Inverse Iteration Research Articles

Numerical Investigation of the Quantum Inverse Algorithm on Small Molecules.

We evaluate the accuracy of the quantum inverse (Q-Inv) algorithm, in which the multiplication of Ĥ-k to the reference wave function is replaced by the Fourier transformed multiplication of e-iλĤ, as a function of the integration parameters and the iteration power k for various systems, including H2, LiH, BeH2 and the notorious H4 molecule at square geometry. We further consider the possibility of employing the Gaussian-quadrature rule as an alternate integration method and compared it to the results employing trapezoidal integration. The Q-Inv algorithm is compared to the inverse iteration method using the Ĥ-1 inverse (I-Iter) and the exact inverse by lower-upper decomposition. Energy values are evaluated as the expectation values of the Hamiltonian. Results suggest that the Q-Inv method provides lower energy results than the I-Iter method up to a certain k, after which the energy increases due to errors in the numerical integration that are dependent on the integration interval. A combined Gaussian-quadrature and trapezoidal integration method proved to be more effective at reaching convergence while decreasing the number of operations. For systems like H4, in which the Q-Inv cannot reach the expected error threshold, we propose a combination of Q-Inv and I-Iter methods to further decrease the error with k at lower computational cost. Finally, we summarize the recommended procedure when treating unknown systems.

Dynamically accelerating the power iteration with momentum

AbstractIn this article, we propose, analyze and demonstrate a dynamic momentum method to accelerate power and inverse power iterations with minimal computational overhead. The method can be applied to real diagonalizable matrices, is provably convergent with acceleration in the symmetric case, and does not require a priori spectral knowledge. We review and extend background results on previously developed static momentum accelerations for the power iteration through the connection between the momentum accelerated iteration and the standard power iteration applied to an augmented matrix. We show that the augmented matrix is defective for the optimal parameter choice. We then present our dynamic method which updates the momentum parameter at each iteration based on the Rayleigh quotient and two previous residuals. We present convergence and stability theory for the method by considering a power‐like method consisting of multiplying an initial vector by a sequence of augmented matrices. We demonstrate the developed method on a number of benchmark problems, and see that it outperforms both the power iteration and often the static momentum acceleration with optimal parameter choice. Finally, we present and demonstrate an explicit extension of the algorithm to inverse power iterations.

Modeling spatial variability of mechanical parameters of layered rock masses and its application in slope optimization at the open-pit mine

Mining engineering with layered structures often obtains extensive geological data to ensure reliability of mining operations. Effectively utilizing these data to characterize the spatial distribution of mechanical parameters in layered rock mass is crucial for slope stability analysis. This study focuses on the Heishan open-pit mine as a case study, leveraging the distinctive characteristics of mining engineering geological data. A three-dimensional fracture network was then generated based on the distribution of fractures in the rocks, and the representative elementary volume of the rock mass was determined by calculating the volumetric joint count (Jv). Statistical characteristics of various geological data are analyzed, and a geological statistical method combining the Kriging method and inverse distance power method is applied to establish a spatial distribution block model for geological strength index (GSI) and rock mechanical properties. Subsequently, a spatial variability model of the mechanical parameters of the layered rock mass was generated based on the Hoek–Brown criterion. Furthermore, the heterogeneous mechanical model is implemented in numerical software for stability analysis and slope angle optimization. The findings indicate that each engineering zone of layered slopes exhibits unique mechanical parameters while demonstrating significant layered distribution patterns at a macro level. Considering spatial variability, the mechanical model significantly impacts slope stability and slope angle optimization outcomes. Our research methodology facilitates accurate quantification of the heterogeneity of mechanical parameters in layered slopes without incurring excessive additional survey costs, enabling more rational explanations of slope landslides and the provision of suitable slope angle optimization designs.

A numerical algorithm for solving the coupled Schrödinger equations using inverse power method

The inverse power method is a numerical algorithm to obtain the eigenvectors of a matrix. In this work, we develop an iteration algorithm, based on the inverse power method, to numerically solve the Schrödinger equation that couples an arbitrary number of components. Such an algorithm can also be applied to the multi-body systems. To show the power and accuracy of this method, we also present an example of solving the Dirac equation under the presence of an external scalar potential and a constant magnetic field, with source code publicly available.

Bayesian hybrid-kernel machine-learning-assisted sensitivity analysis and sensitivity-relevant inverse modeling for groundwater DNAPL contamination

Accurate source characterization and transport parameter estimation is important when seeking to predict the spatiotemporal distribution of dense non-aqueous phase liquid (DNAPL) contaminants in groundwater. However, this is a complex multimodal search problem prone to equifinality and premature convergence, which leads to considerable error. To address this, a sensitivity-relevant dynamic swarm intelligence (SRD-SI) algorithm embedded in a homotopy-variation mechanism is proposed in the present study to rationally balance the inversion processes of sensitivity-varied source characteristics and DNAPL transport parameters. In this approach, global optima are progressively approached in conjunction with the homotopy variation of the search space. Furthermore, to avoid computationally expensive numerical model repetition during the sensitivity analysis and inverse iterations, a Bayesian-based optimization framework that combines multiple kernel functions in a kernel extreme learning machine (KELM) model is designed considering the complex site conditions and statistical characteristics of the input variables, thus creating the Bayesian hybrid KELM (BHK-ELM) model for the reliable surrogate modeling of numerical DNAPL-transport simulations. The results show that the BHK-ELM model recognizes and effectively reconstructs the complex input − output mapping of the numerical model by increasing the determination coefficient R2 to 0.9988 while improving the computational efficiency approximately 4500-fold. Because source characteristics and boundary conditions are far more sensitive than transport parameters to the contaminant distribution, conventional inverse modeling methods struggle to accurately identify these transport parameters. In contrast, the proposed inverse modeling system combining sensitivity analysis, swarm intelligence, and homotopy variation is more stable and provides significantly more accurate estimations for all unknown variables. Compared with the traditional SI algorithm, the homotopy-variation SRD-SI reduced the maximum inversion relative error from 46.22 % to 9.53 %, while the mean inversion relative error was reduced from 11.09 % to 3.90 %.

Data-Driven Inverse Reinforcement Learning Control for Linear Multiplayer Games.

This article proposes a data-driven inverse reinforcement learning (RL) control algorithm for nonzero-sum multiplayer games in linear continuous-time differential dynamical systems. The inverse RL problem in the games is solved by a learner reconstructing the unknown expert players' cost functions from demonstrated expert's optimal state and control input trajectories. The learner, thus, obtains the same control feedback gains and trajectories as the expert, only using data along system trajectories without knowing system dynamics. This article first proposes a model-based inverse RL policy iteration framework that has: 1) policy evaluation step for reconstructing cost matrices using Lyapunov functions; 2) state-reward weight improvement step using inverse optimal control (IOC); and 3) policy improvement step using optimal control. Based on the model-based policy iteration algorithm, this article further develops an online data-driven off-policy inverse RL algorithm without knowing any knowledge of system dynamics or expert control gains. Rigorous convergence and stability analysis of the algorithms are provided. It shows that the off-policy inverse RL algorithm guarantees unbiased solutions while probing noises are added to satisfy the persistence of excitation (PE) condition. Finally, two different simulation examples validate the effectiveness of the proposed algorithms.

Inverse Value Iteration and Q -Learning: Algorithms, Stability, and Robustness.

This article proposes a data-driven model-free inverse Q -learning algorithm for continuous-time linear quadratic regulators (LQRs). Using an agent's trajectories of states and optimal control inputs, the algorithm reconstructs its cost function that captures the same trajectories. This article first poses a model-based inverse value iteration scheme using the agent's system dynamics. Then, an online model-free inverse Q -learning algorithm is developed to recover the agent's cost function only using the demonstrated trajectories. It is more efficient than the existing inverse reinforcement learning (RL) algorithms as it avoids the repetitive RL in inner loops. The proposed algorithms do not need initial stabilizing control policies and solve for unbiased solutions. The proposed algorithm's asymptotic stability, convergence, and robustness are guaranteed. Theoretical analysis and simulation examples show the effectiveness and advantages of the proposed algorithms.

Iterative eigensolver using fixed-point photonic primitive.

Photonic computing has potential advantages in speed and energy consumption yet is subject to inaccuracy due to the limited equivalent bitwidth of the analog signal. In this Letter, we demonstrate a configurable, fixed-point coherent photonic iterative solver for numerical eigenvalue problems using shifted inverse iteration. The photonic primitive can accommodate arbitrarily sized sparse matrix-vector multiplication and is deployed to solve eigenmodes in a photonic waveguide structure. The photonic iterative eigensolver does not accumulate errors from each iteration, providing a path toward implementing scientific computing applications on photonic primitives.

A robust numerical strategy for finding surface waves in flows of non-Newtonian liquids

Gravity-driven flows of liquid films are frequent in nature and industry, such as in landslides, lava flow, cooling of nuclear reactors, and coating processes. In many of these cases, the liquid is non-Newtonian and has particular characteristics. In this paper, we analyze numerically the temporal stability of films of non-Newtonian liquids falling by gravity, on the onset of instability. The liquid flows over an incline, where surface waves appear under certain conditions, and we do not fix a priori its rheological behavior. For that, we made used of the Carreau–Yasuda model without assigning specific values to its constants, and we compute general stability solutions. The numerical strategy is based on expansions of Chebyshev polynomials for discretizing the Orr–Sommerfeld equation and boundary conditions, and a Galerkin method for solving the generalized eigenvalue problem. In addition, an Inverse Iteration method was implemented to increase accuracy and improve computational time. The result is a robust and light numerical tool capable of finding the critical conditions for different types of fluids, which we use to analyze some key fluids. We show that the outputs of the general code match previous solutions obtained for specific computations. Besides increasing our knowledge on surface-wave instabilities in non-Newtonian liquids, our findings provide a new tool for obtaining comprehensive solutions on the onset of instability.

Physics-constrained neural network for solving discontinuous interface K-eigenvalue problem with application to reactor physics

Machine learning-based modeling of reactor physics problems has attracted increasing interest in recent years. Despite some progress in one-dimensional problems, there is still a paucity of benchmark studies that are easy to solve using traditional numerical methods albeit still challenging using neural networks for a wide range of practical problems. We present two networks, namely the Generalized Inverse Power Method Neural Network (GIPMNN) and Physics-Constrained GIPMNN (PC-GIPIMNN) to solve K-eigenvalue problems in neutron diffusion theory. GIPMNN follows the main idea of the inverse power method and determines the lowest eigenvalue using an iterative method. The PC-GIPMNN additionally enforces conservative interface conditions for the neutron flux. Meanwhile, Deep Ritz Method (DRM) directly solves the smallest eigenvalue by minimizing the eigenvalue in Rayleigh quotient form. A comprehensive study was conducted using GIPMNN, PC-GIPMNN, and DRM to solve problems of complex spatial geometry with variant material domains from the field of nuclear reactor physics. The methods were compared with the standard finite element method. The applicability and accuracy of the methods are reported and indicate that PC-GIPMNN outperforms GIPMNN and DRM.

Neural networks based on power method and inverse power method for solving linear eigenvalue problems

Neural networks based on power method and inverse power method for solving linear eigenvalue problems

Musculoskeletal simulation of professional ski jumpers during take-off considering aerodynamic forces.

Introduction: Musculoskeletal simulation has been widely used to analyze athletes' movements in various competitive sports, but never in ski jumping. Aerodynamic forces during ski jumping take-off have been difficult to account for in dynamic simulation. The purpose of this study was to establish an efficient approach of musculoskeletal simulation of ski jumping take-off considering aerodynamic forces and to analyze the muscle function and activity. Methods: Camera-based marker-less motion capture was implemented to measure the take-off kinematics of eight professional jumpers. A suitable full-body musculoskeletal model was constructed for the simulation. A method based on inverse dynamics iteration was developed and validated to estimate the take-off ground reaction force. The aerodynamic forces, which were calculated based on body kinematics and computational fluid dynamics simulations, were exerted on the musculoskeletal model as external forces. The activation and joint torque contributions of lower extremity muscles were calculated through static optimization. Results: The estimated take-off ground reaction forces show similar trend with the results from past studies. Although overall inconsistencies between simulated muscle activation and EMG from previous studies were observed, it is worth noting that the activation of the tibialis anterior, gluteus maximus, and long head of the biceps femoris was similar to specific EMG results. Among lower extremity extensors, soleus, vastus lateralis, biceps femoris long head, gluteus maximus, and semimembranosus showed high levels of activation and joint extension torque contribution. Discussion: Results of this study advanced the understanding of muscle action during ski jumping take-off. The simulation approach we developed may help guide the physical training of jumpers for improved take-off performance and can also be extended to other phases of ski jumping.

A multigrid discretization scheme based on the shifted inverse iteration for the Steklov eigenvalue problem in inverse scattering

Abstract Numerical methods for computing Steklov eigenvalues have attracted the attention of academia for their important physical background and wide applications. In this article we discuss the multigrid discretization scheme based on the shifted inverse iteration for the Steklov eigenvalue problem in inverse scattering, and give the error estimation of the proposed scheme. In addition, on the basis of the a posteriori error indicator, we design an adaptive multigrid algorithm. Finally, we present numerical examples to show the efficiency of the proposed scheme.

Magnetotelluric Deep Learning Forward Modeling and Its Application in Inversion

Magnetotelluric (MT) inversion and forward modeling are closely linked. The optimization and iteration processes of the inverse algorithm require frequent calls to forward modeling. However, traditional numerical simulations for forward modeling are computationally expensive; here, deep learning (DL) networks can simulate forward modeling and significantly improve forward speed. Applying DL for forward modeling in inversion problems requires a high-precision network capable of responding to fine changes in the model to achieve high accuracy in inversion optimization. Most existing MT studies have used a convolutional neural network, but this method is limited by the receptive field and cannot extract global feature information. In contrast, the Mix Transformer has the ability to globally model and extract features. In this study, we used a Mix Transformer to hierarchically extract feature information, adopted a multiscale approach to restore feature information to the decoder, and eliminated the skip connection between the encoder and decoder. We designed a forward modeling network model (MT-MitNet) oriented toward inversion. A sample dataset required for DL forward was established using the forward data generated from the traditional inverse calculation iteration process. The trained network quickly and accurately calculates the forward response. The experimental results indicate a high agreement between the forward results of MT-MitNet and those obtained with traditional methods. When MT-MitNet replaces the forward computation in traditional inversion, the inversion results obtained with it are also highly in agreement with the traditional inversion results. Importantly, under the premise of ensuring high accuracy, the forward speed of MT-MitNet is hundreds of times faster than that of traditional inversion methods in the same process.

Off-policy inverse Q-learning for discrete-time antagonistic unknown systems

This paper proposes a data-driven model-free inverse reinforcement learning (RL) algorithm to reconstruct the unknown cost function of the demonstrated discrete-time (DT) dynamical systems with antagonistic disturbances. We propose an inverse RL policy iteration scheme that uses system dynamics and the input policies, for deriving our main result of a data-driven off-policy inverse Q-learning algorithm using only demonstrated trajectories of the antagonistic system without knowing system dynamics and the control policy gain. This data-driven algorithm consists of Q-function evaluation, state-penalty weight improvement, and action policies update. We guarantee unbiased estimates in the data-driven algorithm when exploration noises exist for the persistence of the excitation. An example verifies the proposed algorithm.

Computing vibrational energy levels using a canonical polyadic tensor method with a fixed rank and a contraction tree.

In this paper, we use the previously introduced Canonical Polyadic (CP)-Multiple Shift Block Inverse Iteration (MSBII) eigensolver [S. D. Kallullathil and T. Carrington, J. Chem. Phys. 155, 234105 (2021)] in conjunction with a contraction tree to compute vibrational spectra. The CP-MSBII eigensolver uses the CP format. The memory cost scales linearly with the number of coordinates. A tensor in CP format represents a wavefunction constrained to be a sum of products (SOP). An SOP wavefunction can be made more accurate by increasing the number of terms, the rank. When the required rank is large, the runtime of a calculation in CP format is long, although the memory cost is small. To make the method more efficient, we break the full problem into pieces using a contraction tree. The required rank for each of the sub-problems is small. To demonstrate the effectiveness of the ideas, we computed vibrational energy levels of acetonitrile (12-D) and ethylene oxide (15-D).

The dependency of spectral gaps on the convergence of the inverse iteration for a nonlinear eigenvector problem

In this paper, we consider the generalized inverse iteration for computing ground states of the Gross–Pitaevskii eigenvector (GPE) problem. For that we prove explicit linear convergence rates that depend on the maximum eigenvalue in magnitude of a weighted linear eigenvalue problem. Furthermore, we show that this eigenvalue can be bounded by the first spectral gap of a linearized Gross–Pitaevskii operator, recovering the same rates as for linear eigenvector problems. With this we establish the first local convergence result for the basic inverse iteration for the GPE without damping. We also show how our findings directly generalize to extended inverse iterations, such as the Gradient Flow Discrete Normalized (GFDN) proposed in [W. Bao and Q. Du, Computing the ground state solution of Bose–Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput. 25 (2004) 1674–1697] or the damped inverse iteration suggested in [P. Henning and D. Peterseim, Sobolev gradient flow for the Gross–Pitaevskii eigenvalue problem: Global convergence and computational efficiency, SIAM J. Numer. Anal. 58 (2020) 1744–1772]. Our analysis also reveals why the inverse iteration for the GPE does not react favorably to spectral shifts. This empirical observation can now be explained with a blow-up of a weighting function that crucially contributes to the convergence rates. Our findings are illustrated by numerical experiments.

Analytical model for tension-induced residual stress of pre-stress filament hoop wound composite rings by inverse iteration method

Analytical model for tension-induced residual stress of pre-stress filament hoop wound composite rings by inverse iteration method

Nonlinear Eigenvalue Methods for Linear Pointwise Stability of Nonlinear Waves

We propose an iterative method to find pointwise exponential growth rates in linear problems posed on essentially one-dimensional domains. Such pointwise growth rates capture pointwise stability and instability in extended systems and arise as spectral values of a family of matrices that depends on a spectral parameter, obtained via a scattering-type problem. Different from methods in the literature that rely on computing determinants of this nonlinear matrix pencil, we propose and analyze an inverse power method that allows one to locate robustly the closest spectral value to a given reference point in the complex plane. The method finds branch points, eigenvalues, and resonance poles without a priori knowledge.

GCGE: a package for solving large scale eigenvalue problems by parallel block damping inverse power method

GCGE: a package for solving large scale eigenvalue problems by parallel block damping inverse power method