Jakub Benda

Three-dimensional super-resolution structured illumination microscopy with maximum a posteriori probability image estimation

We introduce and demonstrate a new high performance image reconstruction method for super-resolution structured illumination microscopy based on maximum a posteriori probability estimation (MAP-SIM). Imaging performance is demonstrated on a variety of fluorescent samples of different thickness, labeling density and noise levels. The method provides good suppression of out of focus light, improves spatial resolution, and allows reconstruction of both 2D and 3D images of cells even in the case of weak signals. The method can be used to process both optical sectioning and super-resolution structured illumination microscopy data to create high quality super-resolution images.

Collisions of electrons with hydrogen atoms II. Low-energy program using the method of the exterior complex scaling ✩

a b s t r a c t While collisions of electrons with hydrogen atoms pose a well studied and in some sense closed problem, there is still no free computer code ready for ‘‘production use’’, that would enable applied researchers to generate necessary data for arbitrary impact energies and scattering transitions directly if absent in online scattering databases. This is the second article on the Hex program package, which describes a new computer code that is, with a little setup, capable of solving the scattering equations for energies ranging from a fraction of the ionization threshold to approximately 100 eV or more, depending on the available computational resources. The program implements the exterior complex scaling method in the B-spline basis.

New version of hex-ecs, the B-spline implementation of exterior complex scaling method for solution of electron–hydrogen scattering

Abstract We provide an updated version of the program hex-ecs originally presented in Comput. Phys. Commun. 185 (2014) 2903–2912. The original version used an iterative method preconditioned by the incomplete LU factorization (ILU), which–though very stable and predictable–requires a large amount of working memory. In the new version we implemented a “separated electrons” (or “Kronecker product approximation”, KPA) preconditioner as suggested by Bar-On et al., Appl. Num. Math. 33 (2000) 95–104. This preconditioner has much lower memory requirements, though in return it requires more iterations to reach converged results. By careful choice between ILU and KPA preconditioners one is able to extend the computational feasibility to larger calculations. Secondly, we added the option to run the KPA preconditioner on an OpenCL device (e.g. GPU). GPUs have generally better memory access times, which speeds up particularly the sparse matrix multiplication. New version program summary Program title: hex-ecs Catalogue identifier: AETI_v2_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AETI_v2_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: MIT License No. of lines in distributed program, including test data, etc.: 73693 No. of bytes in distributed program, including test data, etc.: 520475 Distribution format: tar.gz Programming language: C++11. Computer: Any recent CPU, preferably 64-bit. Computationally intensive parts can be run on GPU (tested on AMD Tahiti and NVidia TitanX models). Operating system: Tested on Windows 10 and various Linux distributions. RAM: Depends on the problem solved and particular setup; KPA test run uses apx. 300 MiB. Classification: 2.4. Catalogue identifier of previous version: AETI_v2_0 Journal reference of previous version: Comput. Phys. Comm. 185 (2014) 2903 External routines: GSL [1], UMFPACK [2], BLAS and LAPACK (ideally threaded OpenBLAS [3]). Does the new version supersede the previous version?: Yes Nature of problem: Solution of the two-particle Schrodinger equation in central field. Solution method: The two-electron states are expanded into angular momentum eigenstates, which gives rise to the coupled bi-radial equations. The bi-radially dependent solution is then represented in a B-spline product basis, which transforms the set of equations into a large matrix equation in this basis. The boundary condition is of Dirichlet type, thanks to the use of the exterior complex scaling method, which extends the coordinates into the complex plane. The matrix equation is then solved by preconditioned conjugated orthogonal conjugate gradient method (PCOCG) [4]. Reasons for new version: The original program has been updated to achieve better performance. Also, some external dependencies have been removed (HDF5, FFTW3), which simplifies deployment. Summary of revisions: We implemented a new preconditioner introduced in [5], both for general CPU and also for an arbitrary OpenCL device (e.g. GPU) conforming to the OpenCL 2.0 specification. Furthermore, many other minor improvements have been made, particularly with the intention of reducing the memory requirements. With appropriate switches the program now does not precompute the used matrices and only calculates their elements on the fly. This is aided also by the vectorized B-spline evaluation function, which can now make use of AVX instructions when a single B-spline is being evaluated at several points. The accompanying tools hex-db and hex-dwba [6] have been also updated to use the shared code base. Running time: KPA test run — apx. 2 minutes on Intel i7-4790K (4 threads) References: [1] Galassi M. et al, GNU Scientific Library: Reference Manual, Network Theory Ltd., 2003. [2] Davis T. A., Algorithm 832: UMFPACK, an unsymmetric-pattern multifrontal method, ACM Trans. Math. Softw. 30 (2004) 196–199. [3] Xianyi Z. et al, Model-driven Level 3 BLAS Performance Optimization on Loongson 3A Processor, 2012 IEEE 18th International Conference on Parallel and Distributed Systems (ICPADS), 17–19 Dec. 2012. [4] van der Vorst H. A., Melissen J. B. M., A Petrov–Galerkin type method for solving A x = b , where A is symmetric complex, IEEE Trans. Magn. 26 (1990) 706–708. [5] Bar-On et al., Parallel solution of the multidimensional Helmholtz/Schroedinger equation using high order methods, Appl. Num. Math. 33 (2000) 95–104. [6] Benda J., Houfek K., Collisions of electrons with hydrogen atoms I. Package outline and high energy code, Comput. Phys. Commun. 185 (2014) 2893–2902.

Reducing the dimensionality of grid based methods for electron-atom scattering calculations below ionization threshold

Abstract For total energies below the ionization threshold it is possible to dramatically reduce the computational burden of the solution of the electron-atom scattering problem based on grid methods combined with the exterior complex scaling. As in the R -matrix method, the problem can be split into the inner and outer problem, where the outer problem considers only the energetically accessible asymptotic channels. The ( N + 1 )-electron inner problem is coupled to the one-electron outer problems for every channel, resulting in a matrix that scales only linearly with size of the outer grid.

Foundations of Quantum Mechanics

In this chapter, we introduce the fundamental principles of quantum mechanics. We commence by discussing the famous Stern-Gerlach experiments for a particle with the spin 1∕2 as several key quantum mechanical phenomena may be well understood thereof. Using these very experiments as an example, we then illustrate how the basic principles are incorporated within the mathematical formalism of quantum mechanics. Subsequently, we generalize this mathematical scheme for more complicated systems. Finally, focusing on the harmonic oscillator as an example, we show the relation between an abstract and a specific approach to the formalism.

Dynamics: The Nonrelativistic Theory

In this chapter, we introduce the basic principles and applications of quantum electrodynamics while describing the motion of a particle within the nonrelativistic approximation. We start by showing how classical electrodynamics may be cast into the Hamilton formalism, the subsequent transition from classical to quantum electrodynamics being then straightforward. Next, we focus on the quantization of a free electromagnetic field wherein plays the abstract solution for harmonic oscillator based on the introduction of non-Hermitian ladder operators a decisive role.

The Hydrogen Atom and Structure of Its Spectral Lines

In this chapter, we focus on hydrogen-like atoms and their spectral structure in great depth. We show that the spectrum consists of a gross structure resulting from the electrostatic interaction between an electron and the nucleus, a fine structure arising from the spin-orbit interaction, and a hyperfine structure stemming from the spin-spin interaction. These structures are not specific solely for the hydrogen-like atoms though, we encounter them in any system which we can describe in the first approximation within the framework of nonrelativistic quantum mechanics. As we will shortly see, the spin-orbit interaction is an effect of relativistic kinematics and the spin-spin interaction is nothing more than a quantum mechanical analogy of the interaction of two magnetic dipoles. In the case of such systems, the effects of relativistic kinematics are minor and likewise the magnetic interaction is substantially smaller in comparison to the electrostatic interaction. Furthermore, we focus on the problem of hydrogen-like atoms to illustrate several methods that we will then systematically develop in the next chapter.

Treasures Hidden in Commutators

In this chapter, we focus on a topic usually called the application of algebraic methods or Lie algebras within quantum mechanics. We have already demonstrated that one can determine very efficiently the spectrum of the harmonic oscillator owing to the closure of the set of three operators, namely the Hamiltonian \(\hat{\mathsf{H}}\) and the ladder operators \(\hat{\mathsf{a}}\) and \(\hat{\mathsf{a}}^{+}\), under the operation of commutation. We now show that this method can be extended to the problem of angular momentum, the addition of angular momenta, the hydrogen atom, and a free particle.

The Helium Atom

In the previous chapters, we thoroughly investigated the simplest atom of all—the hydrogen atom. We have found that, owing to the existence of a sufficient number of integrals of motion, one can solve its energy spectrum exactly. Unfortunately, one cannot determine exactly the spectrum of helium nor of any of the heavier atoms. Nevertheless, we know from Chap. 2 that by means of the variational method we may approximate the solution to any desirable accuracy. We will show that the antisymmetry of the wave function with respect to the exchange of the electrons leads to a so-called exchange interaction. Accounting for this interaction subsequently leads to a qualitatively correct result even when only one two-electron configuration considered. This estimate can be further systematically improved by the inclusion of additional electron configurations. We will see that the symmetries of the helium atom, i.e., the existence of operators commuting with the Hamiltonian, substantially decrease the amount of configurations one needs to include in the calculations.

Approximate Methods in Quantum Mechanics

However light and elegant the mathematical apparatus of quantum mechanics appears, we can solve exactly only very few physically interesting problems with it. Therefore, we need to opt for appropriate approximations when facing the remaining vast majority of quantum-mechanical problems. In this chapter, we will introduce two basic approaches—the variational and perturbation method. Naturally, many other exist (for example the semi-classical approximations). However, those usually focus on a specific class of problems, while we can employ the variational and perturbative methods when facing almost any problem. We will illustrate both methods on the simplest problem one cannot solve analytically—the anharmonic oscillator.