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Regularization Techniques for MR Image Reconstruction
Published in Joseph Suresh Paul, Raji Susan Mathew, Regularized Image Reconstruction in Parallel MRI with MATLAB®, 2019
Joseph Suresh Paul, Raji Susan Mathew
When the system of equations is underdetermined, it has no unique solution. In this case, the prior information that the signal has a sparse representation can be used to optimize the problem. A matrix or a vector is said to be s-sparse when it has only s non-zero coefficients. This means that the actual information content in that signal is less in the sparse representation and requires only a few samples to uniquely represent it in an ideal condition. This is the main idea behind the CS where the number of samples required to represent a signal is viewed as a function of the actual information content rather than the highest frequency assumption in the Nyquist sampling criterion. For an under-determined system, a unique solution cannot be obtained directly. However, consider that x is s-sparse and we know the locations of the non-zero entries (Ω) of x. Now, if s < M, the problem becomes over-determined and the solution can be obtained using: () xΩ=(AΩTAΩ)−1AΩTb,
Metabolism
Published in Markus W. Covert, Fundamentals of Systems Biology, 2017
In linear algebra, the solution to an equation with the form of Equation 9.64 is called the null space of Sb (FBA terminology calls this space the solution space). The operable word here is “space”, illustrated in Figure 9.16a. Usually, the system is underdetermined, meaning that there are many possible solutions to the equations. Assuming that all of the fluxes in our sample system can only take on positive values, the positive orthant (gray) in Figure 9.16 holds all possible combinations of flux values that can be attained by the system, both dynamic and at steady state. The null space of Sb is shown in black (but only schematically, for illustrative purposes; in reality, the solution space is just the straight line vN, in = vN2X = vX, out). This space holds all of the solutions for which d[M]/dt = 0. A particular ODE solution, based on determination of the necessary parameter values as well as an initial condition, will follow a trajectory. In the case illustrated in Figure 9.16b, the trajectory begins at the origin and then travels outside the FBA solution space until the system reaches a steady state, which necessarily falls inside the solution space.
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.
Energy reduction by power loss minimisation through wheel torque allocation in electric vehicles: a simulation-based approach
Published in Vehicle System Dynamics, 2022
Juliette Torinsson, Mats Jonasson, Derong Yang, Bengt Jacobson
Apart from afore-mentioned challenges, control related challenges are introduced with electric vehicles that have not before been present in petrol or diesel vehicles. An increased number of electric motors contribute to over-actuation in ground vehicles which was previously mainly seen in aircraft and marine vessels [1]. Over-actuation means that the motion request from the driver can be attained in several different ways using different actuators. If a vehicle with four electric motors, one for each wheel, is driven and the electric motors could be used for both propulsion and regenerative braking, there is an infinite number of torque distributions that would fulfil the forward motion requested by the driver. If a steering request is added, even more solutions appear as yaw motion can be achieved either through propulsive or braking torques or steering of the wheels. The system of equations of motion is underdetermined, no unique solution exist. This underdetermined problem can be solved by control allocation.
Trajectory-tracking of 6-RSS Stewart-Gough manipulator by feedback-linearization control using a novel inverse dynamic model based on the force distribution algorithm
Published in Mathematical and Computer Modelling of Dynamical Systems, 2020
Zafer Mahmoud, Mohammad Reza Arvan, Vahab Nekoukar, Mohammad Rezaei
Consider the system with , and , where . This equation system is underdetermined because the number of equations is less than the number of the variables . Hence, it has many solutions (if any). However, the minimum-norm solution of this system can be obtained by factoring the Matrix into the product of an orthogonal matrix and un upper triangular matrix : and applying the following algorithm [30]: Solve Set
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: