Project description
Performance and accuracy guarantees of numerical methods for high-complexity problems
Partial differential equations are essential to describing virtually all processes from environmental to industrial ones. Numerical methods are techniques to solve them approximately. However, as the number of independent variables (dimension) increases, specialised numerical methods are required to achieve tractable computational costs. The ERC-funded COCOA project aims to enable better performance and accuracy in solving such highly complex problems. Researchers will focus on mathematically rigorous methods, based in particular on highly non-linear low-rank tensor representations, neural networks, and linear combinations of arbitrary Gaussian functions. The team will also attempt to identify the most suitable methods for specific partial differential equation problems; construct methods to avoid numerical instabilities; and ensure reliability of results in high dimensions.
Objective
The efficient numerical treatment of partial differential equations on high-dimensional spaces often requires approximation methods involving a high degree of nonlinearity, such as low-rank tensor representations or neural networks. By exploiting structural features of solutions, such approaches in many cases promise extremely efficient approximations. However, due to the corresponding greater difficulty of computing highly compressed representations, such results need to be considered in conjunction with the costs of numerical methods for constructing these approximations. A main objective of the project is to address the gap between theoretical complexity bounds and the performance of practical implementations of solvers, with particular focus on low-rank tensor representations, linear combinations of arbitrary Gaussian functions, as well as neural networks and more general compositions of functions. We also aim to understand the relative suitability of particular nonlinear approximation methods for different problem classes, such as problems with many parameters, evolution problems for probability distributions and wave functions, and eigenvalue problems in quantum chemistry. In each case, it is crucial to avoid numerical instabilities in the interaction of nonlinear approximations and differential operators and to ensure reliability of results in high dimensions by suitable computable error bounds.
Fields of science (EuroSciVoc)
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
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Keywords
Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
- Numerical methods for partial differential equations
- high-dimensional problems
- parameter-dependent problems
- evolution problems
- eigenvalue problems
- nonlinear approximation
- adaptive methods
- complexity estimates
- sparse expansions
- low-rank tensor representations
- linear combinations of Gaussian functions
- neural networks
Programme(s)
Multi-annual funding programmes that define the EU’s priorities for research and innovation.
Multi-annual funding programmes that define the EU’s priorities for research and innovation.
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HORIZON.1.1 - European Research Council (ERC)
MAIN PROGRAMME
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Topic(s)
Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
Funding Scheme
Funding scheme (or “Type of Action”) inside a programme with common features. It specifies: the scope of what is funded; the reimbursement rate; specific evaluation criteria to qualify for funding; and the use of simplified forms of costs like lump sums.
Funding scheme (or “Type of Action”) inside a programme with common features. It specifies: the scope of what is funded; the reimbursement rate; specific evaluation criteria to qualify for funding; and the use of simplified forms of costs like lump sums.
HORIZON-ERC - HORIZON ERC Grants
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Call for proposal
Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
(opens in new window) ERC-2024-COG
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Net EU financial contribution. The sum of money that the participant receives, deducted by the EU contribution to its linked third party. It considers the distribution of the EU financial contribution between direct beneficiaries of the project and other types of participants, like third-party participants.
52062 Aachen
Germany
The total costs incurred by this organisation to participate in the project, including direct and indirect costs. This amount is a subset of the overall project budget.