Modified transmission eigenvalues for inverse scattering in a fluid–solid interaction problem

Target signatures are discrete quantities computed from measured scattering data that could potentially be used to classify scatterers or give information about possible defects in the scatterer compared to an ideal object. Here, we study a class of modified interior transmission eigenvalues that are intended to provide target signatures for an inverse fluid–solid interaction problem. The modification is based on an auxiliary problem parametrized by an artificial diffusivity constant. This constant may be chosen strictly positive, or strictly negative. For both choices, we characterize the modified interior transmission eigenvalues by means of a suitable operator so that we can determine their location in the complex plane. Moreover, for the negative sign choice, we also show the existence and discreteness of these eigenvalues. Finally, no matter the choice of the sign, we analyze the approximation of the eigenvalues from far field measurements of the scattered fluid pressure and provide numerical results which show that, even with noisy data, some of the eigenvalues can be determined from far field data.

Analysis of the trajectory of spacecraft return from the lunar surface to a given region of the Earth

The paper describes a method developed for designing the trajectory of a spacecraft flight from the lunar surface to a given area of the Earth’s surface and analyzes a single-pulse flight scheme, in which the trajectory of a take-off lunar rocket is approximated by a single velocity pulse. The characteristics of the spacecraft entry into the Earth’s atmosphere are chosen so as to ensure that the conditions along the entry corridor during the ballistic entry are met and to ensure the landing of the reentry vehicle at a given point on the Earth’s surface. The criterion for optimizing the trajectory of the spacecraft return to the Earth is considered to be the value of the impulse that provides the spacecraft launch from the lunar surface. The method relies on the analysis of an auxiliary problem, the solution of which makes it possible to estimate the main properties of the investigated maneuver and find an initial approximation for the selected characteristics of the optimized trajectory.

Inexact accelerated high-order proximal-point methods

In this paper, we present a new framework of bi-level unconstrained minimization for development of accelerated methods in Convex Programming. These methods use approximations of the high-order proximal points, which are solutions of some auxiliary parametric optimization problems. For computing these points, we can use different methods, and, in particular, the lower-order schemes. This opens a possibility for the latter methods to overpass traditional limits of the Complexity Theory. As an example, we obtain a new second-order method with the convergence rate $$O\left( k^{-4}\right) $$ O k - 4 , where k is the iteration counter. This rate is better than the maximal possible rate of convergence for this type of methods, as applied to functions with Lipschitz continuous Hessian. We also present new methods with the exact auxiliary search procedure, which have the rate of convergence $$O\left( k^{-(3p+1)/ 2}\right) $$ O k - ( 3 p + 1 ) / 2 , where $$p \ge 1$$ p ≥ 1 is the order of the proximal operator. The auxiliary problem at each iteration of these schemes is convex.

Adaptive large neighborhood search for mixed integer programming

Large Neighborhood Search (LNS) heuristics are among the most powerful but also most expensive heuristics for mixed integer programs (MIP). Ideally, a solver adaptively concentrates its limited computational budget by learning which LNS heuristics work best for the MIP problem at hand. To this end, this work introduces Adaptive Large Neighborhood Search (ALNS) for MIP, a primal heuristic that acts as a framework for eight popular LNS heuristics such as Local Branching and Relaxation Induced Neighborhood Search (RINS). We distinguish the available LNS heuristics by their individual search spaces, which we call auxiliary problems. The decision which auxiliary problem should be executed is guided by selection strategies for the multi armed bandit problem, a related optimization problem during which suitable actions have to be chosen to maximize a reward function. In this paper, we propose an LNS-specific reward function to learn to distinguish between the available auxiliary problems based on successful calls and failures. A second, algorithmic enhancement is a generic variable fixing prioritization, which ALNS employs to adjust the subproblem complexity as needed. This is particularly useful for some LNS problems which do not fix variables by themselves. The proposed primal heuristic has been implemented within the MIP solver SCIP. An extensive computational study is conducted to compare different LNS strategies within our ALNS framework on a large set of publicly available MIP instances from the MIPLIB and Coral benchmark sets. The results of this simulation are used to calibrate the parameters of the bandit selection strategies. A second computational experiment shows the computational benefits of the proposed ALNS framework within the MIP solver SCIP.

On the Multiplicity of Solutions for the Discrete Boundary Problem Involving the Singular ϕ -Laplacian

In this paper, we consider the multiplicity of solutions for a discrete boundary value problem involving the singular ϕ -Laplacian. In order to apply the critical point theory, we extend the domain of the singular operator to the whole real numbers. Instead, we consider an auxiliary problem associated with the original one. We show that, if the nonlinear term oscillates suitably at the origin, there exists a sequence of pairwise distinct nontrivial solutions with the norms tend to zero. By our strong maximum principle, we show that all these solutions are positive under some assumptions. Moreover, the solutions of the auxiliary problem are solutions of the original one if the solutions are appropriately small. Lastly, we give an example to illustrate our main results.

Effects of corners in surface superconductivity

We study the Ginzburg–Landau functional describing an extreme type-II superconductor wire with cross section with finitely many corners at the boundary. We derive the ground state energy asymptotics up to o(1) errors in the surface superconductivity regime, i.e., between the second and third critical fields. We show that, compared to the case of smooth domains, each corner provides an additional contribution of order $$ {\mathcal {O}}(1) $$ O ( 1 ) depending on the corner opening angle. The corner energy is in turn obtained from an implicit model problem in an infinite wedge-like domain with fixed magnetic field. We also prove that such an auxiliary problem is well-posed and its ground state energy bounded and, finally, state a conjecture about its explicit dependence on the opening angle of the sector.

Superfast Second-Order Methods for Unconstrained Convex Optimization

In this paper, we present new second-order methods with convergence rate $$O\left( k^{-4}\right) $$ O k - 4 , where k is the iteration counter. This is faster than the existing lower bound for this type of schemes (Agarwal and Hazan in Proceedings of the 31st conference on learning theory, PMLR, pp. 774–792, 2018; Arjevani and Shiff in Math Program 178(1–2):327–360, 2019), which is $$O\left( k^{-7/2} \right) $$ O k - 7 / 2 . Our progress can be explained by a finer specification of the problem class. The main idea of this approach consists in implementation of the third-order scheme from Nesterov (Math Program 186:157–183, 2021) using the second-order oracle. At each iteration of our method, we solve a nontrivial auxiliary problem by a linearly convergent scheme based on the relative non-degeneracy condition (Bauschke et al. in Math Oper Res 42:330–348, 2016; Lu et al. in SIOPT 28(1):333–354, 2018). During this process, the Hessian of the objective function is computed once, and the gradient is computed $$O\left( \ln {1 \over \epsilon }\right) $$ O ln 1 ϵ times, where $$\epsilon $$ ϵ is the desired accuracy of the solution for our problem.

A priori bounds and existence of positive solutions to a p-kirchhoff equations

In this paper, we consider a [Formula: see text]-Kirchhoff problem with Dirichlet boundary problem in a bounded domain. Under suitable conditions, we get a priori estimates for positive solutions to an auxiliary problem by the well-known blow-up argument. As an application, a existence result for positive solutions is proved by the topological degree theory.

Recovery of unknown coefficients in a two-dimensional hyperbolic equation with additional conditions

In this paper, a nonlocal inverse boundary value problem for a two-dimensional hyperbolic equation with overdetermination conditions is studied. To investigate the solvability of the original problem, we first consider an auxiliary inverse boundary value problem and prove its equivalence (in a certain sense) to the original problem. Then using the Fourier method, solving an equivalent problem is reduced to solving a system of integral equations and by the contraction mappings principle the existence and uniqueness theorem for auxiliary problem is proved. Further, on the basis of the equivalency of these problems the uniquely existence theorem for the classical solution of the considered inverse problem is proved and some considerations on the numerical solution for this inverse problem are presented with the examples.

Space Analyticity and Bounds for Derivatives of Solutions to the Evolutionary Equations of Diffusive Magnetohydrodynamics

In 1981, Foias, Guillopé and Temam proved a priori estimates for arbitrary-order space derivatives of solutions to the Navier–Stokes equation. Such bounds are instructive in the numerical investigation of intermittency that is often observed in simulations, e.g., numerical study of vorticity moments by Donzis et al. (2013) revealed depletion of nonlinearity that may be responsible for smoothness of solutions to the Navier–Stokes equation. We employ an original method to derive analogous estimates for space derivatives of three-dimensional space-periodic weak solutions to the evolutionary equations of diffusive magnetohydrodynamics. Construction relies on space analyticity of the solutions at almost all times. An auxiliary problem is introduced, and a Sobolev norm of its solutions bounds from below the size in C3 of the region of space analyticity of the solutions to the original problem. We recover the exponents obtained earlier for the hydrodynamic problem. Moreover, the same approach is followed here to derive and prove similar a priori bounds for arbitrary-order space derivatives of the first-order time derivative of the weak MHD solutions.