Communications on Pure and Applied Mathematics最新文献

This work addresses the question of uniqueness and regularity of the minimizers of a convex but not strictly convex integral functional with linear growth in a two-dimensional setting. The integrand – whose precise form derives directly from the theory of perfect plasticity – behaves quadratically close to the origin and grows linearly once a specific threshold is reached. Thus, in contrast with the only existing literature on uniqueness for functionals with linear growth, that is that which pertains to the generalized least gradient, the integrand is not a norm. We make use of hyperbolic conservation laws hidden in the structure of the problem to tackle uniqueness. Our argument strongly relies on the regularity of a vector field – the Cauchy stress in the terminology of perfect plasticity – which allows us to define characteristic lines and then to employ the method of characteristics. Using the detailed structure of the characteristic landscape evidenced in our preliminary study [5], we show that this vector field is actually continuous, save for possibly two points. The different behaviors of the energy density at zero and at infinity imply an inequality constraint on the Cauchy stress. Under a barrier type convexity assumption on the set where the inequality constraint is saturated, we show that uniqueness holds for pure Dirichlet boundary data devoid of any regularity properties, a stronger result than that of uniqueness for a given trace on the whole boundary since our minimizers can fail to attain the boundary data. We also show a partial regularity result for the minimizer.

Dimer models have been the focus of intense research efforts over the last years. Our paper grew out of an effort to develop new methods to study minimizers or the asymptotic height functions of general dimer models and the geometry of their frozen boundaries. We prove a complete classification of the regularity of minimizers and frozen boundaries for all dimer models for a natural class of polygonal domains, much studied in numerical simulations and elsewhere. In particular, we show that the frozen boundaries are always algebraic curves. Our classification also implies that the Pokrovsky-Talapov law holds for all dimer models at a generic point on the frozen boundary and, in addition, shows a very strong local rigidity of dimer models, which can be interpreted as a geometric universality result. Indeed, we prove a converse result, showing that any geometric situation for any dimer model is, in the simply connected case, realized already by the lozenge model. To achieve these goals we develop a new study on the boundary regularity for a class of Monge–Ampère equations in non-strictly convex domains, of independent interest, as well as a new approach to minimality for a general dimer functional. In the context of polygonal domains, we give the first general results for the existence of gas domains for minimizers.

One of the greatest success stories of randomized algorithms in linear algebra has been the development of fast, randomized solvers for highly overdetermined linear least-squares problems. However, none of the existing algorithms is backward stable, preventing them from being deployed as drop-in replacements for existing QR-based solvers. This paper introduces sketch-and-precondition with iterative refinement (SPIR) and FOSSILS, two provably backward stable randomized least-squares solvers. SPIR and FOSSILS combine iterative refinement with a preconditioned iterative method applied to the normal equations and converge at the same rate as existing randomized least-squares solvers. This work offers the promise of incorporating randomized least-squares solvers into existing software libraries while maintaining the same level of accuracy and stability as classical solvers.

We prove functional inequalities in any dimension controlling the iterated derivatives along a transport of the Coulomb or super-Coulomb Riesz modulated energy in terms of the modulated energy itself. This modulated energy was introduced by the second author and collaborators in the study of mean-field limits and statistical mechanics of Coulomb/Riesz gases, where control of such derivatives by the energy itself is an essential ingredient. In this paper, we extend and improve such functional inequalities, proving estimates which are now sharp in their additive error term, in their density dependence, valid at arbitrary order of differentiation, and localizable to the support of the transport. Our method relies on the observation that these iterated derivatives are the quadratic form of a commutator. Taking advantage of the Riesz nature of the interaction, we identify these commutators as solutions to a degenerate elliptic equation with a right-hand side exhibiting a recursive structure in terms of lower-order commutators and develop a local regularity theory for the commutators, which may be of independent interest. These estimates have applications to obtaining sharp rates of convergence for mean-field limits, quasi-neutral limits, and in proving central limit theorems for the fluctuations of Coulomb/Riesz gases. In particular, we show here the expected $�N^{�\frac{s}{d}�-�1�}$-rate in the modulated energy distance for the mean-field convergence of first-order Hamiltonian and gradient flows.

Extended dynamic mode decomposition (EDMD) is a data-driven algorithm for approximating spectral data of the Koopman operator associated to a dynamical system, combining a Galerkin method with $�N$ functions and a quadrature method with $�M$ quadrature nodes. Spectral convergence of this method subtly depends on an appropriate choice of the space of observables. For chaotic analytic full branch maps of the interval, we derive a constraint between $�M$ and $�N$ guaranteeing spectral convergence of EDMD. In particular, the computed eigenvalues converge exponentially fast (in $�N$) to the eigenvalues of the Koopman operator, taken to act on the dual space of a certain Banach space of analytic functions.

We provide a complete solution to the problem of infinite quantum signal processing (QSP) for the class of Szegő functions, which are functions that satisfy a logarithmic integrability condition and include almost any function that allows for a QSP representation. We do so by introducing a new algorithm called the Riemann–Hilbert–Weiss algorithm, which can compute any individual phase factor independent of all other phase factors. Our algorithm is also the first provably stable numerical algorithm for computing phase factors of any arbitrary Szegő function. The proof of stability involves solving a Riemann–Hilbert factorization problem in nonlinear Fourier analysis using elements of spectral theory.

Recently, a class of algorithms combining classical fixed-point iterations with repeated random sparsification of approximate solution vectors has been successfully applied to eigenproblems with matrices as large as $��{10}^{108}�\times �{10}^{108}�$. So far, a complete mathematical explanation for this success has proven elusive.

The family of methods has not yet been extended to the important case of linear system solves. In this paper, we propose a new scheme based on repeated random sparsification that is capable of solving sparse linear systems in arbitrarily high dimensions. We provide a complete mathematical analysis of this new algorithm. Our analysis establishes a faster-than-Monte Carlo convergence rate and justifies use of the scheme even when the solution is too large to store as a dense vector.

We prove $�\epsilon$-regularity theorems for varifolds with capillary boundary condition in a Riemannian manifold. These varifolds were first introduced by Kagaya–Tonegawa. We establish a uniform first variation control for all such varifolds (and free-boundary varifolds generally) satisfying a sharp density bound and prove that if a capillary varifold has bounded mean curvature and is close to a capillary half-plane with angle not equal to $��\frac{\pi }{2}�$, then it coincides with a $�C^{�1�,�\alpha �}$ properly embedded hypersurface. We apply our theorem to deduce regularity at a generic point along the boundary in the region where the density is strictly less than 1.