We propose two new algebraic reconstruction techniques based on Kaczmarz's method that produce a regularized solution to noisy tomography problems. Tomography problems exhibit semiconvergence when iterative methods are employed, and the aim is therefore to stop near the semiconvergence point. Our approach is based on an error gauge that is constructed by pairing standard down-sweep Kaczmarz's method with its up-sweep version; we stop the iterations when this error gauge is minimal. The reconstructions of the new methods differ from standard Kaczmarz iterates in that our final result is the average of the stopped up-and down-sweeps. Even when Kaczmarz's method is supplied with an oracle that provides the exact error---and is therefore able to stop at the best possible iterate---our methods have a lower two-norm error in the vast majority of our test cases. In terms of computational cost, our methods are a little cheaper than standard Kaczmarz equipped with a statistical stopping rule.

We aim to solve inverse problems in illumination optics by means of optimal control theory. This is done by first formulating geometric optics in terms of Liouville’s equation, which governs the evolution of light distributions on phase space. Choosing a metric that measures how close one distribution is to another, the formal Lagrange method can be applied. We show that this approach has great potential by a simple numerical example of an ideal lens.

Active flux schemes are finite volume schemes that keep track of both point values and averages. The point values are updated using a semi-Lagrangian step, making active flux schemes highly suitable for geometric optics problems on phase space, i.e., to solve Liouville's equation. We use a semi-discrete version of the active flux scheme. Curved optics lead to moving boundaries in phase space. Therefore, we introduce a novel way of defining the active flux scheme on moving meshes. We show, using scaling arguments as well as numerical experiments, that our scheme outperforms the current industry standard, ray tracing. It has higher accuracy as well as a more favourable time scaling. The numerical experiments demonstrate that the active flux scheme is orders of magnitude more accurate and faster than ray tracing.

Root-finders based on full linear multistep methods (LMMs) use previous function values, derivatives and root estimates to iteratively find a root of a nonlinear function. As ODE solvers, full LMMs are typically not zero-stable. However, used as root-finders, the interpolation points are convergent so that such stability issues are circumvented. A general analysis is provided based on inverse polynomial interpolation, which is used to prove a fundamental barrier on the convergence rate of any LMM-based method. We show, using numerical examples, that full LMM-based methods perform excellently. Finally, we also provide a robust implementation based on Brent’s method that is guaranteed to converge.

Root-finders based on full linear multistep methods (LMMs) use previous function values, derivatives and root estimates to iteratively find a root of a nonlinear function. As ODE solvers, full LMMs are typically not zero-stable. However, used as root-finders, the interpolation points are convergent so that such stability issues are circumvented. A general analysis is provided based on inverse polynomial interpolation, which is used to prove a fundamental barrier on the convergence rate of any LMM-based method. We show, using numerical examples, that full LMM-based methods perform excellently. Finally, we also provide a robust implementation based on Brents method that is guaranteed to converge.

Embedded WENO schemes are a new family of weighted essentially nonoscillatory schemes that always utilise all adjacent smooth substencils. This results in increased control over the convex combination of lower-order interpolations. We show that more conventional WENO schemes, such as WENO-JS and WENO-Z (Borges et al., J. Comput. Phys., 2008; Jiang and Shu, J. Comput. Phys., 1996), do not exhibit this feature and as such do not always provide a desirable linear combination of smooth substencils. In a previous work, we have already developed the theory and machinery needed to construct embedded WENO methods and shown some five-point schemes (van Lith et al., J. Comput. Phys., 2016). Here, we construct a seven-point scheme and show that it too performs well using some numerical examples from the one-dimensional Euler equations.

Embedded WENO methods utilise all adjacent smooth substencils to construct a desirable interpolation. Conventional WENO schemes under-use this possibility close to large gradients or discontinuities. We develop a general approach for constructing embedded versions of existing WENO schemes. Embedded methods based on the WENO schemes of Jiang and Shu [1] and on the WENO-Z scheme of Borges et al. [2] are explicitly constructed. Several possible choices are presented that result in either better spectral properties or a higher order of convergence for sufficiently smooth solutions. However, these improvements carry over to discontinuous solutions. The embedded methods are demonstrated to be indeed improvements over their standard counterparts by several numerical examples. All the embedded methods presented have no added computational effort compared to their standard counterparts.

A novel scheme is developed that computes numerical solutions of Liouville's equation with a discontinuous Hamiltonian. It is assumed that the underlying Hamiltonian system has well-defined behaviour even when the Hamiltonian is discontinuous. In the case of geometrical optics such a discontinuity yields the familiar Snell's law or the law of specular reflection. Solutions to Liouville's equation should be constant along curves defined by the Hamiltonian system when the right-hand side is zero, i.e., no absorption or collisions. This consideration allows us to derive a new jump condition, enabling us to construct a first-order accurate scheme. Essentially, the correct physics is built into the solver. The scheme is tested in a two-dimensional optical setting with two test cases, the first using a single jump in the refractive index and the second a compound parabolic concentrator. For these two situations, the scheme outperforms the more conventional method of Monte Carlo ray tracing.

Most of the biological polymers that make up our cells and tissues are hierarchically structured. For biopolymers ranging from collagen, to actin, to fibrin and amyloid fibrils this hierarchy provides vitally important versatility. The structural hierarchy must be encoded in the self-assembly process, from the earliest stages onward, in order to produce the appropriate substructures. In this letter, we explore the kinetics of multistage self-assembly processes in a model system which allows comparison to bulk probes such as light scattering. We apply our model to recent turbidimetry data on the self-assembly of collagen fibrils. Our analysis suggests a connection between diffusion-limited aggregation kinetics and fibril growth, supported by slow, power-law growth at very long time scales.

Most of the biological polymers that make up our cells and tissues are hierarchically structured. For biopolymers ranging from collagen, to actin, to fibrin, this hierarchy provides vitally important versatility, allowing a multitude of structurally and functionally distinct structures to be constructed from a limited set of biomolecular constituents. This structural hierarchy must be encoded in the self-assembly process, from the earliest stages onward, in order to produce the appropriate substructures in the correct sequence. In this Letter, we explore the kinetics of such multi-stage self-assembly processes in a model system which is formulated as a set of discrete master equations capturing the underlying hierarchical molecular-scale process, but which may be homogenized to yield a practical, continuum description in terms of PDEs to compare to bulk experiments such as light scattering or turbidity measurements. We present the general framework, and apply it to recent turbidimetry data on the self-assembly of collagen fibrils. Furthermore, our analysis suggests a connection between diffusion-limited aggregation kinetics and fibril growth, supported by slow, power-law growth at very long timescales observed in both systems.

A virus represents one of the most elegant self-assembling systems, made up of protein subunits which constitute the capsid, the envelop containing genetic material. This viral genome can consist of single-stranded or double-stranded RNA or DNA, which, through various specific and non-specific interaction, can self-assemble together with the capsid proteins to form a complete virus. One of the many aspects of the study of viruses has been the concentration profile inside the capsid, because of the interactions between genome polymer and capsid wall, this is not a homogeneous solution, as shown by experiments. These have shown that the genome polymer is very concentrated on the wall, theory on the subject, however, predicts the concentration profile to vary more slowly and gradually. Since these do not reconcile, the main objective of this project is to use a specific model and find out under what circumstances the genome polymer condensates on the wall. The effects of a container compared to a single wall will be studied as well as the effect of curvature. It will be shown that under good solvent conditions, positive second virial coefficient, there is an optimal amount of material resulting in a lowest energy state for a given radius of the virus and attraction strength. For bad solvent conditions, negative second virial coefficient, it will be shown that there are many minima in the free energy. Typical concentration profiles for a negative virial coefficient exhibit oscillatory behaviour and can feature holes or bulges in the center.