Abstract:
We present an abstract framework for treating the theory of well-posedness of solutions to abstract parabolic partial differential equations on evolving Hilbert spaces. This theory is applicable to variational formulations of PDEs on evolving spatial domains including moving hypersurfaces. We formulate an appropriate time derivative on evolving spaces called the material derivative and define a weak material derivative in analogy with the usual time derivative in fixed domain problems; our setting is abstract and not restricted to evolving domains or surfaces. Then we show well-posedness to a certain class of parabolic PDEs under some assumptions on the parabolic operator and the data.

Abstract:
an iterative method based on picard's approach to odes' initial-value problems is proposed to solve first-order quasilinear pdes with matrix-valued unknowns, in particular, the recently discovered variational pdes for the missing boundary values in hamilton equations of optimal control. as illustrations the iterative numerical solutions are checked against the analytical solutions to some examples arising from optimal control problems for nonlinear systems and regular lagrangians in finite dimension, and against the numerical solution obtained through standard mathematical software. an application to the (n + 1)-dimensional variational pdes associated with the n-dimensional finite-horizon time-variant linear-quadratic problem is discussed, due to the key role the lqr plays in two-degrees-of freedom control strategies for nonlinear systems with generalized costs. mathematical subject classification: primary: 35f30; secondary: 93c10.

Abstract:
We introduce a new concept of sparsity for the stochastic elliptic operator $-{\rm div}\left(a(x,\omega)\nabla(\cdot)\right)$, which reflects the compactness of its inverse operator in the stochastic direction and allows for spatially heterogeneous stochastic structure. This new concept of sparsity motivates a heterogeneous stochastic finite element method ({\bf HSFEM}) framework for linear elliptic equations, which discretizes the equations using the heterogeneous coupling of spatial basis with local stochastic basis to exploit the local stochastic structure of the solution space. We also provide a sampling method to construct the local stochastic basis for this framework using the randomized range finding techniques. The resulting HSFEM involves two stages and suits the multi-query setting: in the offline stage, the local stochastic structure of the solution space is identified; in the online stage, the equation can be efficiently solved for multiple forcing functions. An online error estimation and correction procedure through Monte Carlo sampling is given. Numerical results for several problems with high dimensional stochastic input are presented to demonstrate the efficiency of the HSFEM in the online stage.

Abstract:
First-order jet bundles can be put at the foundations of the modern geometric approach to nonlinear PDEs, since higher-order jet bundles can be seen as constrained iterated jet bundles. The definition of first-order jet bundles can be given in many equivalent ways - for instance, by means of Grassmann bundles. In this paper we generalize it by means of flag bundles, and develop the corresponding theory for higher-oder and infinite-order jet bundles. We show that this is a natural geometric framework for the space of Cauchy data for nonlinear PDEs. As an example, we derive a general notion of transversality conditions in the Calculus of Variations.

Abstract:
A wide variety of communication and information technologies (C&IT) is now available, offering education a broad range of potential benefits, be they educational (Mapp, 1994; Lewis and Merton, 1996; HEFCE, 1997a), economic (HEFCE, 1997a), or in terms of competitiveness with other universities in an increasingly global market (Maier et al, 1997). The uptake and use of these resources is patchy at best (Laurillard et al, 1993; Lewis and Merton, 1996). This mismatch between potential and use has been seen as increasingly important, and incentives and recommendations are leading to an increasing use of C&IT, as illustrated by the recommendations of the recent Dealing report (Dearing et al, 1997) and the priorities of the TLT programme (HEFCE, 1997b). Concerns have been voiced, however, that the push towards a wider embedding of C&IT in education may ignore issues of the appropriate uses of these resources. What is needed is a convincing and practical pedagogically-driven (as opposed to technology-driven) methodology for integrating C&IT into courses.

Abstract:
We propose a global minimal embedding of the Schwarzschild theory in a five-dimensional flat space by using two surfaces. Covariant field equations are deduced for the gravitational forces.

Abstract:
Numerical continuation calculations for ordinary differential equations (ODEs) are, by now, an established tool for bifurcation analysis in dynamical systems theory as well as across almost all natural and engineering sciences. Although several excellent standard software packages are available for ODEs, there are - for good reasons - no standard numerical continuation toolboxes available for partial differential equations (PDEs), which cover a broad range of different classes of PDEs automatically. A natural approach to this problem is to look for efficient gluing computation approaches, with independent components developed by researchers in numerical analysis, dynamical systems, scientific computing and mathematical modelling. In this paper, we shall study several elliptic PDEs (Lane-Emden-Fowler, Lane-Emden-Fowler with microscopic force, Caginalp) via the numerical continuation software pde2path and develop a gluing component to determine a set of starting solutions for the continuation by exploting the variational structures of the PDEs. In particular, we solve the initialization problem of numerical continuation for PDEs via a minimax algorithm to find multiple unstable solution. Furthermore, for the Caginalp system, we illustrate the efficient gluing link of pde2path to the underlying mesh generation and the FEM MatLab pdetoolbox. Even though the approach works efficiently due to the high-level programming language and without developing any new algorithms, we still obtain interesting bifurcation diagrams and directly applicable conclusions about the three elliptic PDEs we study, in particular with respect to symmetry-breaking. In particular, we show for a modified Lane-Emden-Fowler equation with an asymmetric microscopic force, how a fully connected bifurcation diagram splits up into C-shaped isolas on which localized pattern deformation appears towards two different regimes.

Abstract:
Many computer vision and image processing problems can be posed as solving partial differential equations (PDEs). However, designing PDE system usually requires high mathematical skills and good insight into the problems. In this paper, we consider designing PDEs for various problems arising in computer vision and image processing in a lazy manner: \emph{learning PDEs from real data via data-based optimal control}. We first propose a general intelligent PDE system which holds the basic translational and rotational invariance rule for most vision problems. By introducing a PDE-constrained optimal control framework, it is possible to use the training data resulting from multiple ways (ground truth, results from other methods, and manual results from humans) to learn PDEs for different computer vision tasks. The proposed optimal control based training framework aims at learning a PDE-based regressor to approximate the unknown (and usually nonlinear) mapping of different vision tasks. The experimental results show that the learnt PDEs can solve different vision problems reasonably well. In particular, we can obtain PDEs not only for problems that traditional PDEs work well but also for problems that PDE-based methods have never been tried before, due to the difficulty in describing those problems in a mathematical way.

Abstract:
In the first part we present a generalized implicit function theorem for abstract equations of the type $F(\lambda,u)=0$. We suppose that $u_0$ is a solution for $\lambda=0$ and that $F(\lambda,\cdot)$ is smooth for all $\lambda$, but, mainly, we do not suppose that $F(\cdot,u)$ is smooth for all $u$. Even so, we state conditions such that for all $\lambda \approx 0$ there exists exactly one solution $u \approx u_0$, that $u$ is smooth in a certain abstract sense, and that the data-to-solution map $\lambda \mapsto u$ is smooth. In the second part we apply the results of the first part to time-periodic solutions of first-order hyperbolic systems of the type $$ \partial_tu_j + a_j(x,\lambda)\partial_xu_j + b_j(t,x,\lambda,u) = 0, \; x\in(0,1), \;j=1,\dots,n $$ with reflection boundary conditions and of second-order hyperbolic equations of the type $$ \partial_t^2u-a(x,\lambda)^2\partial^2_xu+b(t,x,\lambda,u,\partial_tu,\partial_xu)=0, \; x\in(0,1) $$ with mixed boundary conditions (one Dirichlet and one Neumann). There are at least two distinguishing features of these results in comparison with the corresponding ones for parabolic PDEs: First, one has to prevent small divisors from coming up, and we present explicit sufficient conditions for that in terms of $u_0$ and of the data of the PDEs and of the boundary conditions. And second, in general smooth dependence of the coefficient functions $b_j$ and $b$ on $t$ is needed in order to get smooth dependence of the solution on $\lambda$, this is completely different to what is known for parabolic PDEs.

Abstract:
A general procedure for constructing conservative numerical integrators for time dependent partial differential equations is presented. In particular, linearly implicit methods preserving a time discretised version of the invariant is developed for systems of partial differential equations with polynomial nonlinearities. The framework is rather general and allows for an arbitrary number of dependent and independent variables with derivatives of any order. It is proved formally that second order convergence is obtained. The procedure is applied to a test case and numerical experiments are provided.