Related papers: Approximation theory for Green's functions via the…
Lanczos-type algorithms are efficient and easy to implement. Unfortunately they breakdown frequently and well before convergence has been achieved. These algorithms are typically based on recurrence relations which involve formal orthogonal…
The single particle Green's function provides valuable information on the momentum and energy-resolved spectral properties for a strongly correlated system. In large-scale numerical calculations using quantum Monte Carlo (QMC), dynamical…
We obtain an expression for the error in the approximation of $f(A) \boldsymbol{b}$ and $\boldsymbol{b}^T f(A) \boldsymbol{b}$ with rational Krylov methods, where $A$ is a symmetric matrix, $\boldsymbol{b}$ is a vector and the function $f$…
In this contribution, we present a full overview of the continuous stochastic gradient (CSG) method, including convergence results, step size rules and algorithmic insights. We consider optimization problems in which the objective function…
This work studies approximation based on single-hidden-layer feedforward and recurrent neural networks with randomly generated internal weights. These methods, in which only the last layer of weights and a few hyperparameters are optimized,…
The low rank approximation of matrices is a crucial component in many data mining applications today. A competitive algorithm for this class of problems is the randomized block Lanczos algorithm - an amalgamation of the traditional block…
This paper presents an extended version of the article [Franz, S., Kopteva, N.: J. Differential Equations, 252 (2012)]. The main improvement compared to the latter is in that here we additionally estimate the mixed second-order derivative…
Flexible boundary condition methods couple an isolated defect to bulk through the bulk lattice Green's function. The inversion of the force-constant matrix for the lattice Green's function requires Fourier techniques to project out the…
In this paper we provide a rigorous mathematical foundation for continuous approximations of a class of systems with piece-wise continuous functions. By using techniques from the theory of differential inclusions, the underlying piece-wise…
We present a family of algorithms for the numerical approximation of the Schr\"odinger equation with potential concentrated at a finite set of points. Our methods belong to the so-called fast and oblivious convolution quadrature algorithms.…
We present a calculation of the spectral properties of a single charge doped at a Cu($3d$) site of the Cu-F plane in KCuF$_{3}$. The problem is treated by generating the equations of motion for the Green's function by means of subsequent…
We derive and analyze new diffusion approximations of stationary distributions of Markov chains that are based on second- and higher-order terms in the expansion of the Markov chain generator. Our approximations achieve a higher degree of…
We propose a new model to approximate the wave response of waveguides containing an arbitrary number of small inclusions. The theory is developed to consider any one-dimensional waveguide (longitudinal, flexural, shear, torsional waves or a…
The electromagnetic Green's function is a crucial ingredient for the theoretical study of modern photonic quantum devices, but is often difficult or even impossible to calculate directly. We present a numerically efficient framework for…
Understanding the transport behavior of quantum many-body systems constitutes an important physical endeavor, both experimentally and theoretically. While a reliable classification into normal and anomalous dynamics is known to be…
The theory of Self-Consistent Green's Function (SCGF) is reformulated in an explicit Nambu-covariant fashion for applications to many-body systems at non-zero temperature in symmetry-broken phases. This is achieved by extending the…
The Green's function method has applications in several fields in Physics, from classical differential equations to quantum many-body problems. In the quantum context, Green's functions are correlation functions, from which it is possible…
Lanczos-based methods have become standard tools for tasks involving matrix functions. Progress on these algorithms has been driven by several largely disjoint communities, resulting many innovative and important advancements which would…
We seek to approximate a composite function h(x) = g(f(x)) with a global polynomial. The standard approach chooses points x in the domain of f and computes h(x) at each point, which requires an evaluation of f and an evaluation of g. We…
This work reports the conditions under which weak scattering assumptions can be applied in a beam loaded by multiple resonators supporting both longitudinal and flexural waves. The work derives the equations of motion of a one-dimensional…