. From the perspective of theoretical analysis, our proposed work holds the lower constraint condition. We use multiple scales to reach the goal of accurate navigation. We present a multi-scale gradient descent method to satisfy users’ requirements in WSNs. Formula derivations and simulations show that the method is accurately and efficiently able to solve typical sensor network configuration information navigation problems. Simultaneously, the structure of heat diffusion equation allows more flexibility and adaptability in searching algorithm designs.

After the past decade of active research and field trials, WSNs have started penetrating into many areas of science, engineering, and our daily life. They are also envisioned to be an integral part of cyber-physical systems such as those for alternative energy, transportation, and healthcare. In supporting mission-critical, real-time, closed loop sensing and control, the new WSNs represent a significant departure from traditional WSNs which usually focus on open-loop sensing. The stringent application requirements of cyber-physical system (CPS) make it necessary to rethink WSN design.

Earlier applications [

The novelty of our method is to establish a practical and convenient information potential field that helps clients to discover the local sub-optimal value points such as free parking spots. Our method can guide the users to reach the local area with higher density of information field instead of one extreme value point. The area of higher density of information field will not attract any conflict because this method navigates the car to the higher density of information area that holds several vacancies for cars. It could avoid lower level competition for parking spots. The role of the heat conduction equation is the time when the resource is updated to meet the physical means in the form of pairs of the information on the degree of each node to be updated. Specially, we imitate an information diffusion process by using a heat equation with specified boundary values. Effectively the information potential sets up a smooth ‘hill area’ (information potential field hill) with several local ‘peaks’; almost all nodes on this area are likely to have several ascending neighbors, and thus greater capacity to reach the different definitions. This trick of smoothing out the discrete hop counts by a heat equation can also be applied in other settings where smooth potential fields of information flow need to be maintained. Finally, we note that others [

The rest of the paper is organized as follows. In Section 2, we give the fundamental mathematical formulations of information diffusion and the heat equation. Then, a surface fitting model based on a variation method is introduced and smoothing of a large-scale (global aspect) information field is established. In Section 3, fundamental mathematical formulations of the Laplace equation are described and a continuous small-scale (local aspect) information field is established. In Section 4, several numerical examples are provided. The conclusions are given in the last section.

Information resources are changing dynamically from the time the client receives the guidance signal to the time he/she accesses the target node. It is a process where that client accesses the target node after he receives the guidance information. However, in the whole network, information resources are continuously changing dynamically. Therefore, we can make use of a multi-resolution gradient to accomplish this navigation process; that is to say, we can handle different conditions of navigated targets based upon different levels. First, a customer is required to reach the better area with the higher information level, which means to finish the inaccurate navigation or fuzzy guidance. This configuration navigator is a quick and fuzzy process that cannot guarantee the client will reach the specific parking lot, but it can help the customer reach the certain area that holds some extent information field. Then, it could finish the further guidance to get to the assured point. For the fuzzy configuration navigator, we need to build a large scale and smooth information gradient field (global information gradient) based upon the inaccurate gradient descent method.

The initial information gradient field _{0} (

In this equation, _{0} is a positive constant coefficient that represents the information diffusion rate and ∇^{2} denotes the Laplacian operator. As shown in the second term of

To get the large-scale smooth information field, the gradient descent method is adopted to navigate the car by the information resource. For the above Section 2.1, it is necessary for us to obtain the more accurate grid and fit the information field layer. Thereby, it is natural to introduce energy functional.

Mesh refinement is performed directly. The four vertexes of each grid hold the specific value, and then it can decide a quadrilateral space. Based on the position inside the defined triangle within the quadrilateral space, the information gradient of the new adding vertex can be calculated based on triangular coordinates.

However, after finishing the refinement, the information gradient surface is still a piecewise triangular plane

In the above functional, ^{*}

It is needed to search ^{*}

Set

By the fuzzy configuration navigation information of the large-scale information field, the client reached the higher information field level, and then, the customer needs to be navigated accurately within the small-scale (local) information field. Finally, the client could reach a certain node and finish the demand. To finish the final configuration navigation object, smooth information is built up and it is accurate on each local node. The result of Laplace equation boundary value problem is helpful for this goal:

It is acknowledged that the value of any internal point can be determined uniquely if the values of the information field function on the boundaries of the non-convex area are given. In addition, based on the maximum principle, certain extreme points must lay on the boundary. We are enlightened by the previous work of Gao (see [

Consequently, we can establish a similar Laplace problem in the current node where the local area lies so that we get the small-scale information potential field _{0} (

In this section, by integrating the above mentioned, the process of the multi-scale gradient descent method will be illustrated as follows:

According to the current user’s requirement, the information boundary of the local area of the target node can be set.

Based on the current sensor networks, in terms of the Section 2, the large scale information potential field can be obtained based on the heat equation. Simultaneously, the certain target with the satisfied information level can be reached via the gradient descent method.

From the current user’s node, in terms of the Section 3, the local small scale information potential field can be obtained, thereby; we can reach the node that makes the client satisfactorily depend on the gradient descent method.

The following two examples are given to verify the validity of this method:

Example 1. Single customer requirement. We randomly generate a user node in the network, and start navigation to lead users to reach the destination node which satisfies the information level boundary. The multi-scale gradient descent process of shown in

Example 2. Several users’ requirement. Some clients are randomly generated in the network and competition is considered. The large scale gradients descent process and the final results of navigation are shown in

We estimate information potential fields by simulation in the following aspects: the construction and maintenance costs of the information potential field, robustness to network, the balance of query qualities versus gradient precision cost, as well as the applications of the potential fields in Section 3.

We simulate wireless transmission using ordinary mode in the same radio models of TOSSIM. In the ordinary mode, all nodes within the transmission range can communicate fluently under ideal conditions. We feed our node locations in the TOSSIM radio model and obtain connectivity and link quality for each pair of nodes. At any time slot we can set a percentage of randomly selected links to be not available, throughout all the experiments. The maintenance of the information field is on-demand. We establish the gradient on a neighbor discovery protocol. Gradient maintenance and routing are in the networking layer, and can be integrated with existing protocols that maintain a neighbor list for each node [

Suppose we have a perturbed grid network of size 20 × 20 nodes and consider establishing smooth potential fields with different conditions. First, we generate some random nets which have different topologies and different nodes numbers but have the same density. The numbers of nodes are between 200 and 245. The numbers of iterations for different nets are shown in

What’s more, our method can coordinate the requirements of different users and the competition collisions can be reduced or even eliminated. As shown in

If there is no competition, the lower left moving direction has a closer distance to the users so that it has smaller cost, and

Since

As

Several users’ requirement. Twenty users are randomly generated in the network and competition is considered. The large scale gradient descent process and the final results of navigation are shown in

In this paper, we have proposed a brand new heat diffusion equation to finish the navigation process conveniently and easily. Partitioned scales are used to reach the goal of the accurate navigation. Some theoretical tools such as heat diffusion equation, PDE variational method, and gradient descent methods are adopted in our method. Two smooth potential fields of sensor network are helpful to satisfy the customers’ requirement. Multi-scale gradient descent methods, examples and solid mathematical principles show that the method is accurate and efficient, able to solve typical sensor network configuration information navigation problems. The nonlinear PDE structure allows more flexibility and adaptability in searching algorithm designs.

Compared with the former works depending on the discrete information field, our method ensures a local information field large enough to include appropriate multiple targets and the competition conflicts can be resolved simultaneously. The information level of each node can be updated with a satisfactory physical methodology when resources are dynamically changing. By developing an algebraic structure of heat diffusion equation, we can combine different potentials to enable far greater path diversity and thus provide better performance than it is possible with only one-fold discrete field guidance. The simulation results show that although with much relaxed assumptions, our approach achieves comparable performance with significantly reduced competition collisions. We will further explore this direction in the future.

The authors would like to thank the anonymous reviewers for their constructive feedback and valuable input.

Evolution of information field under the controlling of variable coefficients diffusion equation.

Evolution of potential field under the controlling of variable coefficients diffusion equation.

Results of Laplace problems in several non-convex fields.

Multi-scale gradient descent process of single user.

Relation about density and the numbers of iterations, nodes.

Topology and histogram.

Navigation results of different methods.

Multi-scale gradient descent of one user.

Navigation results of several users.

Navigation results.

−1.5261 | (−1, −1) | (−4, 3) | 54.4739 |

76.4739 | (−6, −9) | (−6, −9) | 76.4739 |

−17.5261 | (−7, −6) | (10, −7) | 61.4739 |

−17.5261 | (−7, 9) | (−9, 10) | 46.4739 |

−17.5261 | (−6, −7) | (−5, −7) | 82.4739 |

−17.5261 | (−4, 7) | (−4, 9) | 70.4739 |

−17.5261 | (−4, 1) | (−4, 2) | 33.4739 |

−17.5261 | (−8, 10) | (−8, 9) | 80.4739 |

70.4739 | (−3, −9) | (−3, −9) | 70.4739 |

35.4739 | (−4, −1) | (−4, −1) | 35.4739 |

−17.5261 | (−5, −8) | (−5, −7) | 82.4739 |

−17.5261 | (−7, 10) | (−9, 10) | 46.4739 |

49.4739 | (−8, −10) | (−8, −10) | 49.4739 |

73.4739 | (−5, 6) | (−5, 6) | 73.4739 |

−17.5261 | (−7, 7) | (−6, 4) | 37.4739 |

−17.5261 | (−6, 8) | (−4, 9) | 70.4739 |

76.4739 | (−6, −9) | (−6, −9) | 76.4739 |

−17.5261 | (−2, −2) | (−5, −3) | 56.4739 |

−17.5261 | (−1, −5) | (−0, −6) | 78.4739 |

58.4739 | (−6, 6) | (−6, 6) | 58.4739 |