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Our latest publications - Computational Mathematics
This list contains the last publications registered in CRIStin by researchers connected to the group in Computational Mathematics, Department of Mathematics, UiO.
The list is not complete.
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Hiltunen, Erik & Rueff, Liora
(2026).
Energy Balance and Optical Theorem for Time-Modulated Subwavelength Resonator Arrays.
Studies in applied mathematics (Cambridge).
ISSN 0022-2526.
156(1).
doi:
10.1111/sapm.70168.
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Hiltunen, Erik Orvehed; Kraisler, Joseph; Schotland, John C. & Weinstein, Michael I.
(2026).
NONLOCAL PARTIAL DIFFERENTIAL EQUATIONS AND QUANTUM OPTICS: BAND STRUCTURE OF PERIODIC ATOMIC SYSTEMS.
SIAM Journal on Mathematical Analysis.
ISSN 0036-1410.
58(3),
p. 2506–2535.
doi:
10.1137/25M1754959.
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Bruijn, Yannick De & Hiltunen, Erik Orvehed
(2026).
Complex Brillouin zone for localised modes in Hermitian and non-Hermitian problems.
Communications in nonlinear science & numerical simulation.
ISSN 1007-5704.
161,
p. 110014–110014.
doi:
10.1016/j.cnsns.2026.110014.
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Hoel, Håkon Andreas & Meo, Johannes Vincent
(2026).
A Priori Adaptive Numerical Methods for Estimating Blow-up Times of Autonomous ODEs.
arXiv.
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We present effective a priori adaptive numerical methods for estimating the blow-up time for solutions of autonomous ODEs. The novelty of our approach is to base our adaptive steps on the sensitivity of an auxiliary hitting time. We provide results on the theoretical error rates, and show that there is a benefit in terms of computational effort in choosing our adaptive algorithm over alternative approaches. Numerical experiments support our theoretical results and show how the methods perform in practice.
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Berzins, Arturs; Radler, Andreas; Volkmann, Eric; Sanokowski, Sebastian; Hochreiter, Sepp & Brandstetter, Johannes
(2025).
Geometry-Informed Neural Networks.
Proceedings of Machine Learning Research (PMLR).
267,
p. 3976–4004.
doi:
https:/proceedings.mlr.press/v267/berzins25a.html.
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Geometry is a ubiquitous tool in computer graphics, design, and engineering. However, the lack of large shape datasets limits the application of state-of-the-art supervised learning methods and motivates the exploration of alternative learning strategies. To this end, we introduce geometry-informed neural networks (GINNs) – a framework for training shape-generative neural fields without data by leveraging user-specified design requirements in the form of objectives and constraints. By adding diversity as an explicit constraint, GINNs avoid mode-collapse and can generate multiple diverse solutions, often required in geometry tasks. Experimentally, we apply GINNs to several problems spanning physics, geometry, and engineering design, showing control over geometrical and topological properties, such as surface smoothness or the number of holes. These results demonstrate the potential of training shape-generative models without data, paving the way for new generative design approaches without large datasets.
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Floater, Michael S.
(2025).
Knot insertion and binary subdivision of splines.
BIT Numerical Mathematics.
ISSN 0006-3835.
65(4).
doi:
10.1007/s10543-025-01093-6.
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In this paper we derive an efficient subdivision algorithm for the binary refinement of non-uniform spline functions or curves represented in the B-spline basis. We recursively insert the new knots into the divided differences that define the B-splines. This creates sequences of intermediate divided differences from which we construct sequences of normalized B-splines. From the refinement equations for these B-splines, the subdivision algorithm follows by induction. The coefficients computed at each step of the algorithm are themselves B-spline coefficients.
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Chatterjee, Neelabja & Risebro, Nils Henrik
(2025).
The Ostrovsky-Hunter equation with a space dependent flux function.
Journal of Numerical Mathematics.
ISSN 1570-2820.
34(2),
p. 177–193.
doi:
10.1515/jnma-2024-0104.
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We study the periodic Ostrovsky–Hunter equation in the case where the flux function may depend on the spatial variable. Our main results are that if the flux function is twice differentiable, then there exists a unique entropy solution. This entropy solution may be constructed as a limit of approximate solutions generated by a finite volume scheme, and the finite volume approximations converge to the entropy solution at a rate 1/2.
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Gazdag, Luca Eva; Antun, Vegard & Hansen, Anders Christian
(2025).
Instability in deep learning – when algorithms cannot compute uncertainty quantifications for neural networks.
European journal of applied mathematics.
ISSN 0956-7925.
37(1),
p. 290–312.
doi:
10.1017/S095679252510017X.
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Floater, Michael S. & Muntingh, Agnar Georg Peder
(2025).
On the injectivity of mean value mappings between quadrilaterals.
Advances in Computational Mathematics.
ISSN 1019-7168.
51(4).
doi:
10.1007/s10444-025-10246-8.
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Mean value coordinates can be used to map one polygon into another, with application to computer graphics and curve and surface modelling. In this paper, we show that if the polygons are quadrilaterals, and if the target quadrilateral is convex, then the mapping is injective.
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Bruijn, Yannick De; Davies, Bryn; Dupuy, Sacha & Hiltunen, Erik Orvehed
(2025).
Spectra and pseudospectra of non-Hermitian Toeplitz operators: Eigenvector decay transitions in banded and dense matrices.
arXiv.
doi:
10.48550/ARXIV.2512.03757.
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Coclite, Giuseppe Maria; Karlsen, Kenneth Aksel Hvistendahl & Risebro, Nils Henrik
(2025).
UPWIND FILTERING OF SCALAR CONSERVATION LAWS.
SIAM Journal on Mathematical Analysis.
ISSN 0036-1410.
57(6),
p. 6119–6143.
doi:
10.1137/25M1730387.
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Rybalko, Yan; Shepelsky, Dmitry & Tian, Shou-Fu
(2025).
The integrable nonlocal nonlinear Schrödinger equation with oscillatory boundary conditions: Long-time asymptotics.
Physica D : Non-linear phenomena.
ISSN 0167-2789.
481.
doi:
10.1016/j.physd.2025.134820.
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Ammari, Habib; Barandun, Silvio; Cao, Jinghao; Davies, Bryn; Hiltunen, Erik Orvehed & Liu, Ping
(2025).
The non-Hermitian skin effect with three-dimensional long-range coupling.
Journal of the European Mathematical Society (Print).
ISSN 1435-9855.
doi:
10.4171/jems/1685.
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We study the non-Hermitian skin effect in a three-dimensional system of finitely many subwavelength resonators with an imaginary gauge potential. We introduce a discrete approximation of the eigenmodes and eigenfrequencies of the system in terms of the eigenvectors and eigenvalues of the so-called gauge capacitance matrix \mathcal{C}_{N}^{\gamma} , which is a dense matrix due to long-range interactions in the system. Based on translational invariance of this matrix and the decay of its off-diagonal entries, we prove the condensation of the eigenmodes at one edge of the structure by showing the exponential decay of its pseudo-eigenvectors. In particular, we consider a range k -approximation to keep the long-range interaction to a certain extent, thus obtaining a k -banded gauge capacitance matrix \mathcal{C}_{N,k}^{\gamma} . Using techniques for Toeplitz matrices and operators, we establish the exponential decay of the pseudo-eigenvectors of \mathcal{C}_{N,k}^{\gamma} and demonstrate that they approximate those of the gauge capacitance matrix \mathcal{C}_{N}^{\gamma} well. Our results are numerically verified. In particular, we show that long-range interactions affect only the first eigenmodes in the system. As a result, a tridiagonal approximation of the gauge capacitance matrix, similar to the nearest-neighbour approximation in quantum mechanics, provides a good approximation for the higher modes.
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Ammari, Habib; Barandun, Silvio; Davies, Bryn; Hiltunen, Erik Orvehed & Uhlmann, Alexander
(2025).
Universal estimates for the density of states for aperiodic block subwavelength resonator systems.
Journal of Physics A: Mathematical and Theoretical.
ISSN 1751-8113.
58(41).
doi:
10.1088/1751-8121/ae0b11.
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We consider the spectral properties of aperiodic block subwavelength resonator systems in one dimension, with a primary focus on the density of states (DoS). We prove that for random block configurations, as the number of blocks M → ∞ , the integrated DoS converges to a non-random, continuous function. We show both analytically and numerically that the DoS exhibits a tripartite decomposition: it vanishes identically within bandgaps; it forms smooth, band-like distributions in shared pass bands (a consequence of constructive eigenmode interactions); and, most notably, it exhibits a distinct fractal-like character in hybridisation regions. We demonstrate that this fractal-like behaviour stems from the limited interaction between eigenmodes within these hybridisation regions. Capitalising on this insight, we introduce an efficient meta-atom approach that enables rapid and accurate prediction of the DoS in these hybridisation regions. This approach is shown to extend to systems with quasiperiodic and hyperuniform arrangements of blocks.
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Fjordholm, U.S. & Mæhlen, Ola Isaac Høgåsen
(2025).
Transport equations for Osgood velocity fields.
Journal of Differential Equations.
ISSN 0022-0396.
443.
doi:
10.1016/j.jde.2025.113566.
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We consider the transport equation with a velocity field satisfying the Osgood condition. The weak formulation is not meaningful in the usual Lebesgue sense, meaning that the usual DiPerna–Lions treatment of the problem is not applicable (in particular, the divergence of the velocity might be unbounded). Instead, we use Riemann–Stieltjes integration to interpret the weak formulation, leading to a well-posedness theory in regimes not covered by existing works. The most general results are for the one-dimensional problem, with generalisations to multiple dimensions in the particular case of log-Lipschitz velocities.
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Haji-Ali, Abdul-Lateef; Hoel, Håkon Andreas & Tempone, Raúl
(2025).
Weak convergence analysis in the particle limit of the McKean–Vlasov equation using stochastic flows of particle systems.
IMA Journal of Applied Mathematics.
ISSN 0272-4960.
90(3),
p. 277–304.
doi:
10.1093/imamat/hxaf015.
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We present a proof showing that the weak error of a system of n interacting stochastic particles approximating the solution of the McKean–Vlasov equation is O(n^-1). Our proof is based on the Kolmogorov backward equation for the particle system and bounds on the derivatives of its solution which we derive more generally using the variations of the stochastic particle system. The convergence rate is verified by numerical experiments which also indicate that the assumptions made here and in the literature can be relaxed.
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Karlsen, Kenneth Aksel Hvistendahl & Mitrovic, Darko
(2025).
A note on conservation laws with discontinuous flux and L^1 initial data.
Monatshefte für Mathematik (Print).
ISSN 0026-9255.
208,
p. 65–83.
doi:
10.1007/s00605-025-02105-x.
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Abstract We study conservation laws with a discontinuous flux function $${\mathfrak {f}}(\textbf{x},\lambda )$$ f ( x , λ ) . The flux function can be expressed as $$g(\beta (\textbf{x},\lambda ))$$ g ( β ( x , λ ) ) , where g is locally Lipschitz, $$\beta (\textbf{x},\lambda )$$ β ( x , λ ) is an increasing function in $$\lambda $$ λ for each fixed $$\textbf{x}$$ x , $$\nabla \beta (\cdot ,0)$$ ∇ β ( · , 0 ) is a finite measure, and $$\beta (\cdot ,0)$$ β ( · , 0 ) is bounded. We consider this problem under the Audusse-Perthame entropy condition and derive a kinetic formulation. Using the kinetic approach, we prove an existence result under the assumption that the initial function belongs to $$L^1$$ L 1 . Uniqueness results are also presented.
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Coclite, Giuseppe Maria; Karlsen, Kenneth Aksel Hvistendahl & Risebro, Nils Henrik
(2025).
Some observations regarding the stationary Buckley–Leverett equation.
Applied Mathematics Letters.
ISSN 0893-9659.
170.
doi:
10.1016/j.aml.2025.109621.
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Ammari, Habib; Hiltunen, Erik & Rueff, Liora
(2025).
Space-time wave localization in systems of subwavelength resonators.
Proceedings of the Royal Society. Mathematical, Physical and Engineering Sciences.
ISSN 1364-5021.
481(2307).
doi:
10.1098/rspa.2024.0698.
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In this article, we study the dynamics of metamaterials composed of high-contrast subwavelength resonators and show the existence of localized modes in such a setting. A crucial assumption in this article is time-modulated material parameters. We prove a so-called capacitance matrix approximation of the wave equation in the form of an ordinary differential equation. These formulas set the ground for the derivation of a first-principle characterization of localized modes in terms of the generalized capacitance matrix. Furthermore, we provide numerical results supporting our analytical results showing for the first time the phenomenon of space–time localized waves in a perturbed time-modulated metamaterial. Such spatio-temporal localization is only possible in the presence of subwavelength resonances in the unperturbed structure. We introduce the time-dependent degree of localization to quantitatively determine the localized modes and provide a variety of numerical experiments to illustrate our formulations and results.
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Alexopoulos, Konstantinos; Davies, Bryn & Hiltunen, Erik Orvehed
(2025).
TOPOLOGICAL INTERFACE MODES IN SYSTEMS WITH DAMPING.
SIAM Journal on Applied Mathematics.
ISSN 0036-1399.
85(4),
p. 1500–1518.
doi:
10.1137/24M1710371.
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Brualdi, R.A. & Dahl, Geir
(2025).
Near-derangements and their polytopes.
The Australasian Journal of Combinatorics.
ISSN 1034-4942.
91(3),
p. 366–386.
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A derangement is a permutation with no fixed point. We refer to permutations of {1, 2, . . . , n} with at most one fixed point as near-derangements, and study the corresponding set of permutation matrices Pn(≤1) . We determine a basis of the linear span of Pn(≤1) consisting of matrices in Pn(≤1). Also, we study the polytope determined by Pn(≤1). In addition, we investigate the polytope of all n × n doubly stochastic matrices with trace at most 1 and determine its extreme points, not all of which are permutation matrices.
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Fjordholm, Ulrik Skre; Karlsen, Kenneth Aksel Hvistendahl & Pang, Ho Cheung
(2025).
Convergent Finite Difference Schemes for Stochastic Transport Equations.
SIAM Journal on Numerical Analysis.
ISSN 0036-1429.
63(1),
p. 149–192.
doi:
10.1137/23M159946X.
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We present difference schemes for stochastic transport equations with low-regularity velocity fields. We establish 𝐿2 stability and convergence of the difference approximations under conditions that are less strict than those required for deterministic transport equations. The 𝐿2 estimate, crucial for the analysis, is obtained through a discrete duality argument and a comprehensive examination of a class of backward parabolic difference schemes.
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Falk, Richard & Winther, Ragnar
(2025).
Local Space-Preserving Decompositions for the Bubble Transform.
Foundations of Computational Mathematics.
ISSN 1615-3375.
doi:
10.1007/s10208-025-09700-2.
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The bubble transform is a procedure to decompose differential forms, which are piecewise smooth with respect to a given triangulation of the domain, into a sum of local bubbles. In this paper, an improved version of a construction in the setting of the de Rham complex previously proposed by the authors is presented. The major improvement in the decomposition is that unlike the previous results, in which the individual bubbles were rational functions with the property that groups of local bubbles summed up to preserve piecewise smoothness, the new decomposition is strictly space-preserving in the sense that each local bubble preserves piecewise smoothness. An important property of the transform is that the construction only depends on the given triangulation of the domain and is independent of any finite element space. On the other hand, all the standard piecewise polynomial spaces are invariant under the transform. Other key properties of the transform are that it commutes with the exterior derivative, is bounded in L2, and satisfies the stable decomposition property.
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Hiltunen, Erik
(2025).
Perturbation theory for dispersion relations of spacetime-periodic materials.
Physical review B (PRB).
ISSN 2469-9950.
111(14).
doi:
10.1103/PhysRevB.111.L140102.
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We consider Bloch states of weak spacetime-periodic perturbations of homogeneous materials in one spatial dimension. The interplay of the periodicity of space and time leads to an infinitely degenerate dispersion relation in the free case. We consider a general perturbation term, and, as consequence of the infinite degeneracy, we show that the effective equations are given by the eigenvalue problem of an infinite matrix. Our method can be viewed as a time-modulated generalization of the nearly-free-electron model. Based on this result, we find that the infinite degeneracy may split into a family of nondegenerate bands. Our results are illustrated with numerical calculations, and we observe close agreement between the perturbation theory and the numerically computed full solution.
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Gottschling, Nina M.; Antun, Vegard; Hansen, Anders C. & Adcock, Ben
(2025).
The Troublesome Kernel: On Hallucinations, No Free Lunches, and the Accuracy-Stability Tradeoff in Inverse Problems.
SIAM Review.
ISSN 0036-1445.
67(1),
p. 73–104.
doi:
10.1137/23M1568739.
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Davies, B.; Barandun, S.; Hiltunen, Erik; Craster, R.V. & Ammari, H.
(2025).
Two-scale effective model for defect-induced localization transitions in non-Hermitian systems.
Physical review B (PRB).
ISSN 2469-9950.
111(3).
doi:
10.1103/PhysRevB.111.035109.
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We illuminate the fundamental mechanism responsible for the transition between the non-Hermitian skin effect and defect-induced localization in the bulk. We study a Hamiltonian with nonreciprocal couplings that exhibits the skin effect (the localization of all eigenvectors at one edge) and add an on-site defect in the center. Using a two-scale asymptotic method, we characterize the long-scale growth and decay of the eigenvectors and derive a simple and intuitive effective model for the transition that occurs when the defect is sufficiently large that one of the modes is localized at the defect site, rather than at the edge of the system.
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Bruijn, Yannick De & Hiltunen, Erik Orvehed
(2025).
Complex Band Structure for Subwavelength Evanescent Waves.
Studies in applied mathematics (Cambridge).
ISSN 0022-2526.
154(2).
doi:
10.1111/sapm.70022.
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We present the mathematical and numerical theory for evanescent waves in subwavelength bandgap materials. We begin in the one-dimensional case, whereby fully explicit formulas for the complex band structure, in terms of the capacitance matrix, are available. As an example, we show that the gap functions can be used to accurately predict the decay rate of the interface mode of a photonic analogue of the Su–Schrieffer–Heeger model. In two dimensions, we derive the bandgap Green's function and characterize the subwavelength gap functions via layer potential techniques. By generalizing existing lattice-summation techniques, we illustrate our results numerically by computing the complex band structure in a variety of settings.
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Karlsen, Kenneth Aksel Hvistendahl; Mitrovic, Darko & Nedeljkov, M.
(2025).
On the viscosity approximation of conservation laws with non-crossing discontinuous flux.
Journal of Differential Equations.
ISSN 0022-0396.
420,
p. 316–334.
doi:
10.1016/j.jde.2024.11.056.
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Fjordholm, Ulrik Skre; Mishra, Siddhartha & Weber, Franziska
(2024).
On the Vanishing Viscosity Limit of Statistical Solutions of the Incompressible Navier–Stokes Equations.
SIAM Journal on Mathematical Analysis.
ISSN 0036-1410.
56(4),
p. 5099–5143.
doi:
10.1137/23M1566261.
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We study statistical solutions of the incompressible Navier–Stokes equation and their vanishing viscosity limit. We show that a formulation using correlation measures as in [U. S. Fjordholm, S. Lanthaler, and S. Mishra, Arch. Ration. Mech. Anal., 226 (2017), pp. 809–849] and moment equations is equivalent to statistical solutions in the Foiaş–Prodi sense. Under the assumption of weak scaling, a weaker version of Kolmogorov’s self-similarity at small scales hypothesis that allows for intermittency corrections, we show that the limit is a statistical solution of the incompressible Euler equations. To pass to the limit, we derive a Kármán–Howarth–Monin relation for statistical solutions and combine it with the weak scaling assumption and a compactness theorem for correlation measures from [U. S. Fjordholm et al., Math. Models Methods Appl. Sci., 30 (2020), pp. 539–609].
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Galimberti, Luca & Karlsen, Kenneth Aksel Hvistendahl
(2024).
Well-Posedness of Stochastic Continuity Equations on Riemannian Manifolds.
Chinese Annals of Mathematics. Series B.
ISSN 0252-9599.
45(1),
p. 81–122.
doi:
10.1007/s11401-024-0005-9.
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Coclite, Giuseppe Maria; Karlsen, Kenneth Aksel Hvistendahl & Risebro, Nils Henrik
(2024).
A nonlocal Lagrangian traffic flow model and the zero-filter limit.
Zeitschrift für Angewandte Mathematik und Physik.
ISSN 0044-2275.
75(2).
doi:
10.1007/s00033-023-02153-z.
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In this study, we start from a Follow-the-Leaders model for traffic flow that is based on a weighted harmonic mean (in Lagrangian coordinates) of the downstream car density. This results in a nonlocal Lagrangian partial differential equation (PDE) model for traffic flow. We demonstrate the well-posedness of the Lagrangian model in the L1 sense. Additionally, we rigorously show that our model coincides with the Lagrangian formulation of the local LWR model in the “zero-filter” (nonlocal-to-local) limit. We present numerical simulations of the new model. One significant advantage of the proposed model is that it allows for simple proofs of (i) estimates that do not depend on the “filter size” and (ii) the dissipation of an arbitrary convex entropy.
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Ammari, Habib; Barandun, Silvio; Davies, Bryn; Hiltunen, Erik & Liu, Ping
(2024).
STABILITY OF THE NON-HERMITIAN SKIN EFFECT IN ONE DIMENSION.
SIAM Journal on Applied Mathematics.
ISSN 0036-1399.
84(4),
p. 1697–1717.
doi:
10.1137/23M1610537.
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This paper shows both analytically and numerically that the skin effect in systems of non-Hermitian subwavelength resonators is robust with respect to random imperfections in the system. The subwavelength resonators are highly contrasting material inclusions that resonate in a low-frequency regime. The non-Hermiticity is due to the introduction of a directional damping term (motivated by an imaginary gauge potential), which leads to a skin effect that is manifested by the system’s eigenmodes accumulating at one edge of the structure. We elucidate the topological protection of the associated (real) eigenfrequencies and illustrate numerically the competition between the two different localization effects present when the system is randomly perturbed: the non-Hermitian skin effect and the disorder-induced Anderson localization. We show numerically that, as the strength of the disorder increases, more and more eigenmodes become localized in the bulk. Our results are based on an asymptotic matrix model for subwavelength physics and can be generalized also to tight-binding models in condensed matter theory.
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Karlsen, Kenneth Aksel Hvistendahl & Rybalko, Yan
(2024).
On the well-posedness of the Cauchy problem for the two-component peakon system in Ck∩Wk,1.
Zeitschrift für Angewandte Mathematik und Physik.
ISSN 0044-2275.
75(3).
doi:
10.1007/s00033-024-02246-3.
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Ammari, Habib; Barandun, Silvio; Davies, Bryn; Hiltunen, Erik; Kosche, Thea & Liu, Ping
(2024).
Exponentially localized interface eigenmodes in finite chains of resonators.
Studies in applied mathematics (Cambridge).
ISSN 0022-2526.
153(4).
doi:
10.1111/sapm.12765.
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This paper studies wave localization in chains of finitely many resonators. There is an extensive theory predicting the existence of localized modes induced by defects in infinitely periodic systems. This work extends these principles to finite-sized systems. We consider one-dimensional, finite systems of subwavelength resonators arranged in dimers that have a geometric defect in the structure. This is a classical wave analog of the Su–Schrieffer–Heeger model. We prove the existence of a spectral gap for defectless finite dimer structures and find a direct relationship between eigenvalues being within the spectral gap and the localization of their associated eigenmode. Then, for sufficiently large-size systems, we show the existence and uniqueness of an eigenvalue in the gap in the defect structure, proving the existence of a unique localized interface mode. To the best of our knowledge, our method, based on Chebyshev polynomials, is the first to characterize quantitatively the localized interface modes in systems of finitely many resonators.
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Lyche, Tom; Manni, Carla & Speleers, Hendrik
(2024).
A Parsimonious Approach to C2 Cubic Splines on Arbitrary Triangulations:
Reduced Macro-Elements on the Cubic Wang–Shi Split.
In Lanini, Martina; Manni, Carla & Schenck, Henry (Ed.),
Approximation Theory and Numerical Analysis
Meet Algebra, Geometry, Topology.
Springer Nature.
ISSN 9789819765072.
p. 265–288.
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Eddargani, Salah; Lyche, Tom; Manni, Carla & Speleers, Hendrik
(2024).
Quadrature rules for C^1 quadratic spline finite elements on the Powell–Sabin 12-split.
Computer Methods in Applied Mechanics and Engineering.
ISSN 0045-7825.
430.
doi:
10.1016/j.cma.2024.117196.
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We present quadrature rules for the space of C1 quadratic splines on triangulations refined according to the Powell–Sabin (PS) 12-split. Focusing on a single triangle, we first provide a symmetric 4-node quadrature rule with positive weights, which is exact on the considered spline space. For its construction we make use of a local simplex spline basis. The rule is shown to be optimal, with the minimum number of nodes. Next, in view of using C1 quadratic splines on triangulations in Galerkin finite element discretizations of differential problems, we design a collection of weighted quadrature rules that are helpful for an efficient formation of the linear systems arising in such discretizations. Lastly, we provide numerical comparisons with elementwise Gaussian rules and we illustrate the performance of the presented quadrature rules in the context of C1 quadratic spline finite elements on triangulations.
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Karlsen, Kenneth Aksel Hvistendahl & Rybalko, Yan
(2024).
On the well-posedness of a nonlocal (two-place) FORQ equation via a two-component peakon system.
Journal of Mathematical Analysis and Applications.
ISSN 0022-247X.
529(1).
doi:
10.1016/j.jmaa.2023.127601.
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We investigate the Cauchy problem for a nonlocal (two-place) FORQ equation. By interpreting this equation as a special case of a two-component peakon system (exhibiting a cubic nonlinearity), we convert the Cauchy problem into a system of ordinary differential equations in a Banach space. Using this approach, we are able to demonstrate local well-posedness in the Sobolev space H8 where s > 5/2. We also establish the continuity properties for the data-to-solution map for a range of Sobolev spaces. Finally, we briefly explore the relationship between the two-component system and the bi-Hamiltonian AKNS hierarchy.
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Andrade, Enide & Dahl, Geir
(2024).
Combinatorial Fiedler theory and graph partition.
Linear Algebra and its Applications.
ISSN 0024-3795.
687,
p. 229–251.
doi:
10.1016/j.laa.2024.02.005.
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Partition problems in graphs are extremely important in applications, as shown in the Data Science and Machine Learning literature. One approach is spectral partitioning based on a Fiedler vector, i.e., an eigenvector corresponding to the second smallest eigenvalue a(G) of the Laplacian matrix LG of the graph G. This problem corresponds to the minimization of a quadratic form associated with LG, under certain constraints involving the ℓ2-norm. We introduce and investigate a similar problem, but using the ℓ1-norm to measure distances. This leads to a new parameter b(G) as the optimal value. We show that a well-known cut problem arises in this approach, namely the sparsest cut problem. We prove connectivity results and different bounds on this new parameter, relate to Fiedler theory and show explicit expressions for b(G) for trees. We also comment on an ℓ∞-norm version of the problem.
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Fjordholm, Ulrik Skre; Mæhlen, Ola Isaac Høgåsen & Ørke, Magnus Christie
(2024).
The particle paths of hyperbolic conservation laws.
Mathematical Models and Methods in Applied Sciences.
ISSN 0218-2025.
34(7),
p. 1205–1234.
doi:
10.1142/S0218202524500209.
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Nonlinear scalar conservation laws are traditionally viewed as transport equations. We take instead the viewpoint of these PDEs as continuity equations with an implicitly defined velocity field. We show that a weak solution is the entropy solution if and only if the ODE corresponding to its velocity field is well-posed. We also show that the flow of the ODE is 1/2-Hölder regular. Finally, we give several examples showing that our results are sharp, and we provide explicit computations in the case of a Riemann problem.
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Mæhlen, Ola Isaac Høgåsen & Seth, Karl Johan Douglas Svensson
(2024).
Asymmetric Traveling Wave Solutions of the Capillary-Gravity Whitham Equation.
SIAM Journal on Mathematical Analysis.
ISSN 0036-1410.
56(6).
doi:
10.1137/24M1632590.
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By a bifurcation argument we prove that the capillary-gravity Whitham equation features asymmetrical periodic travelling wave solutions of arbitrarily small amplitude. Such waves exist only in the weak surface tension regime 0
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Galimberti, Luca; Holden, Helge; Karlsen, Kenneth Aksel Hvistendahl & Pang, Ho Cheung
(2024).
Global existence of dissipative solutions to the Camassa–Holm equation with transport noise.
Journal of Differential Equations.
ISSN 0022-0396.
387,
p. 1–103.
doi:
10.1016/j.jde.2023.12.021.
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We consider a nonlinear stochastic partial differential equation (SPDE) that takes the form of the Camassa–Holm equation perturbed by a convective, position-dependent, noise term. We establish the first global-in-time existence result for dissipative weak martingale solutions to this SPDE, with general finite-energy initial data. The solution is obtained as the limit of classical solutions to parabolic SPDEs. The proof combines model-specific statistical estimates with stochastic propagation of compactness techniques, along with the systematic use of tightness and a.s. representations of random variables on specific quasi-Polish spaces. The spatial dependence of the noise function makes more difficult the analysis of a priori estimates and various renormalisations, giving rise to nonlinear terms induced by the martingale part of the equation and the second-order Stratonovich–Itô correction term.
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Karlsen, Kenneth Aksel Hvistendahl & Towers, John D.
(2024).
Compactness estimates for difference schemes for conservation laws with discontinuous flux.
IMA Journal of Numerical Analysis.
ISSN 0272-4979.
44(6),
p. 3313–3353.
doi:
10.1093/imanum/drad096.
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Ammari, Habib; Davies, Bryn & Hiltunen, Erik
(2024).
Spectral convergence in large finite resonator arrays: The essential spectrum and band structure.
Bulletin of the London Mathematical Society.
ISSN 0024-6093.
doi:
10.1112/blms.13225.
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We show that the resonant frequencies of a system of coupled resonators in a truncated periodic lattice converge to the essential spectrum of the corresponding infinite lattice. We use the capacitance matrix as a model for fully coupled subwavelength resonators with long-range interactions in three spatial dimensions. For one-, two- or three-dimensional lattices embedded in three-dimensional space, we show that the (discrete) density of states (DOS) for the finite system converges in distribution to the (continuous) DOS of the infinite system. We achieve this by proving a weak convergence of the finite capacitance matrix to corresponding (translationally invariant) Toeplitz matrix of the infinite structure. With this characterisation at hand, we use the truncated Floquet transform to introduce a notion of spectral band structure for finite materials. This principle is also applicable to structures that are not translationally invariant and have interfaces. We demonstrate this by considering examples of perturbed systems with defect modes, such as an analogue of the well-known interface Su–Schrieffer–Heeger model.
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Brualdi, R.A. & Dahl, Geir
(2024).
Multipermutations and Stirling Multipermutations.
Graphs and Combinatorics.
ISSN 0911-0119.
40(2).
doi:
10.1007/s00373-024-02751-2.
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We consider multipermutations and a certain partial order, the weak Bruhat order, on this set. This generalizes the Bruhat order for permutations, and is defined in terms of containment of inversions. Different characterizations of this order are given. We also study special multipermutations called Stirling multipermutations and their properties.
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Holden, Helge & Risebro, Nils Henrik
(2024).
The continuum limit of non-local Follow-the-Leader models.
ESAIM: Mathematical Modelling and Numerical Analysis (ESAIM: M2AN).
ISSN 2822-7840.
58(4),
p. 1523–1539.
doi:
10.1051/m2an/2024054.
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We study a generalized Follow-the-Leader model where the driver considers the position of an arbitrary but finite number of vehicles ahead, as well as the position of the vehicle directly behind the driver. It is proved that this model converges to the classical Lighthill–Whitham–Richards model for traffic flow when traffic becomes dense. This also underscores the robustness of the Lighthill–Whitham–Richards model.
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Karlsen, Kenneth Aksel Hvistendahl; Kunzinger, Michael & Mitrovic, Darko
(2024).
A DYNAMIC CAPILLARITY EQUATION WITH STOCHASTIC FORCING ON MANIFOLDS: A SINGULAR LIMIT PROBLEM.
Transactions of the American Mathematical Society.
ISSN 0002-9947.
377(1),
p. 85–166.
doi:
10.1090/tran/9050.
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Hiltunen, Erik & Davies, Bryn
(2024).
Coupled harmonics due to time-modulated point scatterers.
Physical review B (PRB).
ISSN 2469-9950.
110(18).
doi:
10.1103/PhysRevB.110.184102.
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We consider the resonance and scattering properties of a composite medium containing scatterers whose properties are modulated in time. When excited with an incident wave of a single frequency, the scattered field consists of a family of coupled harmonics at frequencies differing by the frequency of temporal modulation. Similarly, the temporal modulation induces coupling between the resonance frequencies, leading to exceptional points at certain modulation amplitudes. Moreover, the lack of energy conservation causes scattering coefficients to blow up when (complex) resonances cross the real axis. We have developed an integral operator approach to characterize the scattering problem and, for high-contrast scatterers, we present small-volume asymptotic formulas analogous to the classical results for the static (unmodulated) case. We conclude the paper with a boundary integral formulation of the time-modulated problem, which gives an efficient numerical approach and corroborates the asymptotic formulas.
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Bendahmane, M.; Karlsen, Kenneth Aksel Hvistendahl & Mroué, F.
(2024).
Stochastic electromechanical bidomain model.
Nonlinearity.
ISSN 0951-7715.
37(7).
doi:
10.1088/1361-6544/ad5132.
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We analyse a system of nonlinear stochastic partial differential equations (SPDEs) of mixed elliptic-parabolic type that models the propagation of electric signals and their effect on the deformation of cardiac tissue. The system governs the dynamics of ionic quantities, intra and extra-cellular potentials, and linearised elasticity equations. We introduce a framework called the active strain decomposition, which factors the material gradient of deformation into an active (electrophysiology-dependent) part and an elastic (passive) part, to capture the coupling between muscle contraction, biochemical reactions, and electric activity. Under the assumption of linearised elastic behaviour and a truncation of the nonlinear diffusivities, we propose a stochastic electromechanical bidomain model, and establish the existence of weak solutions for this model. To prove existence through the convergence of approximate solutions, we employ a stochastic compactness method in tandem with an auxiliary non-degenerate system and the Faedo–Galerkin method. We utilise a stochastic adaptation of de Rham's theorem to deduce the weak convergence of the pressure approximations.
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Floater, Michael S.
(2024).
On best p-norm approximation of discrete data by polynomials.
IMA Journal of Numerical Analysis.
ISSN 0272-4979.
44(5),
p. 2716–2724.
doi:
10.1093/imanum/drad086.
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In this note, we derive a solution to the problem of finding a polynomial of degree at most n that best approximates data at n+2 points in the lp norm. Analogous to a result of de la Vallée Poussin, one can express the solution as a convex combination of the Lagrange interpolants over subsets of n+1 points, and the error oscillates in sign.
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Ren, Panpan; Tang, Hao & Wang, Feng-Yu
(2024).
Distribution-Path Dependent Nonlinear SPDEs with Application to Stochastic Transport Type Equations.
Potential Analysis.
ISSN 0926-2601.
61,
p. 379–407.
doi:
10.1007/s11118-023-10113-5.
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By using a regularity approximation argument, the global existence and uniqueness are derived for a class of nonlinear SPDEs depending on both the whole history and the distribution under strong enough noise. As applications, the global existence and uniqueness are proved for distribution-path dependent stochastic transport type equations, which are arising from stochastic fluid mechanics with forces depending on the history and the environment. In particular, the distribution-path dependent stochastic Camassa-Holm equation with or without Coriolis effect has a unique global solution when the noise is strong enough, whereas for the deterministic model wave-breaking may occur. This indicates that the noise may prevent blow-up almost surely.
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Rached, Nadhir Ben; Hoel, Håkon Andreas & Meo, Johannes Vincent
(2024).
A fast and accurate numerical method for the left tail of sums of independent random variables.
Statistics and computing.
ISSN 0960-3174.
34(6).
doi:
10.1007/s11222-024-10514-5.
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We present a flexible, deterministic numerical method for computing left-tail rare events of sums of non-negative, independent random variables. The method is based on iterative numerical integration of linear convolutions by means of Newtons–Cotes rules. The periodicity properties of convoluted densities combined with the Trapezoidal rule are exploited to produce a robust and efficient method, and the method is flexible in the sense that it can be applied to all kinds of non-negative continuous RVs. We present an error analysis and study the benefits of utilizing Newton–Cotes rules versus the fast Fourier transform (FFT) for numerical integration, showing that although there can be efficiency benefits to using FFT, Newton–Cotes rules tend to preserve the relative error better, and indeed do so at an acceptable computational cost. Numerical studies on problems with both known and unknown rare-event probabilities showcase the method’s performance and support our theoretical findings.
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Floater, Michael S.
(2024).
On the positivity of B-spline Wronskians.
Calcolo.
ISSN 0008-0624.
61(3),
p. 1–9.
doi:
10.1007/s10092-024-00613-0.
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A proof that Wronskians of non-zero B-splines are positive is given, using only recursive formulas for B-splines and their derivatives. This could be used to generalize the de Boor–DeVore geometric proof of the Schoenberg–Whitney conditions and total positivity of B-splines to Hermite interpolation. For Wronskians of maximal order with respect to a given degree, positivity follows from a simple formula.
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Floater, Michael S.
(2024).
Log-concavity of B-splines.
Journal of Approximation Theory.
ISSN 0021-9045.
300,
p. 1–5.
doi:
10.1016/j.jat.2024.106042.
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Curry and Schoenberg showed that a B-spline is log-concave in its support by applying Brunn’s theorem to a simplex. In this note we provide an alternative, ‘analytic’ proof of the log-concave property using only recursion formulas for B-splines and their first and second derivatives.
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Christiansen, Snorre H
(2024).
On the definition of curvature in Regge calculus.
IMA Journal of Numerical Analysis.
ISSN 0272-4979.
44(5),
p. 2698–2715.
doi:
10.1093/imanum/drad095.
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Regge calculus defines a curvature for piecewise constant metrics on simplicial complexes subject to a partial continuity requirement. We prove that if a part of the complex is embedded in a Euclidean space, by a piecewise affine map, and we perform smoothing by convolution there, then the smoothed metrics have a densitized scalar curvature that converges, in the sense of measures, to that defined by Regge, as the smoothing parameter goes to zero.
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Berzins, Arturs
(2023).
Polyhedral Complex Extraction from ReLU Networks using Edge Subdivision.
Proceedings of Machine Learning Research (PMLR).
202,
p. 2234–2244.
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A neural network consisting of piecewise affine building blocks, such as fully-connected layers and ReLU activations, is itself a piecewise affine function supported on a polyhedral complex. This complex has been previously studied to characterize theoretical properties of neural networks, but, in practice, extracting it remains a challenge due to its high combinatorial complexity. A natural idea described in previous works is to subdivide the regions via intersections with hyperplanes induced by each neuron. However, we argue that this view leads to computational redundancy. Instead of regions, we propose to subdivide edges, leading to a novel method for polyhedral complex extraction. A key to this are sign-vectors, which encode the combinatorial structure of the complex. Our approach allows to use standard tensor operations on a GPU, taking seconds for millions of cells on a consumer grade machine. Motivated by the growing interest in neural shape representation, we use the speed and differentiablility of our method to optimize geometric properties of the complex. The code is available at https://github.com/arturs-berzins/relu_edge_subdivision.
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Falk, Richard S. & Winther, Ragnar
(2023).
CONSTRUCTION OF POLYNOMIAL PRESERVING COCHAIN EXTENSIONS BY BLENDING.
Mathematics of Computation.
ISSN 0025-5718.
92(342),
p. 1575–1594.
doi:
10.1090/mcom/3819.
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A classical technique to construct polynomial preserving extensions of scalar functions defined on the boundary of an simplex to the interior is to use so-called rational blending functions. The purpose of this paper is to generalize the construction by blending to the de Rham complex. More precisely, we define polynomial preserving extensions which map traces of k-forms defined on the boundary of the simplex to k-forms defined in the interior. Furthermore, the extensions are cochain maps, i.e., they commute with the exterior derivative.
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Celledoni, Elena; Glöckner, Helge; Riseth, Jørgen Nilsen & Schmeding, Alexander
(2023).
Deep neural networks on diffeomorphism groups for optimal shape reparametrization.
BIT Numerical Mathematics.
ISSN 0006-3835.
63(4),
p. 1–38.
doi:
10.1007/s10543-023-00989-5.
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One of the fundamental problems in shape analysis is to align curves or surfaces before computing geodesic distances between their shapes. Finding the optimal reparametrization realizing this alignment is a computationally demanding task, typically done by solving an optimization problem on the diffeomorphism group. In this paper, we propose an algorithm for constructing approximations of orientation preserving diffeomorphisms by composition of elementary diffeomorphisms. The algorithm is implemented using PyTorch, and is applicable for both unparametrized curves and surfaces. Moreover, we show universal approximation properties for the constructed architectures, and obtain bounds for the Lipschitz constants of the resulting diffeomorphisms.
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Tai, Xue-Cheng; Winther, Ragnar; Zhang, Xiaodi & Zheng, Weiying
(2023).
A uniform preconditioned for a Newton algorithm for total-variation minimisation and minimum-surface problems.
SIAM Journal on Numerical Analysis.
ISSN 0036-1429.
61(5),
p. 2062–2083.
doi:
10.1137/22M1512776.
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Solution methods for the nonlinear PDE of the Rudin–Osher–Fatemi (ROF) and minimum-surface models are fundamental for many modern applications. Many efficient algorithms have been proposed. First-order methods are common. They are popular due to their simplicity and easy implementation. Some second-order Newton-type iterative methods have been proposed, such as Chan–Golub–Mulet method. In this paper, we propose a new Newton–Krylov solver for primal-dual finite element discretization of the ROF model and the minimum-surface model. The method is so simple that we just need to use some diagonal preconditioners during the iterations. Theoretically, the proposed preconditioners are further proved to be robust and optimal with respect to the mesh size, the penalization parameter, the regularization parameter, and the iterative step; essentially, it is a parameter-independent preconditioner. We first discretize the primal-dual system by using mixed finite element methods and then linearize the discrete system by Newton’s method. Exploiting the well-posedness of the linearized problem on appropriate Sobolev spaces equipped with proper norms, we propose block diagonal preconditioners for the corresponding system solved with the minimum residual method. Numerical results are presented to support the theoretical results.
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Tang, Hao
(2023).
On the stochastic Euler-Poincaré equations driven by pseudo-differential/multiplicative noise.
Journal of Functional Analysis.
ISSN 0022-1236.
285(9).
doi:
10.1016/j.jfa.2023.110075.
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The stochastic Euler-Poincaré equations with pseudo-differential/multiplicative noise are considered in this work. We first establish two new cancellation properties on pseudo-differential operators, which considerably extend the previous results for transport type noise only involving gradient operator. Then, we obtain results on local solution, blow-up criterion, and global existence. The interplay between stability on exiting times and continuous dependence of solution on initial data is also studied for the multiplicative noise case.
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Christiansen, Snorre H; Gopalakrishnan, Jay; Guzmán, Johnny & Hu, Kaibo
(2023).
A discrete elasticity complex on three-dimensional Alfeld splits.
Numerische Mathematik.
ISSN 0029-599X.
156,
p. 159–204.
doi:
10.1007/s00211-023-01381-9.
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We construct conforming finite element elasticity complexes on the Alfeld splits of tetrahedra. The complex consists of vector fields and symmetric tensor fields, interlinked via the linearized deformation operator, the linearized curvature operator, and the divergence operator, respectively. The construction is based on an algebraic machinery that derives the elasticity complex from de Rham complexes, and smoother finite element differential forms.
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Hoel, Håkon Andreas & Ragunathan, Sankarasubramanian
(2023).
Higher-order adaptive methods for exit times of Itô diffusions.
IMA Journal of Numerical Analysis.
ISSN 0272-4979.
44(5),
p. 2821–2863.
doi:
10.1093/imanum/drad077.
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Mæhlen, Ola Isaac Høgåsen & Xue, Jun
(2023).
One-sided Hölder regularity of global weak solutions of negative order dispersive equations.
Journal of Differential Equations.
ISSN 0022-0396.
364,
p. 412–455.
doi:
10.1016/j.jde.2023.03.048.
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We prove global existence, uniqueness and stability of entropy solutions with L^2 initial data for a general family of negative order dispersive equations. These weak solutions are found to satisfy one-sided Hölder conditions whose coefficients decay in time. The latter result controls the height of solutions and further provides a way to bound the maximal lifespan of classical solutions from their initial data.
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Floater, Michael S.
(2023).
On a conjecture concerning interpolation by bivariate Bernstein polynomials.
Journal of Approximation Theory.
ISSN 0021-9045.
293.
doi:
10.1016/j.jat.2023.105920.
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In this paper we discuss a conjecture of Schumaker that the principal submatrices of collocation matrices of bivariate Bernstein polynomials over triangular grids have positive determinant. It is easy to show that the conjecture holds for the 2 × 2 submatrices. In this paper we show that it also holds for the 3 × 3 submatrices, working with the equivalent ‘monomial form’ of the conjecture. This result generalizes to a class of 3 × 3 matrices which will be described.
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Musch, Markus; Rupp, Andreas; Aizinger, Vadym & Knabner, Peter
(2023).
Hybridizable discontinuous Galerkin method with mixed-order spaces for non-linear diffusion equations with internal jumps.
GEM - International Journal on Geomathematics.
ISSN 1869-2672.
14(1).
doi:
10.1007/s13137-023-00228-7.
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Abstract We formulate a hybridizable discontinuous Galerkin method for parabolic equations with non-linear tensor-valued coefficients and jump conditions (Henry’s law). The analysis of the proposed scheme indicates the optimal convergence order for mildly non-linear problems. The same order is also obtained in our numerical studies for simplified settings. A series of numerical experiments investigate the effect of choosing different order approximation spaces for various unknowns.
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Christiansen, Snorre H; Halvorsen, Tore Gunnar & Scheid, Claire
(2023).
Convergence of a discretization of the Maxwell–Klein–Gordon equation based on finite element methods and lattice gauge theory.
Numerical Methods for Partial Differential Equations.
ISSN 0749-159X.
39,
p. 3271–3308.
doi:
10.1002/num.23008.
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Abstract The Maxwell–Klein–Gordon equations are a set of coupled nonlinear time‐dependent wave equations, used to model the interaction of an electromagnetic field with a particle. The solutions, expressed with a magnetic vector potential, are invariant under gauge transformations. This characteristic implies a constraint on the solution fields that might be broken at the discrete level. In this article, we propose and study a constraint preserving numerical scheme for this set of equations in dimension 2. At the semidiscrete level, we combine conforming Finite Element discretizations with the so‐called Lattice Gauge Theory to design a compatible gauge invariant discretization, leading to preservation of a discrete constraint. Relying on energy techniques and compactness arguments, we establish the convergence of this semidiscrete scheme, without a priori knowledge of the solution. Finally, at the fully discrete level, we propose a compatible explicit time‐integration strategy of leapfrog type. We implement the resulting fully discrete scheme and provide assessment on academic scenarios.
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Miao, Yingting; Rohde, Christian & Tang, Hao
(2023).
Well-posedness for a stochastic Camassa–Holm type equation with higher order nonlinearities.
Stochastics and Partial Differential Equations: Analysis and Computations.
ISSN 2194-0401.
doi:
10.1007/s40072-023-00291-z.
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Brualdi, Richard A. & Dahl, Geir
(2023).
Multi-alternating sign matrices.
The Electronic Journal of Linear Algebra.
ISSN 1537-9582.
39,
p. 261–282.
doi:
10.13001/ela.2023.7471.
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We introduce a generalization of alternating sign matrices (ASMs) called multiASMs and develop some of their properties. Classes of multiASMs with specified row and column sum vectors R and S extend the classes of (0,1)-matrices with specified R and S. The special case when R=S is a constant vector, in particular all 2's, is treated in more detail. We also investigate the polytope spanned by a class of multiASMs. Finally, we discuss the possibility of defining a Bruhat order on a class of multiASMs.
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Dahl, Geir; Guterman, Alexander & Shteyner, Pavel
(2023).
Majorization for (0,±1)-matrices.
Linear Algebra and its Applications.
ISSN 0024-3795.
663,
p. 200–221.
doi:
10.1016/j.laa.2023.01.009.
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Matrix majorization is a generalization of the classical majorization for vectors. We study several basic questions concerning matrix majorization for (0;±1)-matrices, i.e., matrices whose entries are restricted to 0, 1 and -1. In particular, we characterize when the zero vector is weakly majorized by a matrix, and show related results. Connections to linear programming are discussed. We obtain simpler characterizations of majorization under different assumptions. Also, several results on directional and strong majorization for (0;±1)-matrices are shown.
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Tang, Hao & Yang, Anita
(2023).
Noise effects in some stochastic evolution equations: Global existence and dependence on initial data.
Annales de l'I.H.P. Probabilites et statistiques.
ISSN 0246-0203.
59(1),
p. 378–410.
doi:
10.1214/21-AIHP1241.
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In this paper, we consider the noise effects on a class of stochastic evolution equations including the stochastic Camassa–Holm equations with or without rotation. We first obtain the existence, uniqueness and a blow-up criterion of pathwise solutions in Sobolev space Hs with s>3/2 . Then we prove that strong enough noise can prevent blow-up with probability 1, which justifies the regularization effect of strong nonlinear noise in preventing singularities. Besides, such strengths of noise are estimated in different examples. Finally, for the interplay between regularization effect induced by the noise and the dependence on initial conditions, we introduce and investigate the stability of the exiting time and construct an example to show that the multiplicative noise cannot improve both the stability of the exiting time and the continuity of the dependence on initial data simultaneously. Dans cet article, nous considérons les effets du bruit sur une classe d’équations d’évolution stochastiques y compris les équations stochastiques de Camassa–Holm avec ou sans rotation. Nous obtenons d’abord l’existence, l’unicité et un critère d’explosion de solutions pathwise dans l’espace de Sobolev Hs avec s>3/2. Ensuite, nous prouvons qu’un bruit suffisamment fort peut empêcher l’explosion avec probabilité 1, ce qui justifie l’effet régularisant du bruit non linéaire fort dans la prévention des singularités. De plus, de telles forces de bruit sont estimées dans les différents exemples. Enfin, pour l’interaction entre l’effet de régularisation induit par le bruit et la dépendance par rapport aux conditions initiales, nous introduisons et étudions la stabilité du temps de sortie et construisons un exemple pour montrer que le bruit multiplicatif ne peut pas améliorer simultanément la stabilité du temps de sortie et la continuité de la dépendance par rapport aux données initiales.
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Holden, Helge; Karlsen, Kenneth Aksel Hvistendahl & Pang, Ho Cheung
(2022).
Strong solutions of a stochastic differential equation with irregular random drift.
Stochastic Processes and their Applications.
ISSN 0304-4149.
150,
p. 655–677.
doi:
10.1016/j.spa.2022.05.006.
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We present a well-posedness result for strong solutions of one-dimensional stochastic differential equations (SDEs) of the form where the drift coefficient is random and irregular, with a weak derivative satisfying for some , . The random and regular noise coefficient may vanish. The main contribution is a pathwise uniqueness result under the assumptions that as , and satisfies the one-sided gradient bound , where the process exhibits an exponential moment bound of the form for small times , for some . This study is motivated by ongoing work on the well-posedness of the stochastic Hunter–Saxton equation, a stochastic perturbation of a nonlinear transport equation that arises in the modelling of the director field of a nematic liquid crystal. In this context, the one-sided bound acts as a selection principle for dissipative weak solutions of the stochastic partial differential equation.
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Hoel, Håkon Andreas; Shaimerdenova, Gaukhar & Tempone, Raul
(2022).
Multi-index ensemble Kalman filtering.
Journal of Computational Physics.
ISSN 0021-9991.
470.
doi:
10.1016/j.jcp.2022.111561.
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In this work we combine ideas from multi-index Monte Carlo and ensemble Kalman filtering (EnKF) to produce a highly efficient filtering method called multi-index EnKF (MIEnKF). MIEnKF is based on independent samples of four-coupled EnKF estimators on a multi-index hierarchy of resolution levels, and it may be viewed as an extension of the multilevel EnKF (MLEnKF) method developed by the same authors in 2020. Multi-index here refers to a two-index method, consisting of a hierarchy of EnKF estimators that are coupled in two degrees of freedom: time discretization and ensemble size. Under certain assumptions, when strong coupling between solutions on neighboring numerical resolutions is attainable, the MIEnKF method is proven to be more tractable than EnKF and MLEnKF. Said efficiency gains are also verified numerically in a series of test problems.
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Frid, Hermano; Karlsen, Kenneth Aksel Hvistendahl & Marroquin, Daniel
(2022).
Homogenization of stochastic conservation laws with multiplicative noise.
Journal of Functional Analysis.
ISSN 0022-1236.
283(9),
p. 1–63.
doi:
10.1016/j.jfa.2022.109620.
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We consider the generalized almost periodic homogenization problem for two different types of stochastic conservation laws with oscillatory coefficients and multiplicative noise. In both cases the stochastic perturbations are such that the equation admits special stochastic solutions which play the role of the steady-state solutions in the deterministic case. Specially in the second type, these stochastic solutions are crucial elements in the homogenization analysis. Our homogenization method is based on the notion of stochastic two-scale Young measure, whose existence is established here.
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Lyche, Tom; Manni, Carla & Speleers, Hendrik
(2022).
Construction of C^2 Cubic Splines on Arbitrary Triangulations.
Foundations of Computational Mathematics.
ISSN 1615-3375.
doi:
10.1007/s10208-022-09553-z.
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In this paper, we address the problem of constructing C2 cubic spline functions on a given arbitrary triangulation T. To this end, we endow every triangle of T with a Wang–Shi macro-structure. The C2 cubic space on such a refined triangulation has a stable dimension and optimal approximation power. Moreover, any spline function in this space can be locally built on each of the macro-triangles independently via Hermite interpolation. We provide a simplex spline basis for the space of C2 cubics defined on a single macro-triangle which behaves like a Bernstein/B-spline basis over the triangle. The basis functions inherit recurrence relations and differentiation formulas from the simplex spline construction, they form a nonnegative partition of unity, they admit simple conditions for C2 joins across the edges of neighboring triangles, and they enjoy a Marsden-like identity. Also, there is a single control net to facilitate control and early visualization of a spline function over the macro-triangle. Thanks to these properties, the complex geometry of the Wang–Shi macro-structure is transparent to the user. Stable global bases for the full space of C2 cubics on the Wang–Shi refined triangulation T are deduced from the local simplex spline basis by extending the concept of minimal determining sets.
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Antun, Vegard
(2022).
Recovering Wavelet Coefficients from Binary Samples Using Fast Transforms.
SIAM Journal on Scientific Computing.
ISSN 1064-8275.
44(3),
p. A1315–A1336.
doi:
10.1137/21M1427188.
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Recovering a signal (function) from finitely many binary or Fourier samples is one of the core problems in modern imaging, and by now there exist a plethora of methods for recovering a signal from such samples. Examples of methods which can utilize wavelet reconstruction include generalized sampling, infinite-dimensional compressive sensing, the parameterized-background data-weak (PBDW) method, etc. However, for any of these methods to be applied in practice, accurate and fast modeling of an N×M section of the infinite-dimensional change-of-basis matrix between the sampling basis (Fourier or Walsh--Hadamard samples) and the wavelet reconstruction basis is paramount. Building on the work of Gataric and Poon [SIAM J. Sci. Comput., 38 (2016), pp. A1075--A1099], we derive an algorithm which bypasses the NM storage requirement and the O(NM) computational cost of matrix-vector multiplication with this matrix and its adjoint when using Walsh--Hadamard samples and wavelet reconstruction. The proposed algorithm computes the matrix-vector multiplication in O(NlogN) operations and has a storage requirement of O(2q), where N=2dqM (usually q∈{1,2}) and d=1,2 is the dimension. As matrix-vector multiplications are the computational bottleneck for iterative algorithms used by the mentioned reconstruction methods, the proposed algorithm speeds up the reconstruction of wavelet coefficients from Walsh--Hadamard samples considerably.
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Christiansen, Snorre H & Hu, Kaibo
(2022).
Finite Element Systems for Vector Bundles: Elasticity and Curvature.
Foundations of Computational Mathematics.
ISSN 1615-3375.
doi:
10.1007/s10208-022-09555-x.
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Abstract We develop a theory of finite element systems, for the purpose of discretizing sections of vector bundles, in particular those arising in the theory of elasticity. In the presence of curvature, we prove a discrete Bianchi identity. In the flat case, we prove a de Rham theorem on cohomology groups. We check that some known mixed finite elements for the stress–displacement formulation of elasticity fit our framework. We also define, in dimension two, the first conforming finite element spaces of metrics with good linearized curvature, corresponding to strain tensors with Saint-Venant compatibility conditions. Cochains with coefficients in rigid motions are given a key role in relating continuous and discrete elasticity complexes.
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Colbrook, Matthew J.; Antun, Vegard & Hansen, Anders Christian
(2022).
The difficulty of computing stable and accurate neural networks: On the barriers of deep learning and Smale's 18th problem.
Proceedings of the National Academy of Sciences of the United States of America (PNAS).
ISSN 0027-8424.
119(12).
doi:
10.1073/pnas.2107151119.
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Significance Instability is the Achilles’ heel of modern artificial intelligence (AI) and a paradox, with training algorithms finding unstable neural networks (NNs) despite the existence of stable ones. This foundational issue relates to Smale’s 18th mathematical problem for the 21st century on the limits of AI. By expanding methodologies initiated by Gödel and Turing, we demonstrate limitations on the existence of (even randomized) algorithms for computing NNs. Despite numerous existence results of NNs with great approximation properties, only in specific cases do there also exist algorithms that can compute them. We initiate a classification theory on which NNs can be trained and introduce NNs that—under suitable conditions—are robust to perturbations and exponentially accurate in the number of hidden layers.
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