Project description
Nonlinear Monte Carlo type methods for high-dimensional approximation problems
In many relevant real-world problems it is of fundamental importance to approximately compute evaluations of high-dimensional functions. Standard deterministic approximation methods often suffer in this context from the so-called curse of dimensionality in the sense that the number of computational operations of the approximation method grows at least exponentially in the problem dimension. It is the key objective of the ERC-funded MONTECARLO project to employ multilevel Monte Carlo and stochastic gradient descent type methods to design and analyse algorithms which provably overcome the curse of dimensionality in the numerical approximation of several high-dimensional functions; these include solutions of certain stochastic optimal control problems of some nonlinear partial differential equations and of certain supervised learning problems.
Objective
In a series of relevant real world problems it is of fundamental importance to approximatively compute evaluations of high-dimensional functions. Such high-dimensional approximation problems appear, e.g. in stochastic optimal control problems in operations research, e.g. in supervised learning problems, e.g. in financial engineering where partial differential equations (PDEs) and forward backward stochastic differential equations (FBSDEs) are used to approximatively price financial products, and, e.g. in nonlinear filtering problems where stochastic PDEs are used to approximatively describe the state of a given physical system with only partial information available. Standard approximation methods for such approximation problems suffer from the socalled curse of dimensionality in the sense that the number of computational operations of the approximation method grows at least exponentially in the problem dimension. It is the key objective of this project to design and analyze approximation algorithms which provably overcome the curse of dimensionality in the case of stochastic optimal control problem, nonlinear PDEs, nonlinear FBSDEs, certain SPDEs, and certain supervised learning problems. We intend to solve many of the above named approximation problems by combining different types of multilevel Monte Carlo approximation methods, in particular, multilevel Picard approximation methods, with stochastic gradient descent (SGD) optimization methods. Another chief objective of this project is to prove the conjecture that the SGD optimization method converges in the training of ANNs with ReLU activation. We expect that the outcome of this project will have a significant impact on the way how highdimensional PDEs, FBSDEs, and stochastic optimal control problems are solved in engineering and operations research and on the mathematical understanding of the training of ANNs by means of the SGD optimization methods.
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: https://op.europa.eu/en/web/eu-vocabularies/euroscivoc.
This project's classification has been validated by the project's team.
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: https://op.europa.eu/en/web/eu-vocabularies/euroscivoc.
This project's classification has been validated by the project's team.
- natural sciences computer and information sciences artificial intelligence machine learning supervised learning
- natural sciences mathematics applied mathematics statistics and probability
- natural sciences mathematics pure mathematics mathematical analysis differential equations partial differential equations
- natural sciences mathematics applied mathematics numerical analysis
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)
- information-based complexity
- IBC
- computational stochastics
- Monte Carlo method
- multilevel Monte Carlo method
- numerical analysis
- partial differential equation
- PDE
- backward stochastic differential equation
- BSDE
- stochastic optimal control
- stochastic partial differential equation
- SPDE
- stochastic gradient descent
- SGD
- machine learning
- artificial neural network
- ANN
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)
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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-2021-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.
48149 Muenster
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.