Inverse Coefficient Problem for One-Dimensional Subdiffusion with Data on Disjoint Sets in Time

Concetti Chiave

Investigating the unique recovery of coefficients in subdiffusion models from disjoint time sets.

Sintesi

The article explores an inverse coefficient problem for one-dimensional subdiffusion, aiming to determine coefficients and parameters from lateral Cauchy data. The analysis leverages fractional diffusion memory effects for unique coefficient recovery. Unique solutions are proven under specific conditions, with numerical experiments supporting the findings. The study extends existing results by considering data on disjoint time sets, presenting challenging yet practical scenarios. Theoretical results are complemented by numerical demonstrations using the Levenberg-Marquardt method.

Statistiche

Proposition 3.4 proves that problem (1.2) has a unique weak solution u ∈ L1(0, T; H2(0, ℓ)).
Lemma 3.3 shows that the unique H2(0, ℓ)-solution v satisfies certain properties.
Proposition 3.5 establishes the time-analytic extension of the solution u to (1.2).
Proposition 3.6 demonstrates the unique weak solution w of problem (2.3) and its analytic extension.

Citazioni

Approfondimenti chiave tratti da

Inverse Coefficient Problem for One-Dimensional Subdiffusion with Data on Disjoint Sets in Time

by Siyu Cen,Ban... alle arxiv.org 03-19-2024

https://arxiv.org/pdf/2403.11599.pdf

Domande più approfondite

How does the consideration of data on disjoint time sets impact real-world applications

The consideration of data on disjoint time sets in inverse coefficient problems can have significant implications for real-world applications. In many practical scenarios, it may be challenging or even impossible to simultaneously apply excitation and collect measurements due to various constraints such as equipment limitations, operational restrictions, or safety concerns. By allowing the excitation and measurement to occur at different times, these theoretical results enable more flexible and adaptable approaches in experimental setups.
For example, in fields like medical imaging or environmental monitoring where precise control over timing is crucial, the ability to gather data on disjoint time sets can lead to improved accuracy and reliability of the results. This approach can also enhance the feasibility of conducting experiments in dynamic environments where conditions change rapidly over time.
Furthermore, by accommodating data collection on disjoint time sets, researchers can optimize resources and streamline processes by decoupling the excitation phase from the measurement phase. This separation allows for better planning and coordination of experimental procedures while maintaining scientific rigor and validity.

What potential limitations or challenges could arise when applying these theoretical results practically

While the theoretical results regarding inverse coefficient problems with data on disjoint time sets offer valuable insights into unique recovery possibilities, there are several potential limitations and challenges when applying these findings practically:

Experimental Implementation: Implementing experiments based on these theoretical concepts may require sophisticated instrumentation capable of precisely controlling excitation timings and capturing measurements at specific intervals. Ensuring synchronization between different phases of data collection could pose technical challenges.

Data Processing Complexity: Analyzing datasets obtained from disjoint time sets may introduce complexities in post-processing tasks such as alignment, interpolation, or fusion of information gathered at different instances. Managing disparate datasets effectively while preserving temporal relationships is essential but could be computationally demanding.

Noise Sensitivity: Disjoint data collection methods might be more susceptible to noise interference during measurements due to extended durations between excitation events and observations. Developing robust algorithms that account for noise resilience without compromising result accuracy becomes critical.

Model Assumptions: Theoretical models used for solving inverse coefficient problems often make simplifying assumptions about system behavior that may not fully capture real-world dynamics accurately. Validating model assumptions against empirical data remains a key challenge in practical applications.

How might advancements in computational methods enhance the efficiency of solving similar inverse coefficient problems

Advancements in computational methods play a pivotal role in enhancing the efficiency of solving similar inverse coefficient problems related to subdiffusion models with data on disjoint time sets:

Numerical Simulations: High-performance computing techniques enable researchers to conduct extensive numerical simulations for exploring diverse scenarios efficiently within reasonable time frames.

2 .Optimization Algorithms: Advanced optimization algorithms like genetic algorithms or machine learning-based approaches can help optimize parameter estimation processes involved in solving complex inverse problems.
3 .Parallel Computing: Leveraging parallel computing architectures allows for faster processing speeds when handling large datasets associated with multiple measurements taken at distinct points in time.
4 .Machine Learning Integration: Integrating machine learning methodologies such as deep learning neural networks can aid in pattern recognition within complex datasets generated from disparate temporal sources.
5 .Uncertainty Quantification Techniques: Employing uncertainty quantification methods helps assess confidence levels associated with estimated parameters derived from non-contemporaneous observational inputs.
These computational advancements collectively contribute towards streamlining workflows, improving solution accuracies,
and expediting decision-making processes when addressing inverse coefficient problems involving subdiffusion models
with non-overlapping temporal data collections."