Schedule for: 24w5227 - Modern Methods for Differential Equations of Quantum Mechanics

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Many approximate solutions of the time-dependent Schrödinger equation can be formulated as exact solutions of a nonlinear Schrödinger equation with an effective Hamiltonian operator depending on the state of the system. We show that Heller's thawed Gaussian approximation, Coalson and Karplus's variational Gaussian approximation, and other Gaussian wavepacket dynamics methods fit into this framework if the effective potential is a quadratic polynomial with state-dependent coefficients. We study such a nonlinear Schrödinger equation in full generality: we derive general equations of motion for the Gaussian's parameters, demonstrate the time reversibility and norm conservation, and analyze conservation of the energy, effective energy, and symplectic structure. We also describe efficient, high-order geometric integrators for the numerical solution of this nonlinear Schrödinger equation. The general theory is illustrated by examples of this family of Gaussian wavepacket dynamics, including the variational and nonvariational thawed and frozen Gaussian approximations, and their special limits based on the global harmonic, local harmonic, single-Hessian, local cubic, and local quartic approximations for the potential energy. We also propose a new method by augmenting the local cubic approximation with a single fourth derivative. Without substantially increasing the cost, the proposed "single-quartic" variational Gaussian approximation improves the accuracy over the local cubic approximation and, at the same time, conserves both the effective energy and symplectic structure, unlike the much more expensive local quartic approximation. Most results are presented in both Heller's and Hagedorn's parametrizations of the Gaussian wavepacket. In my talk, I will accompany the theory with applications of the different versions of Gaussian wavepacket dynamics to the calculation of vibrationally resolved electronic spectra as well as with numerical examples demonstrating the geometric properties and fast convergence of the integrators.                 [1] J. Chem. Phys. 159, 014114 (2023)

A practical way to compute time-dependent observables of a non-equilibrium quantum many-body system is to focus on the single-particle Green's function defined on the Keldysh contour. The equation of motion satisfied by such a Green's function is a set of nonlinear integro-differential equations called the Kadanoff-Baym equations. We will describe numerical methods for solving this type of equations and show how to use dynamic mode decomposition and recurrent neural networks to reduce computational complexity.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A high-frequency recovered fully discrete low-regularity integrator is constructed to approximate rough and possibly discontinuous solutions of the semilinear wave equation. The proposed method, with high-frequency recovery techniques, can capture the discontinuities of the solutions correctly without spurious oscillations and approximate rough and discontinuous solutions with a higher convergence rate than pre-existing methods. Rigorous analysis is presented for the convergence rates of the proposed method in approximating solutions such that $(u,\partial_{t}u)\in C([0,T];H^{\gamma}\times H^{\gamma-1})$ for $\gamma\in(0,1]$. For discontinuous solutions of bounded variation in one dimension (which allow jump discontinuities), the proposed method is proved to have almost first-order convergence under the step size condition $\tau \sim N^{-1}$, where $\tau$ and $N$ denote the time step size and the number of Fourier terms in the space discretization, respectively. Numerical examples are presented in both one and two dimensions to illustrate the advantages of the proposed method in improving the accuracy in approximating rough and discontinuous solutions of the semilinear wave equation. The numerical results are consistent with the theoretical results and show the efficiency of the proposed method.

This talk presents a quantum algorithm for the solution of prototypical second-order linear elliptic partial differential equations discretized by d-linear finite elements on Cartesian grids of a bounded d-dimensional domain. The core of the construction is a multilevel BPX preconditioner, which transforms the linear system into a well-conditioned one, essential for the use of quantum computers. We show that suitable functionals of the solution can be computed on a quantum computer up to tolerance tol with complexity proportional to 1/tol, neglecting logarithmic terms. For a fixed physical dimension d, this complexity is comparable to the optimal complexity of the one-dimensional problem and improves existing results. We also present a quantum circuit that implements the algorithm and report some basic numerical results.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

The main idea of this meeting is, of course, to share our research interests, latest achievements and learn from our colleagues about their latest discoveries. All we are looking forward to listening to the interesting talks and to having chats during our free time. This short, "5-10-minute" introductory talk is intended to help us in establishing new research connections or maybe even collaborations!

The main idea of this meeting is, of course, to share our research interests, latest achievements and learn from our colleagues about their latest discoveries. All we are looking forward to listening to the interesting talks and to having chats during our free time. This short, "5-10-minute" introductory talk is intended to help us in establishing new research connections or maybe even collaborations!

The main idea of this meeting is, of course, to share our research interests, latest achievements and learn from our colleagues about their latest discoveries. All we are looking forward to listening to the interesting talks and to having chats during our free time. This short, "5-10-minute" introductory talk is intended to help us in establishing new research connections or maybe even collaborations!

The main idea of this meeting is, of course, to share our research interests, latest achievements and learn from our colleagues about their latest discoveries. All we are looking forward to listening to the interesting talks and to having chats during our free time. This short, "5-10-minute" introductory talk is intended to help us in establishing new research connections or maybe even collaborations!

The main idea of this meeting is, of course, to share our research interests, latest achievements and learn from our colleagues about their latest discoveries. All we are looking forward to listening to the interesting talks and to having chats during our free time. This short, "5-10-minute" introductory talk is intended to help us in establishing new research connections or maybe even collaborations!

The main idea of this meeting is, of course, to share our research interests, latest achievements and learn from our colleagues about their latest discoveries. All we are looking forward to listening to the interesting talks and to having chats during our free time. This short, "5-10-minute" introductory talk is intended to help us in establishing new research connections or maybe even collaborations!

The main idea of this meeting is, of course, to share our research interests, latest achievements and learn from our colleagues about their latest discoveries. All we are looking forward to listening to the interesting talks and to having chats during our free time. This short, "5-10-minute" introductory talk is intended to help us in establishing new research connections or maybe even collaborations!

The main idea of this meeting is, of course, to share our research interests, latest achievements and learn from our colleagues about their latest discoveries. All we are looking forward to listening to the interesting talks and to having chats during our free time. This short, "5-10-minute" introductory talk is intended to help us in establishing new research connections or maybe even collaborations!

The main idea of this meeting is, of course, to share our research interests, latest achievements and learn from our colleagues about their latest discoveries. All we are looking forward to listening to the interesting talks and to having chats during our free time. This short, "5-10-minute" introductory talk is intended to help us in establishing new research connections or maybe even collaborations!

The main idea of this meeting is, of course, to share our research interests, latest achievements and learn from our colleagues about their latest discoveries. All we are looking forward to listening to the interesting talks and to having chats during our free time. This short, "5-10-minute" introductory talk is intended to help us in establishing new research connections or maybe even collaborations!

The main idea of this meeting is, of course, to share our research interests, latest achievements and learn from our colleagues about their latest discoveries. All we are looking forward to listening to the interesting talks and to having chats during our free time. This short, "5-10-minute" introductory talk is intended to help us in establishing new research connections or maybe even collaborations!

The main idea of this meeting is, of course, to share our research interests, latest achievements and learn from our colleagues about their latest discoveries. All we are looking forward to listening to the interesting talks and to having chats during our free time. This short, "5-10-minute" introductory talk is intended to help us in establishing new research connections or maybe even collaborations!

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk, we will introduce our study on the energy minimization model arising in the ensemble Kohn-Sham density functional theory for metallic systems, in which a pseudo-eigenvalue matrix and a general smearing approach are involved. We investigate the invariance of the energy functional and the existence of the minimizer of the ensemble Kohn-Sham model. We propose an adaptive two-parameter step size strategy and the corresponding preconditioned conjugate gradient methods to solve the energy minimization model. Under some mild but reasonable assumptions, we prove the global convergence for the gradients of the energy functional produced by our algorithms. Numerical experiments show that our algorithms are efficient, especially for large scale metallic systems. In particular, our algorithms produce convergent numerical approximations for some metallic systems, for which the traditional self-consistent _eld iterations fail to converge. This is a joint work with Professor Stefano de Gironcoli, Dr. Bin Yang and Prof. Aihui Zhou.

Recently, there has been a significant research interest in using neural networks for solving partial differential equations (PDEs) in general [1], and Schr ̈odinger equations in particular [2]. The use of neural networks was shown to mitigate, or even break the curse of dimensionality [3] encountered in standard numerical methods, such as finite-volume or spectral methods. In the context of quantum mechanics, neural networks were shown to accurately approximate ground-states of molecular systems, while scaling moderately with the dimension of the problem [4]. However, extensions to computing many excited states suffer from convergence issues and remain challenging [5]. To extend the applicability of neural-network-based ansatzes to domains requiring computations of hundreds or even thousands of excited states, such as nuclear-motion problems [6] we propose to learn problem-specific basis sets in L2. In particular, rich families in L2 are produced by pushing forward standard basis sets through differentiable mappings. I show that a bijectivity assumption on the mapping is a necessary condition for the resulting family to be dense in L2 [7]. This allows us to model these mappings using normalizing flows, an important tool from generative machine learning. I present a nonlinear variational framework to approximate molecular wavefunctions in the linear span of these flow-induced families. The framework allowed to compute many eigenstates of various molecular vibrational and electronic systems with orders-of-magnitude improved accuracy over standard linear methods [8]. The present approach can be seen as a nonlinear extension of spectral methods to a spectral learning framework, where basis sets are not predefined but learned in a manor tailored to the problem under consideration [9]. The well-posedness of such a framework and convergence guarantees are discussed. References [1] W. E and B. Yu, “The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems”, Commun. Math. Stat. 6, 1–12 (2018), doi: 10.1007/s40304-018-0127-z. [2] J. Hermann, J. Spencer, K. Choo, A. Mezzacapo, W. Foulkes, D. Pfau, G. Carleo, and F. No ́e, “Ab initio quantum chemistry with neural-network wavefunctions”, Nat. Rev. Chem. 7, 692–709 (2023), doi: https://doi.org/10.1038/s41570-023-00516-8, arXiv: 2208.12590 [physics].

It is common to use Galerkin (or variational) methods to solve the Schroedinger equation to calculate spectra of molecules and rate constants of chemical reactions. For multidimensional problems, the necessary matrices and vectors are large and the calculations are difficult. Iterative eigensolvers, propagation methods, and linear solvers are important tools that make it possible to deal with molecules and reacting systems with about 6 atoms. However, one must also confront the problem of computing the matrices that are fed into the iterative solvers. In this talk, I shall present collocation ideas that obviate the need to compute the matrices requied in a Galerkin method. Collocation is a method for solving differential equations that, like a Galerkin method, uses a basis and represents solutions as linear combinations of basis functions. Unlike Galerkin methods, collocation determines basis function coefficients by demanding that the differential equation be satisfied at points: there are no integrals and no quadratures. New ideas make collocation methods effective for high-dimensional problems. (1) Sparse-grid techniques are used to reduce the size of the basis and the associated point set. (2) It is possible to use a zero-at-points-in-previous-levels basis that spans the same space as any desired basis and which greatly reduces the cost of calculations. (3) It is possible to extract the best possible solutions from a given basis space by using more points than basis functions (rectangular collocation). (4) (1)-(3) can be used in conjunction with the powerful Multiconfiguration Time Dependent Hartree method to obviate the need for a sum-of-products (CP format) potential energy surface. Excellent results are obtained for molecules with as many as six atoms without any need to optimize points. I suspect that methods similar to those I describe will be useful when solving many high-dimensional quantum differential equations.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this work, we present a novel spectral renormalization exponential integrator method for solving gradient flow problems. Our method is specifically designed to simultaneously satisfy discrete analogues of the energy dissipation laws and achieve high-order accuracy in time. To accomplish this, our method first incorporates the energy dissipation law into the target gradient flow equation by introducing a time-dependent spectral renormalization (TDSR) factor. Then, the coupled equations are discretized using the spectral approximation in space and the exponential time differencing (ETD) in time. Finally, the resulting fully discrete nonlinear system is decoupled and solved using the Picard iteration at each time step. Furthermore, we introduce an extra enforcing term into the system for updating the TDSR factor, which greatly relaxes the time step size restriction of the proposed method and enhances its computational efficiency. Extensive numerical tests with various gradient flows are also presented to demonstrate the accuracy and effectiveness of our method as well as its high efficiency when combined with an adaptive time-stepping strategy for long-term simulations.

Although many-body quantum simulations have greatly benefited from high-performance computing facilities, large molecular systems continue to pose formidable challenges. Mixed quantum-classical models, such as Born—Oppenheimer molecular dynamics or Ehrenfest dynamics, have been proposed to overcome the computational costs of fully quantum approaches. However, current mixed quantum-classical models typically suffer from long-standing consistency issues. In this talk, we present a fully Hamiltonian theory of quantum-classical dynamics based on a geometric approach and Koopman wave functions. The resulting model appears to be the first to ensure a series of consistency properties, beyond the positivity of quantum and classical densities. We also exploit Lagrangian trajectories to formulate a finite-dimensional closure scheme for numerical implementations, the "Koopmon method". Numerical experiments demonstrate that the Koopmon method is able to capture effects beyond Ehrenfest dynamics in both the classical and the quantum sectors.

In quantum flows, like liquid helium II at intermediate temperatures between zero and 2.17 K, a normal fluid and a superfluid coexist with independent velocity fields. The most advanced existing models for such systems use the Navier-Stokes equations for the normal fluid and a simplified description of the superfluid, based on the dynamics of quantized vortex filaments, with ad hoc reconnection rules. There was a single attempt [1] to couple Navier-Stokes and Gross-Pitaevskii equations in a global model intended to describe the compressible two-fluid liquid helium II. We present in this contribution a new numerical model to couple a Navier-Stokes incompressible fluid with a Gross-Pitaevskii superfluid [2]. A numerical algorithm based on pseudo-spectral Fourier methods is presented for solving the coupled system of equations. The new numerical system is validated against well-known benchmarks for the evolution in a normal fluid of different types or arrangements of quantized vortices (vortex crystal, vortex dipole and vortex rings). [1] C. Coste, Nonlinear Schrödinger equation and superfluid hydrodynamics, The European Physical Journal B - Condensed Matter and Complex Systems, VOL. 1, P. 245--253, 1998. [2] M. Brachet, G. Sadaka, Z. Zhang, V. Kalt and I. Danaila, Coupling Navier-Stokes and Gross-Pitaevskii equations for the numerical simulation of two-fluid quantum flows, J. Computational Physics, 488, p. 112193(1-17), 2023.

In this talk we discuss a recent research program on the mathematical foundations of spectral methods for time-dependent PDEs, focusing on so-called T systems. These are systems of orthogonal functions which have tridiagonal differentiation matrices, which has computational advantages for the design of spectral methods. Along the way we discuss the key ideas involving Fourier transforms and orthogonal polynomials, approximation of wave packets, and current research in the theory for practical construction of these bases via the differential Krylov subspace iteration. This is joint work with Arieh Iserles (Cambridge).

We propose a novel way to describe numerical methods for ordinary differential equations via the notion of multi-indice. The main idea is to replace rooted trees in Butcher's B-series by multi-indices. The latter were introduced recently in the context of describing solutions of singular stochastic partial differential equations. The combinatorial shift away from rooted trees allows for a compressed description of numerical schemes. Moreover, these multi-indices B-series characterise uniquely the Taylor development of local and affine equivariant maps. This is a joint work with Kurusch Ebrahimi-Fard and Yingtong Hou.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

This talk is based on a recent joint work with Sergio Blanes, Fernando Casas, and Cesáreo González. Our main objective is the extension of modified potential operator splitting methods to specific classes of nonlinear evolution equations. The considered partial differential equations of Schrödinger and parabolic type comprise the Laplacian, a potential acting as multiplication operator, and a cubic nonlinearity. Moreover, we deduce an invariance principle that has a significant impact on the efficient realisation of the resulting modified operator splitting methods for the Schrödinger case. Our numerical experiments for the time-dependent Gross–Pitaevskii equation in the physically most relevant case of three space dimensions and for its parabolic counterpart related to ground state and excited state computations confirm the benefits of the proposed fourth-order modified operator splitting method in comparison with standard splitting methods. To the best of our knowledge, the presented results are novel and of particular interest from both, a theoretical perspective to inspire future investigations of modified operator splitting methods for other classes of nonlinear evolution equations and a practical perspective to advance the reliable and efficient simulation of Gross–Pitaevskii systems in real and imaginary time.

In this talk, I begin with the nonlinear Klein-Gordon equation (NKGE) with weak nonlinearity, which is characterized by  with  a dimensionless parameter. Different numerical methods are applied to discretize the NKGE including finite difference methods, exponential wave integrators and time-splitting methods. Especially, we discretize the NKGE by the second-order time-splitting method in time and combine with the Fourier spectral method in space. By introducing a new technique—Regularity Compensation Oscillation (RCO) which controls the high frequency modes by the regularity of the exact solution and analyzes the low frequency modes by phase cancellation and energy method, we carry out the improved uniform error bounds for the time-splitting methods. The results have been extended to other dispersive PDEs including the (nonlinear) Schrodinger equation and Dirac equation.

We study a filtered Lie splitting scheme for the time integration of the `good' Boussinesq equation with initial data in $H^s$, $s>0$. We are particularly interested in very small values of $s$. The main problem with such low regularity initial data is the following: the stability of the scheme cannot be analyzed by standard Sobolev techniques for $s\le 1/2$, due to the non-validity of the bilinear estimate for such values of $s$. To solve this problem, we reformulate the Boussinesq equation as a first-order evolution equation. The resulting problem has some similarities to a nonlinear Schrödinger equation, and thus the recently introduced framework of discrete Bourgain spaces can be applied and allows us to establish convergence. In particular, for $\tau$ denoting the step size of the method, we prove convergence of order $\tau^{s/2}$ in $L^2$ for $0 < s \le 2$. These analytical results are supported by numerical experiments. This is joint work with Lun Ji, Hang Li, and Chunmei Su.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Splitting methods constitute a powerful tool for the numerical inte- gration of differential equations, either arising directly from dynamical systems or from partial differential equations of evolution previously discretized in space. Efficient high-order schemes have been designed that provide accurate solutions whilst preserving some of the most salient qualitative features of the system. The presence of negative coefficients in methods of order greater than two, however, restricts their application to, e.g., equations defined in semigroups, thus moti- vating the exploration of splitting methods with complex coefficients with positive real part. We provide here preliminary results on a class of methods possessing a new symmetry, and their preserving properties when applied to unitary and Hamiltonian problems. This is an ongoing project in collaboration with Joackim Bernier (Nantes), Sergio Blanes (Valencia) and Alejandro Escorihuela-Tomàs (Castellón).

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We study discretizations of Hamiltonian systems on the probability density manifold equipped with the L2 -Wasserstein metric. Based on discrete optimal transport theory, several Hamiltonian systems, such as the Wasserstein geodesic equation and  Schrodinger equation, on graph (lattice) with different weights are derived, which can be viewed as spatial discretizations to the original Hamiltonian systems. We prove the consistency and provide the approximate orders for those discretizations. By regularizing the system using Fisher information, we deduce an explicit lower bound for the density function, which guarantees that symplectic schemes can be used to discretize in time. Moreover, we show desirable long time behavior of these schemes, and demonstrate their performance on several numerical examples.

Molecular systems and materials containing strongly correlated electrons are not adequately described by the Hartree-Fock mean-field approximation, nor by the Density Functional Theory (DFT) with the approximate exchange-correlation functionals currently available. In this talk, I will present the mathematical formulation of one of the methods used to simulate such systems, namely the CASSCF (complete active space self-consistent field) method, as well as various numerical algorithms to solve the CASSCF problem. If time allows, I will also discuss quantum embedding methods, which can be seen as domain decomposition methods in the Fock space.

“Everything is a Lagrangian submanifold” (symplectic creed by Alan Weinstein) Symplectic geometry and in particular Lagrangian submanifold theory has important interactions with global analysis, mathematical physics and dynamical systems, algebraic geometry, integrable systems, partial differential equations, representation theory, quantization, etc, and also, as we will see, with geometric integration. In this talk, we will use geometric tools, in particular, Lagrangian submanifolds of symplectic manifolds and their generating functions, for deriving, in a systematic way, numerical integrators preserving some relevant geometric structures (such as symplecticity, Poisson structures, symmetry, momentum map, constraints…). References: Barbero-Liñan, M., Marrero, J.C., and Martın de Diego, D. From retraction maps to symplectic-momentum numerical integrators. arXiv:2401.14800 Barbero-Liñán M. and Martın de Diego, D. Retraction maps: a seed of geometric integrators. Found. Comput. Math. Volume 23, pages 1335–1380, (2023) Barbero-Liñán M. , Martın de Diego, D. and Sato Martín de Almagro, Rodrigo T. A new perspective on symplectic integration of constrained mechanical systems via discretization maps. arXiv:2306.06786

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We establish error estimates of various numerical methods for the nonlinear Schr\"{o}dinger equation (NLSE) with low regularity potential and nonlinearity covering purely bounded potential and locally Lipschitz nonlinearity. Typical examples include the square-well potential or step potential, the random or disorder potential, and the non-integer power nonlinearity. New analysis techniques are needed to establish error bounds on classical numerical methods for low regularity potential and nonlinearity. Also, novel accurate, efficient and structure-preserving methods for low regularity potential and nonlinearity need to be developed.

Dispersive nonlinear partial differential equations can be used to describe a range of physical systems, from water waves to spin states in ferromagnetism. The numerical approximation of solutions with limited differentiability (low-regularity) is crucial for simulating fascinating phenomena arising in these systems including emerging structures in random wave fields and dynamics of domain wall states, but it poses a significant challenge to classical algorithms. Recent years have seen the development of tailored low-regularity integrators to address this challenge. Inherited from their description of physicals systems many such dispersive nonlinear equations pos- sess a rich geometric structure, such as a Hamiltonian formulation and conservation laws. To ensure that numerical schemes lead to meaningful results, it is vital to preserve this structure in numerical approxima- tions. This, however, results in an interesting dichotomy: the rich theory of existent structure-preserving algorithms is typically limited to classical integrators that cannot reliably treat low-regularity phenom- ena, while most prior designs of low-regularity integrators break geometric structure in the equation. In this talk, we will outline recent advances incorporating structure-preserving properties into low-regularity integrators. Starting from simple discussions on the nonlinear Schr ̈odinger and the Korteweg–de Vries equation we will discuss the construction of such schemes for a general class of dispersive equations before demonstrating an application to the simulation of low-regularity vortex filaments. This is joint work with Yvonne Alama Bronsard, Valeria Banica, Yvain Bruned and Katharina Schratz.

We propose and analyze a numerical method for time-dependent linear Schrödinger equations with uncertain parameters in both the potential and the initial data. The random parameters are discretized by stochastic collocation on a sparse grid, and the sample solutions in the nodes are approximated with the Strang splitting method. The computational work is reduced by a multilevel strategy, i.e., by combining information obtained from sample solutions computed on different refinement levels of the discretization. We prove new error bounds for the time discretization which take the finite regularity in the stochastic variable into account, and which are crucial to obtain convergence of the multilevel approach. The predicted cost savings of the multilevel stochastic collocation method are verified by numerical examples.

A helium atom interacting with a laser field can be described by the time-dependent Schrödinger equation (TDSE). The construction of accurate numerical solutions for a true two-electron problem is challenging due to the high dimensionality (six) of the TDSE. To reduce it, we resort to the Single Active Electron (SAE) approximation, which effectively decreases the dimensionality to three or even two if the laser light is linearly polarized. In its standard formulation, the SAE includes a model potential that describes the active electron's interaction with the residual ion, and the TDSE describes the evolution of the electron's wavefunction due to the combined Coulomb interactions within the target and the external laser field. Motivated by recent experimental measurements on helium employing the three-sideband RABBITT technique [1], we have increased the accuracy of the SAE approximation by allowing an angular momentum dependence in the model potential. We then solve the TDSE using the midpoint Strang splitting, where the model potential is diagonal for the Coulomb interaction. As a first test, we compare our results on high-harmonic generation with those obtained by the R-matrix with time dependence approach and the standard SAE approximation reported in [2]. This comparison serves as an initial validation of the proposed approach to tackle other phenomena in helium as well as more complex targets. [1] D. Bharti et al., Phys. Rev. A 109, 023110, 2024. [2] A.T. Bondy et al., accepted for publication in Phys. Rev. A, 2024.[4] J. J. Sakurai and J. Napolitano, Modern quantum mechanics; 2nd ed.(2011). [5] A. Dalgarno and J. T. Lewis, Proc. R. Soc. A: Math. Phys. Eng. 233, 1192 (1955).

In this presentation we discuss recent results on discrete minimizers of the Ginzburg-Landau energy in (generalized) finite element spaces. Special focus is given to the influence of the Ginzburg--Landau parameter $\kappa$. This parameter is of physical interest as large values can trigger the appearance of vortex lattices in superconductors. Since the vortices have to be resolved on sufficiently fine computational meshes, it is important to translate the size of $\kappa$ into a mesh resolution condition, which can be done through error estimates that are explicit with respect to $\kappa$ and the spatial mesh width $h$. We present corresponding analytical results for Lagrange finite elements and identify a previously unknown numerical pollution effect. Furthermore, we present some preliminary results on how the approximation properties can be enhanced with multiscale techniques.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk we address the mathematical foundations of spectral methods for time-dependent PDEs. In a deep sense, given a separable Hilbert space structure, a spectral method is tantamount to a choice of an orthonormal basis – yet, such a choice is governed by several analytic and numerical considerations. We will review such choices in different settings, focussing on two specific kinds of orthogonal sequences, T-systems and W-systems. Time allowing, we will discuss applications to dispersive equations of quantum mechanics.

The subject of this talk is the design of efficient and stable spectral methods for time-dependent partial differential equations in unit balls. We commence by sketching the desired features of a spectral method, which is defined by a choice of an orthonormal basis acting in the spatial domain. We continue by considering in detail the choice of a W-function basis in a disc in R^2. This is a nontrivial issue because of a clash between two objectives: skew symmetry of the differentiation matrix that the method is stable and the correct behaviour at the origin. We resolve it by representing the underlying space as an affine space and splitting the underlying functions. This is generalised to any dimension d in a natural manner and the paper is concluded with numerical examples that demonstrate how our choice of basis attains the best outcome out of a number of alternatives. This is a joint work with Prof. Arieh Iserles.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.