388,894 research outputs found
A method for direct calculation of quadratic turning points
For a given one-parameter nonlinear system, the simplest bifurcation is the quadratic turning bifurcation where the Jacobian matrix becomes singular due to rank deficiency 1. To overcome the difficulty in solving the quadratic turning point caused by the singularity of the Jacobian matrix, the conventional Newton method can be applied to the so-called Moore-Spence determination system to solve for the quadratic turning point. However, the Moore-Spence system has much higher dimensions and causes much more complexity in factorisation of the extended Jacobian matrix. In the paper, by introducing an auxiliary variable and an auxiliary linear equation into Newton iterations in solving the Moore-Spence determination system, a matrix reduction technique can be worked out to solve the Moore-Spence extended equations much more efficiently. The high dimensions of the matrix can thus be reduced and the complexity involved in matrix factorisation can be reduced noticeably. The technique is proposed for general nonlinear systems. Formulation is derived for applying this technique to solving quadratic turning points, or say nose points, on load-flow solution curves of power systems. Computer tests on the IEEE 30-busbar system and a 2416-busbar East China power system are reported to show the effectiveness of the suggested technique.published_or_final_versio
A toolbox to solve coupled systems of differential and difference equations
We present algorithms to solve coupled systems of linear differential
equations, arising in the calculation of massive Feynman diagrams with local
operator insertions at 3-loop order, which do {\it not} request special choices
of bases. Here we assume that the desired solution has a power series
representation and we seek for the coefficients in closed form. In particular,
if the coefficients depend on a small parameter \ep (the dimensional
parameter), we assume that the coefficients themselves can be expanded in
formal Laurent series w.r.t.\ \ep and we try to compute the first terms in
closed form. More precisely, we have a decision algorithm which solves the
following problem: if the terms can be represented by an indefinite nested
hypergeometric sum expression (covering as special cases the harmonic sums,
cyclotomic sums, generalized harmonic sums or nested binomial sums), then we
can calculate them. If the algorithm fails, we obtain a proof that the terms
cannot be represented by the class of indefinite nested hypergeometric sum
expressions. Internally, this problem is reduced by holonomic closure
properties to solving a coupled system of linear difference equations. The
underlying method in this setting relies on decoupling algorithms, difference
ring algorithms and recurrence solving. We demonstrate by a concrete example
how this algorithm can be applied with the new Mathematica package
\texttt{SolveCoupledSystem} which is based on the packages \texttt{Sigma},
\texttt{HarmonicSums} and \texttt{OreSys}. In all applications the
representation in -space is obtained as an iterated integral representation
over general alphabets, generalizing Poincar\'{e} iterated integrals
One step hybrid block methods with generalised off-step points for solving directly higher order ordinary differential equations
Real life problems particularly in sciences and engineering can be expressed in differential
equations in order to analyse and understand the physical phenomena. These differential equations involve rates of change of one or more independent variables. Initial value problems of higher order ordinary differential equations are conventionally
solved by first converting them into their equivalent systems of first order ordinary
differential equations. Appropriate existing numerical methods will then be employed to solve the resulting equations. However, this approach will enlarge the number of equations. Consequently, the computational complexity will increase and thus may jeopardise the accuracy of the solution. In order to overcome these setbacks, direct methods were employed. Nevertheless, most of these methods approximate numerical solutions at one point at a time. Therefore, block methods were then introduced with the aim of approximating numerical solutions at many points simultaneously. Subsequently,
hybrid block methods were introduced to overcome the zero-stability barrier occurred in the block methods. However, the existing one step hybrid block methods only focus on the specific off-step point(s). Hence, this study proposed new one step
hybrid block methods with generalised off-step point(s) for solving higher order ordinary
differential equations. In developing these methods, a power series was used as an approximate solution to the problems of ordinary differential equations of order g. The power series was interpolated at g points while its highest derivative was collocated at all points in the selected interval. The properties of the new methods such as order, error constant, zero-stability, consistency, convergence and region of absolute stability were also investigated. Several initial value problems of higher order ordinary
differential equations were then solved using the new developed methods. The numerical results revealed that the new methods produced more accurate solutions than the existing methods when solving the same problems. Hence, the new methods are viable alternatives for solving initial value problems of higher order ordinary differential
equations directly
QR-Factorization Algorithm for Computed Tomography (CT): Comparison With FDK and Conjugate Gradient (CG) Algorithms
[EN] Even though QR-factorization of the system matrix for tomographic devices has been already used for medical imaging, to date, no satisfactory solution has been found for solving large linear systems, such as those used in computed tomography (CT) (in the order of 106 equations). In CT, the Feldkamp, Davis, and Kress back projection algorithm (FDK) and iterative methods like conjugate gradient (CG) are the standard methods used for image reconstruction. As the image reconstruction problem can be modeled by a large linear system of equations, QR-factorization of the system matrix could be used to solve this system. Current advances in computer science enable the use of direct methods for solving such a large linear system. The QR-factorization is a numerically stable direct method for solving linear systems of equations, which is beginning to emerge as an alternative to traditional methods, bringing together the best from traditional methods. QR-factorization was chosen because the core of the algorithm, from the computational cost point of view, is precalculated and stored only once for a given CT system, and from then on, each image reconstruction only involves a backward substitution process and the product of a vector by a matrix. Image quality assessment was performed comparing contrast to noise ratio and noise power spectrum; performances regarding sharpness were evaluated by the reconstruction of small structures using data measured from a small animal 3-D CT. Comparisons of QR-factorization with FDK and CG methods show that QR-factorization is able to reconstruct more detailed images for a fixed voxel size.This work was supported by the Spanish Government under Grant TEC2016-79884-C2 and Grant RTC-2016-5186-1.Rodríguez-Álvarez, M.; Sánchez, F.; Soriano Asensi, A.; Moliner Martínez, L.; Sánchez Góez, S.; Benlloch Baviera, JM. (2018). QR-Factorization Algorithm for Computed Tomography (CT): Comparison With FDK and Conjugate Gradient (CG) Algorithms. IEEE Transactions on Radiation and Plasma Medical Sciences. 2(5):459-469. https://doi.org/10.1109/TRPMS.2018.2843803S4594692
DOSnet as a Non-Black-Box PDE Solver: When Deep Learning Meets Operator Splitting
Deep neural networks (DNNs) recently emerged as a promising tool for
analyzing and solving complex differential equations arising in science and
engineering applications. Alternative to traditional numerical schemes,
learning-based solvers utilize the representation power of DNNs to approximate
the input-output relations in an automated manner. However, the lack of
physics-in-the-loop often makes it difficult to construct a neural network
solver that simultaneously achieves high accuracy, low computational burden,
and interpretability. In this work, focusing on a class of evolutionary PDEs
characterized by having decomposable operators, we show that the classical
``operator splitting'' numerical scheme of solving these equations can be
exploited to design neural network architectures. This gives rise to a
learning-based PDE solver, which we name Deep Operator-Splitting Network
(DOSnet). Such non-black-box network design is constructed from the physical
rules and operators governing the underlying dynamics contains learnable
parameters, and is thus more flexible than the standard operator splitting
scheme. Once trained, it enables the fast solution of the same type of PDEs. To
validate the special structure inside DOSnet, we take the linear PDEs as the
benchmark and give the mathematical explanation for the weight behavior.
Furthermore, to demonstrate the advantages of our new AI-enhanced PDE solver,
we train and validate it on several types of operator-decomposable differential
equations. We also apply DOSnet to nonlinear Schr\"odinger equations (NLSE)
which have important applications in the signal processing for modern optical
fiber transmission systems, and experimental results show that our model has
better accuracy and lower computational complexity than numerical schemes and
the baseline DNNs
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