Starting Vector Research Articles

Backward error analysis of the Lanczos bidiagonalization with reorthogonalization

The k-step Lanczos bidiagonalization reduces a matrix A∈Rm×n into a bidiagonal form Bk∈R(k+1)×k while generating two orthonormal matrices Uk+1∈Rm×(k+1) and Vk+1∈Rn×(k+1). However, any practical implementation of the algorithm suffers from loss of orthogonality of Uk+1 and Vk+1 due to the presence of rounding errors, and several reorthogonalization strategies are proposed to maintain some level of orthogonality. In this paper, we make a backward error analysis of the Lanczos bidiagonalization with reorthogonalization (LBRO) by writing various reorthogonalization strategies in a general form. Our results show that the computed Bk by the k-step LBRO of A with starting vector b is the exact one generated by the k-step Lanczos bidiagonalization of A+E with starting vector b+δb (denoted by LB(A+E,b+δb)), where the 2-norm of perturbation vector/matrix δb and E depend on the roundoff unit and orthogonality levels of Uk+1 and Vk+1. The results also show that the 2-norm of Uk+1−Ūk+1 and Vk+1−V̄k+1 are controlled by the orthogonality levels of Uk+1 and Vk+1, respectively, where Ūk+1 and V̄k+1 are the two orthonormal matrices generated by the k-step LB(A+E,b+δb) in exact arithmetic. Thus, the k-step LBRO is mixed forward–backward stable as long as the orthogonality of Uk+1 and Vk+1 are good enough. We use this result to investigate the backward stability of LBRO based SVD computation algorithm and LSQR algorithm. Numerical experiments confirm our results.

Identifiability of equilibrium constants for receptors with two to five binding sites.

Ligand-gated ion channels (LGICs) are regularly oligomers containing between two and five binding sites for ligands. Neither in homomeric nor heteromeric LGICs the activation process evoked by the ligand binding is fully understood. Here, we show on theoretical grounds that for LGICs with two to five binding sites, the cooperativity upon channel activation can be determined in considerable detail. The main requirements for our strategy are a defined number of binding sites in a channel, which can be achieved by concatenation, a systematic mutation of all binding sites and a global fit of all concentration-activation relationships (CARs) with corresponding intimately coupled Markovian state models. We take advantage of translating these state models to cubes with dimensions 2, 3, 4, and 5. We show that the maximum possible number of CARs for these LGICs specify all 7, 13, 23, and 41 independent model parameters, respectively, which directly provide all equilibrium constants within the respective schemes. Moreover, a fit that uses stochastically varied scaled unitary start vectors enables the determination of all parameters, without any bias imposed by specific start vectors. A comparison of the outcome of the analyses for the models with 2 to 5 binding sites showed that the identifiability of the parameters is best for a case with 5 binding sites and 41 parameters. Our strategy can be used to analyze experimental data of other LGICs and may be applicable to voltage-gated ion channels and metabotropic receptors.

A Study on Implicit Education in Curriculum Civic Education of Universities Based on Differential Evolutionary Algorithm

Abstract In this paper, firstly, the selection of individuals is made according to the selection probability size, and in order to reduce the greediness of selection, the algorithm only selects the starting vectors of base vectors and difference vectors for random sorting selection, and the rest of vectors are still selected according to multi-objective sorting. Secondly, according to the teaching content of Civic Education, the multimodal corpus content needs to take into account the characteristics of topicality, comprehensiveness, and appropriateness, and the differential evolution algorithm is proposed to construct the Civic Education multimodal corpus. The results show that self-efficacy, implicit education, and daily education can be obtained at the level of 0.01, which indicates a significant positive effect on Civic Education. Self-efficacy, and other are all below the level of 0.01, which indicates a significant negative influence on Civic Education in colleges and universities. The combined described implicit education and other education cooperate with each other in order to achieve the development of modern ideological and political education.

The infinite Lanczos method for symmetric nonlinear eigenvalue problems

A new iterative method for solving large scale symmetric nonlinear eigenvalue problems is presented. We firstly derive an infinite dimensional symmetric linearization of the nonlinear eigenvalue problem, then we apply the indefinite Lanczos method to this specific linearization, resulting in a short-term recurrence. We show how, under specific assumption on the starting vector, this method can be carried out in finite arithmetic and how the exploitation of the problem structure leads to improvements in terms of computation time. The eigenpair approximations are extracted with the nonlinear Rayleigh-Ritz procedure combined with a specific choice of the projection space. We illustrate how this extraction technique resolves the instability issues that may occur due to the loss of orthogonality in many standard Lanczos-type methods.

AE and EA versions of X -robustness for interval circulant matrices in max–min algebra

Max–min algebra is defined as a linearly ordered set with two binary operations. Classical addition and multiplication are replaced by maximum and minimum, respectively. A square matrix is called robust, if all its orbits converge to an eigenvector. In practice, matrix inputs and start vector inputs are rather contained in some intervals than exact values. Considering matrices and vectors with interval coefficients is therefore of great practical importance. In this paper, we study a special type of the robustness called the -robustness, the case when a starting vector is limited by a lower bound vector and an upper bound vector . If we consider some components of the interval starting vector with the existential quantifier and others with the general quantifier, two new types of -robustness arise, called -robustness and -robustness. Using a similar quantification of interval matrix elements, we obtain AE/EA--robustness and AE/EA--robustness. A complete characterization of AE/EA--robustness and AE/EA--robustness for a special type of interval matrices, so-called interval circulant matrices, is presented.

A study of defect-based error estimates for the Krylov approximation of \u03c6-functions

Prior recent work, devoted to the study of polynomial Krylov techniques for the approximation of the action of the matrix exponential etAv, is extended to the case of associated φ-functions (which occur within the class of exponential integrators). In particular, a posteriori error bounds and estimates, based on the notion of the defect (residual) of the Krylov approximation are considered. Computable error bounds and estimates are discussed and analyzed. This includes a new error bound which favorably compares to existing error bounds in specific cases. The accuracy of various error bounds is characterized in relation to corresponding Ritz values of A. Ritz values yield properties of the spectrum of A (specific properties are known a priori, e.g., for Hermitian or skew-Hermitian matrices) in relation to the actual starting vector v and can be computed. This gives theoretical results together with criteria to quantify the achieved accuracy on the fly. For other existing error estimates, the reliability and performance are studied by similar techniques. Effects of finite precision (floating point arithmetic) are also taken into account.

Development and Application of a Complete Active Space Spin-Orbit Configuration Interaction Program Designed for Single Molecule Magnets.

We present a spin‐orbit configuration interaction program which has been tailored for the description of the magnetic properties of polynuclear metal complexes with partially filled d‐ and f‐shells. The spin‐orbit operators are directly included in the configuration interaction program based on Slater‐determinants. The lowest states are obtained by a Block‐Davidson‐type diagonalisation. The usage of localised active orbitals enables the construction of start vectors from tensor products of single‐center wave functions that already include spin‐orbit interaction. This allows for an analysis of the role and the interplay of the different metal centres. Furthermore, in case of weak coupling of the metal centres these tensor products are already close to the final wave functions ensuring fast convergence. In combination with a two‐layer hybrid parallelisation, this makes the program highly efficient. Based on the spin‐orbit coupled wave functions, magnetic D‐tensors, g‐tensors and temperature‐dependent susceptibilities can be calculated. The applicability and performance of the program is shown exemplarily on a trinuclear transition metal (CoIIVIICoII) complex.

Global optimization of default phases for parallel transmit coils for ultra-high-field cardiac MRI.

The development of novel multiple-element transmit-receive arrays is an essential factor for improving B1+ field homogeneity in cardiac MRI at ultra-high magnetic field strength (B0 > = 7.0T). One of the key steps in the design and fine-tuning of such arrays during the development process is finding the default driving phases for individual coil elements providing the best possible homogeneity of the combined B1+-field that is achievable without (or before) subject-specific B1+-adjustment in the scanner. This task is often solved by time-consuming (brute-force) or by limited efficiency optimization methods. In this work, we propose a robust technique to find phase vectors providing optimization of the B1-homogeneity in the default setup of multiple-element transceiver arrays. The key point of the described method is the pre-selection of starting vectors for the iterative solver-based search to maximize the probability of finding a global extremum for a cost function optimizing the homogeneity of a shaped B1+-field. This strategy allows for (i) drastic reduction of the computation time in comparison to a brute-force method and (ii) finding phase vectors providing a combined B1+-field with homogeneity characteristics superior to the one provided by the random-multi-start optimization approach. The method was efficiently used for optimizing the default phase settings in the in-house-built 8Tx/16Rx arrays designed for cMRI in pigs at 7T.

AE and EA robustness of interval circulant matrices in max-product algebra

Max-product algebra is defined as a linearly ordered set with two binary operations. Classical addition and multiplication are replaced by maximum and multiplication, respectively. A square matrix A is robust, if for each starting vector x, the orbit stabilizes. If this property holds only for starting vectors from an interval vector X, we say that A is X-robust. We shall deal with the matrices of special type: circulant and the corresponding interval version, so-called interval circulant matrices. The interval approach in this paper is applied in combination with for-all–exists quantification of the values – not all of the considered interval matrix entries have the same importance: whereby, for some more important data, all values of the interval must be taken into account and for some less important data it is sufficient to be considered for at least one value of the interval. In this manner, we define the EA and AE robustness and X-robustness of interval circulant matrices over max-product algebra. Polynomial algorithms for checking these types of robustness and X-robustness are presented.

Feedback-Dubins-RRT Recovery Path Planning of UUV in an Underwater Obstacle Environment

In this paper, a UUV (Unmanned Underwater Vehicle) recovery path planning method of a known starting vector and end vector is studied. The local structure diagram is designed depending on the distance and orientation information about the obstacles. According to the local structure diagram, a Rapidly exploring Random Tree (RRT) method with feedback is used to generate a 3D Dubins path that approaches the target area gradually, and the environmental characteristic of UUV reaching a specific target area is discussed. The simulation results demonstrate that this method can effectively reduce the calculation time and the amount of data storage required for planning. Meanwhile, the smooth spatial path generated can be used further to improve the feasibility of the practical application of UUV.

Admissible and Attainable Convergence Behavior of Block Arnoldi and GMRES

It is well-established that any non-increasing convergence curve is possible for GMRES and a family of pairs $(A,b)$ can be constructed for which GMRES exhibits a given convergence curve with $A$ having arbitrary spectrum. No analog of this result has been established for block GMRES, wherein multiple right-hand sides are considered. By reframing the problem as a single linear system over a ring of square matrices, we develop convergence results for block Arnoldi and block GMRES. In particular, we show what convergence behavior is admissible for block GMRES and how the matrices and right-hand sides producing any admissible behavior can be constructed. Moreover, we show that the convergence of the block Arnoldi method for eigenvalue approximation can be almost fully independent of the convergence of block GMRES for the same coefficient matrix and the same starting vectors.

A Fast Multilevel SVPWM Method Based on the Imaginary Coordinate With Direct Control of Redundant Vectors or Zero Sequence Components

This paper for the first time comprehensively presents a multilevel space vector pulse width modulation (SVPWM) method based on the imaginary coordinate system. This method is probably one of the most efficient methods in fast multilevel SVPWMs. It avoids trigonometric operation, look-up tables and only requires simple logic judgement and arithmetic calculation. A distinct advantage of this method over other coordinate-transformation-based methods is that it does not need to transform from the imaginary coordinate system back to the a-b-c (or α-β) coordinate in order to select the redundant vectors using look-up tables. The redundant vectors or zero-sequence components are directly worked out in the proposed method, e.g., for balancing dc-link capacitor voltages, which avoids identifying the starting vector, vector rotation direction (clockwise or anti-clockwise), vector sequence, etc. as used in conventional methods. This paper entails each step to implement this SVPWM method, which can be replicated for multilevel converters with any number of voltage levels. The presented method has been used and validated by various converters, including a three-level and a five-level neutral-point-clamped converter. The archive value of this paper is that it demonstrates a milestone of fast multilevel SVPWM based on coordinate transformation.

A cross-product approach for low-rank approximations of large matrices

In this paper we present a new type of restarted Krylov method for calculating low-rank approximations of large matrices. In contrast to former Krylov methods, our approach does not apply the Lanczos algorithm, or Lanczos bidiagonalization. This simplifies the basic iteration and allows the introduction of several innovations.An advantage of the proposed method is that it requires a minimal amount of computer storage. The Krylov matrix which is built at each iteration is using an improved starting vector and a new three-term recurrence relation. These modifications lead to fast rate of convergence. Numerical experiments illustrate the usefulness of the proposed approach.

The Use of Time Series to Improve the Prediction Reliability of the Statistical Characteristics of a Controlled Parameter

Based on the analysis of time series compiled from the values of a controlled parameter, we propose options for constructing vectors (vector-neighbors), close to the starting vector. It is shown that the use of the coordinates of the constructed vector-neighbors in conjunction with the coordinates of the starting vector allows one to increase the sample sizes for determining the statistical characteristics during prediction of parameter drift and to increase the reliability of the estimates of these characteristics.

Solving linear equations with messenger-field and conjugate gradient techniques: An application to CMB data analysis

We discuss linear system solvers invoking a messenger-field and compare them with (preconditioned) conjugate gradient approaches. We show that the messenger-field techniques correspond to fixed point iterations of an appropriately preconditioned initial system of linear equations. We then argue that a conjugate gradient solver applied to the same preconditioned system, or equivalently a preconditioned conjugate gradient solver using the same preconditioner and applied to the original system, will in general ensure at least a comparable and typically better performance in terms of the number of iterations to convergence and time-to-solution. We illustrate our conclusions with two common examples drawn from the cosmic microwave background (CMB) data analysis: Wiener filtering and map-making. In addition, and contrary to the standard lore in the CMB field, we show that the performance of the preconditioned conjugate gradient solver can depend significantly on the starting vector. This observation seems of particular importance in the cases of map-making of high signal-to-noise ratio sky maps and therefore should be of relevance for the next generation of CMB experiments.

Critical Exponent of the Anderson Transition Using Massively Parallel Supercomputing

To date the most precise estimations of the critical exponent for the Anderson transition have been made using the transfer matrix method. This method involves the simulation of extremely long quasi one-dimensional systems. The method is inherently serial and is not well suited to modern massively parallel supercomputers. The obvious alternative is to simulate a large ensemble of hypercubic systems and average. While this permits taking full advantage of both OpenMP and MPI on massively parallel supercomputers, a straight forward implementation results in data that does not scale. We show that this problem can be avoided by generating random sets of orthogonal starting vectors with an appropriate stationary probability distribution. We have applied this method to the Anderson transition in the three-dimensional orthogonal universality class and been able to increase the largest $L\times L$ cross section simulated from $L=24$ (New J. Physics, 16, 015012 (2014)) to $L=64$ here. This permits an estimation of the critical exponent with improved precision and without the necessity of introducing an irrelevant scaling variable. In addition, this approach is better suited to simulations with correlated random potentials such as is needed in quantum Hall or cold atom systems.

Multiple Right-Hand Side Techniques in Semi-Explicit Time Integration Methods for Transient Eddy-Current Problems

The spatially discretized magnetic vector potential formulation of magnetoquasistatic field problems is transformed from an infinitely stiff differential algebraic equation system into a finitely stiff ordinary differential equation (ODE) system by application of a generalized Schur complement for nonconducting parts. The ODE can be integrated in time using explicit time integration schemes, e.g. the explicit Euler method. This requires the repeated evaluation of a pseudo-inverse of the discrete curl-curl matrix in nonconducting material by the preconditioned conjugate gradient (PCG) method which forms a multiple right-hand side problem. The subspace projection extrapolation method and proper orthogonal decomposition are compared for the computation of suitable start vectors in each time step for the PCG method which reduce the number of iterations and the overall computational costs.

Tangent bundle geometry from dynamics: Application to the Kepler problem

In this paper, we consider a manifold with a dynamical vector field and enquire about the possible tangent bundle structures which would turn the starting vector field into a second-order one. The analysis is restricted to manifolds which are diffeomorphic with affine spaces. In particular, we consider the problem in connection with conformal vector fields of second-order and apply the procedure to vector fields conformally related with the harmonic oscillator ([Formula: see text]-oscillators). We select one which covers the vector field describing the Kepler problem.

Explicit time integration of transient eddy current problems

SummaryFor time integration of transient eddy current problems, commonly implicit time integration methods are used, where in every time, step 1 or several nonlinear systems of equations have to be linearized with the Newton‐Raphson method because of ferromagnetic materials involved. In this paper, a generalized Schur complement is applied to the magnetic vector potential formulation, which converts a differential‐algebraic equation system of index 1 into a system of ordinary differential equations with reduced stiffness. For the time integration of this ordinary differential equations system of equations, the explicit Euler method is applied. The Courant‐Friedrich‐Levy stability criterion of explicit time integration methods may result in small time steps. Applying a pseudoinverse of the discrete curl‐curl operator in nonconducting regions of the problem is required in every time step. For the computation of the pseudoinverse, the preconditioned conjugate gradient method is used. The cascaded subspace extrapolation method is presented to produce suitable start vectors for these preconditioned conjugate gradient iterations. The resulting scheme is validated using the nonlinear TEAM 10 benchmark problem.

Implicitly Restarted Refined Partially Orthogonal Projection Method with Deflation

AbstractIn this paper we consider the computation of some eigenpairs with smallest eigenvalues in modulus of large-scale polynomial eigenvalue problem. Recently, a partially orthogonal projection method and its refinement scheme were presented for solving the polynomial eigenvalue problem. The methods preserve the structures and properties of the original polynomial eigenvalue problem. Implicitly updating the starting vector and constructing better projection subspace, we develop an implicitly restarted version of the partially orthogonal projection method. Combining the implicit restarting strategy with the refinement scheme, we present an implicitly restarted refined partially orthogonal projection method. In order to avoid the situation that the converged eigenvalues converge repeatedly in the later iterations, we propose a novel explicit non-equivalence low-rank deflation technique. Finally some numerical experiments show that the implicitly restarted refined partially orthogonal projection method with the explicit non-equivalence low-rank deflation technique is efficient and robust.