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A weekly seminar seires with talks in all areas of applied and computational analysis, of broad interest to the mathematical community. For questions about this series or if you have suggestions, please contact: m.colbrook@damtp.cam.ac.uk
URL: http://talks.cam.ac.uk/show/index/9811
Updated: 2 min 27 sec ago

Thu 01 May 15:00: An Explicit Filtered Lie Splitting Scheme for the Original Zakharov System with Low Regularity Error Estimates in All Dimensions

Tue, 22/04/2025 - 22:29
An Explicit Filtered Lie Splitting Scheme for the Original Zakharov System with Low Regularity Error Estimates in All Dimensions

In this talk, we present low-regularity numerical schemes for nonlinear dispersive equations, with a particular focus on the Zakharov system (ZS) and the “good” Boussinesq (GB) equation. These models exhibit strong nonlinear interactions and are known to pose significant analytical and numerical challenges when the solution has limited regularity.

We concentrate on our recent results for the Zakharov system, where we construct and analyze an explicit filtered Lie splitting scheme applied directly to its original coupled form. This method successfully overcomes the essential difficulty of derivative loss in the nonlinear terms, which not only obstructs low-regularity analysis, but has long prevented rigorous error estimates for explicit Lie splitting schemes based directly on the original Zakharov system. By developing multilinear estimates in discrete Bourgain spaces, we rigorously prove the first explicit low-regularity error estimate for the original Zakharov system, and also the first such result for a coupled system within the Bourgain framework. The analytical strategy developed here can also be extended to other dispersive equations with derivative loss, offering a way to overcome both low-regularity difficulties and the fundamental obstacle posed by derivative-loss nonlinearities. Numerical experiments confirm the theoretical predictions.

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Thu 26 Jun 16:00: Title to be confirmed

Mon, 24/03/2025 - 23:53
Title to be confirmed

Abstract not available

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Thu 26 Jun 15:00: Title to be confirmed

Mon, 24/03/2025 - 23:52
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Thu 15 May 15:00: New Insights on High Wave Scattering by Multiple Open Arcs: Exponentially Convergent Methods and Shape Holomorphy

Mon, 24/03/2025 - 18:55
New Insights on High Wave Scattering by Multiple Open Arcs: Exponentially Convergent Methods and Shape Holomorphy

In this talk, we focus on the scattering of time-harmonic acoustic, elastic, and polarized electromagnetic waves by multiple finite-length open arcs in an unbounded two-dimensional domain. We begin by reformulating the corresponding boundary value problems with Dirichlet or Neumann conditions as weakly and hypersingular boundary integral equations (BIEs), respectively. We then introduce a family of fast spectral Galerkin methods for solving these BIEs. The discretization bases are built from weighted Chebyshev polynomials that accurately capture the solutions’ edge behavior. Under the assumption of analyticity of the sources and arc geometries, we show that these bases yield exponential convergence with respect to the polynomial degree.

Numerical examples will illustrate the accuracy and robustness of the proposed methods, with respect to both the number of arcs and the wavenumber. Additionally, we demonstrate that, for general weakly and hypersingular BIEs, the solutions depend holomorphically on perturbations of the arc parametrizations. These results are crucial for establishing the shape holomorphy of domain-to-solution maps arising in boundary integral equations, with applications in uncertainty quantification, inverse problems, and deep learning, among others. They also raise new questions—some of which you may have the answer to!

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Thu 20 Mar 15:00: Transient dynamics under structured perturbations: bridging unstructured and structured pseudospectra

Mon, 17/03/2025 - 13:46
Transient dynamics under structured perturbations: bridging unstructured and structured pseudospectra

As is known, bounds of the resolvent of a matrix in the right complex half-plane yield bounds of solutions of homogeneous and inhomogeneous linear differential equations with this matrix. We ask two basic questions:

- Up to which size of structured perturbations are the resolvent norms of the perturbed matrices within a given bound in the right complex half-plane?

- For a given size of structured perturbations, what is the smallest common bound for the resolvent norms of the perturbed matrices in the right complex half-plane?

This is considered for general linear structures such as complex or real matrices with a given sparsity pattern or with restricted range and corange, or special classes such as Toeplitz or Hankel matrices. Conceptually, we combine unstructured and structured pseudospectra in a joint pseudospectrum, allowing for the use of resolvent bounds as with unstructured pseudospectra and for structured perturbations as with structured pseudospectra. The above questions are addressed by an algorithm which solves eigenvalue optimization problems via suitably discretized rank-1 matrix differential equations. The talk is based on joint work with Nicola Guglielmi.

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Mon 17 Mar 16:00: Two New Developments concerning Noether's Two Theorems

Thu, 06/03/2025 - 14:25
Two New Developments concerning Noether's Two Theorems

In her fundamental 1918 paper, written whilst at Göttingen at the invitation of Klein and Hilbert to help them resolve an apparent paradox concerning the conservation of energy in general relativity, Emmy Noether proved two fundamental theorems relating symmetries and conservation laws of variational problems. Her First Theorem, as originally formulated, relates strictly invariant variational problems and conservation laws of their Euler—Lagrange equations. The Noether correspondence was extended by her student Bessel-Hagen to divergence invariant variational problems. A key issue is when is a divergence invariant variational problem equivalent to a strictly invariant one. Here, I illustrate these issues using a very basic example from her original paper, and then highlight the role of Lie algebra cohomology in resolving this question in general. This part includes some provocative remarks on the role of invariant variational problems in the modern formulation of fundamental physics.

Noether’s Second Theorem concerns variational problems admitting an infinite-dimensional symmetry group depending on an arbitrary function. I first recall the two well-known classes of partial differential equations that admit infinite hierarchies of higher order generalized symmetries: 1) linear and linearizable systems that admit a nontrivial point symmetry group; 2) integrable nonlinear equations such as Korteweg—de Vries, nonlinear Schrödinger, and Burgers’. I will then introduce a new general class: 3) underdetermined systems of partial differential equations that admit an infinite-dimensional symmetry algebra depending on one or more arbitrary functions of the independent variables. An important subclass of the latter are the underdetermined Euler—Lagrange equations arising from a variational principle that admits an infinite-dimensional variational symmetry algebra depending on one or more arbitrary functions of the independent variables. According to Noether’s Second Theorem, the associated Euler—Lagrange equations satisfy Noether dependencies and are hence underdetermined and the conservation laws corresponding to such symmetries are trivial; examples include general relativity, electromagnetism, and parameter-independent variational principles.

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Thu 13 Mar 15:00: An introduction to quantum graphs and their spectral geometry, with some open questions

Thu, 06/03/2025 - 11:39
An introduction to quantum graphs and their spectral geometry, with some open questions

Quantum graphs are (usually self-adjoint) second-order differential operators over collections of intervals that are glued at their endpoints (“metric graphs”). We survey recent and past advances in quantum graph eigenvalue estimates, focusing on how geometry, topology, and diffusion-transport phenomena interact. Specifically, we examine surgery techniques—localized graph modifications—to manipulate eigenvalues. We also explore transport and transport-diffusion equations on directed graphs, highlighting how directional flow and non-symmetric operators impact spectral properties. We discuss adapting surgery techniques to control these effects, emphasizing the link between edge transmission/boundary conditions, diffusion-transport dynamics, and resulting eigenvalues.

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Thu 15 May 15:00: Title to be confirmed

Wed, 19/02/2025 - 21:45
Title to be confirmed

Abstract not available

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Thu 20 Feb 15:00: On Curve Approximation over Nonlinear Domains

Mon, 17/02/2025 - 12:43
On Curve Approximation over Nonlinear Domains

In recent years, progress has been made in constructing methods for approximating curves in various domains, such as manifolds. This talk will explore our recent advances in curve approximation within Wasserstein spaces, covering key concepts, ongoing developments in analyzing our approximation operators, and illustrative examples.

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Mon 17 Mar 15:00: Two New Developments concerning Noether's Two Theorems

Wed, 05/02/2025 - 08:40
Two New Developments concerning Noether's Two Theorems

In her fundamental 1918 paper, written whilst at Göttingen at the invitation of Klein and Hilbert to help them resolve an apparent paradox concerning the conservation of energy in general relativity, Emmy Noether proved two fundamental theorems relating symmetries and conservation laws of variational problems. Her First Theorem, as originally formulated, relates strictly invariant variational problems and conservation laws of their Euler—Lagrange equations. The Noether correspondence was extended by her student Bessel-Hagen to divergence invariant variational problems. A key issue is when is a divergence invariant variational problem equivalent to a strictly invariant one. Here, I illustrate these issues using a very basic example from her original paper, and then highlight the role of Lie algebra cohomology in resolving this question in general. This part includes some provocative remarks on the role of invariant variational problems in the modern formulation of fundamental physics.

Noether’s Second Theorem concerns variational problems admitting an infinite-dimensional symmetry group depending on an arbitrary function. I first recall the two well-known classes of partial differential equations that admit infinite hierarchies of higher order generalized symmetries: 1) linear and linearizable systems that admit a nontrivial point symmetry group; 2) integrable nonlinear equations such as Korteweg—de Vries, nonlinear Schrödinger, and Burgers’. I will then introduce a new general class: 3) underdetermined systems of partial differential equations that admit an infinite-dimensional symmetry algebra depending on one or more arbitrary functions of the independent variables. An important subclass of the latter are the underdetermined Euler—Lagrange equations arising from a variational principle that admits an infinite-dimensional variational symmetry algebra depending on one or more arbitrary functions of the independent variables. According to Noether’s Second Theorem, the associated Euler—Lagrange equations satisfy Noether dependencies and are hence underdetermined and the conservation laws corresponding to such symmetries are trivial; examples include general relativity, electromagnetism, and parameter-independent variational principles.

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Thu 22 May 15:00: Title to be confirmed

Sat, 01/02/2025 - 15:02
Title to be confirmed

Abstract not available

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Thu 06 Feb 15:00: Numerical analysis of high frequency wave scattering via semiclassical analysis: a case study with non-uniform meshes

Fri, 31/01/2025 - 07:49
Numerical analysis of high frequency wave scattering via semiclassical analysis: a case study with non-uniform meshes

In recent years, semiclassical analysis has significantly advanced our understanding of numerical algorithms for high-frequency wave scattering. This talk will begin with an overview of how semiclassical methods have influenced the theory of numerical methods for frequency-domain wave problems. As a case study, we will then focus on the finite element method (FEM), a classical approach for approximating solutions to high-frequency scattering problems. In FEM , the solution is typically approximated using piecewise polynomials of degree p on a mesh of width h. A fundamental question is then: how should h be chosen (as a function of the frequency, k) so that the error in the numerical solution is small? It has been known since the seminal work of Babuska and Ihlenberg that the natural conjecture hk<

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Thu 05 Jun 15:00: Title to be confirmed

Sun, 26/01/2025 - 18:29
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Abstract not available

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Thu 12 Jun 15:00: Title to be confirmed

Tue, 21/01/2025 - 14:18
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Abstract not available

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Thu 23 Jan 15:00: Detecting and Attributing Change in Climate and Complex Systems: Foundations, Green's Functions, and Nonlinear Fingerprints

Sun, 19/01/2025 - 15:12
Detecting and Attributing Change in Climate and Complex Systems: Foundations, Green's Functions, and Nonlinear Fingerprints

Detection and attribution (D&A) studies are cornerstones of climate science, providing crucial evidence for policy decisions. Their goal is to link observed climate change patterns to anthropogenic and natural drivers via the optimal fingerprinting method (OFM). We show that response theory for nonequilibrium systems offers the physical and dynamical basis for OFM , including the concept of causality used for attribution. Our framework clarifies the method’s assumptions, advantages, and potential weaknesses. We use our theory to perform D&A for prototypical climate change experiments performed on an energy balance model and on a low-resolution coupled climate model. We also explain the underpinnings of degenerate fingerprinting, which offers early warning indicators for tipping points. Finally, we extend the OFM to the nonlinear response regime. Our analysis shows that OFM has broad applicability across diverse stochastic systems influenced by time-dependent forcings, with potential relevance to ecosystems, quantitative social sciences, and finance, among others.

Key References V. Lucarini and M. D. Chekroun, Detecting and Attributing Change in Climate and Complex Systems: Foundations, Green’s Functions, and Nonlinear Fingerprints, Phys. Rev. Lett. 133, 244201 (2024) https://doi.org/10.1103/PhysRevLett.133.244201 V. Lucarini and M. D. Chekroun, Theoretical tools for understanding the climate crisis from Hasselmann’s programme and beyond, Nat. Rev. Phys. 5, 744 (2023) https://doi.org/10.1038/s42254-023-00650-8

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Thu 29 May 15:00: Title to be confirmed

Wed, 15/01/2025 - 18:46
Title to be confirmed

Abstract not available

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