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We survey recent developments in the field of complexity of pathwise approximation in $p$-th mean of the solution of a stochastic differential equation at the final time based on finitely many evaluations of the driving Brownian motion. First, we briefly review the case of equations with globally Lipschitz continuous coefficients, for which an error rate of at least $1/2$ in terms of the number of evaluations of the driving Brownian motion is always guaranteed by using the equidistant Euler-Maruyama scheme. Then we illustrate that giving up the global Lipschitz continuity of the coefficients may lead to a non-polynomial decay of the error for the Euler-Maruyama scheme or even to an arbitrary slow decay of the smallest possible error that can be achieved on the basis of finitely many evaluations of the driving Brownian motion. Finally, we turn to recent positive results for equations with a drift coefficient that is not globally Lipschitz continuous. Here we focus on scalar equations with a Lipschitz continuous diffusion coefficient and a drift coefficient that satisfies piecewise smoothness assumptions or has fractional Sobolev regularity and we present corresponding complexity results.

Acoustic wave equation is a partial differential equation (PDE) which describes propagation of acoustic waves through a material. In general, the solution to this PDE is nonunique. Therefore, it is necessary to impose initial conditions in the form of Cauchy conditions for obtaining a unique solution. Theoretically, solving the wave equation is equivalent to representing the wavefield in terms of a radiation source which possesses finite energy over space and time.The radiation source is represented by a forcing term in the right-hand-side of the wave equation. In practice, the source may be represented in terms of normal derivative of pressure or normal velocity over a surface. The pressure wavefield is then calculated by solving an associated boundary-value problem via imposing conditions on the boundary of a chosen solution space. From analytic point of view, this manuscript aims to review typical approaches for obtaining unique solution to the acoustic wave equation in terms of either a volumetric radiation source, or a surface source in terms of normal derivative of pressure or normal velocity. A numerical approximation of the derived formulae will then be explained. The key step for numerically approximating the derived analytic formulae is inclusion of source, and will be studied carefully in this manuscript.

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.

A detailed numerical study of solutions to the Serre-Green-Naghdi (SGN) equations in 2D with vanishing curl of the velocity field is presented. The transverse stability of line solitary waves, 1D solitary waves being exact solutions of the 2D equations independent of the second variable, is established numerically. The study of localized initial data as well as crossing 1D solitary waves does not give an indication of existence of stable structures in SGN solutions localized in two spatial dimensions. For the numerical experiments, an approach based on a Fourier spectral method with a Krylov subspace technique is applied.

In this manuscript, we highlight a new phenomenon of complex algebraic singularity formation for solutions of a large class of genuinely nonlinear partial differential equations (PDEs). We start from a unique Cauchy datum, which is holomorphic ramified around the smooth locus and is sufficiently singular. Then, we expect the existence of a solution which should be holomorphic ramified around the singular locus S defined by the vanishing of the discriminant of an algebraic equation. Notice, moreover, that the monodromy of the Cauchy datum is Abelian, whereas one of the solutions is non-Abelian. Moreover, the singular locus S depends on the Cauchy datum in contrast to the Leray principle (stated for linear problems only). This phenomenon is due to the fact that the PDE is genuinely nonlinear and that the Cauchy datum is sufficiently singular. First, we investigate the case of the inviscid Burgers equation. Later, we state a general conjecture that describes the expected phenomenon. We view this Conjecture as a working programme allowing us to develop interesting new Mathematics. We also state another Conjecture 2, which is a particular case of the general Conjecture but keeps all the flavour and difficulty of the subject. Then, we propose a new algorithm with a map F such that a fixed point of F would give a solution to the problem associated with Conjecture 2. Then, we perform convincing, elaborate numerical tests that suggest that a Banach norm should exist for which the mapping F should be a contraction so that the solution (with the above specific algebraic structure) should be unique. This work is a continuation of Leichtnam (1993).

This paper presents a method for thematic agreement assessment of geospatial data products of different semantics and spatial granularities, which may be affected by spatial offsets between test and reference data. The proposed method uses a multi-scale framework allowing for a probabilistic evaluation whether thematic disagreement between datasets is induced by spatial offsets due to different nature of the datasets or not. We test our method using real-estate derived settlement locations and remote-sensing derived building footprint data.

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.

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.

In an error estimation of finite element solutions to the Poisson equation, we usually impose the shape regularity assumption on the meshes to be used. In this paper, we show that even if the shape regularity condition is violated, the standard error estimation can be obtained if "bad" elements (elements that violate the shape regularity or maximum angle condition) are covered virtually by "good" simplices. A numerical experiment confirms the theoretical result.

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.