Arnoldi Method Research Articles

Wideband Vibro-Acoustic Coupling Investigation in Three Dimensions Using Order-Reduced Isogeometric Finite Element/Boundary Element Method

This study introduces an innovative model-order reduction (MOR) technique that integrates boundary element and finite element methodologies, streamlining the analysis of wideband vibro-acoustic interactions within aquatic and aerial environments. The external acoustic phenomena are efficiently simulated via the boundary element method (BEM), while the finite element method (FEM) adeptly captures the dynamics of vibrating thin-walled structures. Furthermore, the integration of isogeometric analysis within the finite element/boundary element framework ensures geometric integrity and maintains high-order continuity for Kirchhoff–Love shell models, all without the intermediary step of meshing. Foundational to our reduced-order model is the application of the second-order Arnoldi method coupled with Taylor expansions, effectively eliminating the frequency dependence of system matrices. The proposed technique significantly enhances the computational efficiency of wideband vibro-acoustic coupling analyses, as demonstrated through numerical simulations.

Reduced Representation of Spatial Harmonics for Motor Model Order Reduction Using Block Arnoldi Method

Reduced Representation of Spatial Harmonics for Motor Model Order Reduction Using Block Arnoldi Method

Application of data-driven model reduction techniques in reactor neutron field calculations

High-order harmonic techniques can be used to recreate neutron flux distributions in reactor cores using the neutron diffusion equation. However, traditional source iteration and source correction iteration techniques have sluggish convergence rates and protracted calculation periods. The correctness of the implicitly restarted Arnoldi method (IRAM) in resolving the eigenvalue problems of the one-dimensional and two-dimensional neutron diffusion equations was confirmed by computing the benchmark problems SLAB_1D_1G and two-dimensional steady-state TWIGL using IRAM. By integrating Galerkin projection with Proper Orthogonal Decomposition (POD) techniques, a POD-Galerkin reduced-order model was developed and the IRAM model was used as the full-order model. For 14 macroscopic cross-section values, the TWIGL benchmark problem was perturbed within a 20% range. We extracted 100 sample points using the Latin hypercube sampling method, and 70% of the samples were used as the testing set to assess the performance of the reduced-order model The remaining 30% were utilized as the training set to develop the reduced-order model, which was employed to rebuild the TWIGL benchmark problem. The reduced-order model demonstrates good flexibility and can efficiently and accurately forecast the effective multiplication factor and neutron flux distribution in the core. The reduced-order model predicts keff and neutron flux distribution with a high degree of agreement compared to the full-order model. Additionally, the reduced-order model's computation time is only 10.18% of that required by the full-order model.The neutron flux distribution of the steady-state TWIGL benchmark was recreated using the reduced-order model. The obtained results indicate that the reduced-order model can accurately predict the keff and neutron flux distribution of the steady-state TWIGL benchmark.Overall, the proposed technique not only has the potential to accurately project neutron flux distributions in transient settings, but is also relevant for reconstructing neutron flux distributions in steady-state conditions; thus, its applicability is bound to increase in the future.

Reduced order calculation method of subsynchronous oscillation mode of PMSG based wind farm connected with power grid based on Arnoldi method

Large scale wind turbine connected to the power grid is easy to induce subsynchronous oscillation. “Curse of dimensionality” will be encountered when the subsynchronous oscillation mode is analysed and evaluated based on the full-order model of the wind farm. It is urgent to research the fast and effective analysis method of subsynchronous oscillation mode when the wind farm is connected to the power grid. Therefore, a reduced order calculation method of the subsynchronous oscillation mode of PMSG based wind farm connected with a power grid based on the Arnoldi method is proposed. This method can quickly obtain the eigenvalues corresponding to the subsynchronous oscillation mode through iterative approximation when PMSG-based wind farm connected to the power system. This can be used to analyze the subsynchronous oscillation of large-scale wind farms connected to the power grid and provide an effective means for the evaluation of it. By changing system stability, collector network structure, and wind farm interconnection scale, the accuracy of the eigenvalues distribution between the full-order system and the reduced order system are compared. Moreover, the reduced order and computation time are considered. The results present that the proposed method can quickly and accurately obtain the subsynchronous oscillation mode in the system.

Spectral approach with iterative clarification of a radiation boundary conditions for modeling of quasimodes of a gyrotrons open cavities

Purpose. The article presents a new method for numerical simulation of quasi-eigenmode oscillations in open resonators of gyrotrons — powerful vacuum generators of electromagnetic waves in the millimeter and submillimeter ranges. The gyrotron cavity has the shape of a weakly inhomogeneous hollow circular metal waveguide. Methods. The proposed approach uses the inhomogeneous string equation with radiation boundary conditions to formulate a nonlinear spectral boundary value problem describing oscillations in a resonator, neglecting the couplings of waves with different radial indices. By linearizing with respect to frequency the radiation boundary conditions, the boundary value problem is reduced to a linear boundary value problem. To discretize this boundary value problem, the finite difference method is used and a linear generalized matrix eigenvalue problem is formulated. This problem is solved by the Arnoldi method with eigenvalues calculation in a shift-invert mode. An iterative algorithm is proposed that makes it possible to sequentially calculate a given number of frequencies and quality factors of quasi-eigenmodes of oscillations. Results. The computer program was developed written in the Wolfram Language and Fortran using the algorithms proposed in the work. The results of test calculations for real gyrotron resonators are presented, which demonstrate the high accuracy of the obtained values of frequencies, quality factors and field distributions of quasi-eigenmode oscillations in the studied resonators. Conclusion. The methods, algorithms and created program proposed in the article can significantly facilitate the process of developing gyrotrons intended for various practical applications and operating in new frequency ranges. The method of iterative refinement of boundary conditions can be generalized to the case of equations of the linear theory of a gyrotron and used to develop new methods for analyzing the starting conditions for the soft self-excitation in gyrotrons — generators.

Quadrature-based Restarted Arnoldi Method for Fast 3D TEM Forward Modeling of Large-Scale Models

Quadrature-based Restarted Arnoldi Method for Fast 3D TEM Forward Modeling of Large-Scale Models

A BEM broadband topology optimization strategy based on Taylor expansion and SOAR method—Application to 2D acoustic scattering problems

AbstractIn this article, an innovative method is proposed for broadband topology optimization of sound‐absorbing materials adhering to the surface of a sound barrier structure. Helmholtz equation for the acoustic problems is solved using the boundary element method, while sensitivities of the objective function with respect to a large number of design variables are calculated via a sensitivity analysis method originated from the adjoint variable method (AVM), and the optimal solution is obtained by the method of moving asymptotes (MMA). Since the traditional single‐frequency topology optimization is frequency dependent, that is, the optimal solution at a certain frequency may not be optimal at other frequencies, broadband topology optimization of sound‐absorbing materials is conducted in this work. To address the problem of tremendous computational cost due to repetitive calculations at each discrete frequency point during broadband optimization, Taylor expansion of the Hankel function is performed to decouple the frequency dependent and independent terms in the BEM equations. For large‐scale problems, a reduced‐order model that retains the essential structures and key properties of the original model is built using the second‐order Arnoldi (SOAR) method. The accuracy and feasibility of the proposed algorithms are finally validated via some two‐dimensional numerical examples.

Fast and accurate out-of-core PCA framework for large scale biobank data.

Principal component analysis (PCA) is widely used in statistics, machine learning, and genomics for dimensionality reduction and uncovering low-dimensional latent structure. To address the challenges posed by ever-growing data size, fast and memory-efficient PCA methods have gained prominence. In this paper, we propose a novel randomized singular value decomposition (RSVD) algorithm implemented in PCAone, featuring a window-based optimization scheme that enables accelerated convergence while improving the accuracy. Additionally, PCAone incorporates out-of-core and multithreaded implementations for the existing Implicitly Restarted Arnoldi Method (IRAM) and RSVD. Through comprehensive evaluations using multiple large-scale real-world data sets in different fields, we show the advantage of PCAone over existing methods. The new algorithm achieves significantly faster computation time while maintaining accuracy comparable to the slower IRAM method. Notably, our analyses of UK Biobank, comprising around 0.5 million individuals and 6.1 million common single nucleotide polymorphisms, show that PCAone accurately computes the top 40 principal components within 9 h. This analysis effectively captures population structure, signals of selection, structural variants, and low recombination regions, utilizing <20 GB of memory and 20 CPU threads. Furthermore, when applied to single-cell RNA sequencing data featuring 1.3 million cells, PCAone, accurately capturing the top 40 principal components in 49 min. This performance represents a 10-fold improvement over state-of-the-art tools.

Electronic structure calculations with interpolating tensor product wavelet basis.

We introduce a basis set consisting of three-dimensional Deslauriers-Dubuc wavelets and solve numerically the Schrödinger equationsof H and He atoms and molecules H_{2},H_{2}^{+}, and LiH with Hartree-Fock and density functional theory (DFT) methods. We also compute the 2s and 2p excited states of hydrogen. The Coulomb singularity at the nucleus is handled by using a pseudopotential. The eigenvalue problem is solved with Arnoldi and Lanczos methods, Poisson equationwith generalized minimal residual method and conjugate gradient on the normal equationsmethods, and matrix elements are computed using the biorthogonality relations of the interpolating wavelets. Performance is compared with those of CCCBDB and bigdft.

Global stability analysis of elastic aircraft in edge-of-the-envelope flow

Shock buffet on wings is a phenomenon caused by strong shock-wave/boundary-layer interaction resulting first in self-sustained flow unsteadiness and eventually in a detrimental structural response called buffeting. While it is an important aspect of wing design and aircraft certification, particularly for modern transonic air transport, not all of the underlying multidisciplinary physics is thoroughly understood. Building upon a single-discipline shock-buffet stability study, this work now investigates the impact of an elastic structure in these extreme flow conditions. Specifically, a triglobal stability analysis of a fluid–structure coupled system is presented, utilising the implicitly restarted Arnoldi method with a sparse iterative Krylov solver and novel preconditioner. Asymmetry resulting from a static aeroelastic simulation based on a finite-element model of the underlying geometry in a wind tunnel modifies the global modes of the earlier fluid-only symmetric full-span analysis. A flutter stability analysis at wind-tunnel flow conditions below shock-buffet onset finds no instability in the structural degrees-of-freedom, whereas in shock-buffet flow with globally unstable fluid modes additional marginally unstable structural (and fluid) modes emerge. The developed stability tool for coupled analysis is instrumental in identifying those physically relevant and strongly coupled modes where a standard pk-type (p being eigenvalue and k reduced frequency) flutter analysis fails. With the complementary computation of adjoint eigenmodes, the core of the instability is pinpointed to a relatively small wing area which may help to effect the control and delay of this detrimental transonic unsteadiness. We contribute to the question on how the presence of the elastic wing structure impacts on the otherwise pure aerodynamic three-dimensional shock-buffet dynamics.

Solving the time-dependent Klein-Gordon and square-root Klein-Gordon equations with Krylov-subspace methods

For relativistic spinless particles, the standard equation with Lorentz covariance is the Klein-Gordon equation, while the square-root Klein-Gordon equation, which is usually referred to be semirelativistic since it lacks Lorentz covariance, is another frequently used equation. We numerically solve these two equations coupled with time-dependent electromagnetic fields using the non-Hermitian Arnoldi and the Hermitian Lanczos propagator, respectively, in the three-dimensional real space. The non-Hermitian Arnoldi method is shown to be ideally suited for the Klein-Gordon equation. A different method to treat the nonlocal operator in the square-root Klein-Gordon equation is developed, allowing the calculation of the propagator to be more efficient. Furthermore, some strong-field problems such as the photoionization, high harmonic generation, and the pair production are studied using our methods. The results of both equations are compared with those of the nonrelativistic Schr\"odinger equation, which demonstrates the relativistic effects in various regimes. We show that observables of the Klein-Gordon equation and the square-root Klein-Gordon equation are identical up to the first order of relativistic corrections and confirm it numerically.

Using Mixed Precision in Low‐Synchronization Reorthogonalized Block Classical Gram‐Schmidt

AbstractMixed precision hardware has recently become commercially available, and more than 25% of the supercomputers in the TOP500 list now have mixed precision capabilities. Using lower precision in algorithms can be beneficial in terms of reducing both computation and communication costs. There are many current efforts towards developing mixed precision numerical linear algebra algorithms, which can lead to speedups in real applications. Motivated by this, we aim to further the state‐of‐the‐art in developing and analyzing mixed precision variants of iterative methods. In iterative methods based on Krylov subspaces, the orthogonal basis is generated by Arnoldi or Lanczos methods or their variants. In long recurrence methods such as GMRES, one needs to use an explicit orthogonalization scheme such as Gram‐Schmidt to orthonormalize the vectors generated.Krylov subspace methods, such as GMRES, are typically communication‐bound on modern machines; the runtime is usually dominated by the cost of global synchronizations which are required for the necessary orthogonalization. This has motivated the development of various algorithmic variants which attempt to reduce the communication cost while maintaining a stable algorithm. Recent work has focused on the development of low‐synchronization block variants of Gram‐Schmidt orthogonalization, which, when used within communication‐avoiding (s‐step) or block variants of GMRES, can reduce the number of global synchronizations to one per block.In this work, we focus on the block variant of low‐synchronization classical Gram‐Schmidt with reorthogonalization, which we call BCGSI+LS. We demonstrate that the loss of orthogonality produced by this orthogonalization scheme can exceed O(u)κ(𝒳), where u is the unit roundoff and κ(𝒳) is the condition number of the matrix to be orthogonalized, and thus we can not in general expect this to result in a backward stable block GMRES implementation. We then develop a mixed precision variant of this algorithm, called BCGSI+LS‐MP, which uses higher precision in certain parts of the computation. We demonstrate experimentally that for a number of challenging test problems, our mixed precision variant successfully maintains a loss of orthogonality below O(u)κ(𝒳). This indicates that we can achieve a backward stable block GMRES algorithm that requires only one synchronization per iteration.

Adaptively restarted block Krylov subspace methods with low-synchronization skeletons

With the recent realization of exascale performance by Oak Ridge National Laboratory’s Frontier supercomputer, reducing communication in kernels like QR factorization has become even more imperative. Low-synchronization Gram-Schmidt methods, first introduced in Świrydowicz et al. (Numer. Lin. Alg. Appl. 28(2):e2343, 2020), have been shown to improve the scalability of the Arnoldi method in high-performance distributed computing. Block versions of low-synchronization Gram-Schmidt show further potential for speeding up algorithms, as column-batching allows for maximizing cache usage with matrix-matrix operations. In this work, low-synchronization block Gram-Schmidt variants from Carson et al. (Linear Algebra Appl. 638:150–195, 2022) are transformed into block Arnoldi variants for use in block full orthogonalization methods (BFOM) and block generalized minimal residual methods (BGMRES). An adaptive restarting heuristic is developed to handle instabilities that arise with the increasing condition number of the Krylov basis. The performance, accuracy, and stability of these methods are assessed via a flexible benchmarking tool written in MATLAB. The modularity of the tool additionally permits generalized block inner products, like the global inner product.

Exhaustive modal analysis of large-scale power systems using model order reduction

This paper presents an efficient modal analysis methodology that computes all modes of any given large-scale power system in exhaustive manner using the model order reduction techniques. For this, a reduced order model is generated using the Balanced Truncation (BT) method for which the controllability and observability gramians are approximated using the low-rank Cholesky factors. This leads to a rapid identification of classes of coupled dynamic devices of the original system. Next, approximated oscillatory modes are computed for each class. Finally, the exact values of the oscillatory modes of the overall power system are determined by iterative computations (the Modified Arnoldi method) initialized to the approximated modes found at the first step. The proposed methodology is able to put into evidence all coupling modes of any given large-scale power system (containing power electronics or any other specific dynamic devices). No a priori knowledge about the pattern of oscillations is needed. The accuracy and efficiency of the proposed methodology are thoroughly validated on several power systems with different orders, including a large scale model of the interconnected European power system.

A non-intrusive model order reduction boundary element method for frequency sweeps of 2D acoustics

The boundary element method (BEM) is one of the most commonly used prediction technique to deal with acoustic problems governed by the Helmholtz equation. Its use, however, suffers from the disadvantage that the system of equations have to be reformulated for each frequency of interest due to the frequency-dependent property. This undermines the inherent efficiency of frequency sweep analyses of BE models. Additionally, the resulting coefficient matrices are fully-populated, non-symmetric and complex-valued. In order to enable the traditional BEM to be applied for large-scale acoustic problems, a non-intrusive model order reduction boundary element method under the offline–online framework is proposed. In the offline phase, a proper reduced-order basis (ROB) which spans a projection subspace is constructed via the Arnoldi method. A newly developed cutoff technique is incorporated to reduce the involved computational complexity. Once a low-rank ROB is determined, the full-order frequency-decoupled system matrices based on the series expansions are projected onto the spanned subspace column-by-column to favor the memory requirement. In the online phase, fast frequency sweep analyses with a direct solver can be achieved because only the compact reduced-order model needs to be confronted. The high performance of the proposed solution approach in the context of single-frequency and multi-frequency analyses of two-dimensional (2D) BE models with the number of degrees of freedom ranging from a few hundred up to one hundred thousand is demonstrated by various problem settings, including both interior and exterior acoustic fields with different boundary conditions.

Confluent Vandermonde with Arnoldi

In this note, we extend the Vandermonde with Arnoldi method recently advocated by Brubeck et al. (2021) to dealing with the confluent Vandermonde matrix. To apply the Arnoldi process, it is critical to find a Krylov subspace which generates the column space of the confluent Vandermonde matrix. A theorem is established for such Krylov subspaces for any order derivatives. This enables us to compute the derivatives of high degree polynomials to high precision. It also makes many applications involving derivatives possible, as illustrated by numerical examples. We note that one of the approaches orthogonalizes only the function values and is equivalent to the formula given by Brubeck and Trefethen (2022). The other approach orthogonalizes the Hermite data. About which approach is preferable to another, we made the comparison, and the result is problem dependent.

A method for computing a few eigenpairs of large generalized eigenvalue problems

Many methods for the computation of selected eigenpairs of generalized eigenproblems for matrix pairs use a shift-and-invert technique. When applied to large-scale problems, this requires the solution of large linear systems of equations. This paper proposes an application of an Arnoldi method described in Voss (2004) [21] to the computation of a few extreme eigenpairs of a matrix pair. An advantage of this approach, when compared to methods that use the shift-and-invert technique, is that no large systems of equations have to be solved. We compare this approach to using a technique for simultaneously reducing a pair of large matrices to a pair of small matrices by a generalized Arnoldi process described in Li and Ye (2003) [12] and Hoffnung et al. (2006) [6]. The latter technique does not require the solution of large linear systems of equations either. Computed examples show the proposed method to yield approximations of the desired eigenpairs of higher accuracy when using about the same amount of computer storage space.

Efficient Characteristic Mode Analysis for PEC Objects Using Sherman–Morrison–Woodbury Formula-Based Algorithm

In characteristic mode analysis (CMA), solving the characteristic modes (CMs) from the generalized eigenvalue problem takes the main part of the workload. In this letter, the algorithm based on the Sherman–Morrison–Woodbury (SMW) formula is embedded into the implicitly restarted Arnoldi method to improve the efficiency of solving the CMs. In this method, the impedance matrix is compressed into a product of block-diagonal matrices by using the SMW formula with adaptive cross approximation algorithm. Benefiting from the efficient compression, calculating the inverse matrix of the impedance matrix and the matrix vector products related to the solution of the generalized eigenvalue problem is accelerated. Consequently, the computational time and memory requirement in CMA are both reduced. Some numerical examples are demonstrated to show the accuracy and efficiency of the proposed method.

A Second Order Arnoldi Method with Stopping Criterion and Reduced Order Selection for Reducing Second Order Systems

This paper introduces a new algorithm for reducing large dimensional second-order dynamic systems through the Second Order Arnold Reduction (SOAR) procedure, with a stopping criterion to select an acceptable good order for the reduced model based on a new coefficient called the Numerical-Rank Performance Coefficient (NRPC), for efficient early termination and automatic optimal order selection of the reduced model. The key idea of this method is to calculate the NRPC coefficient for each iteration of the SOAR algorithm and measure the dynamic evolution information of the original system, which is added to each vector of the Krylov subspace generated by the SOAR algorithm. When the dynamical tolerance condition is verified, the iterative procedure of the algorithm stops. Three benchmark models were used as numerical examples to check the effectiveness and simplicity of the proposed algorithm.

A Krylov-Schur-like method for computing the best rank-(r1,r2,r3) approximation of large and sparse tensors

The paper is concerned with methods for computing the best low multilinear rank approximation of large and sparse tensors. Krylov-type methods have been used for this problem; here block versions are introduced. For the computation of partial eigenvalue and singular value decompositions of matrices the Krylov-Schur (restarted Arnoldi) method is used. A generalization of this method to tensors is described, for computing the best low multilinear rank approximation of large and sparse tensors. In analogy to the matrix case, the large tensor is only accessed in multiplications between the tensor and blocks of vectors, thus avoiding excessive memory usage. It is proved that if the starting approximation is good enough, then the tensor Krylov-Schur method is convergent. Numerical examples are given for synthetic tensors and sparse tensors from applications, which demonstrate that for most large problems the Krylov-Schur method converges faster and more robustly than higher order orthogonal iteration.