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Sylvester matrix equations are ubiquitous in scientific computing. However, few solution techniques exist for their generalized multiterm version, as they now arise in an increasingly large number of applications. In this work, we consider algebraic parameter-free preconditioning techniques for the iterative solution of generalized multiterm Sylvester equations. They consist in constructing low Kronecker rank approximations of either the operator itself or its inverse. While the former requires solving standard Sylvester equations in each iteration, the latter only requires matrix-matrix multiplications, which are highly optimized on modern computer architectures. Moreover, low Kronecker rank approximate inverses can be easily combined with sparse approximate inverse techniques, thereby enhancing their performance with little or no damage to their effectiveness.

The noncommutative sum-of-squares (ncSoS) hierarchy was introduced by Navascu\'{e}s-Pironio-Ac\'{i}n as a sequence of semidefinite programming relaxations for approximating values of noncommutative polynomial optimization problems, which were originally intended to generalize quantum values of nonlocal games. Recent work has started to analyze the hierarchy for approximating ground energies of local Hamiltonians, initially through rounding algorithms which output product states for degree-2 ncSoS applied to Quantum Max-Cut. Some rounding methods are known which output entangled states, but they use degree-4 ncSoS. Based on this, Hwang-Neeman-Parekh-Thompson-Wright conjectured that degree-2 ncSoS cannot beat product state approximations for Quantum Max-Cut and gave a partial proof relying on a conjectural generalization of Borrell's inequality. In this work we consider a family of Hamiltonians (called the quantum rotor model in condensed matter literature or lattice $O(k)$ vector model in quantum field theory) with infinite-dimensional local Hilbert space $L^{2}(S^{k - 1})$, and show that a degree-2 ncSoS relaxation approximates the ground state energy better than any product state.

The theory of mixed finite element methods for solving different types of elliptic partial differential equations in saddle point formulation is well established since many decades. This topic was mostly studied for variational formulations defined upon the same product spaces of both shape- and test-pairs of primal variable-multiplier. Whenever either these spaces or the two bilinear forms involving the multiplier are distinct, the saddle point problem is asymmetric. The three inf-sup conditions to be satisfied by the product spaces stipulated in work on the subject, in order to guarantee well-posedness, are well known. However, the material encountered in the literature addressing the approximation of this class of problems left room for improvement and clarifications. After making a brief review of the existing contributions to the topic that justifies such an assertion, in this paper we set up finer global error bounds for the pair primal variable-multiplier solving an asymmetric saddle point problem. Besides well-posedness, the three constants in the aforementioned inf-sup conditions are identified as all that is needed for determining the stability constant appearing therein, whose expression is exhibited. As a complement, refined error bounds depending only on these three constants are given for both unknowns separately.

Exceptionally elegant formulae exist for the fractional Laplacian operator applied to weighted classical orthogonal polynomials. We utilize these results to construct a solver, based on frame properties, for equations involving the fractional Laplacian of any power, $s \in (0,1)$, on an unbounded domain in one or two dimensions. The numerical method represents solutions in an expansion of weighted classical orthogonal polynomials as well as their unweighted counterparts with a specific extension to $\mathbb{R}^d$, $d \in \{1,2\}$. We examine the frame properties of this family of functions for the solution expansion and, under standard frame conditions, derive an a priori estimate for the stationary equation. Moreover, we prove one achieves the expected order of convergence when considering an implicit Euler discretization in time for the fractional heat equation. We apply our solver to numerous examples including the fractional heat equation (utilizing up to a $6^\text{th}$-order Runge--Kutta time discretization), a fractional heat equation with a time-dependent exponent $s(t)$, and a two-dimensional problem, observing spectral convergence in the spatial dimension for sufficiently smooth data.

We consider a class of linear Vlasov partial differential equations driven by Wiener noise. Different types of stochastic perturbations are treated: additive noise, multiplicative It\^o and Stratonovich noise, and transport noise. We propose to employ splitting integrators for the temporal discretization of these stochastic partial differential equations. These integrators are designed in order to preserve qualitative properties of the exact solutions depending on the stochastic perturbation, such as preservation of norms or positivity of the solutions. We provide numerical experiments in order to illustrate the properties of the proposed integrators and investigate mean-square rates of convergence.

Spectral methods yield numerical solutions of the Galerkin-truncated versions of nonlinear partial differential equations involved especially in fluid dynamics. In the presence of discontinuities, such as shocks, spectral approximations develop Gibbs oscillations near the discontinuity. This causes the numerical solution to deviate quickly from the true solution. For spectral approximations of the 1D inviscid Burgers equation, nonlinear wave resonances lead to the formation of tygers in well-resolved areas of the flow, far from the shock. Recently, Besse(to be published) has proposed novel spectral relaxation (SR) and spectral purging (SP) schemes for the removal of tygers and Gibbs oscillations in spectral approximations of nonlinear conservation laws. For the 1D inviscid Burgers equation, it is shown that the novel SR and SP approximations of the solution converge strongly in L2 norm to the entropic weak solution, under an appropriate choice of kernels and related parameters. In this work, we carry out a detailed numerical investigation of SR and SP schemes when applied to the 1D inviscid Burgers equation and report the efficiency of shock capture and the removal of tygers. We then extend our study to systems of nonlinear hyperbolic conservation laws - such as the 2x2 system of the shallow water equations and the standard 3x3 system of 1D compressible Euler equations. For the latter, we generalise the implementation of SR methods to non-periodic problems using Chebyshev polynomials. We then turn to singular flow in the 1D wall approximation of the 3D-axisymmetric wall-bounded incompressible Euler equation. Here, in order to determine the blowup time of the solution, we compare the decay of the width of the analyticity strip, obtained from the pure pseudospectral method, with the improved estimate obtained using the novel spectral relaxation scheme.

We present a new method to compute the solution to a nonlinear tensor differential equation with dynamical low-rank approximation. The idea of dynamical low-rank approximation is to project the differential equation onto the tangent space of a low-rank tensor manifold at each time. Traditionally, an orthogonal projection onto the tangent space is employed, which is challenging to compute for nonlinear differential equations. We introduce a novel interpolatory projection onto the tangent space that is easily computed for many nonlinear differential equations and satisfies the differential equation at a set of carefully selected indices. To select these indices, we devise a new algorithm based on the discrete empirical interpolation method (DEIM) that parameterizes any tensor train and its tangent space with tensor cross interpolants. We demonstrate the proposed method with applications to tensor differential equations arising from the discretization of partial differential equations.

This paper introduces an efficient high-order numerical method for solving the 1D stationary Schr\"odinger equation in the highly oscillatory regime. Building upon the ideas from [2], we first analytically transform the given equation into a smoother (i.e. less oscillatory) equation. By developing sufficiently accurate quadratures for several (iterated) oscillatory integrals occurring in the Picard approximation of the solution, we obtain a one-step method that is third order w.r.t. the step size. The accuracy and efficiency of the method are illustrated through several numerical examples.

We propose a new class of finite element approximations to ideal compressible magnetohydrodynamic equations in smooth regime. Following variational approximations developed for fluid models in the last decade, our discretizations are built via a discrete variational principle mimicking the continuous Euler-Poincar\'e principle, and to further exploit the geometrical structure of the problem, vector fields are represented by their action as Lie derivatives on differential forms of any degree. The resulting semi-discrete approximations are shown to conserve the total mass, entropy and energy of the solutions for a wide class of finite element approximations. In addition, the divergence-free nature of the magnetic field is preserved in a pointwise sense and a time discretization is proposed, preserving those invariants and giving a reversible scheme at the fully discrete level. Numerical simulations are conducted to verify the accuracy of our approach and its ability to preserve the invariants for several test problems.

With advances in scientific computing and mathematical modeling, complex scientific phenomena such as galaxy formations and rocket propulsion can now be reliably simulated. Such simulations can however be very time-intensive, requiring millions of CPU hours to perform. One solution is multi-fidelity emulation, which uses data of different fidelities to train an efficient predictive model which emulates the expensive simulator. For complex scientific problems and with careful elicitation from scientists, such multi-fidelity data may often be linked by a directed acyclic graph (DAG) representing its scientific model dependencies. We thus propose a new Graphical Multi-fidelity Gaussian Process (GMGP) model, which embeds this DAG structure (capturing scientific dependencies) within a Gaussian process framework. We show that the GMGP has desirable modeling traits via two Markov properties, and admits a scalable algorithm for recursive computation of the posterior mean and variance along at each depth level of the DAG. We also present a novel experimental design methodology over the DAG given an experimental budget, and propose a nonlinear extension of the GMGP via deep Gaussian processes. The advantages of the GMGP are then demonstrated via a suite of numerical experiments and an application to emulation of heavy-ion collisions, which can be used to study the conditions of matter in the Universe shortly after the Big Bang. The proposed model has broader uses in data fusion applications with graphical structure, which we further discuss.