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Efficient Numerical Methods for Partial Differential Equations

English title Efficient Numerical Methods for Partial Differential Equations
Applicant Sauter Stefan
Number 137146
Funding scheme ProDoc
Research institution Institut für Mathematik Universität Zürich
Institution of higher education University of Zurich - ZH
Main discipline Mathematics
Start/End 01.10.2011 - 31.10.2014
Approved amount 325'074.00
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Lay Summary (English)

Lead
Lay summary
Lead: Many mathematical models underlying numerical simulation in science boil down to
boundary value problems for partial differential equations. Thus, their efficient numerical
solution is of utmost importance and results in an enormous demand for advanced,
problem-adapted numerical methods for tackling the new classes of problems.

Hintergrund: The development of numerical methods for partial differential equations (PDEs)
continues unabated, driven by completely new types of applications. The ProDoc project will
focus on five key topics in this area:

1. Computational wave propagation. Wave phenomena are at the heart of modern communication
technology. From a simulation point of view, they are hard to tackle due to non-local
interactions and resonance phenomena.

2. Resolution of local solution features. Ideally a finite element mesh is adapted to the
solution of a boundary value problem in order to facilitate approximation of its local
features. However, the meshes produced by mesh generators are often
found wanting in this respect. This may be compensated by smartly chosen local trial spaces.

3. Numerical solution of high-dimensional problems. Mathematical models in, e.g., computational
chemistry, quantitative finance, fluid mechanics give rise to PDEs on high-dimensional domains
where classical numerical discretization methods fail by the formidable problem sizes.

4. Numerical methods for stochastic PDEs. The numerical solution of PDEs with random field
input data has attracted much interest and efforts have been made to incorporate randomness
and statistical information into deterministic numerical simulation.

5. Linear algebra for structured problems. High-dimensional PDEs give rise to large scale,
structured problems that exceed the capabilities of standard linear solvers. Advanced
solvers must exploit the structure imposed by the analytical and physical background.

Das Ziel:
The ProDoc training module ``Efficient Numerical Methods for Partial Differential
Equations'' is to meet the demand for high-level education in an array of cutting edge
numerical methods for solving partial differential equations. The graduate students will
be optimally prepared by this training module to conduct the research for their PhD
projects in the key topics which have been outlined above.

Bedeutung:
The increasing number of applications in science and engineering where numerical simulation
becomes a key issue generates an enormous demand for the development, analysis, and
implementation of reliable and efficient numerical computations. Such new methods are
typically highly advanced, problem-adapted for tackling the new classes of problems.
As a consequence the demand for excellently trained and experienced experts is increasing
rapidly. The ProDoc training module has the goal to provide an excellent high-level
education for conducting research in these emerging fields. Last, but not least, the
ProDoc framework will boost cooperation between the participating research groups in
numerical analysis at the host universities.
Direct link to Lay Summary Last update: 21.02.2013

Responsible applicant and co-applicants

Employees

Collaboration

Group / person Country
Types of collaboration
Dr. Andrea Moiola, University of Reading Great Britain and Northern Ireland (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Exchange of personnel
Dr. Bart Vandereycken, Pricton University United States of America (North America)
- in-depth/constructive exchanges on approaches, methods or results
- Exchange of personnel
Prof. Dr. Ilaria Perugia, Universität Wien Austria (Europe)
- in-depth/constructive exchanges on approaches, methods or results

Scientific events

Active participation

Title Type of contribution Title of article or contribution Date Place Persons involved
Workshop on Matrix Equations and Tensor Techniques Poster Low-rank tensor completion by Riemannian optimization 10.10.2013 Lausanne, Switzerland Kressner Daniel;
Dolomites Research Week on Approximation Talk given at a conference Low-rank tensor completion by Riemannian optimization 08.09.2013 Alba di Canazei, Italy Kressner Daniel;


Self-organised

Title Date Place
8th Zurich Summer School 2014 18.08.2014 Zürich, Switzerland
Pro*doc Retreat 2014 13.08.2014 Disentis, Switzerland
Pro*Doc Retreat 2013 14.08.2013 Disentis, Switzerland
7th Zürich Summer School 2012 20.08.2012 Zürich, Switzerland
Pro*Doc Retreat 2012 15.08.2012 Disentis, Switzerland

Awards

Title Year
2014 MASCOT NUM best student presentation award: Jonas Sukys was awarded the Mascot 2014 for his talk on Multi-level Monte Carlo finite volume methods for stochastic systems of hyperbolic conservation laws by the Masnum-UQ consortium. Further information: http://www.ibk.ethz.ch/su/mascotnum2014/index_EN 2014
Housholder award XV (honorable Mention) für den Projektmitarbeiter Cedric Effenberger 2014
2013 ETH Medal for outstanding PhD: Dr. Ulrik Fjordholm was awarded the 2013 ETH Medal for outstanding PhD, High-order accurate entropy stable numercial schemes for hyperbolic conservation laws, PhD Thesis, ETH Diss Nr. 21025. Further information: http://dx.doi.org/10.3929/ethz-a-007622508 2013
GAMM Junior Fellow Christine Tobler was elected to GAMM Junior Fellow. Further information: http://gamm-ev.de/index.php?option=com_content&view=article&id=126&Itemid=202&lang=en 2012

Associated projects

Number Title Start Funding scheme
144342 Advanced Methods for Computational Electromagnetics 01.10.2012 Project funding
140706 Fast Methods for Frequency-Domain Full-Waveform Inversion in Strongly Heterogeneous Media 01.01.2013 Project funding
127483 Robust numerical methods for solving nonlinear eigenvalue problems 01.11.2009 ProDoc
124825 Convergence Analysis for Finite Element Discretizations of HighlyIndefinite Problems and Elliptic Eigenvalue Problems 01.04.2009 ProDoc
127130 Méthodes numériques pour des équations d'ondes stochastiques 01.10.2009 Project funding
122883 Efficient Numerical Methods for Partial Differential Equations 01.10.2008 ProDoc
120290 Partial Differential Equations with Random Input Data - Numerical Analysis and Scientific Computing 01.04.2008 Project funding
121892 Sparse Tensor Approximation Methods for High-Dimensional Transport Problems 01.09.2009 Project funding
132405 Advanced Methods for Computational Electromagnetics 01.10.2010 Project funding
124883 Plane Wave Discontinuous Galerkin Methods 01.04.2009 ProDoc
141754 The AL Basis for the solution of elliptic problems in heterogeneous media 01.09.2012 ProDoc
144973 Computing equipment 01.05.2013 R'EQUIP
124898 Preconditioned methods for large-scale model reduction 01.04.2009 ProDoc
130050 Multiscale analysis and simulation of waves in strongly heterogeneous media 01.06.2010 Project funding
127437 Efficient Numerical Methods for Partial Differential Equations 01.11.2009 ProDoc
127034 Space-Time Adaptive Wavelet Discretization of Nonlinear Parabolic Problems 01.10.2009 ProDoc

Abstract

The rapidly growing number of mathematical models for continually new types of applications such as, e.g., quantitative finance, stochastic models, nano-optics generates an enormous demand for highly advanced, problem-adapted numerical methods which allow their efficient and reliable numerical simulation. As a consequence the demand for excellently trained and experienced experts is increasing substantially. They have to master techniques beyond what is available in commercial numerical simulation software. However, the attraction and the promotion of young scientists in this field is currently not completely satisfactory because of the widening gap between the education in Bachelor programs and the know-how which is necessary to conduct PhD projects in the field of "Efficient Numerical Methods for Partial Differential Equations". We have described PhD projects for the five following challenging research direction in this field:- Computational wave propagation,- Resolution of local solution features,- Numerical solution of high-dimensional problems,- Numerical methods for stochastic partial differential equations,- Linear algebra for structured problems.This ProDoc proposal has the goal to establish a world-class education and training program for graduate students in numerical mathematics so that they are optimally trained for conducting PhD projects in these areas. Due to the long-lasting close scientific cooperations and the quite unique, fruitful mix of joint and complementary research interests of the numerical analysis/computational mathematics groups at ETH, UZH, UniBas, EPFL Lausanne with this ProDoc programme we will coordinate our efforts to achieve this goal. In particular, we apply for an "Ausbildungsmodul" including the support for 1) one guest professorships per semester who a) complete the proposed graduate course program taught by us with special topics courses towards the aim of offering a broad and advanced course program in numerical analysis and computational mathematicsand b) to enable the graduate students to establish scientific ties also with international experts outside Basel, Lausanne, and Zurich,2) one instructor as support for our general teaching duties so that the proposers can develop new courses in these very new areas of numericalmathematics.3) a 30% position for an administrative director (Mrs Myriam Frank). 4) Financial support so that the Pro*Doc students can attend summer schools, workshops and to organize the annual Pro*Doc retreat at Disentis and the biennial Zurich Summer School in Numerical Mathematics.Our goal is that 7-10 Forschungsmoduls will be linked to this Ausbildungsmodul - some of them are currently running under the current Pro*Doc Ausbildungsmodul PDAMP2_122883/1, one will be submitted by 1.3.2011 and some are regular grant proposals which also benefit from the eductational program of the Ausbildungsmodul.For the organization of the ProDoc we may draw on the positive experience of the running Pro*Doc Ausbildungsmodul and of the Zurich Graduate School in Mathematics (ZGSM) which was founded in 2003. We will take over (and adapt) concepts of the organization of the ZGSM such as the recruiting system and the administrative organisation.
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