Volume constrained 2-phase segmentation method utilizing a linear system solver based on the best uniform polynomial approximation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si44.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>

Numerical Linear Algebra App

et al. 2018

Summary In this paper, we consider efficient algorithms for solving the algebraic equation Aαboldu=boldf, 0<α<1, where scriptA is a properly scaled symmetric and positive definite matrix obtained from finite difference or finite element approximations of second‐order elliptic problems in Rd, d=1,2,3. This solution is then written as boldu=Aβ−αboldF with boldF=A−βboldf with β positive integer. The approximate solution method we propose and study is based on the best uniform rational approximation of the function tβ−α for 0

Section: Our Approach and Contributionsmentioning

confidence: 90%

Section: Modified Remez Algorithm For Computing the Buramentioning

confidence: 99%

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Optimal solvers for linear systems with fractional powers of sparse SPD matrices

Harizanov

Lazarov

Numerical Linear Algebra App

et al. 2018

“…In the literature, 20,21 several different numerical quadrature formulas are proposed to approximate integral (11). In our previous studies, 24,25,29 we have used the quadrature formula based on a -dependent graded mesh.…”

Section: Quadrature Methods (M3)mentioning

confidence: 99%

“…Here, y is an argument of a special nonlinear coordinates transformation. The parameter k > 0 controls the accuracy of the approximation of integral (11), that is, the method's transformation error, and the number of discrete 3D elliptic subproblems that should to be solved: M = m 1 + m 2 + 1. It has been proven 21 that this sinc quadrature converges exponentially.…”

Section: Quadrature Methods (M3)mentioning

confidence: 99%

Scalability analysis of different parallel solvers for 3D fractional power diffusion problems

Čiegis

Starikovičius

Concurrency and Computation

et al. 2019

Self Cite

Summary In this paper, we develop and investigate the parallel numerical algorithms for three different state‐of‐the‐art numerical methods for solving the non‐local problems described by fractional powers of elliptic operators. These methods transform the non‐local problem into some local differential problems of elliptic or parabolic type. A two‐level parallelization approach is applied to construct the efficient parallel algorithms using the domain decomposition and master‐slave methods, to deal with the increase in computational complexity. We show and compare the serial and parallel solution times that are required to achieve similar accuracy of the solution using different algorithms. Results of extensive convergence tests are presented solving a three‐dimensional test problem with known decrease of the solution's convergence rate depending on the fractional power coefficient. We analyze and discuss the non‐trivial question, which parallel algorithm is recommended to achieve certain accuracy for the given fractional power coefficient.

Parallel solvers for fractional power diffusion problems

Čiegis

Starikovičius

Concurrency and Computation

et al. 2017

Self Cite

Summary Mathematical models with fractional‐order differential operators are computationally expensive due to the non‐local nature of these operators. In this work, we construct and investigate parallel solvers for problems described by fractional powers of elliptic operators, like fractional diffusion. Three state‐of‐the‐art approaches are used to transform the non‐local fractional‐order differential problem into local partial differential equation problems formulated in a space of higher dimension. Numerical schemes and parallel algorithms are developed for all three approaches. The resulting parallel algorithms have very different properties. We investigate the weak and strong scalability of the developed parallel algorithms and compare their parallel performance.