Mathematics

My mathematical publications are listed below. My current research concentrates on variational discontinuity-preserving approaches to image processing, particularly a discontinuous transport equation for image interpolation, and higher-order regularisation techniques. The earlier work focusses on optimisation methods for finite-dimensional problems applied to clustering, and diff-convexity.

Journal articles

  1. K. Bredies, K. Kunisch and T. Valkonen, Properties of L1-TGV2: The one-dimensional case, Journal of Mathematical Analysis and Applications 398 (2013), 438–454, doi:10.1016/j.jmaa.2012.08.053, [pdf].
  2. T. Valkonen, K. Bredies and F. Knoll, Total generalised variation in diffusion tensor imaging, SIAM Journal on Imaging Sciences 6 (2013), 487–525, doi:10.1137/120867172, [pdf].
  3. T. Valkonen, Strong polyhedral approximation of simple jump sets, Nonlinear Analysis: Theory, Methods, & Applications 75 (2012), 3641–3671, doi:10.1016/j.na.2012.01.022, [pdf].
  4. ———, A primal-dual interior point method for diff-convex problems on symmetric cones, Optimization (2011), doi:10.1080/02331934.2011.585465, [pdf], published online.
  5. ———, Transport equation and image interpolation with SBD velocity fields, Journal de mathématiques pures et appliquées 95 (2011), 459–494, doi:10.1016/j.matpur.2010.10.010, [pdf].
  6. ———, Refined optimality conditions for differences of convex functions, Journal of Global Optimization 48 (2010), 311–321, doi:10.1007/s10898-009-9495-y, [pdf].
  7. T. Valkonen and T. Kärkkäinen, Clustering and the perturbed spatial median, Mathematical and Computer Modelling 52 (2010), 87–106, doi:10.1016/j.mcm.2010.01.018, [pdf].
  8. T. Valkonen, Optimal transportation networks and stations, Interfaces and Free Boundaries 11 (2009), 569–597, doi:10.4171/IFB/223, [pdf].
  9. T. Valkonen and T. Kärkkäinen, Continuous reformulations and heuristics for the Euclidean travelling salesperson problem, ESAIM: Control, Optimization and Calculus of Variations 15 (2009), doi:10.1051/cocv:2008056, [pdf].
  10. R. Glowinski, T. Kärkkäinen, T. Valkonen and A. Ivannikov, Nonsmooth SOR for L1-fitting: Convergence study and discussion of related issues, Journal of Scientific Computing 37 (2008), 103–138, doi:10.1007/s10915-008-9229-1, [pdf].
  11. T. Valkonen, Convergence of a SOR-Weiszfeld type algorithm for incomplete data sets, Numerical Functional Analysis and Optimization 27 (2006), 931–952, doi:10.1080/01630560600791213, (Errata in vol. 29, no 9–10, 2008).

Conference proceedings

  1. T. Valkonen, F. Knoll and K. Bredies, TGV for diffusion tensors: A comparison of fidelity functions, in: Journal of Inverse and Ill-posed problems special issue for IP:M&S 2012, Antalya, Turkey, 2012, doi:10.1515/jip-2013-0005, [pdf], published online.
  2. T. Valkonen and M. Liebmann, GPU-accelerated regularisation of large diffusion tensor volumes, in: Computing special issue for ESCO 2012, Pilsen, Czech Republic, 2012, doi:10.1007/s00607-012-0277-x, [pdf], to appear.
  3. K. Bredies and T. Valkonen, Inverse problems with second-order total generalized variation constraints, in: Proceedings of SampTA 2011 – 9th International Conference on Sampling Theory and Applications, Singapore, 2011, [pdf].

Thesis

  1. T. Valkonen, Diff-convex combinations of Euclidean distances: a search for optima, number 99 in Jyväskylä Studies in Computing, University of Jyväskylä, 2008, [pdf], Ph.D Thesis.

Submitted articles / technical reports

  1. T. Valkonen, A method for weighted projections to the positive definite cone, SFB-Report 2012-016, Karl-Franzens University of Graz (2012), [pdf].