math.NA

arXiv:2501.06886v1 Announce Type: new Abstract: We derive some identities and relations and extremal problems and minimization and Fourier development involving of integral Legendre polynomials.

arXiv:2501.07323v1 Announce Type: new Abstract: This work focuses on developing high-order energy-stable schemes for wave-dominated problems in closed domains using staggered finite-difference summation-by-parts (SBP FD) operators. We extend the previously presented uniform staggered grid SBP FD approach to non-orthogonal curvilinear multi-block grids and derive new higher-order approximations. The combination of Simultaneous-Approximation-Terms (SAT) and projection method is proposed for the treatment of interface conditions on a staggered grid. This reduces approximation stiffness and mitigates stationary wave modes of pure SAT approach. Also, energy-neutral discrete Coriolis terms operators are presented. The proposed approach is tested using the linearized shallow water equations on a rotating sphere, a testbed relevant for ocean and atmospheric dynamics. Numerical experiments show significant improvements in capturing wave dynamics compared to collocated SBP FD methods.

arXiv:2501.06815v1 Announce Type: new Abstract: We propose an arbitrarily high-order globally divergence-free entropy stable nodal discontinuous Galerkin (DG) method to directly solve the conservative form of the ideal MHD equations using appropriate quadrature rules. The method ensures a globally divergence-free magnetic field by updating it at interfaces with a constraint-preserving formulation [5] and employing a novel least-squares reconstruction technique. Leveraging this property, the semi-discrete nodal DG scheme is proven to be entropy stable. To handle the problems with strong shocks, we introduce a novel limiting strategy that suppresses unphysical oscillations while preserving the globally divergence-free property. Numerical experiments verify the accuracy and efficacy of our method.

arXiv:2501.06824v1 Announce Type: new Abstract: Nitsche's method is a numerical approach that weakly enforces boundary conditions for partial differential equations. In recent years, Nitsche's method has experienced a revival owing to its natural application in modern computational methods, such as the cut and immersed finite element methods. This study investigates Nische's methods based on an anisotropic weakly over-penalized symmetric interior penalty method for Poisson and Stokes equations on convex domains. As our primary contribution, we provide a new proof for the consistency term, which allows us to obtain an estimate of the anisotropic consistency error. The key idea of the proof is to apply the relationship between the Crouzeix and Raviart finite element space and the Raviart--Thomas finite element space. We present the error estimates in the energy norm on anisotropic meshes. We compared the calculation results for the anisotropic mesh partitions in the numerical experiments.

arXiv:2501.07117v1 Announce Type: new Abstract: Optimal-order convergence in the $H^1$ norm is proved for an arbitrary Lagrangian-Eulerian interface tracking finite element method for the sharp interface model of two-phase Navier-Stokes flow without surface tension, using high-order curved evolving mesh. In this method, the interfacial mesh points move with the fluid's velocity to track the sharp interface between two phases of the fluid, and the interior mesh points move according to a harmonic extension of the interface velocity. The error of the semidiscrete arbitrary Lagrangian-Eulerian interface tracking finite element method is shown to be $O(h^k)$ in the $L^\infty(0, T; H^1(\Omega))$ norm for the Taylor-Hood finite elements of degree $k \ge 2$. This high-order convergence is achieved by utilizing the piecewise smoothness of the solution on each subdomain occupied by one phase of the fluid, relying on a low global regularity on the entire moving domain. Numerical experiments illustrate and complement the theoretical results.

arXiv:2501.07210v1 Announce Type: new Abstract: This paper studies the low-rank property of the inverse of a class of large-scale structured matrices in the tensor-train (TT) format, which is typically discretized from differential operators. An interesting question that we are concerned with is: Does the inverse of the large-scale structured matrix still admit the low-rank TT representation with guaranteed accuracy? In this paper, we provide a computationally verifiable sufficient condition such that the inverse matrix can be well approximated in a low-rank TT format. It not only answers what kind of structured matrix whose inverse has the low-rank TT representation but also motivates us to develop an efficient TT-based method to compute the inverse matrix. Furthermore, we prove that the inverse matrix indeed has the low-rank tensor format for a class of large-scale structured matrices induced by differential operators involved in several PDEs, such as the Poisson, Boltzmann, and Fokker-Planck equations. Thus, the proposed algorithm is suitable for solving these PDEs with massive degrees of freedom. Numerical results on the Poisson, Boltzmann, and Fokker-Planck equations validate the correctness of our theory and the advantages of our methodology.

arXiv:2501.06621v1 Announce Type: new Abstract: This paper discusses our recent generalized optimal algebraic multigrid (AMG) convergence theory applied to the steady-state Stokes equations discretized using Taylor-Hood elements ($\pmb{ \mathbb{P}}_2/\mathbb{P}_{1}$). The generalized theory is founded on matrix-induced orthogonality of the left and right eigenvectors of a generalized eigenvalue problem involving the system matrix and relaxation operator. This framework establishes a rigorous lower bound on the spectral radius of the two-grid error-propagation operator, enabling precise predictions of the convergence rate for symmetric indefinite problems, such as those arising from saddle-point systems. We apply this theory to the recently developed monolithic smooth aggregation AMG (SA-AMG) solver for Stokes, constructed using evolution-based strength of connection, standard aggregation, and smoothed prolongation. The performance of these solvers is evaluated using additive and multiplicative Vanka relaxation strategies. Additive Vanka relaxation constructs patches algebraically on each level, resulting in a nonsymmetric relaxation operator due to the partition of unity being applied on one side of the block-diagonal matrix. Although symmetry can be restored by eliminating the partition of unity, this compromises convergence. Alternatively, multiplicative Vanka relaxation updates velocity and pressure sequentially within each patch, propagating updates multiplicatively across the domain and effectively addressing velocity-pressure coupling, ensuring a symmetric relaxation. We demonstrate that the generalized optimal AMG theory consistently provides accurate lower bounds on the convergence rate for SA-AMG applied to Stokes equations. These findings suggest potential avenues for further enhancement in AMG solver design for saddle-point systems.

arXiv:2501.06748v1 Announce Type: new Abstract: We begin by addressing the time-domain full-waveform inversion using the adjoint method. Next, we derive the scaled boundary semi-weak form of the scalar wave equation in heterogeneous media through the Galerkin method. Unlike conventional formulations, the resulting system incorporates variable density and two additional terms involving its spatial derivative. As a result, the coefficient matrices are no longer constant and depend on the radial coordinate, rendering the common solution methods inapplicable. Thus, we introduce a radial discretization scheme within the framework of the scaled boundary finite element method. We employ finite difference approximation, yet the choice underlying our ansatz is made for demonstration purposes and remains flexible. Next, we introduce an algorithmic condensation procedure to compute the dynamic stiffness matrices on the fly. Therefore, we maneuver around the need to introduce auxiliary unknowns. As a result, the optimization problem is structured in a two-level hierarchy. We obtain the Fr\'echet kernel by computing the zero-lag cross-correlations of the forward and adjoint wavefields, and solve the minimization problems iteratively by moving downhill on the cost function hypersurface through the limited-memory BFGS algorithm. The numerical results demonstrate the effectiveness and robustness of the new formulation and show that using the simplified differential equation along with the conventional formulation is highly inferior to applying the complete form of the differential equation. This approach effectively decomposes the computational load into independent local problems and a single coupled global system, making the solution method highly parallelizable. We demonstrate that, with a simple OpenMP implementation using 12 threads on a personal laptop, the new formulation outperforms the existing approach in terms of computation time.

arXiv:2501.06615v1 Announce Type: new Abstract: The Poisson-Boltzmann (PB) model is a widely used implicit solvent model in protein simulations. Although variants, such as the size modified PB and nonlocal modified PB models, have been developed to account for ionic size effects and nonlocal dielectric correlations, no existing PB variants simultaneously incorporate both, due to significant modeling and computational challenges. To address this gap, in this paper, a nonlocal size modified PB (NSMPB) model is introduced and solved using a finite element method for a protein with a three-dimensional molecular structure and an ionic solution containing multiple ion species. In particular, a novel solution decomposition is proposed to overcome the difficulties caused by the increased nonlinearity, nonlocality, and solution singularities of the model. It is then applied to the development of the NSMPB finite element solver, which includes an efficient modified Newton iterative method, an effective damping parameter selection strategy, and good selections of initial iterations. Moreover, the construction of the modified Newton iterative method is mathematically justified. Furthermore, an NSMPB finite element package is developed by integrating a mesh generation tool, a protein data bank file retrieval program, and the PDB2PQR package to simplify and accelerate its usage and application. Finally, numerical experiments are conducted on an ionic solution with four species, proteins with up to 11439 atoms, and irregular interface-fitted tetrahedral box meshes with up to 1188840 vertices. The numerical results confirm the fast convergence and strong robustness of the modified Newton iterative method, demonstrate the high performance of the package, and highlight the crucial roles played by the damping parameter and initial iteration selections in enhancing the method's convergence. The package will be a valuable tool in protein simulations.

arXiv:2501.06520v1 Announce Type: new Abstract: In this paper, by using the core EP inverse and the Drazin inverse which are two well known generalized inverses, a new class of matrices entitled core EP Drazin matrices (shortly, CEPD matrices) is introduced. This class contains the set of all EP matrices and also the set of normal matrices. Some algebraic properties of these matrices are also investigated. Moreover, some results about the Drazin inverse and the core EP inverse of partial isometries are derived, and using them, some conditions for which partial isometries are CEPD, are obtained. To illustrate the main results, some numerical examples are given.

arXiv:2501.06539v1 Announce Type: new Abstract: We construct a Neural Network that approximates the matrix multiplication operator for any activation functions for which there exist a Neural Network which can approximate the scalar multiplication function. In particular, we use the Strassen algorithm for reducing the number of weights and layers needed for such Neural Network. This allows us to define another Neural Network for approximating the inverse matrix operator. Also, by relying on the Galerkin method, we apply those Neural Networks for resolving parametric elliptic PDEs for a whole set of parameters at the same time. Finally, we discuss the improvements with respect to prior results.

arXiv:2501.06328v1 Announce Type: new Abstract: Anisotropic mesh adaptation with Riemannian metrics has proven effective for generating straight-sided meshes with anisotropy induced by the geometry of interest and/or the resolved physics. Within the continuous mesh framework, anisotropic meshes are thought of as discrete counterparts to Riemannian metrics. Ideal, or unit, simplicial meshes consist only of simplices whose edges exhibit unit or quasi-unit length with respect to a given Riemannian metric. Recently, mesh adaptation with high-order (i.e., curved) elements has grown in popularity in the meshing community, as the additional flexibility of high-order elements can further reduce the approximation error. However, a complete and competitive methodology for anisotropic and high-order mesh adaptation is not yet available. The goal of this paper is to address a key aspect of metric-based high-order mesh adaptation, namely, the adequacy between a Riemannian metric and high-order simplices. This is done by extending the notions of unit simplices and unit meshes, central to the continuous mesh framework, to high-order elements. The existing definitions of a unit simplex are reviewed, then a broader definition involving Riemannian isometries is introduced to handle curved and high-order simplices. Similarly, the notion of quasi-unitness is extended to curved simplices to tackle the practical generation of high-order meshes. Proofs of concept for unit and (quasi-)isometric meshes are presented in two dimensions.

arXiv:2501.06369v1 Announce Type: new Abstract: We consider the Stokes-Darcy coupled problem, which models the interaction between free-flow and porous medium flow. By enforcing the normal flux continuity interface condition directly within the finite-element spaces, we establish unified well-posedness results for the coupled system under various boundary condition scenarios. Using the operator preconditioning framework, we develop a parameter-robust preconditioner that avoids the use of fractional operators. Numerical experiments employing both $H(\operatorname{div})$-conforming and nonconforming finite-element methods are presented to confirm the theoretical findings and demonstrate the robustness of the proposed block preconditioners with respect to the physical parameters and mesh size.

arXiv:2501.07201v1 Announce Type: new Abstract: We propose an enhanced zeroth-order stochastic Frank-Wolfe framework to address constrained finite-sum optimization problems, a structure prevalent in large-scale machine-learning applications. Our method introduces a novel double variance reduction framework that effectively reduces the gradient approximation variance induced by zeroth-order oracles and the stochastic sampling variance from finite-sum objectives. By leveraging this framework, our algorithm achieves significant improvements in query efficiency, making it particularly well-suited for high-dimensional optimization tasks. Specifically, for convex objectives, the algorithm achieves a query complexity of O(d \sqrt{n}/\epsilon ) to find an epsilon-suboptimal solution, where d is the dimensionality and n is the number of functions in the finite-sum objective. For non-convex objectives, it achieves a query complexity of O(d^{3/2}\sqrt{n}/\epsilon^2 ) without requiring the computation ofd partial derivatives at each iteration. These complexities are the best known among zeroth-order stochastic Frank-Wolfe algorithms that avoid explicit gradient calculations. Empirical experiments on convex and non-convex machine learning tasks, including sparse logistic regression, robust classification, and adversarial attacks on deep networks, validate the computational efficiency and scalability of our approach. Our algorithm demonstrates superior performance in both convergence rate and query complexity compared to existing methods.

arXiv:2501.06388v1 Announce Type: new Abstract: We present a realizability-preserving numerical method for solving a spectral two-moment model to simulate the transport of massless, neutral particles interacting with a steady background material moving with relativistic velocities. The model is obtained as the special relativistic limit of a four-momentum-conservative general relativistic two-moment model. Using a maximum-entropy closure, we solve for the Eulerian-frame energy and momentum. The proposed numerical method is designed to preserve moment realizability, which corresponds to moments defined by a nonnegative phase-space density. The realizability-preserving method is achieved with the following key components: (i) a discontinuous Galerkin (DG) phase-space discretization with specially constructed numerical fluxes in the spatial and energy dimensions; (ii) a strong stability-preserving implicit-explicit (IMEX) time-integration method; (iii) a realizability-preserving conserved to primitive moment solver; (iv) a realizability-preserving implicit collision solver; and (v) a realizability-enforcing limiter. Component (iii) is necessitated by the closure procedure, which closes higher order moments nonlinearly in terms of primitive moments. The nonlinear conserved to primitive and the implicit collision solves are formulated as fixed-point problems, which are solved with custom iterative solvers designed to preserve the realizability of each iterate. With a series of numerical tests, we demonstrate the accuracy and robustness of this DG-IMEX method.

arXiv:2501.06300v1 Announce Type: new Abstract: We present a tensorization algorithm for constructing tensor train representations of functions, drawing on sketching and cross interpolation ideas. The method only requires black-box access to the target function and a small set of sample points defining the domain of interest. Thus, it is particularly well-suited for machine learning models, where the domain of interest is naturally defined by the training dataset. We show that this approach can be used to enhance the privacy and interpretability of neural network models. Specifically, we apply our decomposition to (i) obfuscate neural networks whose parameters encode patterns tied to the training data distribution, and (ii) estimate topological phases of matter that are easily accessible from the tensor train representation. Additionally, we show that this tensorization can serve as an efficient initialization method for optimizing tensor trains in general settings, and that, for model compression, our algorithm achieves a superior trade-off between memory and time complexity compared to conventional tensorization methods of neural networks.

arXiv:2501.06400v1 Announce Type: new Abstract: We define a digital twin (DT) of a physical system governed by partial differential equations (PDEs) as a model for real-time simulations and control of the system behavior under changing conditions. We construct DTs using the Karhunen-Lo\`{e}ve Neural Network (KL-NN) surrogate model and transfer learning (TL). The surrogate model allows fast inference and differentiability with respect to control parameters for control and optimization. TL is used to retrain the model for new conditions with minimal additional data. We employ the moment equations to analyze TL and identify parameters that can be transferred to new conditions. The proposed analysis also guides the control variable selection in DT to facilitate efficient TL. For linear PDE problems, the non-transferable parameters in the KL-NN surrogate model can be exactly estimated from a single solution of the PDE corresponding to the mean values of the control variables under new target conditions. Retraining an ML model with a single solution sample is known as one-shot learning, and our analysis shows that the one-shot TL is exact for linear PDEs. For nonlinear PDE problems, transferring of any parameters introduces errors. For a nonlinear diffusion PDE model, we find that for a relatively small range of control variables, some surrogate model parameters can be transferred without introducing a significant error, some can be approximately estimated from the mean-field equation, and the rest can be found using a linear residual least square problem or an ordinary linear least square problem if a small labeled dataset for new conditions is available. The former approach results in a one-shot TL while the latter approach is an example of a few-shot TL. Both methods are approximate for the nonlinear PDEs.

arXiv:2501.06745v1 Announce Type: new Abstract: The paper at hand presents an in-depth investigation into the fatigue behavior of the high-strength aluminum alloy EN AW-7020 T6 using both experimental and numerical approaches. Two types of specimens are investigated: a dog-bone specimen subjected to cyclic loading in a symmetric strain-controlled regime, and a compact tension specimen subjected to repeated loading and unloading, which leads to damage growth from the notch tip. Experimental data from these tests are used to identify the different phases of fatigue. Subsequently, a plastic-damage model is developed, incorporating J2 plasticity with Chaboche-type mixed isotropic-kinematic hardening. A detailed investigation reveals that the Chaboche model must be blended with a suitable isotropic hardening and combined with a proper damage growth model to accurately describe cyclic fatigue including large plastic strains up to failure. Multiple back-stress components with independent properties are superimposed, and exponential isotropic hardening with saturation effects is introduced to improve alignment with experimental results. For damage, different stress splits are tested, with the deviatoric/volumetric split proving successful in reproducing the desired degradation in peak stress and stiffness. A nonlinear activation function is introduced to ensure smooth transitions between tension and compression. Two damage indices, one for the deviatoric part and one for the volumetric part, are defined, each of which is governed by a distinct trilinear damage growth function. The governing differential equation of the problem is regularized by higher-order gradient terms to address the ill-posedness induced by softening. Finally, the plasticity model is calibrated using finite element simulations of the dog-bone test and subsequently applied to the cyclic loading of the compact tension specimen.

arXiv:2501.06402v1 Announce Type: new Abstract: This paper presents a rigorous theoretical convergence analysis of the Wirtinger Flow (WF) algorithm for Poisson phase retrieval, a fundamental problem in imaging applications. Unlike prior analyses that rely on truncation or additional adjustments to handle outliers, our framework avoids eliminating measurements or introducing extra computational steps, thereby reducing overall complexity. We prove that WF achieves linear convergence to the true signal under noiseless conditions and remains robust and stable in the presence of bounded noise for Poisson phase retrieval. Additionally, we propose an incremental variant of WF, which significantly improves computational efficiency and guarantees convergence to the true signal with high probability under suitable conditions.

arXiv:2501.07439v1 Announce Type: new Abstract: This paper addresses the stabilization of dynamical systems in the infinite horizon optimal control setting using nonlinear feedback control based on State-Dependent Riccati Equations (SDREs). While effective, the practical implementation of such feedback strategies is often constrained by the high dimensionality of state spaces and the computational challenges associated with solving SDREs, particularly in parametric scenarios. To mitigate these limitations, we introduce the Dynamical Low-Rank Approximation (DLRA) methodology, which provides an efficient and accurate framework for addressing high-dimensional feedback control problems. DLRA dynamically constructs a compact, low-dimensional representation that evolves with the problem, enabling the simultaneous resolution of multiple parametric instances in real-time. We propose two novel algorithms to enhance numerical performances: the cascade-Newton-Kleinman method and Riccati-based DLRA (R-DLRA). The cascade-Newton-Kleinman method accelerates convergence by leveraging Riccati solutions from the nearby parameter or time instance, supported by a theoretical sensitivity analysis. R-DLRA integrates Riccati information into the DLRA basis construction to improve the quality of the solution. These approaches are validated through nonlinear one-dimensional and two-dimensional test cases showing transport-like behavior, demonstrating that R-DLRA outperforms standard DLRA and Proper Orthogonal Decomposition-based model order reduction in both speed and accuracy, offering a superior alternative to Full Order Model solutions.