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Numerical method
The adjoint state method is a numerical method for efficiently computing the gradient of a function or operator in a numerical optimization problem. It
Adjoint_state_method
Linear differential equation
interest can be efficiently calculated by solving the adjoint equation. Methods based on solution of adjoint equations are used in wing shape optimization, fluid
Adjoint_equation
Optimization algorithm for artificial neural networks
Pontryagin and others in optimal control theory, especially the adjoint state method, for being a continuous-time version of backpropagation. Hecht-Nielsen
Backpropagation
Process of calculating the causal factors that produced a set of observations
computation of the Jacobian (often called "Fréchet derivatives"): the adjoint state method, proposed by Chavent and Lions, is aimed to avoid this very heavy
Inverse_problem
Relationship between two functors abstracting many common constructions
this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics
Adjoint_functors
Algorithm for solving systems of linear equations
this algorithm does not require the matrix A {\displaystyle A} to be self-adjoint, but instead one needs to perform multiplications by the conjugate transpose
Biconjugate_gradient_method
Linear operator equal to its own adjoint
In mathematics, a self-adjoint operator on a complex vector space V {\displaystyle V} with inner product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot
Self-adjoint_operator
Problem of finding the optimal shape under given conditions
Lagrange multipliers, like the adjoint state method, can work. Shape optimization can be faced using standard optimization methods if a parametrization of the
Shape_optimization
differentiation Adjoint state method — approximates gradient of a function in an optimization problem Euler–Maclaurin formula Numerical methods for ordinary
List of numerical analysis topics
List_of_numerical_analysis_topics
points. In a somewhat opposite manner, the approximation for the costate (adjoint) is performed using a basis of Lagrange polynomials that includes the final
Gauss_pseudospectral_method
Type of vector space in math
A major application of spectral methods is the spectral mapping theorem, which allows one to apply to a self-adjoint operator T any continuous complex
Hilbert_space
Any entity that can be measured
observables correspond to linear self-adjoint operators on a separable complex Hilbert space representing the quantum state space. Observables assign values
Observable
Numerical calculations carrying along derivatives
Greeks by Algorithmic Differentiation Adjoint Algorithmic Differentiation of a GPU Accelerated Application Adjoint Methods in Computational Finance Software
Automatic_differentiation
Method for approximating eigenvalues
approximate the ground-state eigenfunction. In the context of the finite-element method, it is mathematically the same as the Ritz-Galerkin method. In mechanical
Rayleigh–Ritz_method
Infinite series that is not convergent
In applications, the numbers ai are sometimes the eigenvalues of a self-adjoint operator A with compact resolvent, and f(s) is then the trace of A−s. For
Divergent_series
Operation on self-adjoint operators
constructions, of self-adjoint extensions. This problem arises, for example, when one needs to specify domains of self-adjointness for formal expressions
Extensions of symmetric operators
Extensions_of_symmetric_operators
Topological complex vector space
Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra A of continuous linear
C*-algebra
Measure used in functional analysis
function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space. A projection-valued measure (PVM)
Projection-valued_measure
Solution method for linear differential equations
In mathematical physics, the WKB approximation or WKB method is a technique for finding approximate solutions to linear differential equations with spatially
WKB_approximation
Physics phenomenon
This is self-adjoint and positive and has trace 1. Extending the definition of separability from the pure case, we say that a mixed state is separable
Quantum_entanglement
Quantity used to describe the mathematical state of a dynamical system
added and multiplied, are modelled by using self-adjoint elements from a Cstar_algebra, then a state is a normalized positive element of the algebra's
State_variable
Theorem on boundedness of symmetric operators
operators are necessarily self-adjoint, so this theorem can also be stated as follows: an everywhere-defined self-adjoint operator is bounded. The theorem
Hellinger–Toeplitz_theorem
Optimal control equation
to the state equation used in optimal control. It is also referred to as auxiliary, adjoint, influence, or multiplier equation. It is stated as a vector
Costate_equation
Partial differential equations describing diffusion
its adjoint, the Kolmogorov forward equation, are partial differential equations (PDE) that arise in the theory of continuous-time continuous-state Markov
Kolmogorov backward equations (diffusion)
Kolmogorov_backward_equations_(diffusion)
Russian mathematician
considered a great honour. Dissertation: On the method of orthogonal projections for linear self-adjoint equations, 1947[citation needed] Habilitation:
Mark_Vishik
Software for electromagnetic simulations
frequency-domain solver for steady-state fields and eigenmode expansion. The package was subsequently expanded to include an adjoint solver for topology optimization
Meep_(software)
Field of engineering
Adjoint equation Newton's method Steepest descent Conjugate gradient Sequential quadratic programming Hooke-Jeeves pattern search Nelder-Mead method Genetic
Multidisciplinary design optimization
Multidisciplinary_design_optimization
Einstein–Hermitian vector bundle Deformed Hermitian Yang–Mills equation Hermitian adjoint Hermitian connection, the unique connection on a Hermitian manifold that
List of things named after Charles Hermite
List_of_things_named_after_Charles_Hermite
Aerospace engineer, academic, and author
conjugate heat transfer. The key contribution of his work is the coupled-adjoint method, which computes derivatives of coupled systems efficiently to inform
Joaquim_Martins
Physics of many interacting particles
states) is described by a density operator S, which is a non-negative, self-adjoint, trace-class operator of trace 1 on the Hilbert space H describing the
Statistical_mechanics
Algorithm that estimates unknowns from a series of measurements over time
known as the inverse Wiener-Hopf factor. The backward recursion is the adjoint of the above forward system. The result of the backward pass β k {\displaystyle
Kalman_filter
Mathematical way of attaining a desired output from a dynamic system
transversality conditions). The beauty of using an indirect method is that the state and adjoint (i.e., λ {\displaystyle {\boldsymbol {\lambda }}} ) are solved for
Optimal_control
Concept in quantum mechanics
The term "observable" has gained a technical meaning, denoting a self-adjoint operator that represents the possible results of a random variable. The
Observer_(quantum_physics)
Mathematical tool in quantum physics
a convenient representation for the state of this ensemble. This operator is positive semi-definite, self-adjoint, and has trace one. Conversely, it follows
Density_matrix
Method for solving certain nonlinear partial differential equations
function u ( x , t ) {\textstyle u(x,t)} or its derivatives. The self-adjoint operator L {\textstyle L} has a time derivative L t {\textstyle L_{t}}
Inverse_scattering_transform
Process of developing trajectory performance
trajectory optimization problem with an indirect method, you must explicitly construct the adjoint equations and their gradients. This is often difficult
Trajectory_optimization
Determinant of a subsection of a square matrix
adjunct is not adjugate or adjoint. In modern terminology, the "adjoint" of a matrix most often refers to the corresponding adjoint operator. Submatrix Compound
Minor_(linear_algebra)
Mathematical operation
space, B is a square root of T if T = B* B, where B* denotes the Hermitian adjoint of B.[citation needed] According to the spectral theorem, the continuous
Square_root_of_a_matrix
Description of a quantum-mechanical system
momentum, energy, spin – are represented by observables, which are self-adjoint operators acting on the Hilbert space. A wave function can be an eigenvector
Schrödinger_equation
Relativistic quantum mechanical wave equation
^{0}} from the right, the adjoint Dirac equation can be found, with this being the equation of motion for the Dirac adjoint ψ ¯ = ψ † γ 0 {\displaystyle
Dirac_equation
Integral expressing the amount of overlap of one function as it is shifted over another
Reed, Michael; Simon, Barry (1975), Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, New York-London: Academic Press Harcourt
Convolution
{\displaystyle [x,-]:{\mathfrak {g}}\to {\mathfrak {g}}} (known as the adjoint representation) is a nilpotent endomorphism. It is an elementary fact that
Jacobson–Morozov_theorem
Specific quantum state of a quantum harmonic oscillator
are eigenvectors of the non-self-adjoint annihilation operator â). Thus, if the oscillator is in the quantum state | α ⟩ {\displaystyle |\alpha \rangle
Coherent_state
Complex matrix whose conjugate transpose equals its inverse
quantum mechanics, the conjugate transpose is referred to as the Hermitian adjoint of a matrix and is denoted by a dagger ( † {\displaystyle \dagger } )
Unitary_matrix
Type of continuous linear operator
many nonzero eigenvalues. Thus compact self-adjoint operators behave much like finite-dimensional self-adjoint matrices, except that the eigenvalues may
Compact_operator
Quantum mechanics with supersymmetry
which transforms a "spin up" particle into a "spin down" particle. Its adjoint b † {\displaystyle b^{\dagger }} then transforms a spin down particle into
Supersymmetric quantum mechanics
Supersymmetric_quantum_mechanics
Theory of the strong nuclear interactions
{\displaystyle 3} ; ψ ¯ i {\displaystyle {\bar {\psi }}_{i}\,} is the Dirac adjoint of ψ i {\displaystyle \psi _{i}\,} ; D μ {\displaystyle D_{\mu }} is the
Quantum_chromodynamics
Description of physical properties at the atomic and subatomic scale
which are Hermitian (more precisely, self-adjoint) linear operators acting on the Hilbert space. A quantum state can be an eigenvector of an observable,
Quantum_mechanics
Method of hydrodynamics simulation
Smoothed-particle hydrodynamics (SPH) is a computational method used for simulating the mechanics of continuum media, such as solid mechanics and fluid
Smoothed-particle hydrodynamics
Smoothed-particle_hydrodynamics
Subfield of convex optimization
positive semidefinite, for example, positive semidefinite matrices are self-adjoint matrices that have only non-negative eigenvalues. Denote by S n {\displaystyle
Semidefinite_programming
Correspondence between quantum channels and quantum states
{\displaystyle \vert \psi _{VtU}\rangle _{i}} , where t represents the adjoint operation. By applying the generalised gate teleportation scheme, the states
Choi–Jamiołkowski_isomorphism
Systematic procedure of turning a classical theory into a quantum one
an attempt is made to associate a quantum-mechanical observable (a self-adjoint operator on a Hilbert space) with a real-valued function on classical phase
Quantization_(physics)
Mathematical transform that expresses a function of time as a function of frequency
under the proper conditions it may be expected to result from a self-adjoint generator N {\displaystyle N} via F [ ψ ] = e − i t N ψ . {\displaystyle
Fourier_transform
Formulation of classical mechanics in terms of Hilbert spaces
Hilbert-space classical mechanics, observables are represented by commuting self-adjoint operators acting on the Hilbert space of classical wavefunctions. The commutativity
Koopman–von Neumann classical mechanics
Koopman–von_Neumann_classical_mechanics
Operators useful in quantum mechanics
increases the number of particles in a given state by one, and it is the adjoint of the annihilation operator. In many subfields of physics and chemistry
Creation and annihilation operators
Creation_and_annihilation_operators
Universal construction of a complex Lie group from a real Lie group
(1973) gives a method for explicitly computing the elements in the decomposition. For g in GC set h = g*g. This is a positive self-adjoint operator so its
Complexification_(Lie_group)
Random matrix with gaussian entries
the Gaussian ensembles are specific probability distributions over self-adjoint matrices whose entries are independently sampled from the gaussian distribution
Gaussian_ensemble
Quantum mechanical model
approach, we define the operators a ^ {\displaystyle {\hat {a}}} and its adjoint a ^ † {\displaystyle {\hat {a}}^{\dagger }} , a ^ = m ω 2 ℏ ( x ^ + i m
Quantum_harmonic_oscillator
Concepts from linear algebra
eigenstate of H, and E represents the eigenvalue. H is an observable self-adjoint operator, the infinite-dimensional analog of Hermitian matrices. As in
Eigenvalues_and_eigenvectors
Mathematical structures that allow quantum mechanics to be explained
state. The density operator of a mixed state is a trace class, nonnegative (positive semi-definite) self-adjoint operator ρ {\displaystyle \rho } normalized
Mathematical formulation of quantum mechanics
Mathematical_formulation_of_quantum_mechanics
Interaction of a quantum system with a classical observer
possible state of the physical system. The approach codified by John von Neumann represents a measurement upon a physical system by a self-adjoint operator
Measurement in quantum mechanics
Measurement_in_quantum_mechanics
Foundational principle in quantum physics
self-adjoint operators representing observables are subject to similar uncertainty limits. An eigenstate of an observable represents the state of the
Uncertainty_principle
Branch of applied mathematics
interpretation of states, and evolution and measurements in terms of self-adjoint operators on an infinite-dimensional vector space. That is called Hilbert
Mathematical_physics
Immediate emission of neutrons after nuclear fission
fraction of delayed neutrons weighted (over space, energy, and angle) on the adjoint neutron flux. This concept arises because delayed neutrons are emitted
Prompt_neutron
Analog of the continuous Laplace operator
discrete Laplacian on an infinite grid is of key interest; since it is a self-adjoint operator, it has a real spectrum. For the convention Δ = I − M {\displaystyle
Discrete_Laplace_operator
American mathematician
the Theory of Hilbert Space to Partial Differential Equations; the Self-Adjoint Transformations in Hilbert Space Associated with a Formal Partial Differential
John_Williams_Calkin
Governance position
French term for deputy mayor is maire-adjoint or adjoint au maire [fr]. The first deputy mayor is called premier adjoint. This term should not be confused
Deputy_mayor
Integral transform and linear operator
{\displaystyle L^{p}(\mathbb {R} )} . The Hilbert transform is an anti-self adjoint operator relative to the duality pairing between L p ( R ) {\displaystyle
Hilbert_transform
nationale : l'opposition parlementaire accorde le poste de rapporteur adjoint à Ensemble pour la République". Radio Okapi (in French). 21 April 2024
2023 Democratic Republic of the Congo general election
2023_Democratic_Republic_of_the_Congo_general_election
Partial differential equation
the Kolmogorov backward equation can be deduced. If we instead use the adjoint operator of L {\displaystyle {\mathcal {L}}} , L † {\displaystyle {\mathcal
Fokker–Planck_equation
Construction in functional analysis, useful to solve differential equations
the adjoint of an operator T ∈ B(H), not the transpose, and σ(T*) is not σ(T) but rather its image under complex conjugation. For a self-adjoint T ∈ B(H)
Decomposition of spectrum (functional analysis)
Decomposition_of_spectrum_(functional_analysis)
Mathematical method in functional analysis
∗ S = F S {\displaystyle \Delta =S^{*}S=FS} is a positive (hence, self-adjoint) and densely defined operator called the modular operator. The main result
Tomita–Takesaki_theory
Australian and American mathematician (born 1975)
MR 3469428. S2CID 126089972. Zbl 1342.76029. Fuglede, Bent. Commuting self-adjoint partial differential operators and a group theoretic problem. J. Functional
Terence_Tao
Subatomic particle; lightest meson
that these are understood to belong to the triplet representation or the adjoint representation 3 of SU(2). By contrast, the up and down quarks transform
Pion
Conjecture on zeros of the zeta function
that one way to derive the Riemann hypothesis would be to find a self-adjoint operator, from the existence of which the statement on the real parts of
Riemann_hypothesis
Open source analog electronic circuit simulator
circuit parameter) Noise analysis (a small signal analysis done using an adjoint matrix technique, which sums uncorrelated noise currents at a chosen output
SPICE
American mathematician (1930-2009)
Self Adjoint Operators in Hilbert Space ISBN 0-471-60847-5, Part III Spectral Operators ISBN 0-471-60846-7 J. Schwartz (1956). "Riemann's method in the
Jacob_T._Schwartz
Theorem of convex functions
)y{\bigr )}\leq \lambda f(x)+(1-\lambda )f(y)} for every pair of self‐adjoint operators x and y (with spectra in I) and every scalar λ ∈ [ 0 , 1 ] {\displaystyle
Jensen's_inequality
Czech-American mathematician (1932–2023)
University, where he earned his Ph.D. in 1956 under Donald Spencer ("A Non-Self-Adjoint Boundary Value Problem on Pseudo-Kähler Manifolds"). From 1956 to 1957
Joseph_J._Kohn
Physical phenomenon
of maps. This describes the channel in the Schrödinger picture. Taking adjoint maps in the Heisenberg picture, the success condition becomes ⟨ Φ ( ρ ⊗
Quantum_teleportation
zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator. Lindelöf hypothesis that for all ε > 0 {\displaystyle \varepsilon
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Partial differential equation describing the evolution of temperature in a region
Au(x):=\sum _{i,j}\partial _{x_{i}}a_{ij}(x)\partial _{x_{j}}u(x)} is self-adjoint and dissipative, thus by the spectral theorem it generates a one-parameter
Heat_equation
Statistical mechanics of quantum-mechanical systems
operator, the mathematical representation of a quantum state, is a positive semi-definite, self-adjoint operator of trace one acting on the Hilbert space of
Quantum_statistical_mechanics
Type of entropy in quantum theory
operator, the mathematical representation of a quantum state, is a positive semi-definite, self-adjoint operator of trace one acting on the Hilbert space of
Von_Neumann_entropy
Probability problem
(x)} suggests that μ is the spectral measure of a self-adjoint operator. (More precisely stated, μ is the spectral measure for an operator T ¯ {\displaystyle
Hamburger_moment_problem
Modes of vibration in mathematics
conditions. It can be shown, using the spectral theorem for compact self-adjoint operators that the eigenspaces are finite-dimensional and that the Dirichlet
Dirichlet_eigenvalue
Arithmetic operation
S).} This means the functor "exponentiation to the power T " is a right adjoint to the functor "direct product with T ". This generalizes to the definition
Exponentiation
Functional analysis concept
The spectral theorem for (finite-dimensional) self-adjoint matrices generalizes to compact self-adjoint operators on real or complex Hilbert spaces, namely
Compact operator on Hilbert space
Compact_operator_on_Hilbert_space
German mathematician (1862–1943)
area of functional analysis, particularly of the spectral theory of self-adjoint linear operators, that grew up around it during the 20th century. Until
David_Hilbert
Special class of quantum states
non-relativistic dynamical degree of freedom, we define a pair of self-adjoint position and momentum operators q ^ {\displaystyle {\hat {q}}} and p ^
Gaussian_state
Fundamental operation on complex numbers
conjugate transpose (or adjoint) of complex matrices generalizes complex conjugation. Even more general is the concept of adjoint operator for operators
Complex_conjugate
Type theory in logic and mathematics
sense), based on the idea from category theory of improving equivalences to adjoint equivalences. Soon afterwards, Voevodsky proved that the univalence axiom
Homotopy_type_theory
Formulation of the quantum many-body problem
Michael; Simon, Barry (1975). Methods of Modern Mathematical Physics. Volume II: Fourier Analysis, Self-Adjointness. San Diego: Academic Press. p. 328
Second_quantization
Basic circuit in quantum computing
( − φ ) {\displaystyle P^{\dagger }(\varphi )=P(-\varphi )} . The two adjoint (or conjugate transpose) gates S † {\displaystyle S^{\dagger }} and T †
Quantum_logic_gate
Embedding of categories into functor categories
the theory. This approach is akin to (and in fact generalizes) the common method of studying a ring by investigating the modules over that ring. The ring
Yoneda_lemma
Theory of logic to account for observations from quantum theory
article assumes the reader is familiar with the spectral theory of self-adjoint operators on a Hilbert space. However, the main ideas can be understood
Quantum_logic
Raising and lowering operators in quantum mechanics
If N is a Hermitian operator, then c must be real, and the Hermitian adjoint of X obeys the commutation relation [ N , X † ] = − c X † . {\displaystyle
Ladder_operator
Generalized measurement in quantum mechanics
defined on M {\displaystyle M} whose values are positive bounded self-adjoint operators on H {\displaystyle {\mathcal {H}}} such that for every ψ ∈ H
POVM
*-algebra of bounded operators on a Hilbert space
its projections; this is a consequence of the spectral theorem for self-adjoint operators. The projections of a finite factor form a continuous geometry
Von_Neumann_algebra
Imaging technique used in seismology
past several decades since its initial conception. The development of adjoint inversions, which are able to combine several different types of seismic
Seismic_tomography
ADJOINT STATE-METHOD
ADJOINT STATE-METHOD
Boy/Male
Arabic
State; Condition
Boy/Male
Arabic
Power; State
Boy/Male
Hindu, Indian
State; Country
Female
Irish
Variant spelling of Irish Éadan, ÉADAOIN means "face" or perhaps "against" or "opposite."
Girl/Female
Latin
Adroit; skillful.
Boy/Male
Celebrity, Hindu, Indian, Telugu
State
Male
English
English surname transferred to unisex forename use, TATE means "cheerful."
Boy/Male
American, Anglo, Australian, British, Chinese, Christian, English, Finnish, German, Indian, Irish, Norse, Scandinavian
To be Cheerful; Great; Measure of Land; Great Talker
Surname or Lastname
English
English : metonymic occupational name for a slater, from Middle English slate ‘slate’.
Surname or Lastname
English
English : from the Old English personal name TÄta, possibly a short form of various compound names with the obscure first element tÄt, or else a nursery formation. This surname is common and widespread in Britain; the chief area of concentration is northeastern England, followed by northern Ireland.
Boy/Male
Arabic
Leadership; State
Girl/Female
Indian, Kashmiri
State Honour
Boy/Male
English Scandinavian American Irish Native American
Cheerful.
Boy/Male
Arabic
State; Dignity
Surname or Lastname
English
English : unexplained.
Boy/Male
Arabic
Power; State
Girl/Female
English Scandinavian Anglo Saxon Irish
Brings joy.
Girl/Female
American, Anglo, Australian, British, English, Finnish, Irish, Scandinavian
Light Hearted; Cheerful; Pleasant and Bright; Brings Joy; Bright; Great; Measure of Land
Surname or Lastname
English and Irish
English and Irish : variant of Stacey.
Boy/Male
Arabic
Leadership; State
ADJOINT STATE-METHOD
ADJOINT STATE-METHOD
Boy/Male
French, German, Teutonic
Fighter; Army People; Famous Warrior
Boy/Male
Hindu
Heaven
Girl/Female
Indian
Boy/Male
Tamil
Arjwin | à®…à®°à¯à®œà¯à®µà¯€à®¨Â
Boy/Male
Indian
King
Boy/Male
English
Stiles.
Girl/Female
Australian, Chinese, Scandinavian, Vietnamese
Wave Bright; Tall One; Sweet Potato
Surname or Lastname
English
English : patronymic from Maw 1.English : metronymic from a form of Mould 1.
Male
Italian
Italian form of English Harold, AROLDO means "army leader."
Boy/Male
Hindu
Beauty, Son of Ashim
ADJOINT STATE-METHOD
ADJOINT STATE-METHOD
ADJOINT STATE-METHOD
ADJOINT STATE-METHOD
ADJOINT STATE-METHOD
imp. & p. p.
of Adjoin
imp. & p. p.
of State
n.
One who states.
n.
One of several marked phases or periods in the development and growth of many animals and plants; as, the larval stage; pupa stage; zoea stage.
v. t.
To express the particulars of; to set down in detail or in gross; to represent fully in words; to narrate; to recite; as, to state the facts of a case, one's opinion, etc.
n.
See Skate, for the foot.
n.
Estate; state.
n.
Estate, possession.
v. t.
To exhibit upon a stage, or as upon a stage; to display publicly.
v. i.
To lie or be next, or in contact; to be contiguous; as, the houses adjoin.
n.
The state; the general body politic; the common-wealth; the general interest; state affairs.
a.
Recurring at regular time; not occasional; as, stated preaching; stated business hours.
v. t.
To reunite the joints of; to joint anew.
v. t.
To endow with an estate.
a.
Belonging to the state, or body politic; public.
n.
Rank; condition; quality; as, the state of honor.
n.
Any body of men united by profession, or constituting a community of a particular character; as, the civil and ecclesiastical states, or the lords spiritual and temporal and the commons, in Great Britain. Cf. Estate, n., 6.
n.
The bodies that constitute the legislature of a country; as, the States-general of Holland.
v. t.
To cover with slate, or with a substance resembling slate; as, to slate a roof; to slate a globe.
v. t.
To place, as a statue; to form a statue of; to make into a statue.