Quantification of margins and uncertainties

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https://en.wikipedia.org/wiki/Quantification_of_margins_and_uncertainties

Quantification of Margins and Uncertainty (QMU) is a decision support methodology for complex technical decisions. QMU focuses on the identification, characterization, and analysis of performance thresholds and their associated margins for engineering systems that are evaluated under conditions of uncertainty, particularly when portions of those results are generated using computational modeling and simulation. QMU has traditionally been applied to complex systems where comprehensive experimental test data is not readily available and cannot be easily generated for either end-to-end system execution or for specific subsystems of interest. Examples of systems where QMU has been applied include nuclear weapons performance, qualification, and stockpile assessment. QMU focuses on characterizing in detail the various sources of uncertainty that exist in a model, thus allowing the uncertainty in the system response output variables to be well quantified. These sources are frequently described in terms of probability distributions to account for the stochastic nature of complex engineering systems. The characterization of uncertainty supports comparisons of design margins for key system performance metrics to the uncertainty associated with their calculation by the model. QMU supports risk-informed decision-making processes where computational simulation results provide one of several inputs to the decision-making authority. There is currently no standardized methodology across the simulation community for conducting QMU; the term is applied to a variety of different modeling and simulation techniques that focus on rigorously quantifying model uncertainty in order to support comparison to design margins.

History

The fundamental concepts of QMU were originally developed concurrently at several national laboratories supporting nuclear weapons programs in the late 1990s, including Lawrence Livermore National Laboratory, Sandia National Laboratory, and Los Alamos National Laboratory. The original focus of the methodology was to support nuclear stockpile decision-making, an area where full experimental test data could no longer be generated for validation due to bans on nuclear weapons testing. The methodology has since been applied in other applications where safety or mission critical decisions for complex projects must be made using results based on modeling and simulation. Examples outside of the nuclear weapons field include applications at NASA for interplanetary spacecraft and rover development, missile six-degree-of-freedom (6DOF) simulation results, and characterization of material properties in terminal ballistic encounters.

Overview

QMU focuses on quantification of the ratio of design margin to model output uncertainty. The process begins with the identification of the key performance thresholds for the system, which can frequently be found in the systems requirements documents. These thresholds (also referred to as performance gates) can specify an upper bound of performance, a lower bound of performance, or both in the case where the metric must remain within the specified range. For each of these performance thresholds, the associated performance margin must be identified. The margin represents the targeted range the system is being designed to operate in to safely avoid the upper and lower performance bounds. These margins account for aspects such as the design safety factor the system is being developed to as well as the confidence level in that safety factor. QMU focuses on determining the quantified uncertainty of the simulation results as they relate to the performance threshold margins. This total uncertainty includes all forms of uncertainty related to the computational model as well as the uncertainty in the threshold and margin values. The identification and characterization of these values allows the ratios of margin-to-uncertainty (M/U) to be calculated for the system. These M/U values can serve as quantified inputs that can help authorities make risk-informed decisions regarding how to interpret and act upon results based on simulations.

Overview of General QMU Process.

QMU recognizes that there are multiple types of uncertainty that propagate through a model of a complex system. The simulation in the QMU process produces output results for the key performance thresholds of interest, known as the Best Estimate Plus Uncertainty (BE+U). The best estimate component of BE+U represents the core information that is known and understood about the model response variables. The basis that allows high confidence in these estimates is usually ample experimental test data regarding the process of interest which allows the simulation model to be thoroughly validated.

The types of uncertainty that contribute to the value of the BE+U can be broken down into several categories:

• Aleatory uncertainty: This type of uncertainty is naturally present in the system being modeled and is sometimes known as “irreducible uncertainty” and “stochastic variability.” Examples include processes that are naturally stochastic such as wind gust parameters and manufacturing tolerances.
• Epistemic uncertainty: This type of uncertainty is due to a lack of knowledge about the system being modeled and is also known as “reducible uncertainty.” Epistemic uncertainty can result from uncertainty about the correct underlying equations of the model, incomplete knowledge of the full set of scenarios to be encountered, and lack of experimental test data defining the key model input parameters.

The system may also suffer from requirements uncertainty related to the specified thresholds and margins associated with the system requirements. QMU acknowledges that in some situations, the system designer may have high confidence in what the correct value for a specific metric may be, while at other times, the selected value may itself suffer from uncertainty due to lack of experience operating in this particular regime. QMU attempts to separate these uncertainty values and quantify each of them as part of the overall inputs to the process.

QMU can also factor in human error in the ability to identify the unknown unknowns that can affect a system. These errors can be quantified to some degree by looking at the limited experimental data that may be available for previous system tests and identifying what percentage of tests resulted in system thresholds being exceeded in an unexpected manner. This approach attempts to predict future events based on the past occurrences of unexpected outcomes.

The underlying parameters that serve as inputs to the models are frequently modeled as samples from a probability distribution. The input parameter model distributions as well as the model propagation equations determine the distribution of the output parameter values. The distribution of a specific output value must be considered when determining what is an acceptable M/U ratio for that performance variable. If the uncertainty limit for U includes a finite upper bound due to the particular distribution of that variable, a lower M/U ratio may be acceptable. However, if U is modeled as a normal or exponential distribution which can potentially include outliers from the far tails of the distribution, a larger value may be required in order to reduce system risk to an acceptable level.

Ratios of acceptable M/U for safety critical systems can vary from application to application. Studies have cited acceptable M/U ratios as being in the 2:1 to 10:1 range for nuclear weapons stockpile decision-making. Intuitively, the larger the value of M/U, the less of the available performance margin is being consumed by uncertainty in the simulation outputs. A ratio of 1:1 could result in a simulation run where the simulated performance threshold is not exceeded when in actuality the entire design margin may have been consumed. It is important to note that rigorous QMU does not ensure that the system itself is capable of meeting its performance margin; rather, it serves to ensure that the decision-making authority can make judgments based on accurately characterized results.

The underlying objective of QMU is to present information to decision-makers that fully characterizes the results in light of the uncertainty as understood by the model developers. This presentation of results allows decision makers an opportunity to make informed decisions while understanding what sensitivities exist in the results due to the current understanding of uncertainty. Advocates of QMU recognize that decisions for complex systems cannot be made strictly based on the quantified M/U metrics. Subject matter expert (SME) judgment and other external factors such as stakeholder opinions and regulatory issues must also be considered by the decision-making authority before a final outcome is decided.

Verification and validation

Verification and validation (V & V) of a model is closely interrelated with QMU. Verification is broadly acknowledged as the process of determining if a model was built correctly; validation activities focus on determining if the correct model was built. V&V against available experimental test data is an important aspect of accurately characterizing the overall uncertainty of the system response variables. V&V seeks to make maximum use of component and subsystem-level experimental test data to accurately characterize model input parameters and the physics-based models associated with particular sub-elements of the system. The use of QMU in the simulation process helps to ensure that the stochastic nature of the input variables (due to both aleatory and epistemic uncertainties) as well as the underlying uncertainty in the model are properly accounted for when determining the simulation runs required to establish model credibility prior to accreditation.

Advantages and disadvantages

QMU has the potential to support improved decision-making for programs that must rely heavily on modeling and simulation. Modeling and simulation results are being used more often during the acquisition, development, design, and testing of complex engineering systems. One of the major challenges of developing simulations is to know how much fidelity should be built into each element of the model. The pursuit of higher fidelity can significantly increase development time and total cost of the simulation development effort. QMU provides a formal method for describing the required fidelity relative to the design threshold margins for key performance variables. This information can also be used to prioritize areas of future investment for the simulation. Analysis of the various M/U ratios for the key performance variables can help identify model components that are in need of fidelity upgrades to order to increase simulation effectiveness.

A variety of potential issues related to the use of QMU have also been identified. QMU can lead to longer development schedules and increased development costs relative to traditional simulation projects due to the additional rigor being applied. Proponents of QMU state that the level of uncertainty quantification required is driven by certification requirements for the intended application of the simulation. Simulations used for capability planning or system trade analyses must generally model the overall performance trends of the systems and components being analyzed. However, for safety-critical systems where experimental test data is lacking, simulation results provide a critical input to the decision-making process. Another potential risk related to the use of QMU is a false sense of confidence regarding protection from unknown risks. The use of quantified results for key simulation parameters can lead decision makers to believe all possible risks have been fully accounted for, which is particularly challenging for complex systems. Proponents of QMU advocate for a risk-informed decision-making process to counter this risk; in this paradigm, M/U results as well as SME judgment and other external factors are always factored into the final decision.

Poincaré group

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https://en.wikipedia.org/wiki/Poincar%C3%A9_group

Henri Poincaré

The Poincaré group, named after Henri Poincaré (1905), was first defined by Hermann Minkowski (1908) as the isometry group of Minkowski spacetime. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our understanding of the most basic fundamentals of physics.

Overview

The Poincaré group consists of all coordinate transformations of Minkowski space that do not change the spacetime interval between events. For example, if everything were postponed by two hours, including the two events and the path you took to go from one to the other, then the time interval between the events recorded by a stopwatch that you carried with you would be the same. Or if everything were shifted five kilometres to the west, or turned 60 degrees to the right, you would also see no change in the interval. It turns out that the proper length of an object is also unaffected by such a shift.

In total, there are ten degrees of freedom for such transformations. They may be thought of as translation through time or space (four degrees, one per dimension); reflection through a plane (three degrees, the freedom in orientation of this plane); or a "boost" in any of the three spatial directions (three degrees). Composition of transformations is the operation of the Poincaré group, with rotations being produced as the composition of an even number of reflections.

In classical physics, the Galilean group is a comparable ten-parameter group that acts on absolute time and space. Instead of boosts, it features shear mappings to relate co-moving frames of reference.

In general relativity, i.e. under the effects of gravity, Poincaré symmetry applies only locally. A treatment of symmetries in general relativity is not in the scope of this article.

Poincaré symmetry

Poincaré symmetry is the full symmetry of special relativity. It includes:

• translations (displacements) in time and space, forming the abelian Lie group of spacetime translations (P);
• rotations in space, forming the non-abelian Lie group of three-dimensional rotations (J);
• boosts, transformations connecting two uniformly moving bodies (K).

The last two symmetries, J and K, together make the Lorentz group (see also Lorentz invariance); the semi-direct product of the spacetime translations group and the Lorentz group then produce the Poincaré group. Objects that are invariant under this group are then said to possess Poincaré invariance or relativistic invariance.

10 generators (in four spacetime dimensions) associated with the Poincaré symmetry, by Noether's theorem, imply 10 conservation laws:

• 1 for the energy – associated with translations through time
• 3 for the momentum – associated with translations through spatial dimensions
• 3 for the angular momentum – associated with rotations between spatial dimensions
• 3 for a quantity involving the velocity of the center of mass – associated with hyperbolic rotations between each spatial dimension and time

Poincaré group

The Poincaré group is the group of Minkowski spacetime isometries. It is a ten-dimensional noncompact Lie group. The four-dimensional abelian group of spacetime translations is a normal subgroup, while the six-dimensional Lorentz group is also a subgroup, the stabilizer of the origin. The Poincaré group itself is the minimal subgroup of the affine group which includes all translations and Lorentz transformations. More precisely, it is a semidirect product of the spacetime translations group and the Lorentz group,

${\displaystyle {\mathbf{R}}^{1,3}⋊\mathrm{O}(1,3),}$

with group multiplication

${\displaystyle (\alpha ,f)\cdot (\beta ,g)=(\alpha +f\cdot \beta ,f\cdot g)}$.

Another way of putting this is that the Poincaré group is a group extension of the Lorentz group by a vector representation of it; it is sometimes dubbed, informally, as the inhomogeneous Lorentz group. In turn, it can also be obtained as a group contraction of the de Sitter group SO(4, 1) ~ Sp(2, 2), as the de Sitter radius goes to infinity.

Its positive energy unitary irreducible representations are indexed by mass (nonnegative number) and spin (integer or half integer) and are associated with particles in quantum mechanics (see Wigner's classification).

In accordance with the Erlangen program, the geometry of Minkowski space is defined by the Poincaré group: Minkowski space is considered as a homogeneous space for the group.

In quantum field theory, the universal cover of the Poincaré group

${\displaystyle {\mathbf{R}}^{1,3}⋊\mathrm{SL}(2,\mathbf{C}),}$

which may be identified with the double cover

${\displaystyle {\mathbf{R}}^{1,3}⋊\mathrm{Spin}(1,3),}$

is more important, because representations of ${\displaystyle \mathrm{SO}(1,3)}$ are not able to describe fields with spin 1/2; i.e. fermions. Here ${\displaystyle \mathrm{SL}(2,\mathbf{C})}$ is the group of complex ${\displaystyle 2\times 2}$ matrices with unit determinant, isomorphic to the Lorentz-signature spin group ${\displaystyle \mathrm{Spin}(1,3)}$.

Poincaré algebra

Lie groups and Lie algebras

The Poincaré algebra is the Lie algebra of the Poincaré group. It is a Lie algebra extension of the Lie algebra of the Lorentz group. More specifically, the proper ($det\mathrm{\Lambda }=1$), orthochronous (${{\mathrm{\Lambda }}^0}_0\ge 1$) part of the Lorentz subgroup (its identity component), $\mathrm{S}\mathrm{O}(1,3{)}_{+}^{\uparrow }$, is connected to the identity and is thus provided by the exponentiation $\exp \left(i{a}_{\mu }{P}^{\mu }\right)\exp \left(\frac{i}{2}{\omega }_{\mu \nu }{M}^{\mu \nu }\right)$ of this Lie algebra. In component form, the Poincaré algebra is given by the commutation relations:

${\displaystyle \begin{array}{rl}[P_{\mu },P_{\nu }]& =0\\ \frac{1}{i}[M_{\mu \nu },P_{\rho }]& ={\eta }_{\mu \rho }{P}_{\nu }-{\eta }_{\nu \rho }{P}_{\mu }\\ \frac{1}{i}[M_{\mu \nu },M_{\rho \sigma }]& ={\eta }_{\mu \rho }{M}_{\nu \sigma }-{\eta }_{\mu \sigma }{M}_{\nu \rho }-{\eta }_{\nu \rho }{M}_{\mu \sigma }+{\eta }_{\nu \sigma }{M}_{\mu \rho },\end{array}}$

where ${\displaystyle P}$ is the generator of translations, ${\displaystyle M}$ is the generator of Lorentz transformations, and ${\displaystyle \eta }$ is the ${\displaystyle (+,-,-,-)}$ Minkowski metric (see Sign convention).

A diagram of the commutation structure of the Poincaré algebra. The edges of the diagram connect generators with nonzero commutators.

The bottom commutation relation is the ("homogeneous") Lorentz group, consisting of rotations, $J_i=\frac{1}{2}{ϵ}_{imn}{M}^{mn}$, and boosts, $K_i=M_{i0}$. In this notation, the entire Poincaré algebra is expressible in noncovariant (but more practical) language as

${\displaystyle \begin{array}{rl}[J_m,P_n]& =i{ϵ}_{mnk}{P}_k,\\ [J_i,P_0]& =0,\\ [K_i,P_k]& =i{\eta }_{ik}{P}_0,\\ [K_i,P_0]& =-i{P}_i,\\ [J_m,J_n]& =i{ϵ}_{mnk}{J}_k,\\ [J_m,K_n]& =i{ϵ}_{mnk}{K}_k,\\ [K_m,K_n]& =-i{ϵ}_{mnk}{J}_k,\end{array}}$

where the bottom line commutator of two boosts is often referred to as a "Wigner rotation". The simplification $[J_m+i{K}_m,J_n-i{K}_n]=0$ permits reduction of the Lorentz subalgebra to $\mathfrak{s}\mathfrak{u}(2)\oplus \mathfrak{s}\mathfrak{u}(2)$ and efficient treatment of its associated representations. In terms of the physical parameters, we have

${\displaystyle \begin{array}{rl}\left[\mathcal{H},p_i\right]& =0\\ \left[\mathcal{H},L_i\right]& =0\\ \left[\mathcal{H},K_i\right]& =i\hslash c{p}_i\\ \left[p_i,p_j\right]& =0\\ \left[p_i,L_j\right]& =i\hslash {ϵ}_{ijk}{p}_k\\ \left[p_i,K_j\right]& =\frac{i\hslash }{c}\mathcal{H}{\delta }_{ij}\\ \left[L_i,L_j\right]& =i\hslash {ϵ}_{ijk}{L}_k\\ \left[L_i,K_j\right]& =i\hslash {ϵ}_{ijk}{K}_k\\ \left[K_i,K_j\right]& =-i\hslash {ϵ}_{ijk}{L}_k\end{array}}$

The Casimir invariants of this algebra are $P_{\mu }{P}^{\mu }$ and $W_{\mu }{W}^{\mu }$ where $W_{\mu }$ is the Pauli–Lubanski pseudovector; they serve as labels for the representations of the group.

The Poincaré group is the full symmetry group of any relativistic field theory. As a result, all elementary particles fall in representations of this group. These are usually specified by the four-momentum squared of each particle (i.e. its mass squared) and the intrinsic quantum numbers $J^{PC}$, where ${\displaystyle J}$ is the spin quantum number, ${\displaystyle P}$ is the parity and ${\displaystyle C}$ is the charge-conjugation quantum number. In practice, charge conjugation and parity are violated by many quantum field theories; where this occurs, ${\displaystyle P}$ and ${\displaystyle C}$ are forfeited. Since CPT symmetry is invariant in quantum field theory, a time-reversal quantum number may be constructed from those given.

As a topological space, the group has four connected components: the component of the identity; the time reversed component; the spatial inversion component; and the component which is both time-reversed and spatially inverted.

Other dimensions

The definitions above can be generalized to arbitrary dimensions in a straightforward manner. The d-dimensional Poincaré group is analogously defined by the semi-direct product

${\displaystyle \mathrm{IO}(1,d-1):={\mathbf{R}}^{1,d-1}⋊\mathrm{O}(1,d-1)}$

with the analogous multiplication

${\displaystyle (\alpha ,f)\cdot (\beta ,g)=(\alpha +f\cdot \beta ,f\cdot g)}$.

The Lie algebra retains its form, with indices µ and ν now taking values between 0 and d − 1. The alternative representation in terms of J_i and K_i has no analogue in higher dimensions.

Exotic star

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Exotic_star

An exotic star is a hypothetical compact star composed of exotic matter (something not made of electrons, protons, neutrons, or muons), and balanced against gravitational collapse by degeneracy pressure or other quantum properties.

Types of exotic stars include

• quark stars (composed of quarks)
• strange stars (composed of strange quark matter, a condensate of up, down, and strange quarks)
• § Preon stars (speculative material composed of preons, which are hypothetical particles and "building blocks" of quarks and leptons, should quarks be decomposable into component sub-particles).

Of the various types of exotic star proposed, the most well evidenced and understood is the quark star, although its existence is not confirmed.

Quark stars and strange stars

Main articles: Quark star and Strange star

A quark star is a hypothesized object that results from the decomposition of neutrons into their constituent up and down quarks under gravitational pressure. It is expected to be smaller and denser than a neutron star, and may survive in this new state indefinitely, if no extra mass is added. Effectively, it is a single, very large hadron. Quark stars that contain strange matter are called strange stars.

Based on observations released by the Chandra X-Ray Observatory on 10 April 2002, two objects, named RX J1856.5−3754 and 3C 58, were suggested as quark star candidates. The former appeared to be much smaller and the latter much colder than expected for a neutron star, suggesting that they were composed of material denser than neutronium. However, these observations were met with skepticism by researchers who said the results were not conclusive. After further analysis, RX J1856.5−3754 was excluded from the list of quark star candidates.

Electroweak stars

An electroweak star is a hypothetical type of exotic star in which the gravitational collapse of the star is prevented by radiation pressure resulting from electroweak burning; that is, the energy released by the conversion of quarks into leptons through the electroweak force. This proposed process might occur in a volume at the star's core approximately the size of an apple, containing about two Earth masses, and reaching temperatures on the order of 10¹⁵ K (1 PK). Electroweak stars could be identified through the equal number of neutrinos emitted of all three generations, taking into account neutrino oscillation.

Preon stars

A preon star is a proposed type of compact star made of preons, a group of hypothetical subatomic particles. Preon stars would be expected to have huge densities, exceeding 10²³ kg/m³. They may have greater densities than quark stars, and they would be heavier but smaller than white dwarfs and neutron stars.

Boson stars

Conventional stars are formed from mostly protons and electrons, which are fermions, but also contain a large proportion of helium-4 nuclei, which are bosons, and smaller amounts of various heavier nuclei, which can be either. A boson star is a hypothetical astronomical object formed out of particles called bosons For this type of star to exist, there must be a stable type of boson with self-repulsive interaction; one possible candidate particle is the still-hypothetical "axion" (which is also a candidate for the not-yet-detected "non-baryonic dark matter" particles, which appear to compose roughly 25% of the mass of the Universe). It is theorized that unlike normal stars (which emit radiation due to gravitational pressure and nuclear fusion), boson stars would be transparent and invisible. The immense gravity of a compact boson star would bend light around the object, creating an empty region resembling the shadow of a black hole's event horizon. Like a black hole, a boson star would absorb ordinary matter from its surroundings, but because of the transparency, matter (which would probably heat up and emit radiation) would be visible at its center. Rotating boson star models are also possible. Unlike black holes these have quantized angular momentum, and their energy density profiles are torus-shaped, which can be understood as a result of deformation due to centrifugal forces.

There is no significant evidence that such stars exist. However, it may become possible to detect them by the gravitational radiation emitted by a pair of co-orbiting boson stars. GW190521, thought to be the most energetic black hole merger ever recorded, may be the head-on collision of two boson stars. In addition, gravitational wave signals from compact binary boson star mergers can be degenerate with those from black hole mergers, suggesting that some gravitational wave observations interpreted as originating in a black hole binary could really originate in a boson star binary. The invisible companion to a Sun-like star identified by Gaia mission could be a black hole or either a boson star or an exotic star of other types.

Boson stars may have formed through gravitational collapse during the primordial stages of the Big Bang. At least in theory, a supermassive boson star could exist at the core of a galaxy, which may explain many of the observed properties of active galactic cores. However, more recent general-relativistic magnetohydrodynamic simulations, combined with imaging performed by the Event Horizon Telescope, is believed to have largely ruled out the possibility that Sagittarius A*, the supermassive object at the center of the Milky Way, could be a boson star.

Bound states in cosmological bosonic fields have also been proposed as an alternative to dark matter. The dark matter haloes surrounding most galaxies might be viewed as enormous "boson stars."

Compact boson stars and boson shells are often modelled using massive bosonic fields, such as complex scalar fields and U(1) gauge fields, coupled to gravity. The presence of a positive or negative cosmological constant in the theory facilitates a study of these objects in de Sitter and anti-de Sitter spaces.

By changing the potential associated with the matter model, different families of boson star models can be obtained. The so-called solitonic potential, which introduces a degenerate vacuum state at a finite value of the field amplitude, can be used to construct boson star models so compact that they possess a pair of photon orbits, one of which is stable. Because they trap light, such boson stars could mimic much of the observational phenomenology of black holes.

Boson stars composed of elementary particles with spin-1 have been labelled Proca stars.

Planck stars

Main article: Planck star

In loop quantum gravity, a Planck star is a hypothetically possible astronomical object that is created when the energy density of a collapsing star reaches the Planck energy density. Under these conditions, assuming gravity and spacetime are quantized, there arises a repulsive "force" derived from Heisenberg's uncertainty principle. In other words, if gravity and spacetime are quantized, the accumulation of mass-energy inside the Planck star cannot collapse beyond this limit to form a gravitational singularity because it would violate the uncertainty principle for spacetime itself.

Q-stars

Q-stars are hypothetical objects that originate from supernovae or the big bang. They are theorized to be massive enough to bend space-time to a degree such that some, but not all light could escape from its surface. These are predicted to be denser than neutron stars or even quark stars.

Dark stars

In Newtonian mechanics, objects dense enough to trap any emitted light are called dark stars, as opposed to black holes in general relativity. However, the same name is used for hypothetical ancient "stars" which derived energy from dark matter. Quantum effects may prevent true black holes from forming and give rise instead to dense entities called black stars.