scholarly journals Sobolev Gradients for the Möbius Energy

2021 ◽  
Author(s):  
Philipp Reiter ◽  
Henrik Schumacher
Keyword(s):  
Optimization Methods ◽  
Inner Product ◽  
Stationary Points ◽  
Inner Products ◽  
Sobolev Gradients ◽  
Gradient Based ◽  
Standard Techniques ◽  
Sobolev Inner Products

AbstractAiming to optimize the shape of closed embedded curves within prescribed isotopy classes, we use a gradient-based approach to approximate stationary points of the Möbius energy. The gradients are computed with respect to Sobolev inner products similar to the $$W^{3/2,2}$$ W 3 / 2 , 2 -inner product. This leads to optimization methods that are significantly more efficient and robust than standard techniques based on $$L^2$$ L 2 -gradients.

scholarly journals Space-Dependent Sobolev Gradients as a Regularization for Inverse Radiative Transfer Problems

2016 ◽  
Vol 2016 ◽  
pp. 1-18 ◽  
Author(s):  
Y. Favennec ◽  
F. Dubot ◽  
D. Le Hardy ◽  
B. Rousseau ◽  
D. R. Rousse
Keyword(s):  
Cost Function ◽  
Directional Derivative ◽  
Optical Tomography ◽  
Extraction Procedure ◽  
Inner Product ◽  
Pass Filter ◽  
Low Pass Filter ◽  
Inner Products ◽  
Sobolev Gradients ◽  
The Cost

Diffuse optical tomography problems rely on the solution of an optimization problem for which the dimension of the parameter space is usually large. Thus, gradient-type optimizers are likely to be used, such as the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm, along with the adjoint-state method to compute the cost function gradient. Usually, theL2-inner product is chosen within the extraction procedure (i.e., in the definition of the relationship between the cost function gradient and the directional derivative of the cost function) while alternative inner products that act as regularization can be used. This paper presents some results based on space-dependent Sobolev inner products and shows that this method acts as an efficient low-pass filter on the cost function gradient. Numerical results indicate that the use of Sobolev gradients can be particularly attractive in the context of inverse problems, particularly because of the simplicity of this regularization, since a single additional diffusion equation is to be solved, and also because the quality of the solution is smoothly varying with respect to the regularization parameter.

scholarly journals All about the ⊥ with its applications in the linear statistical models

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Augustyn Markiewicz ◽  
Simo Puntanen
Keyword(s):  
Statistical Models ◽  
Inner Product ◽  
Real Matrix ◽  
Inner Products ◽  
Column Space ◽  
The Matrix ◽  
Linear Statistical Models ◽  
Extensive List

Abstract For an n x m real matrix A the matrix A⊥ is defined as a matrix spanning the orthocomplement of the column space of A, when the orthogonality is defined with respect to the standard inner product ⟨x, y⟩ = x'y. In this paper we collect together various properties of the ⊥ operation and its applications in linear statistical models. Results covering the more general inner products are also considered. We also provide a rather extensive list of references

scholarly journals On the semi-inner product in locally convex spaces

1997 ◽  
Vol 20 (2) ◽  
pp. 219-224
Author(s):  
Shih-Sen Chang ◽  
Yu-Qing Chen ◽  
Byung Soo Lee
Keyword(s):  
Inner Product ◽  
Locally Convex Spaces ◽  
Basic Properties ◽  
Inner Products ◽  
Locally Convex ◽  
Convex Spaces

The purpose of this paper is to introduce the concept of semi-inner products in locally convex spaces and to give some basic properties.

An Adjoint Based Approach for Optimal Design of Morphing SMP

2013 ◽  
Author(s):  
Shuang Wang ◽  
John C. Brigham
Keyword(s):  
Shape Change ◽  
Mechanical Energy ◽  
Optimization Methods ◽  
Design Parameters ◽  
Optimization Strategy ◽  
Thermally Activated ◽  
Gradient Based ◽  
Nonlinear Optimization Methods ◽  
Numerical Representations ◽  
Objective Functional

This work presents a strategy to identify the optimal localized activation and actuation for a morphing thermally activated SMP structure or structural component to obtain a targeted shape change or set of shape features, subject to design objectives such as minimal total required energy and time. This strategy combines numerical representations of the SMP structure’s thermo-mechanical behavior subject to activation and actuation with gradient-based nonlinear optimization methods to solve the morphing inverse problem that includes minimizing cost functions which address thermal and mechanical energy, morphing time, and damage. In particular, the optimization strategy utilizes the adjoint method to efficiently compute the gradient of the objective functional(s) with respect to the design parameters for this coupled thermo-mechanical problem.

scholarly journals Implementation of a Hamming distance–like genomic quantum classifier using inner products on ibmqx2 and ibmq_16_melbourne

2020 ◽  
Vol 2 (1) ◽  
Author(s):  
Kunal Kathuria ◽  
Aakrosh Ratan ◽  
Michael McConnell ◽  
Stefan Bekiranov
Keyword(s):  
Hamming Distance ◽  
Training Sample ◽  
General Purpose ◽  
Inner Product ◽  
Matching Problems ◽  
Binary Operations ◽  
Inner Products ◽  
Training Class ◽  
Dot Product ◽  
Product Measures

Abstract Motivated by the problem of classifying individuals with a disease versus controls using a functional genomic attribute as input, we present relatively efficient general purpose inner product–based kernel classifiers to classify the test as a normal or disease sample. We encode each training sample as a string of 1 s (presence) and 0 s (absence) representing the attribute’s existence across ordered physical blocks of the subdivided genome. Having binary-valued features allows for highly efficient data encoding in the computational basis for classifiers relying on binary operations. Given that a natural distance between binary strings is Hamming distance, which shares properties with bit-string inner products, our two classifiers apply different inner product measures for classification. The active inner product (AIP) is a direct dot product–based classifier whereas the symmetric inner product (SIP) classifies upon scoring correspondingly matching genomic attributes. SIP is a strongly Hamming distance–based classifier generally applicable to binary attribute-matching problems whereas AIP has general applications as a simple dot product–based classifier. The classifiers implement an inner product between N = 2n dimension test and train vectors using n Fredkin gates while the training sets are respectively entangled with the class-label qubit, without use of an ancilla. Moreover, each training class can be composed of an arbitrary number m of samples that can be classically summed into one input string to effectively execute all test–train inner products simultaneously. Thus, our circuits require the same number of qubits for any number of training samples and are $O(\log {N})$ O ( log N ) in gate complexity after the states are prepared. Our classifiers were implemented on ibmqx2 (IBM-Q-team 2019b) and ibmq_16_melbourne (IBM-Q-team 2019a). The latter allowed encoding of 64 training features across the genome.

scholarly journals Benchmarking Adaptive Variational Quantum Eigensolvers

2020 ◽  
Vol 8 ◽  
Author(s):  
Daniel Claudino ◽  
Jerimiah Wright ◽  
Alexander J. McCaskey ◽  
Travis S. Humble
Keyword(s):  
Quantum State ◽  
Energy Eigenvalue ◽  
Molecular Size ◽  
Optimization Methods ◽  
Superior Performance ◽  
Prepared State ◽  
Gradient Based ◽  
Structure Problem ◽  
Relevant Finding ◽  
Electronic Ground

By design, the variational quantum eigensolver (VQE) strives to recover the lowest-energy eigenvalue of a given Hamiltonian by preparing quantum states guided by the variational principle. In practice, the prepared quantum state is indirectly assessed by the value of the associated energy. Novel adaptive derivative-assembled pseudo-trotter (ADAPT) ansatz approaches and recent formal advances now establish a clear connection between the theory of quantum chemistry and the quantum state ansatz used to solve the electronic structure problem. Here we benchmark the accuracy of VQE and ADAPT-VQE to calculate the electronic ground states and potential energy curves for a few selected diatomic molecules, namely H2, NaH, and KH. Using numerical simulation, we find both methods provide good estimates of the energy and ground state, but only ADAPT-VQE proves to be robust to particularities in optimization methods. Another relevant finding is that gradient-based optimization is overall more economical and delivers superior performance than analogous simulations carried out with gradient-free optimizers. The results also identify small errors in the prepared state fidelity which show an increasing trend with molecular size.

Non-Probabilistic Based Topology Optimization Under External Load Uncertainty With Eigenvalue-Superposition of Convex Models

2013 ◽  
Author(s):  
Xike Zhao ◽  
Hae Chang Gea ◽  
Wei Song
Keyword(s):  
Topology Optimization ◽  
External Load ◽  
Optimization Problems ◽  
Optimization Methods ◽  
Optimization Method ◽  
Convex Model ◽  
Structure Response ◽  
Gradient Based ◽  
Bounded Convex ◽  
Topology Optimization Method

In this paper the Eigenvalue-Superposition of Convex Models (ESCM) based topology optimization method for solving topology optimization problems under external load uncertainties is presented. The load uncertainties are formulated using the non-probabilistic based unknown-but-bounded convex model. The sensitivities are derived and the problem is solved using gradient based algorithm. The proposed ESCM based method yields the material distribution which would optimize the worst structure response under the uncertain loads. Comparing to the deterministic based topology optimization formulation the ESCM based method provided more reasonable solutions when load uncertainties were involved. The simplicity, efficiency and versatility of the proposed ESCM based topology optimization method can be considered as a supplement to the sophisticated reliability based topology optimization methods.

scholarly journals A ‘Dysonization’ scheme for identifying quasi-particles using non-Hermitian quantum mechanics

2013 ◽  
Vol 371 (1989) ◽  
pp. 20120056 ◽  
Author(s):  
Katherine Jones-Smith
Keyword(s):  
Quantum Mechanics ◽  
High Temperature ◽  
High Temperature Superconductivity ◽  
Spin Waves ◽  
Low Energy ◽  
Inner Product ◽  
Inner Products ◽  
Weakly Interacting

Dyson analysed the low-energy excitations of a ferromagnet using a Hamiltonian that was non-Hermitian with respect to the standard inner product. This allowed for a facile rendering of these excitations (known as spin waves) as weakly interacting bosonic quasi-particles. More than 50 years later, we have the full denouement of the non-Hermitian quantum mechanics formalism at our disposal when considering Dyson’s work, both technically and contextually. Here, we recast Dyson’s work on ferromagnets explicitly in terms of two inner products, with respect to which the Hamiltonian is always self-adjoint, if not manifestly ‘Hermitian’. Then we extend his scheme to doped anti-ferromagnets described by the t – J model, with hopes of shedding light on the physics of high-temperature superconductivity.

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