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Process of calculating the causal factors that produced a set of observations
An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating
Inverse_problem
Computing joint values of a kinematic chain from a known end position
kinematic animation problem uses the kinematics equations to determine the pose given the joint angles. The inverse kinematics problem computes the joint
Inverse_kinematics
Unsolved problem in mathematics
numbers? More unsolved problems in mathematics In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois
Inverse_Galois_problem
In mathematics, the inverse problem for Lagrangian mechanics is the problem of determining whether a given system of ordinary differential equations can
Inverse problem for Lagrangian mechanics
Inverse_problem_for_Lagrangian_mechanics
Problem in applied mathematics
In mathematics and physics, the inverse scattering problem is the problem of determining characteristics of an object, based on data of how it scatters
Inverse_scattering_problem
Special case of the two-body problem
Kepler problem is a special case of the two-body problem, in which the two bodies interact by a central force that varies in strength as the inverse square
Kepler_problem
Measurement of vertical distribution of physical properties of the atmospheric column
the inverse model, f → − 1 {\displaystyle {\vec {f}}^{-1}} . If the problem is linear we can use some type of matrix inverse method—often the problem is
Atmospheric_sounding
The inverse problem in optics (or the inverse optics problem) refers to the fundamentally ambiguous mapping between sources of retinal stimulation and
Inverse_problem_in_optics
Methods in geodesy
Given the coordinates of the two points (Φ1, L1) and (Φ2, L2), the inverse problem finds the azimuths α1, α2 and the ellipsoidal distance s. Calculate
Vincenty's_formulae
Method for the construction of fractals
a solution to a restricted form of the inverse problem using only PIFS; the general form of the inverse problem remains unsolved. By 1995, all fractal
Iterated_function_system
Inverse dynamics is an inverse problem in classical dynamics. Inverse rigid-body dynamics is a method for computing forces and/or moments of force (torques)
Inverse_dynamics
Property of differential equations describing physical phenomena
itself is a smooth function of those parameters. Inverse problems are often ill-posed; for example, the inverse heat equation, deducing a previous distribution
Well-posed_problem
Global electromagnetic resonances, generated and excited by lightning discharges
of the interesting problems in Schumann resonances studies is determining the lightning source characteristics (the "inverse problem"). Temporally resolving
Schumann_resonances
Academic journal
Inverse Problems is a peer-reviewed, broad-based interdisciplinary journal for pure and applied mathematicians and physicists produced by IOP Publishing
Inverse_Problems
Orbital mechanics term
using Mathematica: InverseSeries[Series[ArcSin[Sqrt[t]] - Sqrt[(1 - t) t], {t, 0, 15}]] For most applications, the inverse problem can be computed numerically
Kepler's_equation
Machine learning model training problem
exponentially. The inverse problem, when weight gradients at earlier layers get exponentially larger, is called the exploding gradient problem. Backpropagation
Vanishing_gradient_problem
Technique to solve partial differential equations
Regular PINNs are only able to obtain the solution of a forward or inverse problem on a single geometry. It means that for any new geometry (computational
Physics-informed neural networks
Physics-informed_neural_networks
Inverse problem related to neuroscience
The inverse recovery in EEG is a Calderón-type inverse problem with the goal of recovering source terms and/or conductivity in layers of the human head
Inverse_recovery_in_EEG
Differential calculus on function spaces
directly into the inverse problem of the calculus of variations. By combining the Helmholtz conditions involved in the inverse problem with relations derived
Calculus_of_variations
Mapping brain activity by recording magnetic fields produced by currents in the brain
difficulty is that the inverse problem does not have a unique solution (i.e., there are infinite possible "correct" answers), and the problem of defining the
Magnetoencephalography
interested in determining the Earth's interior structure, to solve the inverse problem: in 1970 three geophysicists improved the previous method and introduced
Resonant ultrasound spectroscopy
Resonant_ultrasound_spectroscopy
Inequality in information theory
was proved independently by Kullback, Csiszár, and Kemperman. A precise inverse of the inequality cannot hold: for every ε > 0 {\displaystyle \varepsilon
Pinsker's_inequality
Ellipse on a spheroid centered on its origin
b{\sqrt {1+e'^{2}\cos ^{2}\alpha _{0}}}} and b {\displaystyle b} .) The "inverse problem" is the determination of s 12 {\displaystyle s_{12}} , α 1 {\displaystyle
Great_ellipse
Probabilistic problem-solving algorithm
moments may not be defined, etc.).[citation needed] When analyzing an inverse problem, obtaining a maximum likelihood model is usually not sufficient, as
Monte_Carlo_method
Method for solving certain nonlinear partial differential equations
In mathematics, the inverse scattering transform (or nonlinear Fourier transform) is a method that solves the initial value problem for a nonlinear partial
Inverse_scattering_transform
Mathematical problem
Zbl 1221.20045. Nathanson, Melvyn B. (1996). Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Graduate Texts in Mathematics. Vol. 165
Zero-sum_problem
Regularization technique for ill-posed problems
method of regularization of ill-posed problem inverse problems. It is particularly useful to mitigate the problem of multicollinearity in linear regression
Ridge_regression
Photomask enhancement technique
optimize photomask design. It is basically an approach to solve an inverse imaging problem: to calculate the shapes of the openings in a photomask ("source")
Inverse_lithography
Type of biological prediction
from primary structure. Structure prediction is different from the inverse problem of protein design. Protein structure prediction is one of the most
Protein_structure_prediction
match with the magnetization that is observed. Advantage Iterative inverse problem solving is faster than Cardiac gating and doesn't involve the patient
Ghosting_(medical_imaging)
Elliptic partial differential equation
smaller than one part in a thousand. Surface reconstruction is an inverse problem. The goal is to digitally reconstruct a smooth surface based on a large
Poisson's_equation
in a material. Methods to fully map out the eigenstrain, called the inverse problem of eigenstrain, are an active area of research. Understanding the total
Eigenstrain
Function's sensitivity to argument change
solving the inverse problem: given f ( x ) = y , {\displaystyle f(x)=y,} one is solving for x, and thus the condition number of the (local) inverse must be
Condition_number
About polynomials in several variables
determinant that is a non-zero constant, then the function has a polynomial inverse. The case n = 2 {\displaystyle n=2} (two variables), also called the plane
Jacobian_conjecture
Special state of wave and quantum systems in physics
symmetries of propagating modes in the continuum. Arise when the eigenvalue problem is solved by the Separation of Variables Method, and the wave function
Bound_state_in_the_continuum
an algorithm to solve ill-posed linear inverse problems, and it has been extended to solve non-linear problems that involve constraints. The method was
Landweber_iteration
Science of measuring the shape, orientation, and gravity of Earth
coordinates of that second point. Second geodetic problem (also known as inverse or reverse geodetic problem): given the coordinates of two points, determine
Geodesy
inverse method based on Bayes' theorem. It is used very commonly in the geosciences, particularly for atmospheric sounding. A matrix inverse problem looks
Optimal_estimation
at a time by any sort of a Kalman filter. The computing load of the inverse problem of an ordinary Kalman recursion is roughly proportional to the cube
Fast_Kalman_filter
American mathematician (born 1940)
differential and integral equations, operator theory, ill-posed and inverse problems, scattering theory, functional analysis, spectral theory, numerical
Alexander_Ramm
Geophysical technique for imaging sub-surface structures
work on regularization of inverse problems also worked on this problem. He explains in detail how to solve the ERT problem in a simple case of 2-layered
Electrical resistivity tomography
Electrical_resistivity_tomography
Noninvasive type of medical imaging
problem was posed by Alberto Calderón, and in the mathematical literature of inverse problems it is often referred to as "Calderón's inverse problem"
Electrical impedance tomography
Electrical_impedance_tomography
Topological index of a molecule
related to the closeness centrality of a vertex in a graph, a quantity inversely proportional to the sum of all distances between the given vertex and
Wiener_index
Plane curved by the intersection of an earth ellipsoid and a plane
of geodetic problems involves skew curves known as geodesics. The inverse problem for earth sections is: given two points, P 1 {\displaystyle P_{1}}
Earth_section_paths
Matrix decomposition
(n-m)\times m} . After some algebra, it can be shown that a solution to the inverse problem can be expressed as: x = Q [ ( R 1 T ) − 1 b 0 ] {\displaystyle \mathbf
QR_decomposition
Statistical modeling method
solve an inverse problem whereby a model is adjusted until its parameters have the greatest consistency with experimental data. Inverse problems are found
Reverse_Monte_Carlo
American mathematician (1897–1965)
and is also known as the soap bubble problem. Douglas also made significant contributions to the inverse problem of the calculus of variations. The American
Jesse_Douglas
Study of how humans produce and perceive sounds
Models that assume movements are planned in extrinsic space run into an inverse problem of explaining the muscle and joint locations which produce the observed
Phonetics
Technique to make a model more generalizable and transferable
inverse problems, regularization is a process that converts the answer to a problem to a simpler one. It is often used in solving ill-posed problems or
Regularization_(mathematics)
Question of whether inductive reasoning leads to definitive knowledge
programming – Learning logic programs from data Intuitive statistics Inverse problem – Process of calculating the causal factors that produced a set of
Problem_of_induction
Russian mathematician
paper "A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems". His work with Arkadi Nemirovski in their 1994 book is the first to
Yurii_Nesterov
Estimate object properties from a finite number of projections
Tomographic reconstruction is a type of multidimensional inverse problem where the challenge is to yield an estimate of a specific system from a finite
Tomographic_reconstruction
Issue in determining wave cycle part
In physics, the phase problem is the problem of loss of information concerning the phase that can occur when making a physical measurement. The name comes
Phase_problem
Visualization method for regularization
L-curve is a graphical tool used in the numerical treatment of linear inverse problems. It displays how the size of a regularized solution varies relative
L-curve
Equation giving the form of a central force
+ V {\displaystyle E=T+V} ). The traditional Kepler problem of calculating the orbit of an inverse square law may be read off from the Binet equation as
Binet_equation
Shortest paths on a bounded deformed sphere-like quadric surface
geodesic problems usually considered are: the direct geodesic problem or first geodesic problem, given A, α1, and s12, determine B and α2; the inverse geodesic
Geodesics_on_an_ellipsoid
that were originally defined and studied in the theory of ill-posed inverse problems (for instance, see) focusing on the inversion of a linear operator
Regularization by spectral filtering
Regularization_by_spectral_filtering
Nanoscale magnetic measurement
desirable, as it reduces the number of variables when solving the inverse electromagnetic problem to determine the coordinates of NP sources in the sample. A
Superparamagnetic_relaxometry
Pattern-recognition performance metrics
by Prevalence). Inverse Precision and Inverse Recall are simply the Precision and Recall of the inverse problem where positive and negative labels are
Precision_and_recall
Area of combinatorics in mathematics
mathematics. One major area of study in additive combinatorics are inverse problems: given the size of the sumset A + B is small, what can we say about
Additive_combinatorics
Theorem in mathematics about plurisubharmonic functions
problem as "problème inverse de Hartogs", and could explain why Levi's problem is occasionally referred to as Hartogs' Inverse Problem. Harrington, Phillip
Oka's_lemma
Electrophysiological method to record electrical activity of the brain
causing an EEG change, there may be a false localization due to the inverse problem. EEG poorly measures neural activity that occurs below the upper layers
Electroencephalography
Envelope of light rays reflected or refracted by a curved surface/object
calculations. Caustic engineering describes the process of solving the inverse problem to computer graphics. That is, given a specific image, to determine
Caustic_(optics)
Chilean mathematician
February 1952, Chile) is a mathematician whose research focuses on inverse problems and imaging, microlocal analysis, partial differential equations and
Gunther_Uhlmann
Generalization of the Gaussian measure using the Besov norm
mathematics — specifically, in the fields of probability theory and inverse problems — Besov measures and associated Besov-distributed random variables
Besov_measure
Open problem on 3x+1 and x/2 functions
Unsolved problem in mathematics For even numbers, divide by 2; For odd numbers, multiply by 3 and add 1. With enough repetition, do all positive integers
Collatz_conjecture
Mathematical function
constructive approximation and in linear inverse problems, and as apodization tapers or window functions in quadratic problems of spectral density estimation.
Slepian_function
utilizes phase images, solves the magnetic field to susceptibility source inverse problem, and generates a three-dimensional susceptibility distribution. Due
Quantitative susceptibility mapping
Quantitative_susceptibility_mapping
term used in the field of electrical impedance tomography, if the inverse problem is the estimation of boundary instead of the conductivity distribution
Boundary_estimation_in_EIT
Electrical capacitance tomography (ECT) is a method for determination of the dielectric permittivity distribution in the interior of an object from external
Electrical capacitance tomography
Electrical_capacitance_tomography
Parameter-free superresolution algorithm
variance) is a parameter-free superresolution algorithm for the linear inverse problem in spectral estimation, direction-of-arrival (DOA) estimation and tomographic
SAMV_(algorithm)
Sum of inverse squares of natural numbers
The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed
Basel_problem
Concept in regression analysis mathematics
least squares (RLS) is a family of methods for solving the least-squares problem while using regularization to further constrain the resulting solution
Regularized_least_squares
Study of geometric properties of sets through measure theory
Borel measure for any K {\displaystyle K} . This then induces the inverse problem: Given a Borel measure on the sphere, is it a Gaussian curvature measure
Geometric_measure_theory
based on numerical methods can be applied. Solution of inverse problem Solution of forward problem Electrical resistivity tomography (ERT) Magnetotellurics
Vertical_electrical_sounding
Problem of inverting exponentiation in groups
the worst case using random self-reducibility. At the same time, the inverse problem of discrete exponentiation is not difficult (it can be computed efficiently
Discrete_logarithm
Imaging technique used in seismology
the upper few meters below the surface. Tomography is solved as an inverse problem. Seismic data are compared to an initial Earth model and the model
Seismic_tomography
Algorithmic determination of wave cycle parts
both 1-D and 2-D cases of the phase retrieval problem, including the phaseless 1-D inverse scattering problem, were proven by Klibanov and his collaborators
Phase_retrieval
Constructing a strictly convex compact surface with specified Gaussian curvature
to the Minkowski problem in Euclidean 3-space: restoration of convex shape over the given Gauss surface curvature. The inverse problem of the short-wave
Minkowski_problem
In atmospheric chemistry, differential optical absorption spectroscopy (DOAS) is used to measure concentrations of trace gases. When combined with basic
Differential optical absorption spectroscopy
Differential_optical_absorption_spectroscopy
Method used to solve integrable many-body quantum systems
In quantum physics, the quantum inverse scattering method, similar to the closely related algebraic Bethe ansatz, is a method for solving integrable models
Quantum inverse scattering method
Quantum_inverse_scattering_method
Physics problem related to laws of motion and gravity
three-body problem in classical mechanics is the helium atom, in which a helium nucleus and two electrons interact according to the inverse-square Coulomb
Three-body_problem
Basic method for pseudo-random number sampling
Inverse transform sampling (also known as inversion sampling, the inverse probability integral transform, the inverse transformation method, or the Smirnov
Inverse_transform_sampling
object by solving a nonlinear inverse problem. The nonlinear inverse problem is converted into a linear inverse problem (i.e., Ax=b where A and b are
Microwave_imaging
Science of characterizing uncertainties
necessary, defining an inverse problem for finding the surrogate model that best approximates the simulations. An approach to inverse uncertainty quantification
Uncertainty_quantification
Laws describing planetary orbits
equation gives M as a function of E. Determining E for a given M is the inverse problem. Iterative numerical algorithms are commonly used. Having computed
Kepler's laws of planetary motion
Kepler's_laws_of_planetary_motion
1947 – May 14, 2021) was a mathematician working in the field of inverse problems for partial differential equations and related topics (potential theory
Victor_Isakov
shapes for inverse rendering. In 2025, researchers extended the topological derivative approach to neural network architecture optimization problem. They developed
Topological_derivative
Physical law
In physical science, an inverse-square law is any scientific law stating that the observed "intensity" of a specified physical quantity (being nothing
Inverse-square_law
American mathematician
Cheney (born 1955) is an American mathematician whose research involves inverse problems. She is Yates Chair and Professor of Mathematics at Colorado State
Margaret Cheney (mathematician)
Margaret_Cheney_(mathematician)
Vector field reconstruction is a method of creating a vector field from experimental or computer-generated data, usually with the goal of finding a differential
Vector_field_reconstruction
Process in multicellular organisms
Based Inverse Problems (Ph.D. thesis). University of Iowa. Coskun H, Li Y, Mackey MA (January 2007). "Ameboid cell motility: a model and inverse problem, with
Cell_migration
French econometrician
econometrics of stochastic processes, causality, frontier estimation, and inverse problems. Jean Pierre Florens was born in Marseille in 1947, France. He completed
Jean-Pierre_Florens
Branch of mathematics
ISBN 978-0-444-50871-3. [Newton] immediately realised that quadrature problems (the inverse problems) could be tackled via infinite series: as we would say nowadays
Calculus
Hungarian–American mathematician (born 1969)
differential equations, microlocal analysis, scattering theory, and inverse problems. He is currently a professor of mathematics at Stanford University
András_Vasy
Old term for the probability distribution of an unobserved variable
probability theory, inverse probability is an old term for the probability distribution of an unobserved variable. Today, the problem of determining an
Inverse_probability
Finnish mathematician
University of Helsinki. Päivärinta's research is mostly in the fields of inverse problems and partial differential equations. Päivärinta received an MSc in mathematics
Lassi_Päivärinta
Globe (IPGP), and author of the book Probabilistic Formulation of Inverse Problems (Tarantola, 1987, 2005). Tarantola was the leader of the Geophysical
Albert_Tarantola
1960s artificial intelligence project
There are many other programs today for solving the mass spectrometry inverse problem, see List of mass spectrometry software, but they are no longer described
Dendral
Love. The estimation of G–renewal process parameters is an ill–posed inverse problem, and therefore, the solution may not be unique and is sensitive to
Generalized_renewal_process
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