Inverse Problems in Boundary Measurements and Differential Equations
Summary
Inverse problems in boundary measurements and differential equations seek to infer internal properties of a domain by probing its boundary. These questions arise across disciplines, from medical imaging and geophysical surveying to nondestructive evaluation of materials. At their core lies the task of recovering coefficients or source terms in partial differential equations through observed fluxes, potentials or field values at the boundary. Such problems are often severely ill-posed: small measurement errors can lead to large uncertainties in reconstructions. Progress depends on establishing uniqueness theorems that guarantee a single solution and stability estimates that quantify sensitivity to noise. Techniques span functional analysis, microlocal analysis and numerical regularisation, with methods such as monotonicity arguments, complex geometrical optics solutions and novel transforms. Recent advances have extended classical results for elliptic equations to fractional and nonlinear settings, improved Lipschitz or Hölder stability under minimal data, and developed real-time imaging algorithms for practical applications. This field remains of global significance, promising refined imaging in healthcare, enhanced resource exploration and reliable monitoring of industrial structures.
Research from Nature Portfolio
No recent Nature Portfolio content available.
Inverse Problems in Boundary Measurements and Differential Equations publication trend
The graph below shows the total number of articles in inverse problems in boundary measurements and differential equations across all publications each year (not limited to Nature Index journals).
Technical terms
Inverse problem: Reconstruction of unknown internal parameters or sources of a differential equation using measurements taken at the boundary.
Dirichlet-to-Neumann map: Operator sending prescribed boundary values (Dirichlet data) to normal derivative measurements (Neumann data) on the boundary.
Calderón problem: Classic inverse problem of determining an electrical conductivity distribution inside a domain from boundary voltage-current measurements.
Magnetoquasistatic limit: Approximation of Maxwell’s equations neglecting displacement currents at low frequencies, relevant in eddy current imaging.
Monotonicity principle: Property that certain input-output maps preserve ordering with respect to the unknown coefficient, used to formulate rigorous imaging algorithms.
Complex geometrical optics (CGO) solutions: Special high-frequency solutions to elliptic equations used to probe internal structures and establish uniqueness in inverse problems.
References
- Monotonicity of the Transfer Function for Eddy Current Tomography. IEEE Sensors Journal (2023).
- The Calderón problem with partial data in two dimensions. Journal of the American Mathematical Society (2010).
- Direct and inverse problems for the nonlinear time-harmonic Maxwell equations in Kerr-type media. Journal of Spectral Theory (2020).
About these summaries
This Nature Research Intelligence Topic summary is created with the cited references and a large language model. We take care to ground generated text with facts, and have systems in place to gain human feedback on the overall quality of the process in line with our AI principles. We strive to create accurate and useful summaries for people unfamiliar with the research topic and that supports this goal. These pages are a beta release and will be updated as we learn how best to help people gain value from a research topic summary.
Turn complex research questions into confident strategic decisions
When you're under pressure to set direction, justify investment, or understand your competitive position, you need more than raw data — you need trusted insights you can act on.
Benchmark your performance against global peers using robust, methodologically sound analysis.
Combine quantitative metrics with qualitative expert insight to uncover strengths, gaps and emerging opportunities.
Gain tailored, decision-ready recommendations aligned to your strategic priorities.
Talk to us to learn more about our data dashboards and bespoke strategy reports.
Grow research skills, confidence and careers with training built for every stage of the research lifecycle.
Developed with Nature Portfolio journal Editors and internationally renowned experts. Discover three ways to learn:
Self-paced, online courses in convenient bite-sized units, covering key skills across scientific writing, publishing, grant writing, data analysis, and more.
Expert trainer-led workshops with hands-on exercises and real-time feedback across core research skills, delivered via interactive group sessions.
Editor-led workshops combining core principles in writing and publishing, personalised 1:1 feedback from Nature Portfolio Editors and hands-on exercises.
Explore course catalogues and workshop agendas, enquire about the options or request institutional pricing.