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Duality in Finite Element Exterior Calculus and Hodge Duality on the Sphere Found. Comput. Math. (IF 2.476) Pub Date : 2021-01-19 Yakov Berchenko-Kogan
Finite element exterior calculus refers to the development of finite element methods for differential forms, generalizing several earlier finite element spaces of scalar fields and vector fields to arbitrary dimension n, arbitrary polynomial degree r, and arbitrary differential form degree k. The study of finite element exterior calculus began with the \({\mathcal {P}}_r\varLambda ^k\) and \({\mathcal
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Metrics, Quantization and Registration in Varifold Spaces Found. Comput. Math. (IF 2.476) Pub Date : 2021-01-07 Hsi-Wei Hsieh, Nicolas Charon
This paper is concerned with the theory and applications of varifolds to the representation, approximation and diffeomorphic registration of shapes. One of its purpose is to synthesize and extend several prior works which, so far, have made use of this framework mainly in the context of submanifold comparison and matching. In this work, we instead consider deformation models acting on general varifold
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Primary Ideals and Their Differential Equations Found. Comput. Math. (IF 2.476) Pub Date : 2021-01-05 Yairon Cid-Ruiz, Roser Homs, Bernd Sturmfels
An ideal in a polynomial ring encodes a system of linear partial differential equations with constant coefficients. Primary decomposition organizes the solutions to the PDE. This paper develops a novel structure theory for primary ideals in a polynomial ring. We characterize primary ideals in terms of PDE, punctual Hilbert schemes, relative Weyl algebras, and the join construction. Solving the PDE
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Computing the Homology of Semialgebraic Sets. II: General Formulas Found. Comput. Math. (IF 2.476) Pub Date : 2021-01-04 Peter Bürgisser, Felipe Cucker, Josué Tonelli-Cueto
We describe and analyze a numerical algorithm for computing the homology (Betti numbers and torsion coefficients) of semialgebraic sets given by Boolean formulas. The algorithm works in weak exponential time. This means that outside a subset of data having exponentially small measure, the cost of the algorithm is single exponential in the size of the data. This extends the work in Part I to arbitrary
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Tensor-Free Proximal Methods for Lifted Bilinear/Quadratic Inverse Problems with Applications to Phase Retrieval Found. Comput. Math. (IF 2.476) Pub Date : 2021-01-04 Robert Beinert, Kristian Bredies
We propose and study a class of novel algorithms that aim at solving bilinear and quadratic inverse problems. Using a convex relaxation based on tensorial lifting, and applying first-order proximal algorithms, these problems could be solved numerically by singular value thresholding methods. However, a direct realization of these algorithms for, e.g., image recovery problems is often impracticable
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Exact Splitting Methods for Semigroups Generated by Inhomogeneous Quadratic Differential Operators Found. Comput. Math. (IF 2.476) Pub Date : 2021-01-04 Joackim Bernier
We introduce some general tools to design exact splitting methods to compute numerically semigroups generated by inhomogeneous quadratic differential operators. More precisely, we factorize these semigroups as products of semigroups that can be approximated efficiently, using, for example, pseudo-spectral methods. We highlight the efficiency of these new methods on the examples of the magnetic linear
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Homological Algebra for Persistence Modules Found. Comput. Math. (IF 2.476) Pub Date : 2021-01-04 Peter Bubenik, Nikola Milićević
We develop some aspects of the homological algebra of persistence modules, in both the one-parameter and multi-parameter settings, considered as either sheaves or graded modules. The two theories are different. We consider the graded module and sheaf tensor product and Hom bifunctors as well as their derived functors, Tor and Ext, and give explicit computations for interval modules. We give a classification
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The Reeb Graph Edit Distance is Universal Found. Comput. Math. (IF 2.476) Pub Date : 2020-12-29 Ulrich Bauer, Claudia Landi, Facundo Mémoli
We consider the setting of Reeb graphs of piecewise linear functions and study distances between them that are stable, meaning that functions which are similar in the supremum norm ought to have similar Reeb graphs. We define an edit distance for Reeb graphs and prove that it is stable and universal, meaning that it provides an upper bound to any other stable distance. In contrast, via a specific construction
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Convergence Rates for Discretized Monge–Ampère Equations and Quantitative Stability of Optimal Transport Found. Comput. Math. (IF 2.476) Pub Date : 2020-12-14 Robert J. Berman
In recent works—both experimental and theoretical—it has been shown how to use computational geometry to efficiently construct approximations to the optimal transport map between two given probability measures on Euclidean space, by discretizing one of the measures. Here we provide a quantitative convergence analysis for the solutions of the corresponding discretized Monge–Ampère equations. This yields
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Function Values Are Enough for $$L_2$$ L 2 -Approximation Found. Comput. Math. (IF 2.476) Pub Date : 2020-12-07 David Krieg, Mario Ullrich
We study the \(L_2\)-approximation of functions from a Hilbert space and compare the sampling numbers with the approximation numbers. The sampling number \(e_n\) is the minimal worst-case error that can be achieved with n function values, whereas the approximation number \(a_n\) is the minimal worst-case error that can be achieved with n pieces of arbitrary linear information (like derivatives or Fourier
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A Variational Finite Element Discretization of Compressible Flow Found. Comput. Math. (IF 2.476) Pub Date : 2020-11-02 Evan S. Gawlik, François Gay-Balmaz
We present a finite element variational integrator for compressible flows. The numerical scheme is derived by discretizing, in a structure-preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the associated variational principles. Given a triangulation on the fluid domain, the discrete group of diffeomorphisms is defined as a certain subgroup of the group of linear
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Computing Higher Leray–Serre Spectral Sequences of Towers of Fibrations Found. Comput. Math. (IF 2.476) Pub Date : 2020-10-27 Andrea Guidolin, Ana Romero
The higher Leray–Serre spectral sequence associated with a tower of fibrations represents a generalization of the classical Leray–Serre spectral sequence of a fibration. In this work, we present algorithms to compute higher Leray–Serre spectral sequences leveraging the effective homology technique, which allows to perform computations involving chain complexes of infinite type associated with interesting
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Correction To: The Complexity of Factors of Multivariate Polynomials Found. Comput. Math. (IF 2.476) Pub Date : 2020-10-06 Peter Bürgisser
Vladimir Lysikov kindly pointed out an error in the proof of Theorem 5.7. We provide here a corrected statement and its proof.
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Constraint-Preserving Hybrid Finite Element Methods for Maxwell’s Equations Found. Comput. Math. (IF 2.476) Pub Date : 2020-09-28 Yakov Berchenko-Kogan, Ari Stern
Maxwell’s equations describe the evolution of electromagnetic fields, together with constraints on the divergence of the magnetic and electric flux densities. These constraints correspond to fundamental physical laws: the nonexistence of magnetic monopoles and the conservation of charge, respectively. However, one or both of these constraints may be violated when one applies a finite element method
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Stable Rank-One Matrix Completion is Solved by the Level 2 Lasserre Relaxation Found. Comput. Math. (IF 2.476) Pub Date : 2020-09-25 Augustin Cosse, Laurent Demanet
This paper studies the problem of deterministic rank-one matrix completion. It is known that the simplest semidefinite programming relaxation, involving minimization of the nuclear norm, does not in general return the solution for this problem. In this paper, we show that in every instance where the problem has a unique solution, one can provably recover the original matrix through the level 2 Lasserre
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A Unifying Representer Theorem for Inverse Problems and Machine Learning Found. Comput. Math. (IF 2.476) Pub Date : 2020-09-25 Michael Unser
Regularization addresses the ill-posedness of the training problem in machine learning or the reconstruction of a signal from a limited number of measurements. The method is applicable whenever the problem is formulated as an optimization task. The standard strategy consists in augmenting the original cost functional by an energy that penalizes solutions with undesirable behavior. The effect of regularization
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Algebraic Boundaries Among Typical Ranks for Real Binary Forms of Arbitrary Degree Found. Comput. Math. (IF 2.476) Pub Date : 2020-09-24 Maria Chiara Brambilla, Giovanni Staglianò
We show that the algebraic boundaries of the regions of real binary forms with fixed typical rank are always unions of dual varieties to suitable coincident root loci.
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Local Approximation from Spline Spaces on Box Meshes Found. Comput. Math. (IF 2.476) Pub Date : 2020-07-07 Andrea Bressan, Tom Lyche
This paper analyzes the approximation properties of spaces of piecewise tensor product polynomials over box meshes with a focus on application to isogeometric analysis. Local and global error bounds with respect to Sobolev or reduced seminorms are provided. Attention is also paid to the dependence on the degree, and exponential convergence is proved for the approximation of analytic functions in the
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Iterative Potts Minimization for the Recovery of Signals with Discontinuities from Indirect Measurements: The Multivariate Case Found. Comput. Math. (IF 2.476) Pub Date : 2020-07-06 Lukas Kiefer, Martin Storath, Andreas Weinmann
Signals and images with discontinuities appear in many problems in such diverse areas as biology, medicine, mechanics and electrical engineering. The concrete data are often discrete, indirect and noisy measurements of some quantities describing the signal under consideration. A frequent task is to find the segments of the signal or image which corresponds to finding the discontinuities or jumps in
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Towards Massively Parallel Computations in Algebraic Geometry Found. Comput. Math. (IF 2.476) Pub Date : 2020-07-06 Janko Böhm, Wolfram Decker, Anne Frühbis-Krüger, Franz-Josef Pfreundt, Mirko Rahn, Lukas Ristau
Introducing parallelism and exploring its use is still a fundamental challenge for the computer algebra community. In high-performance numerical simulation, on the other hand, transparent environments for distributed computing which follow the principle of separating coordination and computation have been a success story for many years. In this paper, we explore the potential of using this principle
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Moment Identifiability of Homoscedastic Gaussian Mixtures Found. Comput. Math. (IF 2.476) Pub Date : 2020-07-06 Daniele Agostini, Carlos Améndola, Kristian Ranestad
We consider the problem of identifying a mixture of Gaussian distributions with the same unknown covariance matrix by their sequence of moments up to certain order. Our approach rests on studying the moment varieties obtained by taking special secants to the Gaussian moment varieties, defined by their natural polynomial parametrization in terms of the model parameters. When the order of the moments
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Numerical Performance of Optimized Frolov Lattices in Tensor Product Reproducing Kernel Sobolev Spaces Found. Comput. Math. (IF 2.476) Pub Date : 2020-07-06 Christopher Kacwin, Jens Oettershagen, Mario Ullrich, Tino Ullrich
In this paper, we deal with several aspects of the universal Frolov cubature method, which is known to achieve optimal asymptotic convergence rates in a broad range of function spaces. Even though every admissible lattice has this favorable asymptotic behavior, there are significant differences concerning the precise numerical behavior of the worst-case error. To this end, we propose new generating
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Error estimates of a Fourier integrator for the cubic Schrödinger equation at low regularity Found. Comput. Math. (IF 2.476) Pub Date : 2020-07-06 Alexander Ostermann, Frédéric Rousset, Katharina Schratz
We present a new filtered low-regularity Fourier integrator for the cubic nonlinear Schrödinger equation based on recent time discretization and filtering techniques. For this new scheme, we perform a rigorous error analysis and establish better convergence rates at low regularity than known for classical schemes in the literature so far. In our error estimates, we combine the better local error properties
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On the Convergence of Time Splitting Methods for Quantum Dynamics in the Semiclassical Regime Found. Comput. Math. (IF 2.476) Pub Date : 2020-06-30 François Golse, Shi Jin, Thierry Paul
By using the pseudo-metric introduced in Golse and Paul (Arch Ration Mech Anal 223:57–94, 2017), which is an analogue of the Wasserstein distance of exponent 2 between a quantum density operator and a classical (phase-space) density, we prove that the convergence of time splitting algorithms for the von Neumann equation of quantum dynamics is uniform in the Planck constant \(\hbar \). We obtain explicit
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Sparse Harmonic Transforms: A New Class of Sublinear-Time Algorithms for Learning Functions of Many Variables Found. Comput. Math. (IF 2.476) Pub Date : 2020-06-24 Bosu Choi, Mark A. Iwen, Felix Krahmer
In this paper we develop fast and memory efficient numerical methods for learning functions of many variables that admit sparse representations in terms of general bounded orthonormal tensor product bases. Such functions appear in many applications including, e.g., various Uncertainty Quantification (UQ) problems involving the solution of parametric PDE that are approximately sparse in Chebyshev or
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Noncommutative Polynomials Describing Convex Sets Found. Comput. Math. (IF 2.476) Pub Date : 2020-06-24 J. William Helton, Igor Klep, Scott McCullough, Jurij Volčič
The free closed semialgebraic set \({\mathcal {D}}_f\) determined by a hermitian noncommutative polynomial \(f\in {\text {M}}_{{\delta }}({\mathbb {C}}\mathop {<}x,x^*\mathop {>})\) is the closure of the connected component of \(\{(X,X^*)\mid f(X,X^*)\succ 0\}\) containing the origin. When L is a hermitian monic linear pencil, the free closed semialgebraic set \({\mathcal {D}}_L\) is the feasible set
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The Grassmannian of affine subspaces Found. Comput. Math. (IF 2.476) Pub Date : 2020-05-27 Lek-Heng Lim, Ken Sze-Wai Wong, Ke Ye
The Grassmannian of affine subspaces is a natural generalization of both the Euclidean space, points being 0-dimensional affine subspaces, and the usual Grassmannian, linear subspaces being special cases of affine subspaces. We show that, like the Grassmannian, the affine Grassmannian has rich geometrical and topological properties: It has the structure of a homogeneous space, a differential manifold
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Weak Convergence Rates for Euler-Type Approximations of Semilinear Stochastic Evolution Equations with Nonlinear Diffusion Coefficients Found. Comput. Math. (IF 2.476) Pub Date : 2020-05-18 Arnulf Jentzen, Ryan Kurniawan
Strong convergence rates for time-discrete numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the literature. Weak convergence rates for time-discrete numerical approximations of such SEEs have, loosely speaking, been investigated since 2003 and are far away from being well understood: roughly speaking, no essentially
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A Neural Network-Based Policy Iteration Algorithm with Global $$H^2$$H2 -Superlinear Convergence for Stochastic Games on Domains Found. Comput. Math. (IF 2.476) Pub Date : 2020-05-18 Kazufumi Ito, Christoph Reisinger, Yufei Zhang
In this work, we propose a class of numerical schemes for solving semilinear Hamilton–Jacobi–Bellman–Isaacs (HJBI) boundary value problems which arise naturally from exit time problems of diffusion processes with controlled drift. We exploit policy iteration to reduce the semilinear problem into a sequence of linear Dirichlet problems, which are subsequently approximated by a multilayer feedforward
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Topological Properties of the Set of Functions Generated by Neural Networks of Fixed Size Found. Comput. Math. (IF 2.476) Pub Date : 2020-05-14 Philipp Petersen, Mones Raslan, Felix Voigtlaender
We analyze the topological properties of the set of functions that can be implemented by neural networks of a fixed size. Surprisingly, this set has many undesirable properties. It is highly non-convex, except possibly for a few exotic activation functions. Moreover, the set is not closed with respect to \(L^p\)-norms, \(0< p < \infty \), for all practically used activation functions, and also not
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Perturbations of Christoffel–Darboux Kernels: Detection of Outliers Found. Comput. Math. (IF 2.476) Pub Date : 2020-04-20 Bernhard Beckermann, Mihai Putinar, Edward B. Saff, Nikos Stylianopoulos
Two central objects in constructive approximation, the Christoffel–Darboux kernel and the Christoffel function, encode ample information about the associated moment data and ultimately about the possible generating measures. We develop a multivariate theory of the Christoffel–Darboux kernel in \(\mathbb {C}^d\), with emphasis on the perturbation of Christoffel functions and their level sets with respect
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Real Quadratic Julia Sets Can Have Arbitrarily High Complexity Found. Comput. Math. (IF 2.476) Pub Date : 2020-04-16 Cristobal Rojas, Michael Yampolsky
We show that there exist real parameters \(c\in (-2,0)\) for which the Julia set \(J_{c}\) of the quadratic map \(z^{2} + c\) has arbitrarily high computational complexity. More precisely, we show that for any given complexity threshold T(n), there exist a real parameter c such that the computational complexity of computing \(J_{c}\) with n bits of precision is higher than T(n). This is the first known
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Wavenumber-Explicit hp -FEM Analysis for Maxwell’s Equations with Transparent Boundary Conditions Found. Comput. Math. (IF 2.476) Pub Date : 2020-04-06 Jens M. Melenk, Stefan A. Sauter
The time-harmonic Maxwell equations at high wavenumber k are discretized by edge elements of degree p on a mesh of width h. For the case of a ball as the computational domain and exact, transparent boundary conditions, we show quasi-optimality of the Galerkin method under the k-explicit scale resolution condition that (a) kh/p is sufficient small and (b) \(p/\ln k\) is bounded from below.
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On the Complexity Exponent of Polynomial System Solving Found. Comput. Math. (IF 2.476) Pub Date : 2020-03-19 Joris van der Hoeven, Grégoire Lecerf
We present a probabilistic Las Vegas algorithm for solving sufficiently generic square polynomial systems over finite fields. We achieve a nearly quadratic running time in the number of solutions, for densely represented input polynomials. We also prove a nearly linear bit complexity bound for polynomial systems with rational coefficients. Our results are obtained using the combination of the Kronecker
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Analysis of Artifacts in Shell-Based Image Inpainting: Why They Occur and How to Eliminate Them Found. Comput. Math. (IF 2.476) Pub Date : 2020-03-16 L. Robert Hocking, Thomas Holding, Carola-Bibiane Schönlieb
In this paper we study a class of fast geometric image inpainting methods based on the idea of filling the inpainting domain in successive shells from its boundary inwards. Image pixels are filled by assigning them a color equal to a weighted average of their already filled neighbors. However, there is flexibility in terms of the order in which pixels are filled, the weights used for averaging, and
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Data Analysis from Empirical Moments and the Christoffel Function Found. Comput. Math. (IF 2.476) Pub Date : 2020-03-16 Edouard Pauwels, Mihai Putinar, Jean-Bernard Lasserre
Spectral features of the empirical moment matrix constitute a resourceful tool for unveiling properties of a cloud of points, among which, density, support and latent structures. This matrix is readily computed from an input dataset, and its eigen decomposition can then be used to identify algebraic properties of the support or density/support estimates with the Christoffel function. It is already
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Effective Models and Numerical Homogenization for Wave Propagation in Heterogeneous Media on Arbitrary Timescales Found. Comput. Math. (IF 2.476) Pub Date : 2020-03-16 Assyr Abdulle, Timothée Pouchon
A family of effective equations for wave propagation in periodic media for arbitrary timescales \(\mathcal {O}(\varepsilon ^{-\alpha })\), where \(\varepsilon \ll 1\) is the period of the tensor describing the medium, is proposed. The well-posedness of the effective equations of the family is ensured without requiring a regularization process as in previous models (Benoit and Gloria in Long-time homogenization
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Wilkinson’s Bus: Weak Condition Numbers, with an Application to Singular Polynomial Eigenproblems Found. Comput. Math. (IF 2.476) Pub Date : 2020-03-10 Martin Lotz, Vanni Noferini
We propose a new approach to the theory of conditioning for numerical analysis problems for which both classical and stochastic perturbation theories fail to predict the observed accuracy of computed solutions. To motivate our ideas, we present examples of problems that are discontinuous at a given input and even have infinite stochastic condition number, but where the solution is still computed to
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Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation Found. Comput. Math. (IF 2.476) Pub Date : 2020-03-10 Robert I McLachlan, Christian Offen
We show that symplectic integrators preserve bifurcations of Hamiltonian boundary value problems and that non-symplectic integrators do not. We provide a universal description of the breaking of umbilic bifurcations by nonysmplectic integrators. We discover extra structure induced from certain types of boundary value problems, including classical Dirichlet problems, that is useful to locate bifurcations
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On the Minimal Displacement Vector of Compositions and Convex Combinations of Nonexpansive Mappings Found. Comput. Math. (IF 2.476) Pub Date : 2020-02-19 Heinz H. Bauschke, Walaa M. Moursi
Monotone operators and (firmly) nonexpansive mappings are fundamental objects in modern analysis and computational optimization. It was shown in 2012 that if finitely many firmly nonexpansive mappings have or “almost have” fixed points, then the same is true for compositions and convex combinations. More recently, sharp information about the minimal displacement vector of compositions and of convex
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A Proof via Finite Elements for Schiffer’s Conjecture on a Regular Pentagon Found. Comput. Math. (IF 2.476) Pub Date : 2020-02-19 Nilima Nigam, Bartłomiej Siudeja, Benjamin Young
A modified version of Schiffer’s conjecture on a regular pentagon states that Neumann eigenfunctions of the Laplacian do not change sign on the boundary. In a companion paper by Bartłomiej Siudeja it was shown that eigenfunctions which are strictly positive on the boundary exist on regular polygons with at least 6 sides, while on equilateral triangles and cubes it is not even possible to find an eigenfunction
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Propagation of One- and Two-Dimensional Discrete Waves Under Finite Difference Approximation Found. Comput. Math. (IF 2.476) Pub Date : 2020-01-28 Umberto Biccari, Aurora Marica, Enrique Zuazua
We analyze the propagation properties of the numerical versions of one- and two-dimensional wave equations, semi-discretized in space by finite difference schemes. We focus on high-frequency solutions whose propagation can be described, both at the continuous and at the semi-discrete levels, by micro-local tools. We consider uniform and non-uniform numerical grids as well as constant and variable coefficients
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Stability of Low-Rank Tensor Representations and Structured Multilevel Preconditioning for Elliptic PDEs Found. Comput. Math. (IF 2.476) Pub Date : 2020-01-23 Markus Bachmayr, Vladimir Kazeev
Folding grid value vectors of size \(2^L\) into Lth-order tensors of mode size \(2\times \cdots \times 2\), combined with low-rank representation in the tensor train format, has been shown to result in highly efficient approximations for various classes of functions. These include solutions of elliptic PDEs on nonsmooth domains or with oscillatory data. This tensor-structured approach is attractive
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Linking Combinatorial and Classical Dynamics: Conley Index and Morse Decompositions Found. Comput. Math. (IF 2.476) Pub Date : 2020-01-23 Bogdan Batko, Tomasz Kaczynski, Marian Mrozek, Thomas Wanner
We prove that every combinatorial dynamical system in the sense of Forman, defined on a family of simplices of a simplicial complex, gives rise to a multivalued dynamical system F on the geometric realization of the simplicial complex. Moreover, F may be chosen in such a way that the isolated invariant sets, Conley indices, Morse decompositions and Conley–Morse graphs of the combinatorial vector field
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Reconstruction and Interpolation of Manifolds. I: The Geometric Whitney Problem Found. Comput. Math. (IF 2.476) Pub Date : 2019-11-13 Charles Fefferman, Sergei Ivanov, Yaroslav Kurylev, Matti Lassas, Hariharan Narayanan
We study the geometric Whitney problem on how a Riemannian manifold (M, g) can be constructed to approximate a metric space \((X,d_X)\). This problem is closely related to manifold interpolation (or manifold reconstruction) where a smooth n-dimensional submanifold \(S\subset {{\mathbb {R}}}^m\), \(m>n\) needs to be constructed to approximate a point cloud in \({{\mathbb {R}}}^m\). These questions are
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Computing the Interleaving Distance is NP-Hard Found. Comput. Math. (IF 2.476) Pub Date : 2019-11-11 Håvard Bakke Bjerkevik, Magnus Bakke Botnan, Michael Kerber
We show that computing the interleaving distance between two multi-graded persistence modules is NP-hard. More precisely, we show that deciding whether two modules are 1-interleaved is NP-complete, already for bigraded, interval decomposable modules. Our proof is based on previous work showing that a constrained matrix invertibility problem can be reduced to the interleaving distance computation of
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A Polynomial-Time Algorithm to Compute Turaev–Viro Invariants $$\mathrm {TV}_{4,q}$$ TV 4 , q of 3-Manifolds with Bounded First Betti Number Found. Comput. Math. (IF 2.476) Pub Date : 2019-11-11 Clément Maria, Jonathan Spreer
In this article, we introduce a fixed-parameter tractable algorithm for computing the Turaev–Viro invariants \(\mathrm {TV}_{4,q}\), using the first Betti number, i.e. the dimension of the first homology group of the manifold with \(\mathbb {Z}_2\)-coefficients, as parameter. This is, to our knowledge, the first parameterised algorithm in computational 3-manifold topology using a topological parameter
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Projective Geometry of Wachspress Coordinates Found. Comput. Math. (IF 2.476) Pub Date : 2019-11-11 Kathlén Kohn, Kristian Ranestad
We show that there is a unique hypersurface of minimal degree passing through the non-faces of a polytope which is defined by a simple hyperplane arrangement. This generalizes the construction of the adjoint curve of a polygon by Wachspress (A rational finite element basis, Academic Press, New York, 1975). The defining polynomial of our adjoint hypersurface is the adjoint polynomial introduced by Warren
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On the Convergence of the Spectral Viscosity Method for the Two-Dimensional Incompressible Euler Equations with Rough Initial Data Found. Comput. Math. (IF 2.476) Pub Date : 2019-11-08 Samuel Lanthaler, Siddhartha Mishra
We propose a spectral viscosity method to approximate the two-dimensional Euler equations with rough initial data and prove that the method converges to a weak solution for a large class of initial data, including when the initial vorticity is in the so-called Delort class, i.e., it is a sum of a signed measure and an integrable function. This provides the first convergence proof for a numerical method
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Inverse Potential Problems for Divergence of Measures with Total Variation Regularization Found. Comput. Math. (IF 2.476) Pub Date : 2019-11-07 L. Baratchart, C. Villalobos Guillén, D. P. Hardin, M. C. Northington, E. B. Saff
We study inverse problems for the Poisson equation with source term the divergence of an \({{\mathbb {R}}}^3\)-valued measure, that is, the potential \(\varPhi \) satisfies $$\begin{aligned} \Delta \varPhi = \nabla \cdot {{\varvec{\mu }}}, \end{aligned}$$ and \({{\varvec{\mu }}}\) is to be reconstructed knowing (a component of) the field \(\, {\mathrm{grad}}\,\varPhi \) on a set disjoint from the support
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Tropical Combinatorial Nullstellensatz and Sparse Polynomials Found. Comput. Math. (IF 2.476) Pub Date : 2019-10-25 Dima Grigoriev; Vladimir V. Podolskii
Tropical algebra emerges in many fields of mathematics such as algebraic geometry, mathematical physics and combinatorial optimization. In part, its importance is related to the fact that it makes various parameters of mathematical objects computationally accessible. Tropical polynomials play a fundamental role in this, especially for the case of algebraic geometry. On the other hand, many algebraic
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Orthogonal Systems with a Skew-Symmetric Differentiation Matrix Found. Comput. Math. (IF 2.476) Pub Date : 2019-10-10 Arieh Iserles; Marcus Webb
In this paper, we explore orthogonal systems in \(\mathrm {L}_2({\mathbb R})\) which give rise to a real skew-symmetric, tridiagonal, irreducible differentiation matrix. Such systems are important since they are stable by design and, if necessary, preserve Euclidean energy for a variety of time-dependent partial differential equations. We prove that there is a one-to-one correspondence between such
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Symbolic Computations of First Integrals for Polynomial Vector Fields Found. Comput. Math. (IF 2.476) Pub Date : 2019-09-27 Guillaume Chèze; Thierry Combot
In this article, we show how to generalize to the Darbouxian, Liouvillian and Riccati case the extactic curve introduced by J. Pereira. With this approach, we get new algorithms for computing, if it exists, a rational, Darbouxian, Liouvillian or Riccati first integral with bounded degree of a polynomial planar vector field. We give probabilistic and deterministic algorithms. The arithmetic complexity
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Error Estimates for Spectral Convergence of the Graph Laplacian on Random Geometric Graphs Toward the Laplace–Beltrami Operator Found. Comput. Math. (IF 2.476) Pub Date : 2019-09-24 Nicolás García Trillos; Moritz Gerlach; Matthias Hein; Dejan Slepčev
We study the convergence of the graph Laplacian of a random geometric graph generated by an i.i.d. sample from a m-dimensional submanifold \({\mathcal {M}}\) in \(\mathbb {R}^d\) as the sample size n increases and the neighborhood size h tends to zero. We show that eigenvalues and eigenvectors of the graph Laplacian converge with a rate of \(O\Big (\big (\frac{\log n}{n}\big )^\frac{1}{2m}\Big )\)
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Second-Order Models for Optimal Transport and Cubic Splines on the Wasserstein Space Found. Comput. Math. (IF 2.476) Pub Date : 2019-08-08 Jean-David Benamou; Thomas O. Gallouët; François-Xavier Vialard
On the space of probability densities, we extend the Wasserstein geodesics to the case of higher-order interpolation such as cubic spline interpolation. After presenting the natural extension of cubic splines to the Wasserstein space, we propose a simpler approach based on the relaxation of the variational problem on the path space. We explore two different numerical approaches, one based on multimarginal
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On the Sample Complexity of the Linear Quadratic Regulator Found. Comput. Math. (IF 2.476) Pub Date : 2019-08-05 Sarah Dean; Horia Mania; Nikolai Matni; Benjamin Recht; Stephen Tu
This paper addresses the optimal control problem known as the linear quadratic regulator in the case when the dynamics are unknown. We propose a multistage procedure, called Coarse-ID control, that estimates a model from a few experimental trials, estimates the error in that model with respect to the truth, and then designs a controller using both the model and uncertainty estimate. Our technique uses
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Convergence of Finite Volume Schemes for the Euler Equations via Dissipative Measure-Valued Solutions Found. Comput. Math. (IF 2.476) Pub Date : 2019-08-05 Eduard Feireisl; Mária Lukáčová-Medvid’ová; Hana Mizerová
The Cauchy problem for the complete Euler system is in general ill-posed in the class of admissible (entropy producing) weak solutions. This suggests that there might be sequences of approximate solutions that develop fine-scale oscillations. Accordingly, the concept of measure-valued solution that captures possible oscillations is more suitable for analysis. We study the convergence of a class of
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Mathematics of Smoothed Particle Hydrodynamics: A Study via Nonlocal Stokes Equations Found. Comput. Math. (IF 2.476) Pub Date : 2019-08-05 Qiang Du; Xiaochuan Tian
Smoothed particle hydrodynamics (SPH) is a popular numerical technique developed for simulating complex fluid flows. Among its key ingredients is the use of nonlocal integral relaxations to local differentiations. Mathematical analysis of the corresponding nonlocal models on the continuum level can provide further theoretical understanding of SPH. We present, in this part of a series of works on the
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Lie–Poisson Methods for Isospectral Flows Found. Comput. Math. (IF 2.476) Pub Date : 2019-08-05 Klas Modin; Milo Viviani
The theory of isospectral flows comprises a large class of continuous dynamical systems, particularly integrable systems and Lie–Poisson systems. Their discretization is a classical problem in numerical analysis. Preserving the spectrum in the discrete flow requires the conservation of high order polynomials, which is hard to come by. Existing methods achieving this are complicated and usually fail
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Computing Tropical Varieties Over Fields with Valuation Found. Comput. Math. (IF 2.476) Pub Date : 2019-08-05 Thomas Markwig; Yue Ren
We show how the tropical variety of an ideal \(I\unlhd K[x_1,\ldots ,x_n]\) over a field K with non-trivial discrete valuation can always be traced back to the tropical variety of an ideal \(\pi ^{-1}I\unlhd R\llbracket t\rrbracket [x_1,\ldots ,x_n]\) over some dense subring R in its ring of integers. We show that this connection is compatible with the Gröbner polyhedra covering them. Combined with