The learning or approximation of manifold-valued functions has attracted a lot of attention in recent years. The non-linear structure of the functions’ range requires techniques and methods which significantly differ from standard approximation schemes. In this paper, we concentrate on the approximation of functions with values in the class of positive definite, symmetric matrices. Though not a compact manifold, its specific structure allows us to globally employ the log-Euclidean approach to define a Hilbert space structure on these matrices. Combining this approach with a kernel-based approximation method for symmetric matrices leads to a complete theory, including a thorough error analysis for deterministic approximations.
We give an algorithm to compute in polynomial time the roots of a fractional ideal of an order R. We take care not to assume R is Dedekind, since the maximal order of a number field is generally inaccessible in polynomial time. Consequently, the output of such an algorithm is no longer uniquely defined. For it to be a satisfying algorithm we additionally require it be functorial, i.e., isomorphisms on the inputs should induce isomorphisms on the outputs. To adhere to these two constraints, we generalize results from Dade–Taussky–Zassenhaus, and Ge and Buchmann–Eisenbrand.
. We consider an interior penalty hybridized discontinuous Galerkin (IP-HDG) method for a model of second-order elliptic problem on a convex polygonal or polyhedral domain. Our first main goal is to prove the best and the almost best approximation properties of the errors in the W1,infinity and L infinity norms by an interior error estimate introduced in this paper. The second main contribution is an application of the approximation property in the L infinity norm to the convergence analysis for IP-HDG methods for an optimal control problem with point values of the state in the objective functional. Since the low regularity induced by the Dirac function on the right-hand side of the adjoint state equation, the standard L2-projection and Lagrange interpolation can only get the convergence rates O(h2-d/2-E) and O(h2-d/2) for piecewise constant and variational discretizations for the L2 norm of the control, where E is a positive constant that can be arbitrarily small. However, with the help of the approximation property in the L infinity norm, the convergence rate for d = 3 can be improved to be O(h| ln h|). Finally, two numerical examples in two dimensions are provided to validate the theoretical analysis.
In this paper, we investigate the isoparametric finite element semidiscretization for a parabolic problem on a curvilinear polyhedral domain S2 subset of RN with homogeneous Dirichlet boundary condition. The domain S2 may include nonconvex corners, i.e., with edge openings possibly greater than pi. We establish the analyticity and maximal regularity of the discrete semigroup by employing a transformation method to address the domain perturbation effect S2 not equal S2h. As an application of the logarithmically quasi-maximal L infinity- regularity, we derive a quasi-optimal maximum-norm error estimate for the semidiscrete isoparametric finite element methods on curvilinear polyhedral domains with edge openings smaller than pi, which includes a term of quasioptimal order due to domain perturbation.
We establish the convergence of a speed-up version of the Halpern iteration with adaptive anchoring parameters in the general geodesic setting of Hadamard spaces, generalizing a recent result by He, Xu, Dong and Mei [Math. Comp. 93 (2024), pp. 327-345] from a linear to a nonlinear setting. In particular, our results extend the fast rates of asymptotic regularity obtained by these authors for the first time to a nonlinear setting. Our approach relies on a quantitative study of these previous results in the linear setting, combined with certain optimizations and an elimination of the weak compactness arguments employed crucially in the linear setting, which not only allows for the lift of the result to a nonlinear setting but also streamlines the previous convergence analysis considerably. This work is set in the context of recent developments in proof mining, and as a byproduct of our approach, we further obtain quantitative information in the form of highly uniform rates of metastability of low complexity, which are new already in the context of Hilbert spaces.
It is well known that a numerical scheme for solving a hyperbolic conservation law, when convergent, may not always converge to a weak solution. The remedy is the famous Lax-Wendroff theorem, stating that a conservative numerical scheme, when convergent, always converges to a weak solution of the conservation law. In this paper, we address the same issue for numerically solving degenerate non-linear diffusion or convection-diffusion equations. We start with an example showing that a conservative scheme, in the sense of that for conservation laws, may still converge to a function which is not a weak solution of the degenerate non-linear diffusion equation. We then introduce a stronger form of conservative schemes, which we term as doublyconservative (DoC) schemes, that would allow the proof of a Lax-Wendroff type theorem, namely if a DoC scheme converges, then it will converge to the weak solution of the degenerate non-linear diffusion or convection-diffusion equation. The concept of DoC schemes is applicable to high order schemes, on one-dimensional non-uniform meshes as well as on two-dimensional Cartesian meshes and unstructured triangular meshes. Finally, we design a new DoC local discontinuous Galerkin scheme that remains semi-discrete stable even when the diffusion coefficient degenerates. Numerical experiments demonstrate optimal convergence rates of this new scheme and validate our theoretical results.
. A forbidden substructure theorem characterizes a class of mathematical objects within an overclass through a list of forbidden substructures. One typical example of such a theorem is: a graph is bipartite if and only if it does not contain cycles of odd length. At the conference on Artificial Intellisubstructure theorem for lattices, found using a combination of AITP tools and proofs by hand; he concluded by conjecturing that it should be possible to develop a fully autonomous AITP tool capable of conjecturing and proving forbidden substructure theorems without human intervention. The goal of this paper is to answer that conjecture in the affirmative. After introducing the algorithm behind our tool, we demonstrate its power by proving many new forbidden substructure theorems. In particular, a recent paper provides the partially ordered set of all lazy magma (groupoid) varieties. Our tool, working in fully autonomous mode, managed to derive a forbidden substructure theorem for each inclusion in that paper. This tool is available to any mathematician on a free online website. The paper ends with a list of open problems.
. In this paper, we propose and study first- and second-order (in time) stabilized linear finite element schemes for the incompressible NavierStokes (NS) equations. The energy, momentum, and angular momentum conserving (EMAC) formulation has emerged as a promising approach for conserving energy, momentum, and angular momentum of the NS equations, while the exponential scalar auxiliary variable (ESAV) has become a popular technique for designing linear energy-stable numerical schemes. Our method leverages the EMAC formulation and the Taylor-Hood element with grad-div stabilization for spatial discretization. We adopt the implicit-explicit backward differential formulas (BDFs) coupled with a novel stabilized ESAV approach for time stepping. For the solution process, we develop an efficient decoupling technique for the resulting fully-discrete systems so that only one linear Stokes solve is needed at each time step, which is similar to the cost of classic implicit-explicit BDF schemes for the NS equations. Robust optimal error estimates are successfully derived for both velocity and pressure for the two proposed schemes, with Gronwall constants that are particularly independent of the viscosity. Furthermore, it is rigorously shown that the grad-div stabilization term can greatly alleviate the viscosity-dependence of the mesh size constraint, which is required for error estimation when such a term is not present in the schemes. Various numerical experiments are conducted to verify the theoretical results and demonstrate the effectiveness and efficiency of the grad-div and ESAV stabilization strategies and their combination in the proposed numerical schemes, especially for problems with high Reynolds numbers.
There is an error in the statement and proof of Theorem 4.8 of the authors' paper, 'A unifying theory for metrical results on regular continued fraction convergents and mediants' [Math. Comp. 94 (2025), pp. 3101-3144]. This theorem (and subsequent, dependent results) are reformulated here with correct hypotheses. We note that our applications of the original results satisfy these stronger hypotheses and thus remain valid. Moreover, we present a counterexample to the original statement of Theorem 4.8.
. Testing the unimodular equivalence of two full-dimensional integral simplices can be reduced to testing unimodular permutation (UP) equivalence of two nonsingular matrices. We conduct a systematic study of UPequivalence, which leads to the first average-case quasi-polynomial time algorithm, called HEM, for deciding the unimodular equivalence of d-dimensional integral simplices, as well as achieving a polynomial-time complexity with a failure probability less than 2.5 & times; 10-7. A key ingredient is the introduction of the permuted Hermite normal form and its associated pattern group, which streamlines the UP-equivalence test by comparing canonical forms derived from induced coset representatives. We also present an acceleration strategy based on Smith normal forms. As a theoretical by-product, we prove that two full-dimensional integral simplices are unimodularly equivalent if and only if their n-dimensional pyramids are unimodularly equivalent. This resolves an open question posed by Abney-McPeek et al.
We propose a deterministic algorithm based on Coppersmith's method that employs a rank-3 lattice to address factoring-related problems. An interesting aspect of our approach is that we utilize the second vector in the Lenstra-Lenstra-Lovasz (LLL)-reduced basis to avoid trivial collisions in the Baby-step Giant-step method, rather than the shortest vector as is commonly used in the literature. Our results are as follows:- Compared to the result by Harvey and Hittmeir [Math. Comp. 91 (2022), (N1/5 log16/5 N) pp. 1367-1379], who achieved a complexity of O for factoring (loglog N)3/5 a semiprime N = pq, we demonstrate that in the balanced p and q case, the ( N1/5 log13/5 N) complexity can be improved to O . (log log N)3/5-For factoring sums and differences of powers, i.e., numbers of the form N = an +/- bn, we improve Hittmeir's result [Math. Comp. 86 (2017), pp. ( ) 2947-2954] from O(N1/4 log3/2 N) to O N1/5 log13/5 N .- For the problem of finding r-power divisors, i.e., finding all integers p such that pr | N, Harvey and Hittmeir [Res. Number Theory 8 (2022)] recently directly applied Coppersmith's method and achieved a complexity of ( N1/4r log10+epsilon N) O r3 . By using faster LLL-type algorithm and sieving on small (N1/4r log7+3 epsilon N) primes, we improve their result to O (log logN-log 4r)r2+epsilon . The worst case running time for their algorithm occurs when N = prq with q = Theta(N1/2). By focusing on this case and employing our rank-3 lattice approach, we achieve a ( ) complexity of O r1/4N1/4r log5/2 N . In conclusion, we offer a new perspective on these problems, which we hope will provide further insights.
