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Projection Radiography and Computed Tomography
Published in Bethe A. Scalettar, James R. Abney, Cyan Cowap, Introductory Biomedical Imaging, 2022
Bethe A. Scalettar, James R. Abney, Cyan Cowap
For specified Rk, these equations can, in principle, be solved for the ai (HW 11.16). Unfortunately, this approach becomes impractical for the very large data sets acquired in CT because hundreds of thousands of equations are involved. In addition, an overdetermined system typically will not have a solution that satisfies all equations simultaneously (i.e., the equations are inconsistent). For simplicity, we assume consistency here.
Linear Algebra
Published in Erchin Serpedin, Thomas Chen, Dinesh Rajan, Mathematical Foundations for SIGNAL PROCESSING, COMMUNICATIONS, AND NETWORKING, 2012
Fatemeh Hamidi Sepehr, Erchin Serpedin
It is quite common for underdetermined systems of equations to present an infinite number of solutions, while for overdetermined systems to lack any exact solution. In general, the system of linear equations just introduced in (3.57) is expressed in the compact matrix form: () Ax=b,
Linear Algebra
Published in Brian Vick, Applied Engineering Mathematics, 2020
In addition to square systems of equations containing the same number of equations as unknowns (m = n), it is possible to have overdetermined systems containing more equations than unknowns (m > n) and underdetermined systems with more unknowns than equations (n > m). Examples of these are shown in the panels of Figure 5.12.
Some Green’s functions for steady-state heat conduction in anisotropic plane media and their application to thermoelastic boundary element analysis
Published in Journal of Thermal Stresses, 2023
Chyanbin Hwu, Meng-Ling Hsieh, Cheng-Lin Huang
In this paper, when the BIE of heat conduction is singular, we add an additional constraint to the system of equations to obtain a unique solution. This additional constraint is the conservation of heat flux, i.e., where is heat flux normal to the boundary. With (5.9), one additional equation is added to the system (5.7), which now has more equations than unknowns. To solve this overdetermined system of equations, linear least square method is applied. First, we move all the unknown values to the left-hand side and rewrite (5.7) to where v is a vector of unknown nodal values, and p is a vector of known values. With linear least square, the unknown vector is solved by [31]
Multi-Hinge Failure Mechanisms of Masonry Arches Subject to Self-Weight as Derived from Minimum Thickness Analysis
Published in International Journal of Architectural Heritage, 2022
Orsolya Gáspár, István Sajtos, András A. Sipos
Note that y0 and γ/H are uniquely determined if these are set. Furthermore, it can be quickly derived that both D1 and D2 are zero for both cases, deductible from constraint (1) and the consideration that c(0) = y0 by definition. As highlighted above, any constraint within this geometrical approach is linked to the arrangement of the hinges. Now let us assume that there was one more internal hinge. Observe that one more internal hinge would result in two more constraints versus one more unknown. This would lead to an overdetermined system of equations, which is generally not solvable. We denote below any arch with an r reference line as described in Section 2 and corresponding to a 5- or 6 -hinge mechanism at its limit state, a regular arch.
GPU parameter tuning for tall and skinny dense linear least squares problems
Published in Optimization Methods and Software, 2020
Benjamin Sauk, Nikolaos Ploskas, Nikolaos Sahinidis
This paper addresses the problem of using DFO and SO to determine optimal tuning parameters for solving linear least squares problems (LLSPs) with GPUs. Dense LLSPs are solved in a wide range of fields, such as curve fitting, modelling of noisy data, signal processing, parameter estimation, and machine learning, including best subset selection and the lasso [17,25]. LLSPs arise when solving an overdetermined system of equations Ax = b, where , , and . In modelling an input-output system, the A matrix contains information about input variables. One common element in LLSPs is that A is usually tall and skinny (TS), i.e. . The vector x denotes the LLSP solution, and b is the vector of output measurements. In an overdetermined system (m>n), there may not exist an exact solution. The LLSP finds x that minimizes the difference between b and Ax or, more formally, . It can be shown that the optimal LLSP solution satisfies the normal equation.