A bstract Building on the duality between Krylov complexity and geodesic length in Jackiw-Teitelboim and sine-dilaton gravity, we develop a precise holographic dictionary for quantities in the Krylov subspace of the double-scaled Sachdev-Ye-Kitaev model (DSSYK). First, we demonstrate that the growth rate of Krylov state complexity corresponds to the wormhole velocity, and show that its expectation value in coherent states serves as a boundary diagnostic of firewall-like structures via bulk reconstruction. We also delineate an alternative bulk description in terms of the proper momentum of an infalling particle at early times, establishing a threefold duality between the Krylov complexity growth rate, wormhole velocity, and proper momentum, with clear regimes of validity. Beyond the first moments, we argue that higher-order Krylov complexities capture connected bulk contributions encoded by replica wormholes, while the logarithmic variant probes the replica saddle structure. Finally, within a third-quantized setting incorporating baby universes, we show that the Krylov entropy equals the von Neumann entropy of the parent-geometry density matrix obtained after tracing out baby universes, thereby quantifying information flow into the baby universe sector. Together, these results elevate Krylov-space observables to sharp probes of bulk dynamics and topology in ensemble-averaged 2D gravity.

Abstract We consider iterative methods for solving linear ill-posed problems with compact operator and right-hand side only available via noise-polluted measurements. Conjugate gradients (CGs) applied to the normal equations with an appropriate stopping rule and CG applied to the system solving for a Tikhonov-regularized solution ( CGtikh ) ( A ∗ A + c I X ) x ( δ , c ) = A ∗ y δ are closely related regularization methods that build iterates from the same Krylov subspaces. In this work, we show that the CGtikh iterate can be expressed as x m ( δ , c ) = ∑ i = 1 m γ i ( m ) ( c ) z i ( m ) v i , where { γ i ( m ) ( c ) } i = 1 m are functions of the Tikhonov parameter c and x m ( δ ) = ∑ i = 1 m z i ( m ) v i is the m th CG iterate. We call these functions Lanczos filters , and they can be shown to have decay properties as c → ∞ with the speed of decay increasing with i . This has the effect of filtering out the contribution of the later terms of the CG iterate. The filters can be constructed using quantities defined via recursions at each iteration. We demonstrate with numerical experiments that good parameter choices correspond to appropriate damping of the Lanczos vectors. The filtration approach also provides a platform for further development of parameter choice rules, and similar representations may hold for other hybrid iterative schemes.

Reduced order modeling of commensurate fractional order systems using Arnoldi–Krylov subspace optimization technique

The HOC-Krylov Solver is a Fortran-based computational tool designed to efficiently solve Poisson’s equation using high-order compact (HOC) finite difference schemes integrated with Krylov subspace iterative methods. This combination enhances both the accuracy and convergence speed of solutions in two-dimensional domains. The solver is particularly suited for applications in computational fluid dynamics and electrostatics, where precise and rapid solutions are essential. • Integrates high-order compact finite difference schemes with multiple Krylov solvers to deliver highly accurate and efficient solutions of large Poisson systems. • Supports flexible sparse matrix storage (CSR, MSR, ELLPACK-ITPACK, DIA) with optimized MVM routines, enabling systematic performance evaluation across solver–storage– preconditioner combinations. • Modular Fortran architecture enables easy extension, facilitating research in CFD, electrostatics, and vorticity–streamfunction formulations while providing benchmarking and post-processing capabilities.

Speeding up an unsteady flow simulation by adaptive BDDC and Krylov subspace recycling

We describe an approach for analyzing biological networks using rows of the Krylov subspace of the adjacency matrix. Specifically, we explore the scenario where the Krylov subspace matrix is computed via power iteration using a non-random and potentially non-uniform initial vector that captures a specific biological state or perturbation. In this case, the rows the Krylov subspace matrix (i.e., Krylov trajectories) carry important functional information about the network nodes in the biological context represented by the initial vector. We demonstrate the utility of this approach for community detection and perturbation analysis using the C. Elegans neural network.

In complex terrain modern day geophysical electromagnetic work or microwave engineering needs a good model of the electro magnetic field to succeed. The outdated numerical calculations relying equally distributed reguar grid finite diff, and un-even grounds like waves, faults, all sorts of messed up 3D anomalous distortions, it would greatly warp the local EM field response data points in ALL but maybe some last steps before 'inverted', and trying to interpret that result. In order to solve the bottleneck problem, we explore and realize a 3D vector finite element forward modeling method of electromagnetism field with unstructured tetrahedron grids. Utilizes very closely conformed unstructured grids to closely approximate complicated geological model and also has vector basic function based on edge element to remove these kinds of pseudo solution that you have when you do normal node FEM method. This article describes how to obtain the weak form integral equation of Maxwell’s equation of the electric field curl and introduces a pre-process Krylov subspace iteration solution strategy for solving big sparse complex linear equations. It is also the proof for truth of both its accuracy and strength in converging, it has experienced a lot of easy numerical check upon simple halfspace, roughest terrain and even most complicated bodies with deeply buried 3D high conductivity targets. The article can be considered true 3dfoward Model Theory Support and algorithm Basis To Get True Data Refinements And Exact Location Of Any Complex Geologic Body Within China while carrying out an Electromagnetic Investigation.

Abstract We introduce Sparse Matrix Preconditioner for Iterative Refinement and Inversion Toolkit (SPIRIT), a general-purpose sparse matrix solver with a novel preconditioner designed for the large, sparse, and ill-conditioned Jacobian systems that arise in r -process nucleosynthesis simulations. The preconditioner employs a dual-threshold filtering strategy applied before factorization, yielding highly sparse yet stable approximations. Coupled with the Krylov subspace solver biconjugate gradient-stabilized method, SPIRIT dramatically accelerates the matrix inversion step during reaction network evolution. In tests on 500 Jacobian matrices of dimension 7836 × 7836 generated in an r -process run, SPIRIT outperformed traditional incomplete LU preconditioners by several orders of magnitude in both runtime and residual accuracy. Benchmarks against Intel Math Kernel Library (Intel MKL) show that SPIRIT reduced matrix inversion costs from 68% to 16% of total runtime while maintaining agreement with MKL solutions to within 10 −6 error. Final abundance distributions matched those from standard solvers across all astrophysical relevant nuclear species. SPIRIT also performed reliably for smaller test cases, including trivial and small networks, as well as X-ray burst scenarios (with minor deviations). We have integrated SPIRIT into the network code SkyNet, providing a fast, robust, and open-source alternative for high-performance simulations in nuclear astrophysics and other scientific applications requiring efficient solutions of large, sparse, and ill-conditioned linear systems.

With the increasing demand for both accuracy and efficiency in transient electromagnetic (TEM) simulations, conventional 3-D forward modeling methods face growing challenges. This study presents a high-accuracy and high-efficiency 3-D forward modeling approach that combines the spectral-element method (SEM) with a model order reduction (MOR) scheme. High-order orthogonal basis functions are employed, and the computational domain is discretized in a finite-element manner to improve modeling accuracy. During element-level analysis, a reduced-integration strategy is introduced to enhance the sparsity of the double-curl and conductivity matrices, thereby reducing the computational time and memory consumption required for matrix assembly. For temporal treatment, a shift-and-invert Krylov (SAI-Krylov) subspace algorithm is adopted: the basis and projection matrices are constructed using only one matrix factorization and tens of back-substitutions, after which low-dimensional matrix exponential functions are evaluated to efficiently obtain electromagnetic responses at arbitrary times. Comparisons with other numerical methods demonstrate the superior efficiency and accuracy of the proposed approach. Finally, simulations on a 3-D sulfide ore-body model are performed to investigate TEM field propagation for both galvanic and loop sources, confirming the capability of the method to model electromagnetic responses in complex geological settings.

In order to overcome the computational challenges associated with block preconditioners for Krylov subspace methods, particularly those arising from Schur complement systems, this paper proposes an improved new block (INB) preconditioner for solving 3 × 3 block saddle point problems. A detailed semi-convergence analysis of the iterative scheme induced by the INB preconditioner is provided. Moreover, the spectral properties of the preconditioned matrix are analyzed, revealing strong eigenvalue clustering around one. Efficient formulas for selecting quasi-optimal parameters are derived based on Frobenius-norm minimization. Extensive numerical experiments demonstrate that the proposed INB preconditioner significantly reduces iteration counts and CPU time compared with several existing block preconditioners.

The Krylov subspace expansion is a workhorse method for sparse numerics that has been increasingly explored as source of physical insight into many-body dynamics in recent years. In this work we revisit the venerable Anderson model of localization in dimensions $d=1, 2, 3, 4$ to construct local integrals of motion (LIOM) in Krylov space. These appear as zero eigenvalue edge states of an effective hopping problem in the Krylov superoperator subspace and can be analytically constructed given the Lanczos coefficients. We exploit this idea, focusing on $d=3$, to study the manifestation of the disorder driven Anderson transition in the anatomy of LIOMs. We find that the increasing complexity of the Krylov operators results in a suppression of the fluctuations of the Lanczos coefficients. As such, one can study the phenomenology of the integrals of motion in the disorder averaged Krylov chain. We find edge states localized on vanishing fraction of Krylov space (of dimension $D_K=V^2$ for cubes of volume $V$), both in localized and extended phases. Importantly, in the localized phase, disorder induces powerlaw decaying dimerization in the (Krylov) hopping problem, producing stretched exponential decay of the LIOMs in Krylov space with a stretching exponent $1/2d$. Metallic LIOMs are completely delocalized albeit across only $\propto \sqrt{D_K}$ states. Critical LIOMs exhibit powerlaw decay with an exponent matching the expected value of $0.29$.

The high complexity of many-body quantum dynamics means that essentially all analytical or numerical approaches either exploit special structure or are approximate in nature. One such approach—the memory function formalism—involves a carefully chosen split into “fast” and “slow” modes. An approximate model for the fast modes can then be used to solve for Green’s functions G ( z ) of the slow modes, and the success of this approach depends on the accuracy of the fast space approximation. Using a formulation in operator Krylov space known as the recursion method, we prove the emergence of a universal random matrix description of the fast mode dynamics. This is captured by the “level- n Green’s function” G n ( z ) , which we show approaches universal scaling forms in the “fast limit” n → ∞ . Notably, this emergent universality can occur in both chaotic and nonchaotic systems, provided their spectral functions are sufficiently smooth. This universality of G n ( z ) turns out to be precisely analogous to the universality of eigenvalue correlations in random matrix theory (RMT), even though there is present in the Hamiltonian. Concretely, at finite z we show that G n ( z ) approaches the Wigner semicircle law, while if G ( z ) is the Green’s function of certain hydrodynamical variables, we show that at low frequencies G n ( z ) is instead governed by the Bessel universality class from RMT. As an application of this universality, we give a new numerical method, the , for approximating spectral functions, including hydrodynamic transport data, from a finite number of Lanczos coefficients. Our proof involves a map to a Riemann-Hilbert problem which we solve using a steepest-descent-type method, rigorously controlled in the n → ∞ limit. Via the steepest-descent procedure, we are led to a related Coulomb gas optimization problem, and we discuss how a recent conjecture—the “Operator Growth Hypothesis”—implies that chaotic operator dynamics can generically be identified with the critical point of a confinement transition in this Coulomb gas. These results elevate the recursion method from a useful numerical technique to a theoretically principled framework with universal content.

ABSTRACT This study proposes a Jacobian‐free Newton‐Krylov approach for finite‐volume solid mechanics. Traditional Newton‐based approaches require explicit Jacobian matrix formation and storage, which can be computationally expensive and memory‐intensive. In contrast, Jacobian‐free Newton‐Krylov methods approximate the Jacobian's action using finite differences, combined with Krylov subspace solvers such as the generalised minimal residual method (GMRES), enabling seamless integration into existing segregated finite‐volume frameworks without major code refactoring. This work proposes and benchmarks the performance of a compact‐stencil Jacobian‐free Newton‐Krylov method against a conventional segregated approach on a suite of test cases that span varying geometric dimensions, nonlinearities, dynamic responses and material behaviours. Key metrics, including computational cost, memory efficiency and robustness, are evaluated, along with the influence of preconditioning strategies and stabilisation scaling. Results show that the proposed Jacobian‐free Newton‐Krylov method outperforms the segregated approach in all linear and nonlinear elastic cases, achieving order‐of‐magnitude speedups in many instances; however, divergence is observed in elastoplastic cases, highlighting areas for further development. It is found that preconditioning choice affects performance: a LU direct solver is fastest for small to moderately sized cases, while a multigrid method is more effective for larger problems. The findings demonstrate that Jacobian‐free Newton‐Krylov methods are promising for advancing finite‐volume solid mechanics simulations, particularly for existing segregated frameworks where minimal modifications enable their adoption. The described implementations are available in the solids4foam toolbox for OpenFOAM, inviting the community to explore, extend and compare these procedures.

We study the acceleration and accuracy resulting from the application of different Rosenbrock–Euler exponential integrators of orders 2, 3, and 4 for temporal integration of partial differential equations corresponding to the quantitative phase-field modeling of pure elements solidification. The exact Jacobian of differential operators is computed analytically using the continuous differentiation approach. After spatial discretization, we have large-scale nonlinear ordinary differential equations. Compared to linearly implicit time integration approaches that require at least solving one linear system of equations at each time step, our schemes do not include any implicit phase. Instead, it is essential to compute at least one matrix function vector product per time step, which is extremely computationally and memory demanding using direct approaches. An efficient implementation of the Krylov subspace method is exploited to approximate the action of a matrix function on a desired vector to address this challenge. The time step size is adaptively calculated based on posteriori error estimations. Numerical experiments confirm the convergence rates of our proposed schemes. According to our numerical results, these exponential time-differencing schemes reveal superior computational performance compared to the Euler explicit method, which is the most popular approach for time integration of phase-field solidification equations. Moreover, when a medium to high level of accuracy is desired, these schemes are orders of magnitude faster than this scheme. In addition, using exponential time differencing with adaptive time stepping allows us to use time step size of more than two orders of magnitude larger than that of the stable Euler explicit method, while having either accuracy of the same level or superior.

ABSTRACT We introduce the Tensorized-and-Restricted Krylov (TReK) method, a simple and efficient algorithm for estimating covariance tensors with large observational sizes. TReK extends the Krylov subspace method to incorporate range restrictions, enabling its use in a variety of covariance smoothing applications. By leveraging tensor-matrix operations, it achieves significant improvements in both computational speed and memory cost, improving over existing methods by an order of magnitude. TReK ensures finite-step convergence in the absence of rounding errors and converges fast in practice, making it well-suited for large-scale problems. The algorithm is tensor-free and highly flexible, supporting a wide range of forward and projection tensors.

The fermionic t-J model has been widely recognized as a canonical model for broad range of strongly correlated phases, particularly the high-T_{c} superconductor. Simulating this model with controllable quantum platforms offers new possibilities to probe high-T_{c} physics, yet suffers challenges. Here we propose a novel scheme to realize a highly tunable extended t-J model in a programmable Rydberg-dressed tweezer array. Through engineering the Rydberg-dressed dipole-dipole interaction and intertweezer couplings, the fermionic t-J model with independently tunable exchange and hopping couplings is achieved. With the high tunability, we explore quantum many-body dynamics in the large J/t limit, a regime well beyond the conventional optical lattices and cuprates, and predict an unprecedented many-body self-pinning effect related to the local quantum entanglement with emergent conserved quantities. The self-pinning effect leads to novel nonthermal quantum many-body dynamics, which violates eigenstate thermalization hypothesis in Krylov subspace. Our prediction opens a new horizon in exploring exotic quantum many-body physics with t-J model, and shall also make a step toward simulating the high-T_{c} physics in neutral atom systems.

Conventional three-dimensional (3D) inversion algorithms for the transient electromagnetic method (TEM) require time-domain discretization in both forward and adjoint modeling. These algorithms start from the initial time and compute the electric field iteratively at each new time step, which involve repeatedly solving large-scale systems of linear equations and thus cause a substantial computational cost. In contrast, the Krylov subspace-based model order reduction technique computes the 3D time-domain electromagnetic response with just a one-time reduction process, allowing for the electric field at any time to be obtained by solving only a small-scale matrix exponential problem, significantly improving the efficiency of forward modeling. However, this approach has not yet been widely utilized in 3D TEM inversion. In this study, the rational Krylov subspace-based model order reduction technique is applied to both forward and adjoint modeling. The data misfit residuals at each observation time channel are treated as virtual sources for the adjoint problem, enabling the use of the rational Krylov subspace method to solve the corresponding adjoint field. The gradient is evaluated using results from both the forward and adjoint modeling, and the model is updated via the L-BFGS method. Throughout the gradient calculation, only two matrix factorizations and a limited number of backward substitutions for right-hand sides are required in total for all necessary forward and adjoint modeling, which substantially reduces the computation burden. The proposed algorithm is validated using synthetic data. Results show that the presented 3D TEM inversion algorithm using the rational Krylov subspace technique achieves satisfactory inversion accuracy while significantly improving computational efficiency compared to inversion methods that employ conventional time-stepping approaches. Therefore, this work can provide valuable insights for further application of rational Krylov subspace methods in TEM data inversion.

Abstract This research investigates the improvement of expansive soils by incorporating fly ash, focusing on its impact on soil engineering properties such as plasticity index (PI), swelling potential (SP), California bearing ratio (CBR), maximum dry density (MDD), and unconfined compressive strength (UCS). The different mix proportions comprised of CM (control mix), FSM 1, FSM 2, FSM 3, FSM 4, and FSM 5 (where FSM is fly ash stabilized mix), were subjected to tests after a curing period of 7, 14, and 28 days. Using the experimental result, FSM 4 was determined to be the most effective mix regarding UCS, PI, SP, CBR, and MDD. The response surface methodology (RSM) was used to model the five properties of the most effective mix, producing R 2 values of 0.9331 for UCS, 0.9189 for CBR, 0.9368 for PI, 0.9286 for SP, and 0.9421 for MDD in order to increase prediction accuracy. The hybrid quantum neural network–Krylov subspace optimization model (QNN‐KSO) was introduced and proved to be the best out of the RSM, deep neural network–grey wolf optimization, and random forest–artificial bee colony out of all six criteria, as it resulted in more reliable and accurate prediction of UCS, CBR, PI, SP, and MDD. The hybrid QNN‐KSO model produced excellent performance while minimizing the root mean squared error while achieving an R 2 value of 0.99 making this an improved modeling technique for soil stabilization.

Model-based reconstruction plays a key role in compressed sensing (CS) MRI, as it incorporates effective image regularizers to improve the quality of reconstruction. The Plug-and-Play and Regularization-by-Denoising frameworks leverage advanced denoisers (e.g., convolutional neural network (CNN)-based denoisers) and have demonstrated strong empirical performance. However, their theoretical guarantees remain limited, as practical CNNs often violate key assumptions. In contrast, gradient-driven denoisers achieve competitive performance, and the required assumptions for theoretical analysis are easily satisfied. However, solving the associated optimization problem remains computationally demanding. To address this challenge, we propose a generalized Krylov subspace method (GKSM) to solve the optimization problem efficiently. Moreover, we also establish rigorous convergence guarantees for GKSM in nonconvex settings. Numerical experiments on CS MRI reconstruction with spiral and radial acquisitions validate both the computational efficiency of GKSM and the accuracy of the theoretical predictions. The proposed optimization method is applicable to any linear inverse problem.

In this paper, we present a finite element method (FEM) framework enhanced by an operator-adapted wavelet decomposition algorithm designed for the efficient analysis of multiscale electromagnetic problems. Usual adaptive FEM approaches, while capable of achieving the desired accuracy without requiring a complete re-meshing of the computational domain, inherently couple different resolution levels. This coupling requires recomputation of coarser-level solutions whenever finer details are added to improve accuracy, resulting in substantial computational overhead. Our proposed method addresses this issue by <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">decoupling</i> the different resolution levels. This feature enables independent computations at each scale that can be incorporated into the solutions to improve accuracy whenever needed, without requiring re-computation of coarser-level solutions. The main algorithm is hierarchical, constructing solutions from finest to coarser levels through a series of sparse matrix-vector multiplications. Due to its sparse nature, the overall computational complexity of the algorithm is nearly linear. Moreover, Krylov subspace iterative solvers are employed to solve the final linear equations, with ILU preconditioners that enhance solver convergence and maintain overall computational efficiency. The proposed algorithm is applicable to both structured and unstructured meshes. Several two-dimensional numerical experiments demonstrate its high precision and nearly <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(N)</i> computational complexity.