CARMA RETREAT. 19 July An overview of Pure mathematical research within CARMA. Brailey Sims

Transcription

1 CARMA RETREAT 1 19 July 2011 An overview of Pure mathematical research within CARMA Brailey Sims

2 Pure mathematics in CARMA is represented by: 2 Principal researchers - Laureate Professor Jonathan Borwein (Director) Dr Murray Elder Dr Jeffery Hogan Associate Professor Brailey Sims Professor George Willis Associate Professor Wadim Zudilin

3 Conjoints 3 Prof. Brian Alspach (Newcastle) Prof. Richard Brent (ANU) Prof. Roger Eggleton (Newcastle) A/Prof. John Giles (Emeritus, Newcastle) Dr James MacDougall (Emeritus, Newcastle) Prof. Sinai Robins (Nanyang Technological University) Prof. Steven Wright (University of Wisconsin)

4 Associates 4 Dr Miroslav Bacak (former CARMA Post Doc) Dr O-Yeat Chan (former CARMA Post Doc) Prof. Andrew Eberhard (RMIT) Prof.Mirka Miller (Computer Science, Newcastle) Dr Joseph Ryan (Computer Science, Newcastle) Dr Warren Moors (University of Auckland) Post Doctoral fellows Dr Judy-Anne Osborn Dr Francisco Aragón Artacho

5 5 Plus six Graduate Research Higher Degree students

6 I now briefly discuss some current research programs. Although the focus will be on the pure mathematics we are very conscious of potential applications and intent on pursuing them. 6 Firstly I ll look at work by Jeff Hogan, who unfortunately is unable to be here today. Then I ll look at some of the work I m currently involved in.

7 Hyper-complex signal processing 7 Jeff Hogan (Newcastle), Mark Craddock (UTS), Andrew Morris (PhD student) Fourier transforms and other signal processing tools such as wavelet analysis and various algorithms of time-frequency analysis, have proven to be spectacularly successful in the analysis and processing of single-channel signals such as an acoustic signal.

8 Multichannel signals such as a colour image have been analysed by applying these tools to each channel separately, thereby ignoring the often significant interchannel correlations and so yielding less-than-optimal algorithms. 8 Rather than modelling a multichannel signal as an ensemble of one-channel signals, we view it as taking values in an appropriate Clifford algebra, the simplest non-trivial example of which (and the one which arises when considering colour images) is the familiar algebra of quaternions.

9 Using the Clifford generalisation of the Hermite operator, it is possible to define a natural Clifford Fourier transfrom, just as can be done in the one-dimensional case. However, to use it in practice, we need to know its integral kernel. The two-dimensional (quaternionic) kernel is known and recently it was shown that all evendimensional kernels are explicitly computable. 9 Work on the odd dimensional case continues. Sampling in Paley-Wiener and shift-invariant spaces

10 10 Non-commutativity of the algebra in which the signals and kernels take values means the usual convolution theorem for the Fourier transform cannot apply. We have developed an effective and elegant substitute and used it to investigate analogues of the important operators of harmonic analysis such as the Hilbert transform.

11 Bandlimitedness is a paradime of signal analysis. That it is compatible with the conversion of an analog signal to a sequence of digital samples is the miracle of the Shannon sampling theorem. 11 But, the sampling theorem is not without flaws; computing a local representation of a bandlimited signal from its uniform samples requires a knowledge of all the samples.

12 A replacement for spaces of bandlimited signals are shift-invariant spaces (spaces invariant under integral shifts). Algorithms exploiting the redundant information resulting from oversampling have been developed to solve the synchronisation problem (that is, to determine the shift). 12

13 Nonlinear analysis: Metric fixed point theory 13 Borwein, sims, RHD students and colleagues

14 A (nonlinear) mapping normed spaces and is nonexpansive if, between subsets of 14 Isometries such as the right shift operator, strict contractions, closest point projections onto closed convex sets in a Hilbert space, and resolvents of monotone (accretive) mappings are all nonexpansive. Besides representing a border case between the Banach contraction mapping principle and Brouwer s / Schauder s theorem, the fixed point theory of nonexpansive maps is significant because of its close connections with the theory of accretive/monotone operators, variational inequalities and hence optimization and nonlinear analysis in general.

15 We say the Banach space X has the: fpp if every nonempty norm closed and bounded convex subset C has the fpp, 15 τ - fpp if every nonempty τ relatively compact, norm closed and bounded, convex subset C has the fpp, where, τ is a linear topology on X. Typically τ is the weak (ω) topolgy on X, or when X is the dual of a given Banach space, X = Y*, the weak* [ω* = σ(x,y) ] topology, or in the case of many function spaces the topology of local convergence in measure. We seek widely applicable, easily verified conditions which ensure a Banach space has the fpp or the τ - fpp, or can be equivalently renormed to have these properties.

16 We also study 16 Effective iterative schemes for approximating fixed points of nonexpansive maps Leray-Schauder alternatives for nonexpansive non-self mappings on closed convex sets. The structure of the fixed point set, Fix(T), and approximate fixed point sets of a nonexpansive map T, building on the seminal work of Bruck, 73. Common fixed points of families, particularlysemi-groups, of nonexpansive maps. Extensions to classes of mappings wider than nonexpansive (asymptotically nonexpansive, uniformly Lipschitzian, and to multifunctions. The stability of the (ω - ) fpp under renormings.

17 Alternating projection and Douglas Rachford type algorithms for finding feasible points under convex constraints. 17 A B Norm convergent to a point in A B Von Neumann s alternating projection algorithm (1932)

18 Which may be extended to convex sets and parallelized; Divide and Concur, so that without loss of generality one need only consider the case of two sets 18 A x B

19 19 Restricting to Hilbert space: the Douglas - Rachford algorithm (reflectreflect and average) for finding a point in the intersection of two closed convex sets A, B is an over relaxation of alternating projections.

20 x 20 P A x R A x O P A x - x

21 21 A T B

22 The Douglas-Rachford algorithm is the iterative scheme 22 where, Provided A and B are convex and have nonempty intersection the Douglas - Rachford algorithm was shown to converge weakly by Lions and Mercier using the fact that closest point projections are firmly nonexpansive and consequently reflections are nonexpansive and hence T is also nonexpansive and asymptotically regular. That its iterates weakly converge to a fixed point of T then followed by a well known result initially due to Opial. But, what happens when one or more of the sets is non-convex? Borwein, Sims, Skerrett

23 Veit Esler and a 25-by-25 Sudoku 23 For this Veit Elser (Cornell 2006) used divide and concur in a discrete space of = dimensions

24 Alternating projections and Douglas Rachford (which proves more stable and to have better convergence properties) are very frequently successfully applied to solve inverse problems in many areas including protein folding and signal/image reconstruction problems 24 For example; the phase-reconstruction problem which arises in areas such as tomography and optical image reconstruction. Estimate a signal, x in L := L 2 [R N, C], from a priori knowledge that the support of x is contained in some subset D of R N ; x in A := {y in L : y 1 R N \ D = 0}, an affine constraint And the modulus of its Fourier transform; x in C := { y in L : F(y) = m}, a non-convex constraint. So we seek, x in A C,

25 The algorithm has been routinely used by engineers to solve a variety of inverse problems, including image reconstruction problems; for example, those encountered in modern telescope systems, including Hubble. 25 Unfortunately in a great many of these situations one or more of the constraints involved are non-convex. Typically B is the boundary of a convex set; for instance, as with Hubble, B = {x: Mx = b}. The situation begs for a theoretical under pinning which is currently missing.

26 We illustrate with the following prototypical situation. 26 Let A be the line y = h and let B be the unit circle in 2-dimensional Hilbert space (more generally A is a line and B the unit sphere in a finite dimensional Hilbert space). Here, and where

27 What do we know, even in this simple case? Well? If A and B fail to intersect the iterates diverge 27

28 If A a is tangent to b they converge, but not to a feasible point. 28

29 When A and B intersect, applying Perron s theorem from the stability theory for difference equations, at each of the two feasible points we have local (exponential) convergence of the iterates. But, computer experiments clearly show that the scheme converges from any starting point in either open half-plane ( x > 0, or x < 0). 29

30 30 Computer experimentation (using a combination of Maple and Cinderella) also demonstrates the effectiveness of the algorithm in a much wider variety of situations involving different sets and configurations - so where s the global theory??????

31 (CONVEX) ANALYSIS IN THE ABSENCE OF LINEARITY 31 Bacak, Borwein, Sims. Searston Metric spaces allow applications in situations where no natural linear structure is present. Instances of where this occurs include: State spaces it often makes sense to measure how near one state is to another, but adding or scaling a state may make no sense (E.g. the position states of a robot ). Some cognitive models of recognition the type of an object is identified with that of its nearest prototype; Voronoi cells/tessellations.

32 CAT(0) spaces [So named by Gromov in honour of Cartan, Alexandrov, and Toponogov], 32 Provide a rich and natural setting for this endeavour. Many Hilbert space results and techniques remain true in such spaces. They admit: a rich fixed point theory [W. Art Kirk] nonlinear optimization theory alternating projection algorithms a metric analogue of convex analysis

33 33 CAT(0) Spaces x 2 x 2 ' x 1 x 3 x 1 ' x 3 '

34 34 CAT(0) Spaces p x 2 p' x 2 ' x 1 q x 3 x 1 ' q' x 3 '

35 35 A CAT(0) space is a geodesic (= Menger convex) metric space in which every triangle satisfies the CAT(0) inequality. We will also usually assume the space is complete, in which case it is sometimes referred to as a Hadamard space.

36 Metric spaces 36 Banach Spaces Hilbert spaces CAT(0) spaces Hyperconvex metric spaces

37 37 [x,z] x z y

38 38 X is a star-like subset of a Hilbert space

39 One of the difficulties in extending results from Hilbert spaces into CAT(0) spaces is the lack of a weak topology, and hence weak-compactness. However many arguments involving weakcompactness can be replaced by asymptotic centre arguments because of the following. 39 For any bounded sequence (x n ), its asymptotic radius about x is, its asymptotic radius is, and, its asymptotic centre is, Proposition: In any complete CAT(0) space the asymptotic centre of any bounded sequence consists of a single point.

40 40 Definition: Δ- converges to x,, if x is the unique asymptotic centre of every subsequence of Proposition: Every bounded sequence of a complete CAT(0) space has a Δ- convergent subsequence. Note: In a Hilbert space Δ- convergence and weak convergence coincide. This is a simple consequence of closed balls being weak compact and Opial s property: for any weakly null sequence and we have,

41 Δ- convergence is also equivalent to Sosov s notion of Ø-convergence [2004]: A sequence (x n ) is Ø-convergent to x if for every geodesic G through x we have, 41 where is the closest point projection onto G. P G (x n ) x φ G (x n ) G x n