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On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations

arXiv

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arXiv 2017年

作者： Weinan, E. Hutzenthaler, Martin Jentzen, Arnulf Kruse, Thomas Department of Mathematics and Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544-1000 United States BICMR and School of Mathematical Sciences Peking University Beijing100870 China Faculty of Mathematics University of Duisburg-Essen Essen45117 Germany Seminar für Angewandte Mathematik ETH Zurich Zürich8092 Switzerland

Parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) are key ingredients in a number of models in physics and financial engineering. In particular, parabolic PDEs and BSDEs are fundamental tools in the state-of-the-art pricing and hedging of financial derivatives. The PDEs and BSDEs appearing in such applications are often high-dimensional and nonlinear. Since explicit solutions of such PDEs and BSDEs are typically not available, it is a very active topic of research to solve such PDEs and BSDEs approximately. In the recent article [E, W., Hutzenthaler, M., Jentzen, A., & Kruse, T. Linear scaling algorithms for solving high-dimensional nonlinear parabolic differential equations. arXiv:1607.03295 (2017)] we proposed a family of approximation methods based on Picard approximations and multilevel Monte Carlo methods and showed under suitable regularity assumptions on the exact solution for semilinear heat equations that the computational complexity is bounded by O(d ϵ-(4+δ)) for any δ 2 (0,∞), where d is the dimensionality of the problem and ϵ ∈ (0,∞) is the prescribed accuracy. In this paper, we test the applicability of this algorithm on a variety of 100-dimensional nonlinear PDEs that arise in physics and finance by means of numerical simulations presenting approximation accuracy against runtime. The simulation results for these 100-dimensional example PDEs are very satisfactory in terms of accuracy and speed. In addition, we also provide a review of other approximation methods for nonlinear PDEs and BSDEs from the literature. Copyright © 2017, The Authors. All rights reserved.

关键词： Monte Carlo methods

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Stability of stationary inverse transport equation in diffusion scaling

arXiv

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arXiv 2017年

作者： Chen, Ke Li, Qin Wang, Li Mathematics Department University of Wisconsin-Madison 480 Lincoln Dr. MadisonWI53705 United States Department of Mathematics Computational and Data-Enabled Science and Engineering Program State University of New York at Buffalo 244 Mathematics Building BuffaloNY14206 United States

We consider the inverse problem of reconstructing the optical parameters for stationary radiative transfer equation (RTE) from velocity-averaged measurement. The RTE often contains multiple scales characterized by the magnitude of a dimensionless parameter-the Knudsen number (Kn). In the diffusive scaling (Kn 1), the stationary RTE is well approximated by an elliptic equation in the forward setting. However, the inverse problem for the elliptic equation is acknowledged to be severely ill-posed as compared to the well-posedness of inverse transport equation, which raises the question of how uniqueness being lost as Kn → 0. We tackle this problem by examining the stability of inverse problem with varying Kn. We show that, the discrepancy in two measurements is amplified in the reconstructed parameters at the order of Knp (p = 1 or 2), and as a result lead to ill-posedness in the zero limit of Kn. Our results apply to both continuous and discrete settings. Some numerical tests are performed in the end to validate these theoretical findings. Copyright © 2017, The Authors. All rights reserved.

关键词： Parameter estimation

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A class of null space conditions for sparse recovery via nonconvex, non-separable minimizations

arXiv

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arXiv 2017年

作者： Tran, Hoang Webster, Clayton Department of Computational and Applied Mathematics Oak Ridge National Laboratory 1 Bethel Valley Road Oak RidgeTN37831-6164 Department of Mathematics University of Tennessee 1403 Circle Drive KnoxvilleTN37916 United States

For the problem of sparse recovery, it is widely accepted that nonconvex minimizations are better than 1penalty in enhancing the sparsity of solution. However, to date, the theory verifying that nonconvex penalties outperform (or are at least as good as) 1minimization in exact, uniform recovery has mostly been limited to separable cases. In this paper, we establish general recovery guarantees through null space conditions for nonconvex, non-separable regularizations, which are slightly less demanding than the standard null space property for 1minimization. Copyright © 2017, The Authors. All rights reserved.

关键词： Recovery

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3d normal coordinate systems for cortical areas

arXiv

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arXiv 2018年

作者： Ratnanather, J. Tilak Arguillère, Sylvain Kutten, Kwame S. Kral, Andrej Younes, Laurent Center for Imaging Science Institute for Computational Medicine Department of Biomedical Engineering Johns Hopkins University BaltimoreMD21218 United States CNRS Universite Claude Bernard Lyon 1 CNRS UMR 5208 Institut Camille Jordan Lyon France Center for Imaging Science Johns Hopkins University BaltimoreMD21218 United States Institute of AudioNeuroTechnology Department of Experimental Otology Hannover Medical School Hannover Germany Center for Imaging Science Institute for Computational Medicine Department of Applied Mathematics and Statistics Johns Hopkins University BaltimoreMD21218 United States

A surface-based diffeomorphic algorithm to generate 3D coordinate grids in the cortical ribbon is described. In the grid, normal coordinate lines are generated by the diffeomorphic evolution from the grey/white (inner) surface to the grey/csf (outer) surface. Specifically, the cortical ribbon is described by two triangulated surfaces with open boundaries. Conceptually, the inner surface sits on top of the white matter structure and the outer on top of the gray matter. It is assumed that the cortical ribbon consists of cortical columns which are orthogonal to the white matter surface. This might be viewed as a consequence of the development of the columns in the embryo. It is also assumed that the columns are orthogonal to the outer surface so that the resultant vector field is orthogonal to the evolving surface. Then the distance of the normal lines from the vector field such that the inner surface evolves diffeomorphically towards the outer one can be construed as a measure of thickness. Applications are described for the auditory cortices in human adults and cats with normal hearing or hearing loss. The approach offers great potential for cortical morphometry. Copyright © 2018, The Authors. All rights reserved.

关键词： Audition

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The Effect of Mission Duration on LISA Science Objectives

arXiv

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arXiv 2021年

作者： Seoane, Pau Amaro Sedda, Manuel Arca Babak, Stanislav Berry, Christopher P. L. Berti, Emanuele Bertone, Gianfranco Blas, Diego Bogdanović, Tamara Bonetti, Matteo Breivik, Katelyn Brito, Richard Caldwell, Robert Capelo, Pedro R. Caprini, Chiara Cardoso, Vitor Carson, Zack Chen, Hsin-Yu Chua, Alvin J. K. Dvorkin, Irina Haiman, Zoltan Heisenberg, Lavinia Isi, Maximiliano Karnesis, Nikolaos Kavanagh, Bradley J. Littenberg, Tyson B. Mangiagli, Alberto Marcoccia, Paolo Maselli, Andrea Nardini, Germano Pani, Paolo Peloso, Marco Pieroni, Mauro Ricciardone, Angelo Sesana, Alberto Tamanini, Nicola Toubiana, Alexandre Valiante, Rosa Vretinaris, Stamatis Weir, David J. Yagi, Kent Zimmerman, Aaron Institute of Multidisciplinary Mathematics Universitat Politécnica de Valéncia Spain DESY Zeuthen Germany Institute of Applied Mathematics Academy of Mathematics and Systems Science CAS Beijing China Kavli Institute for Astronomy and Astrophysics Beijing China Astronomisches Rechen-Institut Zentrüm für Astronomie Universität Heidelberg Mönchofstr. 12-14 Heidelberg Germany Université de Paris CNRS Astroparticule et Cosmologie ParisF-75006 France Moscow Institute of Physics and Technology Moscow region Dolgoprudny Russia Department of Physics and Astronomy Northwestern University 1800 Sherman Ave EvanstonIL60201 United States SUPA School of Physics and Astronomy University of Glasgow Kelvin Building University Ave GlasgowG12 8QQ United Kingdom Department of Physics and Astronomy Johns Hopkins University 3400 N. Charles St BaltimoreMD21218 United States Institute for Theoretical Physics University of Amsterdam Science Park 904 Amsterdam1098 XH Netherlands Theoretical Particle Physics and Cosmology Group Department of Physics King's College London Strand LondonWC2R 2LS United Kingdom Grup de Física Teòrica Departament de Física Universitat Autònoma de Barcelona Bellaterra 08193 Spain The Barcelona Institute of Science and Technology Campus UAB Bellaterra 08193 Spain Center for Relativistic Astrophysics School of Physics Georgia Institute of Technology AtlantaGA30332 United States Università degli Studi di Milano-Bicocca Piazza della Scienza 3 Milano20126 Italy Center for Computational Astrophysics Flatiron Institute New YorkNY10010 United States CENTRA Departamento de Física Instituto Superior Técnico - IST Universidade de Lisboa - UL Avenida Rovisco Pais 1 Lisboa 1049 France Department of Physics and Astronomy HB6127 Wilder Lab Dartmouth College HanoverNH03755 United States Center for Theoretical Astrophysics and Cosmology Institute for Computational Science University of Zurich Winterthurerstrasse 190 ZürichCH-8057

The science objectives of the LISA mission have been defined under the implicit assumption of a 4-yr continuous data stream. Based on the performance of LISA Pathfinder, it is now expected that LISA will have a duty cycle of ≈ 0.75, which would reduce the effective span of usable data to 3 yr. This paper reports the results of a study by the LISA Science Group, which was charged with assessing the additional science return of increasing the mission lifetime. We explore various observational scenarios to assess the impact of mission duration on the main science objectives of the mission. We find that the science investigations most affected by mission duration concern the search for seed black holes at cosmic dawn, as well as the study of stellar-origin black holes and of their formation channels via multi-band and multi-messenger observations. We conclude that an extension to 6 yr of mission operations is recommended. Copyright © 2021, The Authors. All rights reserved.

关键词： Black holes

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Coulomb impurities in two-dimensional topological insulators

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Physical Review B 2017年第12期95卷 125431-125431页

作者： Jia-Lin Zhu Guo Li Ning Yang Department of Physics Tsinghua University Beijing 100084 China and State Key Laboratory of Low-Dimensional Quantum Physics Tsinghua University Beijing 100084 China Institute of Applied Physics and Computational Mathematics Beijing 100088 China

Introducing a powerful method, we obtain the exact solutions for a Coulomb impurity in two-dimensional infinite and finite topological insulators. The level order and zero-energy degeneracy of the spectra are found to be quite different between topological trivial and nontrivial phases. For quantum dots of topological insulator, the variation of the edge and Coulomb states with dot size, Coulomb potential, and magnetic field are clearly shown. It is found that for small dots the edge states can be strongly coupled with the Coulomb states and for large dots the edge states are insensitive to the Coulomb fields but sensitive to the magnetic fields.

关键词： Edge states Impurities Topological phases of matter Quantum dots Series expansions & exact enumeration

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Macroeconomics and fintech: Uncovering latent macroeconomic effects on peer-To-peer lending

arXiv

引用

arXiv 2017年

作者： Foo, Jessica Lim, Lek-Heng Wong, Ken Sze-Wai Department of Statistics University of Chicago 5747 South Ellis Avenue ChicagoIL60637 United States Computational and Applied Mathematics Initiative Department of Statistics University of Chicago

Peer-To-peer (P2P) lending is a fast growing financial technology (FinTech) trend that is displacing traditional retail banking. Studies on P2P lending have focused on predicting individual interest rates or default probabilities. However, the relationship between aggregated P2P interest rates and the general economy will be of interest to investors and borrowers as the P2P credit market matures. We show that the variation in P2P interest rates across grade types are determined by three macroeconomic latent factors formed by Canonical Correlation Analysis (CCA)-macro default, investor uncertainty, and the fundamental value of the market. However, the variation in P2P interest rates across term types cannot be explained by the general *** Codes 62H25, 62P20 Copyright © 2017, The Authors. All rights reserved.

关键词： Correlation methods

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Optimal jittered sampling for two points in the unit square

arXiv

引用

arXiv 2017年

作者： Pausinger, Florian Rachh, Manas Steinerberger, Stefan TU Munich Germany Applied Mathematics Program Yale University New HavenCT06510 United States Department of Mathematics Yale University New HavenCT06510 United States

Jittered Sampling is a refinement of the classical Monte Carlo sampling method. Instead of picking n points randomly from [0, 1]2, one partitions the unit square into n regions of equal measure and then chooses a point randomly from each partition. Currently, no good rules for how to partition the space are available. In this paper, we present a solution for the special case of subdividing the unit square by a decreasing function into two regions so as to minimize the expected squared L2−discrepancy. The optimal partitions are given by a highly nonlinear integral equation for which we determine an approximate solution. In particular, there is a break of symmetry and the optimal partition is not into two sets of equal measure. We hope this stimulates further interest in the construction of good partitions.65C05 (primary) and 93E20 (secondary). Copyright © 2017, The Authors. All rights reserved.

关键词： Monte Carlo methods

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EFFICIENT EVALUATION OF ELLIPSOIDAL HARMONICS FOR POTENTIAL MODELING

arXiv

引用

arXiv 2017年

作者： Klotz, Thomas S. Bardhan, Jaydeep P. Knepley, Matthew G. Department of Computational and Applied Mathematics Rice University HoustonTX77005 United States Department of Mechanical Engineering Northeastern University BostonMA02115 United States

Ellipsoidal harmonics are a useful generalization of spherical harmonics but present additional numerical challenges. One such challenge is in computing ellipsoidal normalization constants which require approximating a singular integral. In this paper, we present results for approximating normalization constants using a well-known decomposition and applying tanh-sinh quadrature to the resulting integrals. Tanh-sinh has been shown to be an effective quadrature scheme for a certain subset of singular integrands. To support our numerical results, we prove that the decomposed integrands lie in the space of functions where tanh-sinh is optimal and compare our results to a variety of similar change-of-variable *** 65D32, 33C50;Secondary 78M16, 65Y20 Copyright © 2017, The Authors. All rights reserved.

关键词： Harmonic analysis

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On the complexity of exact counting of dynamically irreducible polynomials

arXiv

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arXiv 2017年

作者： Gómez-Pérez, Domingo Mérai, László Shparlinski, Igor E. Department of Mathematics University of Cantabria Santander39005 Spain Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences Altenberger Straße 69 LinzA-4040 Austria School of Mathematics and Statistics University of New South Wales SydneyNSW2052 Australia

We give an efficient algorithm to enumerate all sets of r ě 1 quadratic polynomials over a finite field, which remain irreducible under iterations and compositions. Copyright © 2017, The Authors. All rights rese...

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