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Synthetic Magnetic Resonance (MR) imaging predicts images at new design parameter settings from a few observed MR scans. Model-based methods, that use both the physical and statistical properties underlying the MR signal and its acquisition, can predict images at any setting from as few as three scans, allowing it to be used in individualized patient- and anatomy-specific contexts. However, the estimation problem in model-based synthetic MR imaging is ill-posed and so regularization, in the form of correlated Gaussian markov random fields, is imposed on the voxel-wise spin-lattice relaxation time, spin-spin relaxation time and the proton density underlying the MR image. We develop theoretically sound but computationally practical matrix-free estimation methods for synthetic MR imaging. Our evaluations demonstrate superior performance of our methods in currently-used clinical settings when compared to existing model-based and deep learning methods. Moreover, unlike deep learning approaches, our fast methodology can synthesize needed images during patient visits, with good estimation and prediction accuracy and consistency. An added strength of our model-based approach, also developed and illustrated here, is the accurate estimation of standard errors of regional contrasts in the synthesized images. A R package symr implements our methodology. for this article are available online. [ABSTRACT FROM AUTHOR]

We construct bivariate orthogonal polynomials (OPs) on algebraic curves of the form ym=ϕ(x)$y^{m} = \phi (x)$ in R2${\mathbb {R}}^2$ where m=1,2$m = 1, 2$ and ϕ is a polynomial of arbitrary degree d, in terms of univariate semiclassical OPs. We compute connection coefficients that relate the bivariate OPs to a polynomial basis that is itself orthogonal and whose span contains the OPs as a subspace. The connection matrix is shown to be banded and the connection coefficients and Jacobi matrices for OPs of degree 0,...,N$0, \ldots , N$ are computed via the Lanczos algorithm in O(Nd4)$\mathcal {O}(Nd^4)$ operations. [ABSTRACT FROM AUTHOR]

The approximation of trace(f(Ω)), where f is a function of a symmetric matrix Ω, can be challenging when Ω is exceedingly large. In such a case even the partial Lanczos decomposition of Ω is computationally demanding and the stochastic method investigated by Bai et al. (J. Comput. Appl. Math. 74:71–89, 1996) is preferred. Moreover, in the last years, a partial global Lanczos method has been shown to reduce CPU time with respect to partial Lanczos decomposition. In this paper we review these techniques, treating them under the unifying theory of measure theory and Gaussian integration. This allows generalizing the stochastic approach, proposing a block version that collects a set of random vectors in a rectangular matrix, in a similar fashion to the partial global Lanczos method. We show that the results of this technique converge quickly to the same approximation provided by Bai et al. (J. Comput. Appl. Math. 74:71–89, 1996), while the block approach can leverage the same computational advantages as the partial global Lanczos. Numerical results for the computation of the Von Neumann entropy of complex networks prove the robustness and efficiency of the proposed block stochastic method. [ABSTRACT FROM AUTHOR]

The generalized eigenvalue problem for a symmetric definite matrix pencil obtained from finite-element modeling of electroelastic materials is solved numerically by the Lanczos algorithm. The mass matrix is singular in the considered problem, and therefore the process proceeds with the semi-inner product defined by this matrix. The shift-and-invert Lanczos algorithm is used to find multiple eigenvalues closest to some shift and the corresponding eigenvectors. The results of the numerical experiments are presented. [ABSTRACT FROM AUTHOR]

Identifying the most important nodes according to specific centrality indices is an important issue in network analysis. Node metrics based on the computation of functions of the adjacency matrix of a network were defined by Estrada and his collaborators in various papers. This paper describes a MATLAB toolbox for computing such centrality indices using efficient numerical algorithms based on the connection between the Lanczos method and Gauss-type quadrature rules. [ABSTRACT FROM AUTHOR]

We show constructively that, under certain regularity assumptions, any system of coupled linear differential equations with variable coefficients can be tridiagonalized by a time-dependent Lanczos-like method. The proof we present formally establishes the convergence of the so-called ⁎-Lanczos algorithm and yields a full characterization of algorithmic breakdowns. From there, the solution of the original differential system is available in a finite and treatable number of scalar integral equations. This is a key piece in evaluating the elusive ordered exponential function both formally and numerically. [ABSTRACT FROM AUTHOR]

The time-ordered exponential of a time-dependent matrix A(t) is defined as the function of A(t) that solves the first-order system of coupled linear differential equations with non-constant coefficients encoded in A(t). The authors have recently proposed the first Lanczos-like algorithm capable of evaluating this function. This algorithm relies on inverses of time-dependent functions with respect to a non-commutative convolution-like product, denoted by *. Yet, the existence of such inverses, crucial to avoid algorithmic breakdowns, still needed to be proved. Here we constructively prove that *-inverses exist for all non-identically null, smooth, separable functions of two variables. As a corollary, we partially solve the Green's function inverse problem which, given a distribution G, asks for the differential operator whose fundamental solution is G. Our results are abundantly illustrated by examples. [ABSTRACT FROM AUTHOR]

This paper presents a new regularization for Extreme Learning Machines (ELMs). ELMs are Randomized Neural Networks (RNNs) that are known for their fast training speed and good accuracy. Nevertheless the complexity of ELMs has to be selected, and regularization has to be performed in order to avoid underfitting or overfitting. Therefore, a novel Regularization is proposed using a modified Lanczos Algorithm: Iterative Lanczos Extreme Learning Machine (Lan-ELM). As summarized in the experimental Section, the computational time is on average divided by 4 and the Normalized MSE is on average reduced by 11%. In addition, the proposed method can be intuitively parallelized, which makes it a very valuable tool to analyze huge data sets in real-time. [ABSTRACT FROM AUTHOR]

This technical note proposes a novel profile likelihood method for estimating the covariance parameters in exploratory factor analysis of high-dimensional Gaussian datasets with fewer observations than number of variables. An implicitly restarted Lanczos algorithm and a limited-memory quasi-Newton method are implemented to develop a matrix-free framework for likelihood maximization. Simulation results show that our method is substantially faster than the expectation-maximization solution without sacrificing accuracy. Our method is applied to fit factor models on data from suicide attempters, suicide ideators, and a control group. for this article are available online. [ABSTRACT FROM AUTHOR]

We study the Lanczos algorithm where the initial vector is sampled uniformly from Sn-1. Let A be an n \times n Hermitian matrix. We show that when run for few iterations, the output of Lanczos on A is almost deterministic. More precisely, we show that for any ε ∊ (0, 1) there exists c > 0 depending only on ε and a certain global property of the spectrum of A (in particular, not depending on n) such that when Lanczos is run for at most c log n iterations, the output Jacobi coefficients deviate from their medians by t with probability at most exp(-nεt²) for t < ∥ A∥. We directly obtain a similar result for the Ritz values and vectors. Our techniques also yield asymptotic results: Suppose one runs Lanczos on a sequence of Hermitian matrices An ∊ Mn(C) whose spectral distributions converge in Kolmogorov distance with rate O(n-ε) to a density µ for some ε > 0. Then we show that for large enough n, and for k = O(√logn), the Jacobi coefficients output after k iterations concentrate around those for µ. The asymptotic setting is relevant since Lanczos is often used to approximate the spectral density of an infinite-dimensional operator by way of the Jacobi coefficients; our result provides some theoretical justification for this approach. In a different direction, we show that Lanczos fails with high probability to identify outliers of the spectrum when run for at most c' log n iterations, where again c' depends only on the same global property of the spectrum of A. Classical results imply that the bound c' log n is tight up to a constant factor. [ABSTRACT FROM AUTHOR]

This article discusses an extension of the singular vector (SV) method in the context of an initial perturbation generator for an ensemble prediction system (EPS). In general, multiple SVs targeted at different regions are computed in operational EPSs to extract growing modes with focus on different parts of the EPS domain. However, significant computational cost is associated with running all the procedures of the SV computations multiple times. In this study, the Lanczos algorithm used for SV computation was extended to allow simultaneous computation of multiple targeted SV sets. Algebraic calculations in the algorithm, such as orthonormalizations and eigenvalue problem resolution, are separately implemented for multiple "subdomains" incorporating different targeting areas, and SV sets are computed for individual subdomains. However, forward and backward linearized propagation runs through the whole domain, shared among the SV sets from all the subdomains. As such algebraic operation accounts for a relatively small part of all computation, the computational cost increment brought by the algorithm extension is also small in relation to single SV computations. For verifications, consistency between SVs produced with the original and extended algorithms was examined. Both SV sets spanned the same subspaces with similar linear growth rates, except those derived on subdomain boundaries, where SVs produced using the extended algorithm were truncated by the boundary. To avoid such truncation, it is necessary to set a subdomain large enough to cover the target area and its surrounding region. In this article, some applications of this algorithm in operational situations are suggested. Also, an application of subdomain to the wave‐number space is described. [ABSTRACT FROM AUTHOR]

This study develops a scalable matrix-free h-likelihood method for spatial-temporal Gaussian state-space models. The state vectors are constructed in such a way that they follow spatial-temporal Gaussian autoregressions that are consistent with the conditional formulation of auto-normal spatial fields. The proposed h-likelihood method provides the same inferences as those obtained from the Kalman filter and residual maximum likelihood analyses. However, for data from a large number of spatial sites, our method has significant computational advantages. Furthermore, we describe inferences in small time steps and indicate how the proposed method can be adapted to other complex spatial-temporal dynamical models based on stochastic partial differential equations. The proposed method applies to data with regularly or irregularly sampled spatial locations. Lastly, we illustrate our method by means of a simulation study and a data example on atmospheric concentrations of total nitrate across eastern North America. [ABSTRACT FROM AUTHOR]

Krylov subspace spectral (KSS) methods are explicit time-stepping methods for partial differential equations that are designed to extend the advantages of Fourier spectral methods, when applied to constant-coefficient problems, to the variable-coefficient case. This paper presents a convergence analysis of a first-order KSS method applied to a system of coupled equations for modeling first-order photobleaching kinetics. The analysis confirms what has been observed in numerical experiments—that the method is unconditionally stable and achieves spectral accuracy in space. Further analysis shows that this unconditional stability is not limited to the case in which the leading coefficient is constant. [ABSTRACT FROM AUTHOR]

Scar eigenstates in a many-body system refers to a small subset of non-thermal finite energy density eigenstates embedded into an otherwise thermal spectrum. This novel non-thermal behaviour has been seen in recent experiments simulating a one-dimensional PXP model with a kinetically-constrained local Hilbert space realised by a chain of Rydberg atoms. We probe these small sets of special eigenstates starting from particular initial states by computing the spread complexity associated to time evolution of the PXP hamiltonian. Since the scar subspace in this model is embedded only loosely, the scar states form a weakly broken representation of the Lie algebra. We demonstrate why a careful usage of the forward scattering approximation (FSA), instead of any other method, is required to extract the most appropriate set of Lanczos coefficients in this case as the consequence of this approximate symmetry. Only such a method leads to a well defined notion of a closed Krylov subspace and consequently, that of spread complexity. We show this using three separate initial states, namely|Z2⟩,|Z3⟩and the vacuum state, due to the disparate classes of scar states hosted by these sectors. We also discuss systematic methods of remedying the imperfections in the FSA setup stemming from these approximate symmetries., (Creative Commons Attribution license.)

Minimization algorithms are singular components in four-dimensional variational data assimilation (4DVar). In this paper, the convergence and application of the conjugate gradient algorithm (CGA), which is based on the Lanczos iterative algorithm and the Hessian matrix derived from tangent linear and adjoint models using a non-hydrostatic framework, are investigated in the 4DVar minimization. First, the influence of the Gram-Schmidt orthogonalization of the Lanczos vector on the convergence of the Lanczos algorithm is studied. The results show that the Lanczos algorithm without orthogonalization fails to converge after the ninth iteration in the 4DVar minimization, while the orthogonalized Lanczos algorithm converges stably. Second, the convergence and computational efficiency of the CGA and quasi-Newton method in batch cycling assimilation experiments are compared on the 4DVar platform of the Global/Regional Assimilation and Prediction System (GRAPES). The CGA is 40% more computationally efficient than the quasi-Newton method, although the equivalent analysis results can be obtained by using either the CGA or the quasi-Newton method. Thus, the CGA based on Lanczos iterations is better for solving the optimization problems in the GRAPES 4DVar system. [ABSTRACT FROM AUTHOR]

We consider the problem of efficiently solving ensembles of variational data assimilations in the context of numerical weather prediction. Running several assimilations notably allows us to initialize ensemble prediction systems and to more accurately represent background‐error statistics, but is computationally expensive, limiting ensemble size. We propose a new class of algorithms for speeding up the minimization of the ensemble of data assimilations. It consists of using block Krylov methods to simultaneously perform the minimization for all members of the ensemble, instead of performing each minimization separately. We develop preconditioned block Krylov versions of the Full Orthogonal Method and of the Lanczos algorithm in both primal and dual space. The latter works in observation space that is usually of smaller dimension than the state space, thus giving an advantage in terms of memory requirements and computational cost. We describe and compare several parallelization strategies for speeding up the minimization and limiting the communications. These methods have been tested on a quasi‐geostrophic system, consisting of a simplified atmospheric circulation model equipped with an ensemble of 3D‐Var schemes tuned to mimic some features of a limited‐area numerical weather prediction system. Experimentation shows that the number of iterations needed to converge is drastically reduced by the block Krylov approaches. We indicate that, while working in primal space does not save significant computational time, working in the dual space may reduce the computational time by a factor of 2 to 5 (depending on ensemble size) compared to standard Krylov methods, making our approach attractive for operational use. Illustration of the two workload distributions implemented in this work on the AROME‐France domain for four MPI processes and two members in the ensemble. (a) Workload geographical distribution (no distribution by member), as used in the SEQ version of block B‐FOM. (b) Workload distribution by member, combined with an underlying geographical distribution, as used in the MPI and MPIstored versions of block B‐FOM, and in B‐FOM. [ABSTRACT FROM AUTHOR]

We solve the first order 2-D reaction–diffusion equations which describe binding-diffusion kinetics using the photobleaching scanning profile of a confocal laser scanning microscope, approximated by a Gaussian laser profile. We show how to solve the first-order photobleaching kinetics partial differential equations (PDEs) using a time-stepping method known as a Krylov subspace spectral (KSS) method. KSS methods are explicit methods for solving time-dependent variable-coefficient partial differential equations. They approximate Fourier coefficients of the solution using Gaussian quadrature rules in the spectral domain. In this paper, we show how a KSS method can be used to obtain not only an approximate numerical solution, but also an approximate analytical solution when using initial conditions that come from pre-bleach steady states and also general initial conditions, to facilitate asymptotic analysis. Analytical and numerical results are presented. It is observed that although KSS methods are explicit, it is possible to use a time step that is far greater than what the CFL condition would indicate. [ABSTRACT FROM AUTHOR]