241 papers
This paper presents an error analysis and a new regularization parameter selection strategy for the iterated Golub-Kahan-Tikhonov method applied to discretized ill-posed operator equations, demonstrating its superior accuracy over both standard Golub-Kahan-Tikhonov and iterated Arnoldi-Tikhonov methods, particularly for non-symmetric matrices.
This paper establishes the quasi-optimality of the Crouzeix-Raviart finite element method for nonlinear -Laplace-type problems by proving that its error is bounded by the best-approximation error and a data oscillation term, while also deriving a novel localized a priori error estimate for the conforming lowest-order Lagrange FEM.
This paper introduces a novel computational framework that utilizes Legendre time-dimensional reduction to transform the inverse initial-data problem for compressible anisotropic Navier-Stokes equations into a solvable system of elliptic equations, enabling the robust and accurate reconstruction of initial velocity fields from noisy boundary observations.
This paper establishes the rigorous convergence of hyperbolic approximations to various higher-order PDEs, including the KdV and Kuramoto-Sivashinsky equations, by proving that weak solutions of the approximations converge to smooth solutions of the limiting problems, a result supported by numerical evidence.
This paper presents a quadrature-based model reduction framework for Lagrangian hydrodynamics that utilizes a strongly energy-conservative variant of the empirical quadrature procedure to achieve near machine-precision total energy conservation while maintaining accuracy comparable to standard methods.
This paper demonstrates that the union of randomly digitally shifted Korobov polynomial lattice point sets achieves a star discrepancy with an inverse that depends linearly on the dimension, thereby narrowing the search for explicit constructions from a continuum to a finite set of candidates.
This paper proposes an auto-adaptive sampling method for Physics Informed Neural Networks (PINNs) that utilizes arbitrary problem-specific heuristics to accurately resolve interfacial regions in Allen-Cahn equations without post-hoc resampling, demonstrating superior performance over residual-adaptive frameworks.
This paper presents numerically stable, closed-form algorithms for computing the eigenvalues of real, diagonalizable $3 \times 3$ matrices using four specific invariants, demonstrating that these methods are both more accurate for repeated eigenvalues and approximately ten times faster than the LAPACK library while maintaining comparable precision.
This paper establishes a minimax theory for operator learning in Hilbert spaces, proving that even with higher regularity assumptions, learning generic Lipschitz operators from noisy samples inevitably suffers from a curse of sample complexity where the risk cannot decay at any algebraic rate.
This paper introduces a novel numerical method that combines a functional equation for the Laplace-Stieltjes transform with a moment-matching technique to efficiently and stably compute the density of the Kesten-Stigum limit for supercritical Galton-Watson processes with arbitrary offspring laws.
This paper introduces a novel Geometric Structure-Preserving Interpolation (-SPIN) method that utilizes geodesic elements and a projection-based relaxation of rotation-deformation coupling to achieve stable, locking-free finite-strain Cosserat micropolar elasticity simulations, particularly in the asymptotic couple-stress limit.
This paper introduces and validates two new structure-preserving discontinuous Galerkin methods, SWIPD-L and SIPGD-L, for the Cahn-Hilliard-Navier-Stokes equations with degenerate mobility, demonstrating that they achieve optimal convergence, preserve key physical properties like mass conservation and energy dissipation, and offer significant computational savings on adaptive meshes compared to existing approaches.
This paper introduces a non-separable progressive multivariate WENO scheme tailored for cell-average data that achieves high-order accuracy and non-oscillatory stability in image processing, demonstrating superior performance over linear Lagrange reconstruction through theoretical analysis and numerical experiments.
This paper proposes a post-processing method that significantly improves the accuracy of physics-informed neural networks (PINNs) by finding the best approximation in a function space associated with the network, achieving errors four to five orders of magnitude lower than standard PINNs while enabling transfer learning and providing a metric for optimal basis function selection.
This paper introduces a time-discrete Variational Physics-Informed Neural Network (VPINN) that minimizes the residual's dual norm at each time step to effectively model the temperature-dependent heat conduction dynamics of industrial coffee extract freezing.
This paper proposes an explicit truncated Euler-Maruyama scheme with time and space truncation to approximate the invariant probability measures of super-linear stochastic functional differential equations with infinite delay, establishing strong convergence and proving that the numerical invariant measure converges to the exact one in Wasserstein distance with an explicit rate.
This paper introduces a multilevel training framework for Kolmogorov-Arnold Networks (KANs) that leverages their structural equivalence to multichannel MLPs and the properties of spline basis functions to create a properly nested hierarchy of models, resulting in orders-of-magnitude improvements in training accuracy and speed, particularly for physics-informed neural networks.
This paper establishes the regularizing properties of quantum relative entropy for solving the inverse problem of quantum state tomography in high or infinite dimensions, providing the necessary theoretical foundations and computational tools to apply iterative convex optimization algorithms to practical examples like PINEM and optical homodyne tomography.
This paper proves the -convergence of the time-splitting scheme to the unique global strong solution of the nonlinear Dirac equation in 1+1 dimensions by establishing pointwise estimates, constructing a modified Glimm-type functional for stability, and demonstrating the precompactness of the approximate solutions.
This paper presents a general framework for deriving Nitsche Finite Element Methods to enforce equality and inequality constraints in solid mechanics by reformulating stabilized saddle point problems into a minimization form suitable for nonlinear applications and automatic differentiation, while validating the approach through numerical convergence studies.