Stochastic-Deterministic Fusion: Unifying Monte Carlo, Spectral, and Finite Element Methods for High-Dimensional Analysis Article Swipe
Revista, Zen
,
MATH, 10
·
YOU?
·
· 2025
· Open Access
·
· DOI: https://doi.org/10.5281/zenodo.17838585
YOU?
·
· 2025
· Open Access
·
· DOI: https://doi.org/10.5281/zenodo.17838585
High-dimensional problems are ubiquitous in science and engineering, posing significant computational challenges for traditional numerical methods. This paper proposes a novel Stochastic-Deterministic Fusion framework that unifies Monte Carlo (MC), Spectral, and Finite Element (FE) methods to efficiently and accurately tackle such problems. The approach leverages the strengths of each method: Monte Carlo for robust exploration of high-dimensional parameter spaces and uncertainty quantification, spectral methods for effective dimensionality reduction and representation of stochastic processes, and finite element methods for precise deterministic solutions on reduced-dimensional or surrogate domains. We outline a methodology that integrates these techniques, beginning with MC sampling to gather initial insights into the parameter space, followed by spectral decomposition (e.g., Polynomial Chaos Expansion or Karhunen-Loève Expansion) to identify dominant modes and construct low-dimensional surrogate models. Finally, FE methods are applied to solve the underlying physical equations within this reduced-order framework. This fusion strategy aims to mitigate the curse of dimensionality, enhance computational efficiency, and improve solution accuracy for a broad class of high-dimensional problems, particularly those involving partial differential equations with random inputs or outputs.
Related Topics
Concepts
Polynomial chaos
Curse of dimensionality
Monte Carlo method
Uncertainty quantification
Finite element method
Mathematical optimization
Applied mathematics
Computer science
Representation (politics)
Dimensionality reduction
Spectral method
Mathematics
Partial differential equation
Algorithm
Importance sampling
Surrogate model
Monte Carlo integration
Particle filter
Sensor fusion
Numerical analysis
Statistical physics
Polynomial
Markov chain Monte Carlo
Reduction (mathematics)
Sampling (signal processing)
Stochastic process
Metadata
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- article
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https://openalex.org/W7109697997Canonical identifier for this work in OpenAlex
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https://doi.org/10.5281/zenodo.17838585Digital Object Identifier
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Stochastic-Deterministic Fusion: Unifying Monte Carlo, Spectral, and Finite Element Methods for High-Dimensional AnalysisWork title
- Type
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articleOpenAlex work type
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2025Year of publication
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2025-12-06Full publication date if available
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Revista, Zen, MATH, 10List of authors in order
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https://doi.org/10.5281/zenodo.17838585Publisher landing page
- Open access
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YesWhether a free full text is available
- OA status
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greenOpen access status per OpenAlex
- OA URL
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https://doi.org/10.5281/zenodo.17838585Direct OA link when available
- Concepts
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Polynomial chaos, Curse of dimensionality, Monte Carlo method, Uncertainty quantification, Finite element method, Mathematical optimization, Applied mathematics, Computer science, Representation (politics), Dimensionality reduction, Spectral method, Mathematics, Partial differential equation, Algorithm, Importance sampling, Surrogate model, Monte Carlo integration, Particle filter, Sensor fusion, Numerical analysis, Statistical physics, Polynomial, Markov chain Monte Carlo, Reduction (mathematics), Sampling (signal processing), Stochastic processTop concepts (fields/topics) attached by OpenAlex
- Cited by
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0Total citation count in OpenAlex
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