Our latest publications - Computational Mathematics

• Ørke, Magnus Christie (2024). Highest waves for fractional Korteweg–De Vries and Degasperis–Procesi equations. Arkiv för matematik. ISSN 0004-2080. 62(1), p. 153–190. doi: 10.4310/ARKIV.2024.v62.n1.a9.
• Bendahmane, M.; Karlsen, Kenneth Aksel Hvistendahl & Mroué, F. (2024). Stochastic electromechanical bidomain model. Nonlinearity. ISSN 0951-7715. 37(7). doi: 10.1088/1361-6544/ad5132.
• Karlsen, Kenneth Aksel Hvistendahl & Rybalko, Ya. (2024). On the well-posedness of the Cauchy problem for the two-component peakon system in Ck∩Wk,1. Zeitschrift für Angewandte Mathematik und Physik. ISSN 0044-2275. 75(3). doi: 10.1007/s00033-024-02246-3.
• Floater, Michael S. (2024). Log-concavity of B-splines. Journal of Approximation Theory. ISSN 0021-9045. 300. doi: 10.1016/j.jat.2024.106042.
• Fjordholm, Ulrik Skre; Mæhlen, Ola Isaac Høgåsen & Ørke, Magnus Christie (2024). The particle paths of hyperbolic conservation laws. Mathematical Models and Methods in Applied Sciences. ISSN 0218-2025. doi: 10.1142/S0218202524500209. Full text in Research Archive
• Coclite, Giuseppe Maria; Karlsen, Kenneth Aksel Hvistendahl & Risebro, Nils Henrik (2024). A nonlocal Lagrangian traffic flow model and the zero-filter limit. Zeitschrift für Angewandte Mathematik und Physik. ISSN 0044-2275. 75(2). doi: 10.1007/s00033-023-02153-z.
• Galimberti, Luca & Karlsen, Kenneth Aksel Hvistendahl (2024). Well-Posedness of Stochastic Continuity Equations on Riemannian Manifolds. Chinese Annals of Mathematics. Series B. ISSN 0252-9599. 45(1), p. 81–122. doi: 10.1007/s11401-024-0005-9.
• Karlsen, Kenneth Aksel Hvistendahl; Kunzinger, Michael & Mitrovic, Darko (2024). A DYNAMIC CAPILLARITY EQUATION WITH STOCHASTIC FORCING ON MANIFOLDS: A SINGULAR LIMIT PROBLEM. Transactions of the American Mathematical Society. ISSN 0002-9947. 377(1), p. 85–166. doi: 10.1090/tran/9050.
• Karlsen, Kenneth Aksel Hvistendahl & Rybalko, Ya. (2024). On the well-posedness of a nonlocal (two-place) FORQ equation via a two-component peakon system. Journal of Mathematical Analysis and Applications. ISSN 0022-247X. 529(1). doi: 10.1016/j.jmaa.2023.127601.
• Galimberti, Luca; Holden, Helge; Karlsen, Kenneth Aksel Hvistendahl & Pang, Ho Cheung (2024). Global existence of dissipative solutions to the Camassa–Holm equation with transport noise. Journal of Differential Equations. ISSN 0022-0396. 387, p. 1–103. doi: 10.1016/j.jde.2023.12.021. Full text in Research Archive
• Mæhlen, Ola Isaac Høgåsen; Varholm, Kristoffer & Ehrnstrom, Mats (2024). On the precise cusped behaviour of extreme solutions to Whitham-type equations. Annales de l'Institut Henri Poincare. Analyse non linéar. ISSN 0294-1449.
• Manni, Carla; Speleers, Hendrik & Lyche, Tom (2024). A local simplex spline basis for C3 quartic splines on arbitrary triangulations. Applied Mathematics and Computation. ISSN 0096-3003. 462. doi: 10.1016/j.amc.2023.128330. Full text in Research Archive
• Christiansen, Snorre H; Gopalakrishnan, Jay; Guzmán, Johnny & Hu, Kaibo (2023). A discrete elasticity complex on three-dimensional Alfeld splits. Numerische Mathematik. ISSN 0029-599X. 156, p. 159–204. doi: 10.1007/s00211-023-01381-9. Full text in Research Archive
• Hoel, Håkon Andreas & Ragunathan, Sankarasubramanian (2023). Higher-order adaptive methods for exit times of Itô diffusions. IMA Journal of Numerical Analysis. ISSN 0272-4979. doi: 10.1093/imanum/drad077. Full text in Research Archive
• Berzins, Arturs (2023). Polyhedral Complex Extraction from ReLU Networks using Edge Subdivision. Proceedings of Machine Learning Research (PMLR). ISSN 2640-3498. 202, p. 2234–2244. Full text in Research Archive
• Tai, Xue-Cheng; Winther, Ragnar; Zhang, Xiaodi & Zheng, Weiying (2023). A uniform preconditioned for a Newton algorithm for total-variation minimisation and minimum-surface problems. SIAM Journal on Numerical Analysis. ISSN 0036-1429. 61(5), p. 2062–2083. doi: 10.1137/22M1512776. Full text in Research Archive
• Celledoni, Elena; Glöckner, Helge; Riseth, Jørgen Nilsen & Schmeding, Alexander (2023). Deep neural networks on diffeomorphism groups for optimal shape reparametrization. BIT Numerical Mathematics. ISSN 0006-3835. 63(4), p. 1–38. doi: 10.1007/s10543-023-00989-5. Full text in Research Archive Show summary

  One of the fundamental problems in shape analysis is to align curves or surfaces before computing geodesic distances between their shapes. Finding the optimal reparametrization realizing this alignment is a computationally demanding task, typically done by solving an optimization problem on the diffeomorphism group. In this paper, we propose an algorithm for constructing approximations of orientation preserving diffeomorphisms by composition of elementary diffeomorphisms. The algorithm is implemented using PyTorch, and is applicable for both unparametrized curves and surfaces. Moreover, we show universal approximation properties for the constructed architectures, and obtain bounds for the Lipschitz constants of the resulting diffeomorphisms.

• Tang, Hao (2023). On the stochastic Euler-Poincaré equations driven by pseudo-differential/multiplicative noise. Journal of Functional Analysis. ISSN 0022-1236. 285(9). doi: 10.1016/j.jfa.2023.110075. Full text in Research Archive
• Musch, Markus; Rupp, Andreas; Aizinger, Vadym & Knabner, Peter (2023). Hybridizable discontinuous Galerkin method with mixed-order spaces for non-linear diffusion equations with internal jumps. GEM - International Journal on Geomathematics. ISSN 1869-2672. 14(1). doi: 10.1007/s13137-023-00228-7. Full text in Research Archive
• Floater, Michael S. (2023). On a conjecture concerning interpolation by bivariate Bernstein polynomials. Journal of Approximation Theory. ISSN 0021-9045. 293. doi: 10.1016/j.jat.2023.105920. Full text in Research Archive
• Brualdi, Richard A. & Dahl, Geir (2023). Multi-alternating sign matrices. The Electronic Journal of Linear Algebra. ISSN 1537-9582. 39, p. 261–282. doi: 10.13001/ela.2023.7471. Full text in Research Archive
• Lyche, Tom ; Mørken, Knut Martin & Reif, Ulrich (2023). On the condition of cubic B-splines. Journal of Approximation Theory. ISSN 0021-9045. 289. doi: 10.1016/j.jat.2023.105883.
• Mæhlen, Ola Isaac Høgåsen & Xue, Jun (2023). One-sided Hölder regularity of global weak solutions of negative order dispersive equations. Journal of Differential Equations. ISSN 0022-0396. 364, p. 412–455. doi: 10.1016/j.jde.2023.03.048. Full text in Research Archive Show summary

  We prove global existence, uniqueness and stability of entropy solutions with L^2 initial data for a general family of negative order dispersive equations. These weak solutions are found to satisfy one-sided Hölder conditions whose coefficients decay in time. The latter result controls the height of solutions and further provides a way to bound the maximal lifespan of classical solutions from their initial data.

• Miao, Yingting; Rohde, Christian & Tang, Hao (2023). Well-posedness for a stochastic Camassa–Holm type equation with higher order nonlinearities. Stochastics and Partial Differential Equations: Analysis and Computations. ISSN 2194-0401. doi: 10.1007/s40072-023-00291-z. Full text in Research Archive
• Christiansen, Snorre H; Halvorsen, Tore Gunnar & Scheid, Claire (2023). Convergence of a discretization of the Maxwell–Klein–Gordon equation based on finite element methods and lattice gauge theory. Numerical Methods for Partial Differential Equations. ISSN 0749-159X. 39, p. 3271–3308. doi: 10.1002/num.23008. Full text in Research Archive
• Falk, Richard S. & Winther, Ragnar (2023). CONSTRUCTION OF POLYNOMIAL PRESERVING COCHAIN EXTENSIONS BY BLENDING. Mathematics of Computation. ISSN 0025-5718. 92(342), p. 1575–1594. doi: 10.1090/mcom/3819. Full text in Research Archive
• Tang, Hao & Yang, Anita (2023). Noise effects in some stochastic evolution equations: Global existence and dependence on initial data. Annales de l'I.H.P. Probabilites et statistiques. ISSN 0246-0203. 59(1), p. 378–410. doi: 10.1214/21-AIHP1241. Full text in Research Archive
• Tang, Hao & Wang, Zhi-An (2023). Strong solutions to a nonlinear stochastic aggregation-diffusion equation. Communications in Contemporary Mathematics. ISSN 0219-1997. doi: 10.1142/S0219199722500730.
• Dahl, Geir; Guterman, Alexander & Shteyner, Pavel (2023). Majorization for (0,±1)-matrices. Linear Algebra and its Applications. ISSN 0024-3795. 663, p. 200–221. doi: 10.1016/j.laa.2023.01.009. Full text in Research Archive
• Falk, Richard S. & Winther, Ragnar (2022). The Bubble Transform and the de Rham Complex. Foundations of Computational Mathematics. ISSN 1615-3375. doi: 10.1007/s10208-022-09589-1. Full text in Research Archive
• Holden, Helge; Karlsen, Kenneth Aksel Hvistendahl & Pang, Ho Cheung (2022). Global well-posedness of the viscous Camassa–Holm equation with gradient noise. Discrete and Continuous Dynamical Systems. Series A. ISSN 1078-0947. 43(2), p. 568–618. doi: 10.3934/dcds.2022163. Full text in Research Archive
• Chada, Neil; Hoel, Håkon Andreas; Jasra, Ajay & Zouraris, Georgios (2022). Improved Efficiency of Multilevel Monte Carlo for Stochastic PDE through Strong Pairwise Coupling. Journal of Scientific Computing. ISSN 0885-7474. 93(3). doi: 10.1007/s10915-022-02031-2. Full text in Research Archive
• Hoel, Håkon Andreas; Shaimerdenova, Gaukhar & Tempone, Raul (2022). Multi-index ensemble Kalman filtering. Journal of Computational Physics. ISSN 0021-9991. 470. doi: 10.1016/j.jcp.2022.111561. Show summary

  In this work we combine ideas from multi-index Monte Carlo and ensemble Kalman filtering (EnKF) to produce a highly efficient filtering method called multi-index EnKF (MIEnKF). MIEnKF is based on independent samples of four-coupled EnKF estimators on a multi-index hierarchy of resolution levels, and it may be viewed as an extension of the multilevel EnKF (MLEnKF) method developed by the same authors in 2020. Multi-index here refers to a two-index method, consisting of a hierarchy of EnKF estimators that are coupled in two degrees of freedom: time discretization and ensemble size. Under certain assumptions, when strong coupling between solutions on neighboring numerical resolutions is attainable, the MIEnKF method is proven to be more tractable than EnKF and MLEnKF. Said efficiency gains are also verified numerically in a series of test problems.

• Antun, Vegard (2022). Recovering Wavelet Coefficients from Binary Samples Using Fast Transforms. SIAM Journal on Scientific Computing. ISSN 1064-8275. 44(3), p. A1315–A1336. doi: 10.1137/21M1427188. Full text in Research Archive
• Fjordholm, Ulrik Skre; Musch, Markus & Risebro, Nils Henrik (2022). Well-Posedness and Convergence of a Finite Volume Method for Conservation Laws on Networks. SIAM Journal on Numerical Analysis. ISSN 0036-1429. 60(2), p. 606–630. doi: 10.1137/21M145001X. Full text in Research Archive
• Bendahmane, Mostafa & Karlsen, Kenneth Aksel Hvistendahl (2022). Martingale solutions of stochastic nonlocal cross-diffusion systems. Networks and Heterogeneous Media. ISSN 1556-1801. 17(5), p. 719–752. doi: 10.3934/nhm.2022024. Full text in Research Archive
• Opitz, Thomas; Bakka, Haakon; Huser, Raphaël & Lombardo, Luigi (2022). High-resolution Bayesian mapping of landslide hazard with unobserved trigger event. Annals of Applied Statistics. ISSN 1932-6157. 16(3), p. 1653–1675. doi: 10.1214/21-AOAS1561.
• Frid, Hermano; Karlsen, Kenneth Aksel Hvistendahl & Marroquin, Daniel (2022). Homogenization of stochastic conservation laws with multiplicative noise. Journal of Functional Analysis. ISSN 0022-1236. 283(9), p. 1–63. doi: 10.1016/j.jfa.2022.109620. Full text in Research Archive
• Holden, Helge; Karlsen, Kenneth Aksel Hvistendahl & Pang, Ho Cheung (2022). Strong solutions of a stochastic differential equation with irregular random drift. Stochastic Processes and their Applications. ISSN 0304-4149. 150, p. 655–677. doi: 10.1016/j.spa.2022.05.006. Full text in Research Archive
• Musch, Markus; Fjordholm, Ulrik Skre & Risebro, Nils Henrik (2022). Well-posedness theory for nonlinear scalar conservation laws on networks. Networks and Heterogeneous Media. ISSN 1556-1801. 17(1), p. 101–128. doi: 10.3934/nhm.2021025. Full text in Research Archive
• Lyche, Tom ; Manni, Carla & Speleers, Hendrik (2022). Construction of C^2 Cubic Splines on Arbitrary Triangulations. Foundations of Computational Mathematics. ISSN 1615-3375. doi: 10.1007/s10208-022-09553-z. Full text in Research Archive
• Christiansen, Snorre H & Hu, Kaibo (2022). Finite Element Systems for Vector Bundles: Elasticity and Curvature. Foundations of Computational Mathematics. ISSN 1615-3375. doi: 10.1007/s10208-022-09555-x. Full text in Research Archive
• Colbrook, Matthew J.; Antun, Vegard & Hansen, Anders Christian (2022). The difficulty of computing stable and accurate neural networks: On the barriers of deep learning and Smale's 18th problem. Proceedings of the National Academy of Sciences of the United States of America. ISSN 0027-8424. 119(12). doi: 10.1073/pnas.2107151119. Full text in Research Archive
• Fjordholm, Ulrik Skre & Lye, Kjetil Olsen (2022). Convergence Rates of Monotone Schemes for Conservation Laws for Data with Unbounded Total Variation. Journal of Scientific Computing. ISSN 0885-7474. 91. doi: 10.1007/s10915-022-01806-x. Full text in Research Archive
• Kirkeby, Adrian (2022). Exact and approximate solutions to the Helmholtz, Schrödinger and wave equation in R3 with radial data. Wave motion. ISSN 0165-2125. 108. doi: 10.1016/j.wavemoti.2021.102841. Full text in Research Archive
• Andrade, Enide; Ciardo, Lorenzo & Dahl, Geir (2022). Perron values and classes of trees. Linear Algebra and its Applications. ISSN 0024-3795. 639, p. 135–158. doi: 10.1016/j.laa.2022.01.005. Full text in Research Archive
• Chatterjee, Neelabja & Fjordholm, Ulrik Skre (2020). A convergent finite volume method for the Kuramoto equation and related nonlocal conservation laws. IMA Journal of Numerical Analysis. ISSN 0272-4979. 40(1), p. 405–421. doi: 10.1093/imanum/dry074.

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• Lyche, Tom ; Muntingh, Georg & Ryan, Øyvind (2020). Exercises in Numerical Linear Algebra and Matrix Factorizations. Springer Nature. ISBN 978-3-030-59788-7. 255 p.
• Lyche, Tom (2020). Numerical Linear Algebra and Matrix Factorizations. Springer Nature. ISBN 978-3-030-36467-0. 363 p.
• Ryan, Øyvind (2019). Linear Algebra, Signal Processing, and Wavelets - a Unified Approach. Python version. Springer Nature. ISBN 978-3-030-02939-5. 364 p.
• Ryan, Øyvind (2019). Linear Algebra, Signal Processing, and Wavelets - a Unified Approach. MATLAB version. Springer Nature. ISBN 978-3-030-01811-5. 360 p.

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• Pang, Ho Cheung (2023). Convergence of stochastic integrals with weakly convergent integrands: applications to SPDEs.
• Pang, Ho Cheung (2023). The stochastic compactness method in SPDEs.
• Pang, Ho Cheung (2022). Second order commutator estimates in renormalisation theory for SPDEs with gradient-type noise.
• Antun, Vegard (2022). Hvorfor hallusinerer kunstig intelligens?
• Antun, Vegard (2022). Can all AI systems be computed?
• Antun, Vegard (2022). Can all AI systems be computed? -- Mathematical paradox demonstrates the limits of AI.
• Antun, Vegard (2022). AI-generated hallucinations: Why do they happen when stable/accurate NNs exist?
• Antun, Vegard (2022). Still no free lunch – On AI generated hallucinations and the accuracy-stability trade-off in inverse problems.
• Antun, Vegard (2022). Robustness of Deep Learning in Image Reconstruction.
• Antun, Vegard; Colbrook, Matthew & Hansen, Anders Christian (2022). Mathematical paradoxes unearth the boundaries of AI. TheScienceBreaker. ISSN 2571-9262. doi: 10.25250/thescbr.brk653.
• Antun, Vegard; Colbrook, Matthew J. & Hansen, Anders Christian (2022). Proving existence is not enough: Mathematical paradoxes unravel the limits of neural networks in AI. SIAM News. ISSN 0036-1437. 55(4), p. 1–4.
• Esmaeili, Morteza; Antun, Vegard; Vettukattil, Muhammad Riyas; BANITALEBI, HASAN; Krogh, Nina R. & Geitung, Jonn Terje (2021). Evaluation of Automated Brain Tumor Localization by Explainable Deep Learning Methods.
• Antun, Vegard (2021). AI generated hallucinations in the sciences - On the stability accuracy trade-off in deep learning.
• Antun, Vegard & Colbrook, Matthew (2021). On the barriers of AI and the trade-off between stability and accuracy in deep learning. Show summary

  Deep learning (DL) has had unprecedented success and is now entering scientific computing with full force. This presents exciting new opportunities, and there is currently a profound optimism on the impact of DL and AI. However, there is also growing evidence across numerous applications that modern DL has an Achilles' heel: instability. The current situation in AI is comparable to the situation in mathematics in the early 20th century and Hilbert's optimism (e.g. his 10th problem) regarding provability and algorithms. Hilbert's optimism was turned upside down by Gödel and Turing, who established limitations on what mathematics can prove and which problems computers can solve (however, without limiting the impact of mathematics and computer science), leading to modern logic and computer science. Similarly, the present situation calls for a program on the foundations of AI and DL to discover the boundaries of what can be achieved (Smale's 18th problem). This series of talks will address the stability and accuracy trade-off, and the resulting barriers of AI, in deep learning applications to imaging. For example, we present basic standard problems in scientific computing where one can prove the existence of stable neural networks with great approximation qualities; however, such networks are not computed by the current training approaches. In fact, no algorithm (even randomised) can compute such a network to even 1-digit of accuracy with a probability greater than 1/2. We discuss suitable stability tests in imaging (e.g. adversarial examples and their construction), current methods aimed at fixing instabilities and improving robustness, impossibility results in computational optimisation and inverse problems, the popular method of algorithm unrolling, and a unified theory for compressed sensing and DL which leads to sufficient conditions for the existence of algorithms that compute stable and accurate (e.g. exponential convergence in the number of layers) neural networks. The results and methods discussed point towards a potentially vast classification theory, which goes beyond imaging, and describes conditions under which an algorithm can compute (stable) neural networks with a given accuracy.

• Antun, Vegard & Seres, Silvija (2021). #1100: BIGDATA: Vegard Antun: AI & Deep learning som applikasjon og verktøy. [Internet]. Podcast.
• Antun, Vegard; Gottschling, Nina; Hansen, Anders Christian & Adcock, Ben (2021). Deep learning in scientific computing: Understanding the instability mystery. SIAM News. ISSN 0036-1437. 54(2), p. 3–5.
• Antun, Vegard (2019). On instabilities of deep learning in image reconstruction - Does AI come at a cost?
• Antun, Vegard (2019). On instabilities of deep learning in image reconstruction - Does AI come at a cost?
• Antun, Vegard (2019). On Instabilities of Deep Learning in Image Reconstruction - Part I.
• Antun, Vegard (2019). On instabilities of deep learning in image reconstruction.
• Lyche, Tom ; Merrien, Jean-Louis & Sauer, Tomas (2019). Polynomial Splines with Arbitrary Smoothness on Plane Triangulations. Show summary

  Given a triangulation on a polygonal domain. We consider polynomial splines of global smoothness $r$ on a refined triangulation, where each triangle is split into sub patches. Examples are the WangShi split and the Powell-Sabin 6 and 12 splits. We propose a partition of unity simplex spline basis of degree 3r and smoothness r on the Clough-Tocher split. This is joint work with Jean-Louis Merrein and Tomas Sauer.

• Lyche, Tom ; Merrien, Jean-Louis & Sauer, Tomas (2019). Simplex--Splines on the Clough--Tocher Split with Arbitrary Smoothness,
• Lyche, Tom ; Manni, Carla & Speleers, Hendrik (2019). Interesting Splits II.
• Lyche, Tom ; Manni, Carla & Speleers, Hendrik (2019). Interesting splits.
• Ruf, Adrian Montgomery (2019). Second-order numerical methods for nonlocal conservation laws.
• Ruf, Adrian Montgomery (2019). The optimal convergence rate of monotone schemes for conservation laws in the Wasserstein distance.
• Ruf, Adrian Montgomery (2019). A second-order scheme for a class of nonlocal conservation laws.
• Tellefsen, Cathrine Wahlstrøm & Mørken, Knut Martin (2019). Programmering i naturfag.
• Coclite, Giuseppe Maria; di Ruvo, Lorenzo & Karlsen, Kenneth Hvistendahl (2018). The initial-boundary-value problem for an Ostrovsky- Hunter type equation, Non-linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis: The Helge Holden Anniversary Volume. European Mathematical Society (EMS) Press. ISSN 978-3-03719-186-6. p. 97–109.
• Tellefsen, Cathrine Wahlstrøm & Mørken, Knut Martin (2018). ProFag: realfaglig programmering.
• Tellefsen, Cathrine Wahlstrøm & Mørken, Knut Martin (2018). Programmering er mer enn koding.
• Tellefsen, Cathrine Wahlstrøm & Mørken, Knut Martin (2018). Programmering i naturfaget.
• Mørken, Knut Martin (2018). Computing in Science Education.
• Mørken, Knut Martin (2018). Emneutvikling og systematisk integrering av generiske ferdigheter i utdanningsprogrammene.
• Mørken, Knut Martin (2018). Utdanning og utveksling .
• Mørken, Knut Martin (2018). Programmering i skolen: muligheter og utfordringer.
• Mørken, Knut Martin (2018). Programmering i alle fag!
• Mørken, Knut Martin (2018). Om fagfornyelsen i matematikk.
• Mørken, Knut Martin & Sølna, Hanne (2018). An Evolving Culture for Learning in Practice.
• Mørken, Knut Martin (2018). Educational Development: Computing in Science Education in a Wider Perspective.
• Mørken, Knut Martin (2018). Some reflections on how computing can enhance the learning of mathematics.
• Solem, Susanne; Fjordholm, Ulrik Skre & Carrillo, José A (2018). A second-order numerical method for the aggregation equations.
• Muntingh, Agnar Georg Peder & Lyche, Tom (2018). B-spline-like simplex spline bases on the Powell-Sabin 12-split.
• Lyche, Tom (2018). Tchebycheffian B-splines. Show summary

  We give a short survey of Tchebycheffian Splines and Tchebycheffian B-splines. These piecewise functions are natural generalizations of polynomial splines and polynomial B-splines. Each segment of the spline belongs to a piecewise Ex- tended Complete Tchebycheff (ECT)-space. We discuss various ways to define them. And chose a recursive definition, allowing different spaces on the seg- ments. Several examples will be given to illustrate the concepts.

• Lyche, Tom (2018). Tchebycheffian Splines and Tchebycheffian B-splines . Show summary

  We give a survey of Tchebycheffian Splines and Tchebycheffian B-splines. These piecewise functions are natural generalizations of polynomial splines and polynomial B-splines. Each segment of the spline belongs to a piecewise Extended Complete Tchebycheff (ECT)-space. An ECT-space can also be considered as the null space of a linear differential operator with variable coefficients defined on a suitable interval. ECT-spaces can either be spanned by a set of basis functions that are a natural generalization of the polynomial power basis or an alternative basis with similar properties to the Bernstein basis for polynomials. We define Tchebycheffian B-splines recursively, allowing different spaces on the segments. If time allows we also consider Tchebycheffian divided differences. Several examples will be given to illustrate the concepts.

• Ruf, Adrian Montgomery (2018). A second-order method for nonlocal conservation laws.
• Dahl, Geir; Andrade, Enide & Ciardo, Lorenzo (2018). Combinatorial Perron Parameters and Trees.
• Ruf, Adrian Montgomery (2018). The Ostrovsky-Hunter equation with Dirichlet boundary conditions.
• Antun, Vegard (2020). Stability and accuracy in compressive sensing and deep learning. Universitetet i Oslo. Full text in Research Archive Show summary

  There are currently two paradigm shifts happening in society and scientific computing: (1) Artificial Intelligence (AI) is replacing humans in problem solving, and, (2) AI is replacing the standard algorithms in computational science and engineering. Since reliable numerical calculations are paramount, algorithms for computational science are traditionally based on two pillars: accuracy and stability. Notably, this is true for image reconstruction, which is a mainstay of computational science, providing fundamental tools in medical, scientific and industrial imaging. In this thesis, we demonstrate that the stability pillar is typically absent in current deep learning and AI-based algorithms for image reconstruction, and we present a solution to why this phenomenon occurs for AI-based methods applied both to image reconstruction and to classification in general. This raises two fundamental questions: how reliable are such algorithms when applied in society, and do AI-based algorithms have the unavoidable Achilles heel of instability? We investigate these phenomena, and we introduce a framework designed to demonstrate, investigate and ultimately answer these fundamental questions.

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