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In these notes we propose and analyze an inertial type method for obtaining stable approximate solutions to nonlinear ill-posed operator equations. The method is based on the Levenberg-Marquardt (LM) iteration. The main obtained results are: monotonicity and convergence for exact data, stability and semi-convergence for noisy data. Regarding numerical experiments we consider: i) a parameter identification problem in elliptic PDEs, ii) a parameter identification problem in machine learning; the computational efficiency of the proposed method is compared with canonical implementations of the LM method.

The Bregman-Kaczmarz method is an iterative method which can solve strongly convex problems with linear constraints and uses only one or a selected number of rows of the system matrix in each iteration, thereby making it amenable for large-scale systems. To speed up convergence, we investigate acceleration by heavy ball momentum in the so-called dual update. Heavy ball acceleration of the Kaczmarz method with constant parameters has turned out to be difficult to analyze, in particular no accelerated convergence for the L2-error of the iterates has been proven to the best of our knowledge. Here we propose a way to adaptively choose the momentum parameter by a minimal-error principle similar to a recently proposed method for the standard randomized Kaczmarz method. The momentum parameter can be chosen to exactly minimize the error in the next iterate or to minimize a relaxed version of the minimal error principle. The former choice leads to a theoretically optimal step while the latter is cheaper to compute. We prove improved convergence results compared to the non-accelerated method. Numerical experiments show that the proposed methods can accelerate convergence in practice, also for matrices which arise from applications such as computational tomography.

Electrical manipulation of antiferromagnetic states, a cornerstone of antiferromagnetic spintronics, is a great challenge, requiring novel material platforms. Here we report the full control over antiferromagnetic states by voltage pulses in the insulating Co$_3$O$_4$ spinel. We show that the strong linear magnetoelectric effect emerging in its antiferromagnetic state is fully governed by the orientation of the N\'eel vector. As a unique feature of Co$_3$O$_4$, the magnetoelectric energy can easily overcome the weak magnetocrystalline anisotropy, thus, the N\'eel vector can be manipulated on demand, either rotated smoothly or reversed suddenly, by combined electric and magnetic fields. We succeed with switching between antiferromagnetic states of opposite N\'eel vectors by voltage pulses within a few microsecond in macroscopic volumes. These observations render quasi-cubic antiferromagnets, like Co$_3$O$_4$, an ideal platform for the ultrafast (pico- to nanosecond) manipulation of microscopic antiferromagnetic domains and may pave the way for the realization of antiferromagnetic spintronic devices., Comment: 7 pages, 3 figures

We propose a new randomized method for solving systems of nonlinear equations, which can find sparse solutions or solutions under certain simple constraints. The scheme only takes gradients of component functions and uses Bregman projections onto the solution space of a Newton equation. In the special case of euclidean projections, the method is known as nonlinear Kaczmarz method. Furthermore, if the component functions are nonnegative, we are in the setting of optimization under the interpolation assumption and the method reduces to SGD with the recently proposed stochastic Polyak step size. For general Bregman projections, our method is a stochastic mirror descent with a novel adaptive step size. We prove that in the convex setting each iteration of our method results in a smaller Bregman distance to exact solutions as compared to the standard Polyak step. Our generalization to Bregman projections comes with the price that a convex one-dimensional optimization problem needs to be solved in each iteration. This can typically be done with globalized Newton iterations. Convergence is proved in two classical settings of nonlinearity: for convex nonnegative functions and locally for functions which fulfill the tangential cone condition. Finally, we show examples in which the proposed method outperforms similar methods with the same memory requirements.

The cubic spinel FeCr2S4 has been receiving immense research interest because of its emergent phases and the interplay of spin, orbital and lattice degrees of freedom. Despite the intense research, several fundamental questions are yet to be answered, such as the refinement of the crystal structure in the different magnetic and orbital ordered phases. Here, using high-resolution synchrotron powder diffraction on stoichiometric crystals of FeCr2S4 we resolved the long sought-after cubic to tetragonal transition at ~65 K, reducing the lattice symmetry to I41/amd. With further lowering the temperature, at ~9 K, the crystal structure becomes polar, hence the compound becomes multiferroic. The elucidation of the lattice symmetry throughout different phases of FeCr2S4 provides a basis for the understanding this enigmatic system and also highlights the importance of structural deformation in correlated materials., Comment: 10 pages, 6 figures

The Extended Randomized Kaczmarz method is a well known iterative scheme which can find the Moore-Penrose inverse solution of a possibly inconsistent linear system and requires only one additional column of the system matrix in each iteration in comparison with the standard randomized Kaczmarz method. Also, the Sparse Randomized Kaczmarz method has been shown to converge linearly to a sparse solution of a consistent linear system. Here, we combine both ideas and propose an Extended Sparse Randomized Kaczmarz method. We show linear expected convergence to a sparse least squares solution in the sense that an extended variant of the regularized basis pursuit problem is solved. Moreover, we generalize the additional step in the method and prove convergence to a more abstract optimization problem. We demonstrate numerically that our method can find sparse least squares solutions of real and complex systems if the noise is concentrated in the complement of the range of the system matrix and that our generalization can handle impulsive noise.

Geometrical frustration among interacting spins combined with strong quantum fluctuations destabilize long-range magnetic order in favour of more exotic states such as spin liquids. By following this guiding principle, a number of spin liquid candidate systems were identified in quasi-two-dimensional (quasi-2D) systems. For 3D, however, the situation is less favourable as quantum fluctuations are reduced and competing states become more relevant. Here we report a comprehensive study of thermodynamic, magnetic and dielectric properties on single crystalline and pressed-powder samples of PbCuTe$_2$O$_6$, a candidate material for a 3D frustrated quantum spin liquid featuring a hyperkagome lattice. Whereas the low-temperature properties of the powder samples are consistent with the recently proposed quantum spin liquid state, an even more exotic behaviour is revealed for the single crystals. These crystals show ferroelectric order at $T_{\text{FE}} \approx 1\,\text{K}$, accompanied by strong lattice distortions, and a modified magnetic response -- still consistent with a quantum spin liquid -- but with clear indications for quantum critical behaviour., Comment: 59 pages, 15 figures, This version of the article has been accepted for publication, after peer review but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online

The stochastic inverse eigenvalue problem aims to reconstruct a stochastic matrix from its spectrum. While there exists a large literature on the existence of solutions for special settings, there are only few numerical solution methods available so far. Recently, Zhao et al. (2016) proposed a constrained optimization model on the manifold of so-called isospectral matrices and adapted a modified Polak-Ribi\`ere-Polyak conjugate gradient method to the geometry of this manifold. However, not every stochastic matrix is an isospectral one and the model from Zhao et al. is based on the assumption that for each stochastic matrix there exists a (possibly different) isospectral, stochastic matrix with the same spectrum. We are not aware of such a result in the literature, but will see that the claim is at least true for $3 \times 3$ matrices. In this paper, we suggest to extend the above model by considering matrices which differ from isospectral ones only by multiplication with a block diagonal matrix with $2 \times 2$ blocks from the special linear group $SL(2)$, where the number of blocks is given by the number of pairs of complex-conjugate eigenvalues. Every stochastic matrix can be written in such a form, which was not the case for the form of the isospectral matrices. We prove that our model has a minimizer and show how the Polak-Ribi\`ere-Polyak conjugate gradient method works on the corresponding more general manifold. We demonstrate by numerical examples that the new, more general method performs similarly as the one from Zhao et al.