scholarly journals Power1D: a Python toolbox for numerical power estimates in experiments involving one-dimensional continua

2017 ◽  
Vol 3 ◽  
pp. e125 ◽  
Author(s):  
Todd C. Pataky
Keyword(s):  
Analytical Solutions ◽  
Approximate Solutions ◽  
Random Field Theory ◽  
One Dimensional ◽  
Signal Noise ◽  
Noise Modeling ◽  
Flexible Modeling ◽  
Broad Signal ◽  
The One ◽  
High Level

The unit of experimental measurement in a variety of scientific applications is the one-dimensional (1D) continuum: a dependent variable whose value is measured repeatedly, often at regular intervals, in time or space. A variety of software packages exist for computing continuum-level descriptive statistics and also for conducting continuum-level hypothesis testing, but very few offer power computing capabilities, where ‘power’ is the probability that an experiment will detect a true continuum signal given experimental noise. Moreover, no software package yet exists for arbitrary continuum-level signal/noise modeling. This paper describes a package called power1d which implements (a) two analytical 1D power solutions based on random field theory (RFT) and (b) a high-level framework for computational power analysis using arbitrary continuum-level signal/noise modeling. First power1d’s two RFT-based analytical solutions are numerically validated using its random continuum generators. Second arbitrary signal/noise modeling is demonstrated to show how power1d can be used for flexible modeling well beyond the assumptions of RFT-based analytical solutions. Its computational demands are non-excessive, requiring on the order of only 30 s to execute on standard desktop computers, but with approximate solutions available much more rapidly. Its broad signal/noise modeling capabilities along with relatively rapid computations imply that power1d may be a useful tool for guiding experimentation involving multiple measurements of similar 1D continua, and in particular to ensure that an adequate number of measurements is made to detect assumed continuum signals.

scholarly journals Optimising Matrix Product State Simulations of Shor's Algorithm

Quantum ◽  
2019 ◽  
Vol 3 ◽  
pp. 116 ◽  
Author(s):  
Aidan Dang ◽  
Charles D. Hill ◽  
Lloyd C. L. Hollenberg
Keyword(s):  
Dimensional Structure ◽  
Matrix Product ◽  
Product State ◽  
One Dimensional ◽  
Matrix Product State ◽  
Shor's Algorithm ◽  
The One ◽  
High Level ◽  
Space Requirements ◽  
Factoring Algorithm

We detail techniques to optimise high-level classical simulations of Shor's quantum factoring algorithm. Chief among these is to examine the entangling properties of the circuit and to effectively map it across the one-dimensional structure of a matrix product state. Compared to previous approaches whose space requirements depend on r, the solution to the underlying order-finding problem of Shor's algorithm, our approach depends on its factors. We performed a matrix product state simulation of a 60-qubit instance of Shor's algorithm that would otherwise be infeasible to complete without an optimised entanglement mapping.

Including lateral velocity variations into true-amplitude common-shot wave-equation migration

Geophysics ◽  
2010 ◽  
Vol 75 (5) ◽  
pp. S175-S186 ◽  
Author(s):  
Daniela Amazonas ◽  
Rafael Aleixo ◽  
Gabriela Melo ◽  
Jörg Schleicher ◽  
Amélia Novais ◽  
...  
Keyword(s):  
Wave Equations ◽  
Heterogeneous Media ◽  
Synthetic Data ◽  
Approximate Solutions ◽  
Vertical Gradient ◽  
Correct Description ◽  
Lateral Velocity ◽  
One Dimensional ◽  
Velocity Variations ◽  
The One

In heterogeneous media, standard one-way wave equations describe only the kinematic part of one-way wave propagation correctly. For a correct description of amplitudes, the one-way wave equations must be modified. In media with vertical velocity variations only, the resulting true-amplitude one-way wave equations can be solved analytically. In media with lateral velocity variations, these equations are much harder to solve and require sophisticated numerical techniques. We present an approach to circumvent these problems by implementing approximate solutions based on the one-dimensional analytic amplitude modifications. We use these approximations to show how to modify conventional migration methods such as split-step and Fourier finite-difference migrations in such a way that they more accurately handle migration amplitudes. Simple synthetic data examples in media with a constant vertical gradient demonstrate that the correction achieves the recovery of true migration amplitudes. Applications to the SEG/EAGE salt model and the Marmousi data show that the technique improves amplitude recovery in the migrated images in more realistic situations.

scholarly journals Construction of a global solution for the one dimensional singularly-perturbed boundary value problem

2017 ◽  
Vol 8 (1-2) ◽  
pp. 52
Author(s):  
Samir Karasuljic ◽  
Enes Duvnjakovic ◽  
Vedad Pasic ◽  
Elvis Barakovic
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Boundary Value Problem ◽  
Uniform Convergence ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Singularly Perturbed ◽  
One Dimensional ◽  
The One ◽  
Theoretical Results

We consider an approximate solution for the one--dimensional semilinear singularly--perturbed boundary value problem, using the previously obtained numerical values of the boundary value problem in the mesh points and the representation of the exact solution using Green's function. We present an \(\varepsilon\)--uniform convergence of such gained the approximate solutions, in the maximum norm of the order \(\mathcal{O}\left(N^{-1}\right)\) on the observed domain. After that, the constructed approximate solution is repaired and we obtain a solution, which also has \(\varepsilon\)--uniform convergence, but now of order \(\mathcal{O}\left(\ln^2N/N^2\right)\) on \([0,1]\). In the end a numerical experiment is presented to confirm previously shown theoretical results.

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