Algorithm 937

Choi, Sou‐Cheng T.; Saunders, Michael A.

doi:10.1145/2527267

Cited by 24 publications

(7 citation statements)

References 21 publications

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“…The weak point of this type of penumbra function is the discrete nature in describing the penumbra curve. On the contrary, the LSRE is based on an inverse square root function [12, 13] and is a continuous equation that can fit the entire dose profile along the x axis (Figure 9).…”

Section: Discussionmentioning

confidence: 99%

An Empirical Model for Describing the Small Field Penumbra in Radiation Therapy

Tang

Jen

2019

BioMed Research International

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Purpose We developed a mathematic empirical model for describing the small field penumbra in order to analyze the potential dose perturbation caused by overlapping field to avoid the dose calculation errors in linear accelerator-based radiosurgery. Materials and methods A ball phantom was fabricated for measuring penumbra at 4 different gantry angles in the coplanar plane. A least square root estimation (LSRE) Model was created to fit the measured penumbra dose profile and to predict the penumbra dose profile at any gantry angles. The Sum of Squared Errors (SSE) was used for finding the parameters n and t for the best fitting of the LSRE model. Geometric and mathematical methods were used to derive the model parameters. Results The results showed that the larger the gantry angle of the field, the more the expansion of the penumbra dose profile. The least square root estimation model for describing small field penumbra is as follows: PenumbraDš=T·1/2·1−š/n+š2+t where PenumbraD(š) denotes the dose profile D(š) at the penumbra region, T is the penumbra height (usually in scalar 100), n is the parameter for curvature, š = x − W d/2 (x and š are the values in cm on x-axis), and t is the radiation transmission of the collimator. Geometric analysis establishes the correlation between the penetration depth of the exposure and its effect on the penumbra region in ball phantom. The penumbra caused by an exposure at any arbitrary angles can be geometrically derived by using a one-variable quadratic equation. Conclusion The dose distribution in penumbra region of small field can be created by the LSRE model and the potential overdosage or underdosage owing to overlapping field perturbation can be estimated.

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Section: Discussionmentioning

confidence: 99%

An Empirical Model for Describing the Small Field Penumbra in Radiation Therapy

Tang

Jen

2019

BioMed Research International

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“…We initialize the RBM with a set of random parameters W, generally using α = 4, and search for the ground state of the Hamiltonian (2) by minimizing the energy expectation value of the RBM state (1) using the stochastic reconfiguration (SR) method to update the RBM wavefunction [1,26,27,29,42]. In Table I, we compare the results we obtain from exact diagonalization (ED) to those from the RBM ansatz for various system sizes.…”

Section: Restricted Boltzmann Machine Statesmentioning

confidence: 99%

“…for dw. We adopt the MINRES-QLP algorithm [42], which is an iterative linear solver based on the Lanczos method. Lanczos tridiagonalization requires the calculation of the Krylov space which involves matrix-vector multiplications of the form S w v, where v is a generic vector with 2N w entries.…”

Section: A2 Efficient Calculation Of Step In Parameter Spacementioning

confidence: 99%

Chiral topological phases from artificial neural networks

2018

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Motivated by recent progress in applying techniques from the field of artificial neural networks (ANNs) to quantum many-body physics, we investigate as to what extent the flexibility of ANNs can be used to efficiently study systems that host chiral topological phases such as fractional quantum Hall (FQH) phases. With benchmark examples, we demonstrate that training ANNs of restricted Boltzmann machine type in the framework of variational Monte Carlo can numerically solve FQH problems to good approximation. Furthermore, we show by explicit construction how n-body correlations can be kept at an exact level with ANN wavefunctions exhibiting polynomial scaling with power n in system size. Using this construction, we analytically represent the paradigmatic Laughlin wavefunction as an ANN state. arXiv:1710.04713v2 [cond-mat.str-el]

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“…The MINRES method is then a special variant of GMRES, which makes use of the tridiagonal structure. The table below, adopted from [3], gives a comparison of computational complexities of MINRES and GMRES without preconditioning for solution of a linear system Ax = b with a symmetric m × m matrix A in terms of memory storage required by working vectors in the solvers and the number of floating-point operations. By t P we denote the work needed for evaluating a i (V ).…”

Section: Algorithm 1 Preconditioned Gmres Without Restartsmentioning

confidence: 99%

“…MINRES requires symmetric positive definite preconditioners such as in [12]. In our MINRES simulations, although not reported in Section 5 in details, we use the preconditioned MINRES-QLP method from [3].…”

Section: Storagementioning

confidence: 99%

Preconditioned Continuation Model Predictive Control

Knyazev¹,

Fujii²,

Malyshev³

2015

2015 Proceedings of the Conference on Control and Its Applications

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Model predictive control (MPC) anticipates future events to take appropriate control actions. Nonlinear MPC (NMPC) describes systems with nonlinear models and/or constraints. A Continuation/GMRES Method for NMPC, suggested by T. Ohtsuka in 2004, uses the GMRES iterative algorithm to solve a forward difference approximation Ax = b of the Continuation NMPC (CNMPC) equations on every time step. The coefficient matrix A of the linear system is often illconditioned, resulting in poor GMRES convergence, slowing down the on-line computation of the control by CNMPC, and reducing control quality. We adopt CNMPC for challenging minimum-time problems, and improve performance by introducing efficient preconditioning, utilizing parallel computing, and substituting MINRES for GMRES. IntroductionModel predictive control (MPC) is used in many applications to control complex dynamical systems. Examples of such systems include production lines, car engines, robots, other numerically controlled machining, and power generators. The MPC is based on optimization of the operation of the system over a future finite time-horizon, subject to constraints, and implementing the control only over the current time step.Model predictive controllers rely on dynamic models of the process, most often linear empirical models, in which case the MPC is linear. Nonlinear MPC (NMPC), which describes systems with nonlinear models and constraints, is often more realistic, compared to the linear MPC, but computationally more difficult. Similar to the linear MPC, the NMPC requires solving optimal control problems on a finite prediction horizon, generally not convex, which poses computational challenges.

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Algorithm 937

Cited by 24 publications

References 21 publications

An Empirical Model for Describing the Small Field Penumbra in Radiation Therapy

An Empirical Model for Describing the Small Field Penumbra in Radiation Therapy

Chiral topological phases from artificial neural networks

Preconditioned Continuation Model Predictive Control

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