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In this thesis, we consider the planning problem for first-order mean-field games (MFG). These games degenerate into optimal transport when there is no coupling between players. Our aim is to extend the concept of displacement convexity from optimal transport to MFGs. This extension gives new estimates for solutions of MFGs. First, we introduce the Monge-Kantorovich problem and examine related results on rearrangement maps. Next, we present the concept of displacement convexity. Then, we derive first-order MFGs, which are given by a system of a Hamilton-Jacobi equation coupled with a transport equation. Finally, we identify a large class of functions, that depend on solutions of MFGs, which are convex in time. Among these, we find several norms. This convexity gives bounds for the density of solutions of the planning problem.

Interpolation is essential in digital image processing, especially magnification. Many different approaches to interpolation specific to magnification have been developed in an effort to overcome the shortcomings of bilinear and bicubic interpolation. One of these approaches, Constraint-Based Interpolation, produces an image that is free of jaggies and has less blurring than bilinear or bicubic interpolation. Although Constraint-Based Interpolation produces a visually pleasing image, there are user-chosen parameters that make the algorithm difficult to use. In this thesis we propose a method for automatic selection of those parameters and an extension of Constraint-Based Interpolation to other forms of image manipulation, such as skew, rotation, warp, or any other invertable image transformation. By extending Constaint-Based Interpolation the same improvements observed in magnification could be observed in these other image transformations.

Yes / In CAD, when considering the question of new designs of complex mechanical parts, such as engine pistons, a parametric representation of the design is usually defined. However, in general there is a lack of efficient tools to create and manipulate such parametrically defined shapes. In this paper, we show how the geometry of complex mechanical parts can be parameterised efficiently enabling a designer to create and manipulate such geometries within an interactive environment. For surface generation we use the PDE method which allows surfaces to be defined in terms of a relatively small number of design parameters. The PDE method effectively creates surfaces by using the information contained at the boundaries (edges) of the surface patch. An interactively defined parameterisation can then be introduced on the boundaries (which are defined by means of space curves) of the surface. Thus, we show how complex geometries of mechanical parts, such as engine pistons, can be efficiently parameterised for geometry manipulation allowing a designer to create alternative designs.

No / This paper presents a surface generation tool designed for the construction of aircraft geometry. The software generates complex geometries which can be crafted or modified by the user in real time. The surface generation is based on partial differential equations (PDEs). The PDE method can produce different configurations of aircraft shapes interactively. Each surface is generated by a number of curves representing the character lines of a given part of the aircraft shape that can be manipulated in real time. Different surfaces then blend to create the full shape of the airplane. An important function of the proposed tool is its ability to change the aircraft shape through the adjustments of parameters associated with the initial curves. The user can apply linear transformations to the curves generating the airplane through simple input from the computer keyboard and the mouse. The updated curves can then be used to generate the surface leading to different configurations of a given airplane shape. The work presents detailed descriptions on the PDE method, parametric design and manipulation of aircrafts along with graphical demonstrations of its abilities and a series of examples to illustrate the capacity of the methodology implemented.

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Yes / This paper presents a method to extend PDE surfaces to high dimensional spaces. We review a common existing analytic solution, and show how it can be used straightforwardly to increase the dimension of the space the surface is embedded within. We then further develop a numerical scheme suitable for increasing the number of variables that parametrise the surface, and investigate some of the properties of this solution with a view to future work.

No / We propose an efficient alternative to commonly used parametric surfaces such as NURBS surfaces for definition of complex geometry in shared virtual spaces. Our mathematical model allows to define objects by only providing coordinates of the section curves in 3-space. The resulting parametric functions allow fast calculation of the coordinates of the points on the surface of the objects. We devise an algorithm which evaluates the coefficients of these functions in real time. Given the small size of the resulting formulas and interactive rates for their calculation, we are able to efficiently use such PDE-based models for making virtual objects in shared virtual spaces. We describe the modeling framework and illustrate the proposed theoretical concepts with our function-based extension of VRML and X3D.

Pharmaceutical tablets have been the most dominant form for drug delivery and they need to be strong enough to withstand external stresses due to packaging and loading conditions before use. The strength of the produced tablets, which is characterised by their compressibility and compactibility, is usually deter-mined through a physical prototype. This process is sometimes quite expensive and time consuming. Therefore, simulating this process before hand can over-come this problem. A technique for shape modelling of pharmaceutical tablets based on the use of Partial Differential Equations is presented in this thesis. The volume and the sur-face area of the generated parametric tablet in various shapes have been es-timated numerically. This work also presents an extended formulation of the PDE method to a higher dimensional space by increasing the number of pa-rameters responsible for describing the surface in order to generate a solid tab-let. The shape and size of the generated solid tablets can be changed by ex-ploiting the analytic expressions relating the coefficients associated with the PDE method. The solution of the axisymmetric boundary value problem for a finite cylinder subject to a uniform axial load has been utilised in order to model a displace-ment component of a compressed PDE-based representation of a flat-faced round tablet. The simulation results, which are analysed using the Heckel model, show that the developed model is capable of predicting the compressibility of pharmaceutical powders since it fits the experimental data accurately. The opti-mal design of pharmaceutical tablets with particular volume and maximum strength has been obtained using an automatic design optimisation which is performed by combining the PDE method and a standard method for numerical optimisation.