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"description": "The Johann Radon Institute for Computational and Applied Mathematics (RICAM) of the Austrian Academy of Sciences focuses on basic research in applied mathematics.\nThe institute is based in Austria’s third-largest city and industrial hub, Linz.\nResearchers from all around the globe collaborate on common core areas in mathematical modeling, simulation, inverse problems, and optimization.\nRICAM stands for excellence in research, as can be seen from a high level of publications and the popularity of the Institute’s Special Semesters within the academic community.\nThe work groups at RICAM provide a broad field of expertise over a whole range of different subjects, and together they create an exciting atmosphere to carry out research in applied mathematics.",
"identifiers": [],
"references": [
{
"type": "Article",
"id": "bib-bib1",
"authors": [],
"title": "\nB. Azmi, D. Kalise and K. Kunisch,\nOptimal feedback law recovery by gradient-augmented sparse polynomial regression.\nJournal of Machine Learning Research, 22, 1–32 (2021)\n"
},
{
"type": "Article",
"id": "bib-bib2",
"authors": [],
"title": "\nJ. Berner, M. Dablander and P. Grohs,\nNumerically solving parametric families of high-dimensional Kolmogorov partial differential equations via deep learning.\nIn Advances in Neural Information Processing Systems,\nvolume 33, pages 16615–16627,\nedited by H. Larochelle, M. Ranzato, R. Hadsell, M. F. Balcan and H. Lin,\nCurran Associates, Inc. (2020)\n"
},
{
"type": "Article",
"id": "bib-bib3",
"authors": [],
"title": "\nJ. Berner, P. Grohs and A. Jentzen,\nAnalysis of the generalization error: empirical risk minimization over deep artificial neural networks overcomes the curse of dimensionality in the numerical approximation of Black–Scholes partial differential equations.\nSIAM J. Math. Data Sci.2, 631–657 (2020)\n"
},
{
"type": "Article",
"id": "bib-bib4",
"authors": [],
"title": "\nE. F. Brunner, I. Shatokhina, M. F. Shirazi, W. Drexler, C. K. F. Hitzenberger, R. A. Leitgeb, R. Ramlau and M. Pircher,\nIn-vivo demonstration of AO-OCT with a 3-sided pyramid wavefront sensor\n(Conference Presentation).\nIn Ophthalmic Technologies XXX, volume 11218,\nedited by F. Manns, A. Ho and P. G. Söderberg,\nInternational Society for Optics and Photonics, SPIE (2020)\n"
},
{
"type": "Article",
"id": "bib-bib5",
"authors": [],
"title": "\nR. Davies, J. Alves, R. Ramlau, R. Wagner, V. Hutterer, J. Niebsch et al.,\nThe MICADO first light imager for the ELT: overview, operation, simulation.\nIn Proceedings of the SPIE Astronomical Telescopes and Instrumentation Conference, Vol. 10702 (Austin, Texas, June 2018): Ground-based and Airborne Instrumentation for Astronomy VII, edited by C. J. Evans, L. Simard and H. Takami, SPIE, 570–581 (2018)\n"
},
{
"type": "Article",
"id": "bib-bib6",
"authors": [],
"title": "\nS. Dolgov, D. Kalise and K. K. Kunisch,\nTensor decomposition methods for high-dimensional Hamilton–Jacobi–Bellman equations.\nSIAM J. Sci. Comput.43, A1625–A1650 (2021)\n"
},
{
"type": "Article",
"id": "bib-bib7",
"authors": [],
"title": "\nA. Effland, E. Kobler, K. Kunisch and T. Pock,\nVariational networks: an optimal control approach to early stopping variational methods for image restoration.\nJ. Math. Imaging Vision62, 396–416 (2020)\n"
},
{
"type": "Article",
"id": "bib-bib8",
"authors": [],
"title": "\nH. W. Engl and R. Ramlau.\nRegularization of inverse problems.\nIn Encyclopedia of Applied and Computational Mathematics,\nedited by B. Engquist,\nSpringer, Heidelberg (2015)\n"
},
{
"type": "Article",
"id": "bib-bib9",
"authors": [],
"title": "\nP. Grohs and L. Herrmann,\nDeep neural network approximation for high-dimensional parabolic Hamilton–Jacobi–Bellman equations.\narXiv:2103.05744 (2021)\n",
"url": "https://arxiv.org/abs/2103.05744"
},
{
"type": "Article",
"id": "bib-bib10",
"authors": [],
"title": "\nP. Grohs, F. Hornung, A. Jentzen and P. Von Wurstemberger.\nA proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black–Scholes partial differential equations.\nMemoirs of the AMS, to appear (2020)\n"
},
{
"type": "Article",
"id": "bib-bib11",
"authors": [],
"title": "\nV. Hutterer, R. Ramlau and I. Shatokhina,\nReal-time adaptive optics with pyramid wavefront sensors: part I.\nA theoretical analysis of the pyramid sensor model.\nInverse Problems35, 045007 (2019)\n"
},
{
"type": "Article",
"id": "bib-bib12",
"authors": [],
"title": "\nJ. A. Iglesias and G. Mercier,\nInfluence of dimension on the convergence of level-sets in total variation regularization.\nESAIM Control Optim. Calc. Var.26, Paper No. 52 (2020)\n"
},
{
"type": "Article",
"id": "bib-bib13",
"authors": [],
"title": "\nK. Kunisch and D. Walter,\nSemiglobal optimal feedback stabilization of autonomous systems via deep neural network approximation.\nESAIM Control Optim. Calc. Var.27, Paper No. 16 (2021)\n"
},
{
"type": "Article",
"id": "bib-bib14",
"authors": [],
"title": "\nR. Ramlau and O. Scherzer (eds.),\nThe Radon transform. The First 100 Years and Beyond.\nRadon Ser. Comput. Appl. Math. 22,\nDe Gruyter, Berlin (2019)\n"
},
{
"type": "Article",
"id": "bib-bib15",
"authors": [],
"title": "\nM. Scherbela, R. Reisenhofer, L. Gerard, P. Marquetand and P. Grohs,\nSolving the electronic Schrödinger equation with weight-sharing deep neural networks for multiple nuclear geometries.\narXiv:2105.08351 (2021)\n",
"url": "https://arxiv.org/abs/2105.08351"
}
],
"title": "RICAM, the Johann Radon Institute for Computational and Applied Mathematics",
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" (RICAM) was founded in 2003 by Heinz Engl, who is now acting as the Rector of the University of Vienna, with the goal of establishing an internationally visible and successful research institute in the field of applied mathematics.\nSince then, RICAM has carried out basic research in computational and applied mathematics according to highest international standards and has emphasized interdisciplinary cooperation between its workgroups and with institutions with similar scope and universities around the globe.\nThe researchers also cooperate with other disciplines, in particular within the framework of Special Semesters on topics of major current interest.\nOne of the institute’s goals is to support young scientists.\nIndeed, the positions at the institute are usually for PhD students and PostDocs, and most of the members are young scientists who are in a stage of their career between the very beginning of their doctorate and the final step of obtaining a permanent position at a university or another research institution.\nThe leaders of the work groups, which usually comprise 2–10 members, are typically university professors in Austrian universities.\nThrough its position as the biggest mathematical non-university institute in Austria, through its work, and by its efforts in public outreach, RICAM promotes the role of mathematics in science, industry, and society."
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"Computational Methods for PDEs"
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" led by Ulrich Langer/Herbert Egger (Johannes Kepler University Linz),"
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"Geometry in Simulations"
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" led by Bert Jüttler (Johannes Kepler University Linz),"
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}
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"Inverse Problems and Mathematical Imaging"
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" led by Otmar Scherzer (University of Vienna),"
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" led by Luca Gerardo-Giorda (Johannes Kepler University Linz),"
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" led by Josef Schicho (Johannes Kepler University Linz),"
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}
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"Multiscale Modeling and Simulation of Crowded Transport in the Life and Social Sciences"
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" led by Marie-Therese Wolfram (University of Warwick), being phased out in 2021,"
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}
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},
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"Prof. Heinz Engl was the first Director of the Institute until the end of 2011, when he became Rector of the University of Vienna."
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},
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{
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"content": [
"Left to right:\nManaging Director Ronny Ramlau,\nDeputy Director Karl Kunisch,\nand the founder of RICAM, Heinz Engl.\nPhotos by Claudia Börner."
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}
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"The institute is housed in one of the Science Park buildings on the campus of JKU Linz.\nPhoto by Claudia Börner."
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"2 Scientific machine learning"
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"The usage of (data-driven) machine learning methods to tackle model-based problems, for example from the natural sciences, has recently evolved into a promising research area often termed “scientific machine learning”.\nIn this section we describe partly related efforts in this direction by the two research groups “Mathematical Data Science” and “Optimization and Optimal Control”.\n"
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"Empirical evidence shows that the total computational cost for solving parametric Black–Scholes PDEs by the deep learning based algorithm of [",
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"Potential energy surface of H10 chain computed by the deep learning based algorithm of [",
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"The group “Mathematical Data Science” represents the area of data science, particularly in relation to different aspects of applied mathematics.\nIts research is motivated by interdisciplinary applications and ranges from theory over algorithm development to the solution of real-world problems.\nA particular focus is currently put on the use and analysis of deep learning methods for the numerical solution of mostly high dimensional partial differential equations (PDEs) that are burdened by the curse of dimensionality in the sense that the computational complexity of most known algorithms scales at least exponentially in the underlying problem dimension.\nProminent examples include the Black–Scholes (BS) PDE from computational finance, the Hamilton–Jacobi–Bellmann (HJB) PDEs arising from stochastic optimal control problems, or the many-electron Schrödinger equation from computational chemistry.\nFor BS PDEs and certain nonlinear HJB PDEs we were recently able to prove that neural networks are capable of representing their solutions without incurring the curse of dimensionality [",
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"], and that such solutions can be numerically found by solving an empirical risk minimization (ERM) problem of a size scaling only polynomially in the problem dimension [",
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"].\nWhile the analysis of the computational complexity of the ERM problem remains wide open, there are some empirical results suggesting that its scaling does not suffer from the curse of dimensionality either, see [",
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"] and Figure ",
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".\nIn another direction, we investigate the numerical solution of the many-electron Schrödinger equation using a deep learning based Quantum Monte Carlo ansatz.\nWe show that a judicious weight sharing strategy enables the efficient computation of potential energy surfaces with chemical accuracy, see [",
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"] and Figure ",
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".\nWe see that the use of modern machine learning methods for problems in scientific computing has the potential to overcome the curse of dimensionality for several important problems arising for example in computational finance, computational chemistry or optimal control.\nIt therefore presents exciting opportunities to render previously intractable problems in these fields feasible.\nThe extent to which this potential can be fully realized will constitute a central topic in future work."
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"] a tensor decomposition approach for the solution of high-dimensional, fully nonlinear HJB equations arising in deterministic optimal feedback control of nonlinear dynamics is presented.\nIt combines a tensor train approximation for the value function with a Newton-like iterative method for the solution of the resulting nonlinear system.\nIn numerical tests the tensor approximation leads to a polynomial scaling with respect to the dimension, thus partially circumventing the curse of dimensionality.\nIn an alternative approach [",
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"], rather than obtaining the feedback from the HJB equation directly, the feedback gains are approximated by neural networks, which are trained by open loop optimal controls.\nA third promising approach for the computation of high-dimensional optimal feedback laws is based on sparse regression exploiting the control-theoretical link between HJB equations and first-order optimality conditions via Pontryagin’s Maximum Principle [",
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"].\nCombined with model reduction techniques these methods have the potential of solving closed loop optimal control for complex partial differential equations.\nIn another line of research [",
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"], variational formulations of mathematical imaging problems are combined with deep learning techniques by introducing a data-driven total deep variation regularizer.\nWe take advantage of the well-known phenomenon that typically the best image quality is achieved when the gradient flow process is stopped before converging to a stationary point.\nThis paradox originates from a trade-off between optimization and modeling errors of the underlying variational model.\nAn optimal stopping time is introduced, which is learned from data by means of an optimal control approach."
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"].\nA well known example is Computerized Tomography (CT), where the measured damping of x-rays passing through a body is used to reconstruct the density distribution of the body.\nIn this case, the connection between the density and the data is given by the Radon transform ",
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"The “Transfer Group” is involved in several research projects with industry as well as other branches of science, with the goal of developing state of the art mathematical methods for applications.\nThe group has a very tight collaboration with the “Inverse Problems and Imaging Group” within the special research program (SFB) ",
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", which is formed by a group of researchers from mathematics, physics, astronomy and medical engineering.\nThe Transfer Group has been particularly active in the field of Adaptive Optics (AO).\nAO was initially invented for the correction of degraded images from astronomical images, caused by turbulences in the atmosphere above earth-bound telescopes.\nTo this end, incoming wavefront data from reference guide stars is measured by a wavefront sensor, and the reconstructed wavefront is used to compute an appropriate shape of deformable mirrors that are employed to correct the science images of the telescope (Figure ",
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"Top: ESO’s Very Large Telescope in the Atacama desert (Chile) using Laser guide stars for a tomography of the atmosphere (Source: ESO).\nBottom: Sketch of an AO system.\nThe light from the scientific object is collected by the main mirror M1 and then corrected by the deformable mirrors DM1 and DM2."
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"More recently, AO systems have also been used for image improvement in Ophthalmology and Microscopy.\nWithin the aforementioned SFB, the Transfer Group works, in cooperation with the Medical University of Vienna, on the development of methods for AO systems in ophthalmic OCT systems that achieve an improved imaging quality of, e.g., the human retina, see Figure ",
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"As visible from the research work presented above, RICAM is more than just the sum of its individual work groups; cooperations between the groups and an atmosphere of creativity and innovation are among the strengths of the institute.\nAnother is the special mix of mathematical disciplines present at RICAM.\nApart from the groups highlighted in the previous sections, there is a group working on computational methods for PDEs, which has been led by Ulrich Langer until this year, and which is now in a transition phase after which it will be taken over by Langer’s successor, Herbert Egger.\nThe group will focus on the development, analysis, and efficient realization of numerical methods for the simulation of coupled and multiscale phenomena in physics, engineering, and material and life sciences.\nTight collaborations and synergies with several existing RICAM groups can be expected.\nIn particular, this group is strongly connected to the “Geometry in Simulations” group, led by Bert Jüttler."
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"What is more, in contrast to several other large mathematics institutes, RICAM has a very strong and productive group working on symbolic computation, algebra, and combinatorics.\nA recent addition to the work groups is a team led by Luca Gerardo-Giorda focusing on applications to the life sciences.\nFurther teams working on cryptography, simulation of crowded transport, and quasi-Monte Carlo methods round off the picture."
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" attract world leaders in their respective research fields, and are the starting point for many international cooperations with RICAM scientists.\nFurthermore, they contribute significantly to the international visibility of the institute and to the exchange of the newest trends and developments.\nIn general, at least one group leader, one other member of a group, and one member of the administrative staff are responsible for the organization of each Special Semester, frequently in cooperation with leading scientists from international research institutions.\nRICAM Special Semesters regularly attract around 200–300 guests.\nThese may be either short term guests, who usually attend selected workshops, or long-term guests staying for longer periods during the Special Semester.\nThe next Special Semester, “Tomography Across the Scales”, will be held in the fall of 2022, subject to positive developments with respect to the current global pandemic."
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"\nPhilipp Grohs is professor at the University of Vienna and group leader at RICAM.\nHis main research interests are in numerical analysis, signal processing, and mathematics of machine learning.\nHe was awarded the ETH Latsis Prize and selected for an Alexander-von-Humboldt Professorship award.\n",
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"\nPeter Kritzer is leading the RICAM work group “Multivariate Algorithms and Quasi-Monte Carlo Methods”.\nHis research interests include high-dimensional numerical integration, function approximation, and information-based complexity.\nHe received the Prize for Achievement in IBC in 2015.\n",
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"\nKarl Kunisch is professor emeritus at the Karl-Franzens University Graz and group leader at RICAM.\nHe is an applied mathematician whose main research interests lie in control and optimization in the context of partial differential equations.\nHe was awarded an ERC advanced grant, and received the W. T. and Adalia Reid Prize.\n",
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"\nRonny Ramlau is professor for Industrial Mathematics at the Johannes Kepler University Linz and director of RICAM.\nHis main research is on inverse problems and imaging, with a focus on astronomical and medical applications.\nAdditionally, he is involved in several cooperations with industrial partners.\n",
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"\nOtmar Scherzer is professor at the University of Vienna and group leader at RICAM.\nHe is working in applied mathematics, with a special focus on inverse problems and computational microscopic imaging.\nFurthermore, he is the editor-in-chief of the flagship journal “Inverse Problems”.\n",
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