China
Africa
Explore chapters and articles related to this topic
Compressive Sensing and Its Application in Wireless Sensor Networks
Published in Fei Hu, Qi Hao, Intelligent Sensor Networks, 2012
Jae-Gun Choi, Sang-Jun Park, Heung-No Lee
where the measurement vector is y∈Rm, with m < n, and the measurement matrix A∈Rm×n :. Our goal is to recover x from the measurement vector y. We note that Equation 15.3 is an underdetermined system because it has fewer equations than unknowns; thus, it does not have a unique solution in general. However, the theory of CS asserts that, if the vector x is sufficiently sparse, an underdetermined system is guaranteed with high probability to have a unique solution.
Chapter 11: Miscellaneous Topics Used for Engineering Problems
Published in Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene, Modern Engineering Mathematics, 2017
Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene
An underdetermined system of linear equations has more unknowns than equations and generally has an infinite number of solutions. In order to choose a solution to such a system, one must impose extra constraints or conditions (such as smoothness) as appropriate. In compressed sensing, one adds the constraint of sparsity, allowing only solutions which have a small number of non-zero coefficients. Not all underdetermined systems of linear equations have sparse solutions. However, if there is a unique sparse solution to the underdetermined system, then the compressed sensing framework allows the recovery of that solution.
Sensitivity Coefficient Evaluation of an Accelerator-Driven System Using ROM-Lasso Method
Published in Nuclear Science and Engineering, 2022
Ryota Katano, Akio Yamamoto, Tomohiro Endo
In the estimation of the sensitivity coefficients, Eq. (2) is a simultaneous linear equation whose solution is the sensitivity coefficient vector . It is worth noting that becomes an N × N full-rank diagonal matrix in the direct method, and it is easy to solve Eq. (2). When M is smaller than N, the number of equations is fewer than the unknowns; thus, the solution of Eq. (2) is not mathematically unique. For such an underdetermined system, certain restrictions are required to uniquely determine the solution. As a restriction, penalized linear regression determines the solution as follows:
Incoherent source localization in random acoustic waveguides
Published in Waves in Random and Complex Media, 2020
L. Borcea, E. Karasmani, C. Tsogka
We are interested in ranges beyond the equipartition distance, , and in this case, we have where is the jth component of the eigenvector . Thus, we can write Equation (12) in the following form where is the matrix with entries and have been defined in Equation (9). Remark that the matrix Q is the identity when the array spans the whole depth of the waveguide, due to the orthogonality of the modes. We observe that it is impossible to recover from Equation (13). Nevertheless, we can extract the weighted average θ from our processed data and write Equation (13) as Considering now M different sub-bands with central circular frequencies , number of propagating modes and , we can obtain from Equation (14) the following system This is an underdetermined system, since it has more unknowns than equations, . Note that as we add sub-bands, we increase the number of equations and therefore we can hope for an improvement in our estimate of the unknowns.