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- W1416158096 abstract "This thesis analyzes the problem of localizing sensors in wireless ad hoc networks. Wireless ad hoc networks can be found in several practical applications, among which the most known is the climatic monitoring. Every sensor in such a network has the ability to register specific factors. These can be forwarded (if necessary) to other sensors within the network until reaching a so-called anchorpoint. Due to this special form, these networks are applied for example to predict forest fires, earthquakes or climatic changes. The Sensor Network Localization Problem (SNLP) is a NP-hard problem and is solved by the approximate solution of a non-linear least squares problem in practice. Leaning on the work of Ye et al. [5] a semidefinite program (SDP) formulation is proposed: begin{equation*} min { Cbullet X mid mathcal{A} (X) =b,; Xin mathcal{S}^n_{+} }, end{equation*} where the variable is $Xin mathcal{S}^n$, the space of real symmetric matrices of dimension $n$. The vector $binR^m$, the linear mapping $mathcal{A}:mathcal{S}^nto R^m$ and the matrix $Cin mathcal{S}^n$ are given. The cone $mathcal{S}^n_+$ is the set of all real symmetric positive semidefinite matrices of dimension $n$, i.e. $mathcal{S}^n_+ := { Min mathcal{S}^n mid Ssucceq 0}$. The notation $Cbullet X$ denotes the inner product of the symmetric matrices $C$ and $X$ and is given by $$ Cbullet X = tr(C, X) = underset{i=1}{overset{n}{sum}}, underset{j=1}{overset{n}{sum}} , C_{ij}, X_{ij}. $$ This SDP formulation is a relaxation of the original problem (in the model proposed in this thesis, the exact solution of the original problem has to fulfil additionally a rank 1 condition). The great advantage of employing SDPs is their good numerical treatment via interior point methods. For special cases (so-called uniquely'' localizable networks) is shown, that the presented SDP provides the exact solution for the SNLP. The interesting problem variant, which is common in practice, incorporates measurement errors due to interferences between the sensors in a network. This yields a disturbed localization problem. In this context a local error bound for the disturbed solution is presented. To improve the solution of the relaxed problem, several heuristics are presented and compared numerically. The so-called Curvature Descent method achieves global and local quadratic convergence." @default.
- W1416158096 created "2016-06-24" @default.
- W1416158096 creator A5085304860 @default.
- W1416158096 date "2009-01-01" @default.
- W1416158096 modified "2023-10-02" @default.
- W1416158096 title "A Rank 1 SDP Approach for the Sensor Network Localization Problem" @default.
- W1416158096 hasPublicationYear "2009" @default.
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