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Smoothness and continuity of cost functionals for ECG mismatch computation

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arXiv 2022年

作者： Grandits, Thomas Pezzuto, Simone Plank, Gernot Institute of Mathematics and Scientific Computing University of Graz Graz Austria Gottfried Schatz Research Center Division of Biophysics Medical University of Graz Graz Austria Center for Computational Medicine in Cardiology Euler Institute Università della Svizzera italiana Lugano Switzerland BioTechMed-Graz Graz Austria

The field of cardiac electrophysiology tries to abstract, describe and finally model the electrical characteristics of a heartbeat. With recent advances in cardiac electrophysiology, models have become more powerful and descriptive as ever. However, to advance to the field of inverse electrophysiological modeling, i.e. creating models from electrical measurements such as the ECG, the less investigated field of smoothness of the simulated ECGs w.r.t. model parameters need to be further explored. The present paper discusses smoothness in terms of the whole pipeline which describes how from physiological parameters, we arrive at the simulated ECG. Employing such a pipeline, we create a test-bench of a simplified idealized left ventricle model and demonstrate the most important factors for efficient inverse modeling through smooth cost functionals. Such knowledge will be important for designing and creating inverse models in future optimization and machine learning methods. Copyright © 2022, The Authors. All rights reserved.

关键词： Electrocardiograms

TAYLOR-HOOD LIKE FINITE ELEMENTS FOR NEARLY INCOMPRESSIBLE STRAIN GRADIENT ELASTICITY PROBLEMS

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arXiv 2021年

作者： Liao, Yulei Ming, Pingbing Xu, Yun LSEC Institute of Computational Mathematics and Scientific/Engineering Computing AMSS Chinese Academy of Sciences No. 55 East Road Zhong-Guan-Cun Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Beijing100088 China

We propose a family of mixed finite elements that are robust for the nearly incompressible strain gradient model, which is a fourth-order singular perturbed elliptic system. The element is similar to [C. Taylor and P. Hood, Comput. & Fluids, 1(1973), 73-100] in the Stokes flow. Using a uniform discrete B-B inequality for the mixed finite element pairs, we show the optimal rate of convergence that is robust in the incompressible limit. By a new regularity result that is uniform in both the materials parameter and the incompressibility, we prove the method converges with 1/2 order to the solution with strong boundary layer effects. Moreover, we estimate the convergence rate of the numerical solution to the unperturbed second-order elliptic system. Numerical results for both smooth solutions and the solutions with sharp layers confirm the theoretical *** Codes 65N15, 65N30 (Primary), 74K20 (Secondary) Copyright © 2021, The Authors. All rights reserved.

关键词： Elasticity

Quantum Circuits for the Heat Equation with Physical Boundary Conditions Via Schrödingerisation

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SSRN

SSRN 2024年

作者： Jin, Shi Liu, Nana Yu, Yue School of Mathematical Sciences Institute of Natural Sciences MOE-LSC Shanghai Jiao Tong University Shanghai200240 China University of Michigan Shanghai Jiao Tong University Joint Institute Shanghai200240 China School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education National Center for Applied Mathematics in Hunan Xiangtan University Hunan Xiangtan411105 China

This paper explores the explicit design of quantum circuits for quantum simulation of partial differential equations (PDEs) with physical boundary conditions. These equations and/or their discretized forms usually do not evolve via unitary dynamics, thus are not suitable for quantum simulation. Boundary conditions (either time-dependent or independent) make the problem more difficult. To tackle this challenge, the Schrödingerisation method can be employed, which converts linear partial and ordinary differential equations with non-unitary dynamics into systems of Schrödinger-type equations, via the so-called warped phase transformation that maps the equation into one higher dimension. Despite advancements in Schrödingerisation techniques, the explicit implementation of quantum circuits for solving general PDEs, especially with physical boundary conditions, remains underdeveloped. We present two methods for handling the inhomogeneous terms arising from time-dependent physical boundary conditions. One approach utilizes Duhamel's principle to express the solution in integral form and employs linear combination of unitaries (LCU) for coherent state preparation. Another method applies an augmentation to transform the inhomogeneous problem into a homogeneous one. We then apply the quantum simulation technique from [CJL23] to transform the resulting non-autonomous system to an autonomous system in one higher dimension. We provide detailed implementations of these two methods and conduct a comprehensive complexity analysis in terms of queries to the time evolution input oracle. © 2024, The Authors. All rights reserved.

关键词： Boundary conditions

Quantum Circuits for the heat equation with physical boundary conditions via Schrödingerisation

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arXiv 2024年

This paper explores the explicit design of quantum circuits for quantum simulation of partial differential equations (PDEs) with physical boundary conditions. These equations and/or their discretized forms usually do not evolve via unitary dynamics, thus are not suitable for quantum simulation. Boundary conditions (either time-dependent or independent) make the problem more difficult. To tackle this challenge, the Schrödingerisation method can be employed, which converts linear partial and ordinary differential equations with non-unitary dynamics into systems of Schrödinger-type equations, via the so-called warped phase transformation that maps the equation into one higher dimension. Despite advancements in Schrödingerisation techniques, the explicit implementation of quantum circuits for solving general PDEs, especially with physical boundary conditions, remains underdeveloped. We present two methods for handling the inhomogeneous terms arising from time-dependent physical boundary conditions. One approach utilizes Duhamel’s principle to express the solution in integral form and employs linear combination of unitaries (LCU) for coherent state preparation. Another method applies an augmentation to transform the inhomogeneous problem into a homogeneous one. We then apply the quantum simulation technique from [CJL23] to transform the resulting non-autonomous system to an autonomous system in one higher dimension. We provide detailed implementations of these two methods and conduct a comprehensive complexity analysis in terms of queries to the time evolution input oracle. © 2024, CC BY-NC-ND.

关键词： Ordinary differential equations

Why is Grain Growth Not Curvature Flow?

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arXiv 2024年

作者： Qiu, Caihao Srolovitz, David J. Rohrer, Gregory S. Han, Jian Salvalaglio, Marco Department of Materials Science and Engineering City University of Hong Kong Hong Kong Department of Mechanical Engineering The University of Hong Kong Pokfulam Road Hong Kong The University of Hong Kong Shenzhen China Department of Materials Science and Engineering Carnegie Mellon University PittsburghPA United States Institute of Scientific Computing TU Dresden Dresden01062 Germany Dresden Center for Computational Materials Science TU Dresden Dresden01062 Germany

Grain growth describes the increase in the mean grain size with time during the annealing of a polycrystal;it is widely accepted that this is driven by capillarity (surface tension). Although classically modeled and interpreted as mean curvature flow, a growing body of evidence has demonstrated that this is overly simplistic and inconsistent with experimental observations. Despite many proposals for the source of this failure, it remains a subject of ongoing debate. By considering the fundamental microscopic mechanism of grain boundary (GB) migration, we demonstrate that internal stress (shear coupling) associated with GB motion is the primary factor causing deviations in grain growth from mean curvature flow. This is achieved through the application of a two-dimensional bicrystallography-respecting continuum model for GB migration to grain growth, implemented in a diffuse-interface approach. These results are validated by comparison with atomistic simulations and the scatter in the curvature-velocity relationship is shown to be in excellent agreement with experimental results for grain growth in 3D polycrystals (not possible based on other proposed explanations). This work also demonstrates how to account for the mechanical effects that are integral to grain growth in crystalline solids. Copyright © 2024, The Authors. All rights reserved.

关键词： Polycrystals

Towards Efficient Large-Scale Network Slicing: An LP Dynamic Rounding-and-Refinement Approach

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arXiv 2021年

作者： Chen, Wei-Kun Liu, Ya-Feng Liu, Fan Dai, Yu-Hong Luo, Zhi-Quan School of Mathematics and Statistics Beijing Key Laboratory on MCAACI Beijing Institute of Technology Beijing100081 China State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China Department of Electrical and Electronic Engineering Southern University of Science and Technology Shenzhen518055 China Shenzhen Research Institute of Big Data The Chinese University of Hong Kong Shenzhen518172 China

In this paper, we propose an efficient algorithm for the network slicing problem which attempts to map multiple customized virtual network requests (also called services) to a common shared network infrastructure and allocate network resources to meet diverse service requirements. The problem has been formulated as a mixed integer linear programming (MILP) formulation in the literature. We first propose a novel linear programming (LP) relaxation of the MILP formulation. We show that compared with a natural LP relaxation of the MILP formulation, the novel LP relaxation is much more compact in terms of smaller numbers of variables and constraints, and much stronger in terms of providing a better LP bound, which makes it particularly suitable to be embedded in an LP relaxation based algorithm. Then we design an efficient two-stage LP dynamic rounding-and-refinement algorithm based on this novel LP relaxation. In the first stage, the proposed algorithm uses an LP dynamic rounding procedure to place the virtual network functions of all services into cloud nodes while taking traffic routing of all services into consideration;in the second stage, the proposed algorithm uses an iterative LP refinement procedure to obtain a solution for traffic routing of all services with their end-to-end delay constraints being satisfied. Compared with the existing algorithms which either have an exponential complexity or return a low-quality solution, our proposed algorithm achieves a better trade-off between the solution quality and the computational complexity. In particular, the worst-case complexity of our proposed algorithm is polynomial, which makes it suitable for solving large-scale problems. Numerical results demonstrate the effectiveness and efficiency of our proposed algorithm. Copyright © 2021, The Authors. All rights reserved.

关键词： Iterative methods

k-commutativity and measurement reduction for expectation values

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arXiv 2023年

作者： DalFavero, Ben Sarkar, Rahul Camps, Daan Sawaya, Nicolas LaRose, Ryan Center for Quantum Computing Science and Engineering Michigan State University East LansingMI48823 United States Institute for Computational & Mathematical Engineering Stanford University United States National Energy Research Scientific Computing Center Lawrence Berkeley National Laboratory BerkeleyCA United States HPI Biosciences OaklandCA94608 United States Azulene Labs Inc. San FranciscoCA94115 United States

We introduce a notion of commutativity between operators on a tensor product space, nominally Pauli strings on qubits, that interpolates between qubit-wise commutativity and (full) commutativity. We apply this notion, which we call k-commutativity, to measuring expectation values of observables in quantum circuits and show a reduction in the number measurements at the cost of increased circuit depth. Last, we discuss the asymptotic measurement complexity of k-commutativity for several families of n-qubit Hamiltonians, showing examples with O(1), O(√n), and O(n) scaling. © 2023, CC BY.

关键词： Qubits

Adversarially robust clustering with optimality guarantees

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arXiv 2023年

作者： Jana, Soham Yang, Kun Kulkarni, Sanjeev The Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre DameIN United States The Data Science Institute Center For Research Computing Southern Methodist University DallasTX United States The Department of Operations Research and Financial Engineering Princeton University PrincetonNJ United States

We consider the problem of clustering data points coming from sub-Gaussian mixtures. Existing methods that provably achieve the optimal mislabeling error, such as the Lloyd algorithm, are usually vulnerable to outliers. In contrast, clustering methods seemingly robust to adversarial perturbations are not known to satisfy the optimal statistical guarantees. We propose a simple robust algorithm based on the coordinatewise median that obtains the optimal mislabeling rate even when we allow adversarial outliers to be present. Our algorithm achieves the optimal error rate in constant iterations when a weak initialization condition is satisfied. In the absence of outliers, in fixed dimensions, our theoretical guarantees are similar to that of the Lloyd algorithm. Extensive experiments on various simulated and public datasets are conducted to support the theoretical guarantees of our *** Codes 62G30, 62G35, 62H30 Copyright © 2023, The Authors. All rights reserved.

关键词： Clustering algorithms

Automatic 3D Registration of Dental CBCT and Face Scan Data using 2D Projection Images

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arXiv 2023年

作者： Park, Hyoung Suk Hyun, Chang Min Lee, Sang-Hwy Seo, Jin Keun Jeon, Kiwan National Institute for Mathematical Sciences Daejeon34047 Korea Republic of School of Mathematics and Computing Computational Science and Engineering Yonsei University Seoul03722 Korea Republic of Department of Oral and Maxillofacial Surgery Oral Science Research Center College of Dentistry Yonsei University Seoul03722 Korea Republic of

This paper presents a fully automatic registration method of dental cone-beam computed tomography (CBCT) and face scan data. It can be used for a digital platform of 3D jaw-teeth-face models in a variety of applications, including 3D digital treatment planning and orthognathic surgery. Difficulties in accurately merging facial scans and CBCT images are due to the different image acquisition methods and limited area of correspondence between the two facial surfaces. In addition, it is difficult to use machine learning techniques because they use face-related 3D medical data with radiation exposure, which are difficult to obtain for training. The proposed method addresses these problems by reusing an existing machine-learning-based 2D landmark detection algorithm in an open-source library and developing a novel mathematical algorithm that identifies paired 3D landmarks from knowledge of the corresponding 2D landmarks. A main contribution of this study is that the proposed method does not require annotated training data of facial landmarks because it uses a pre-trained facial landmark detection algorithm that is known to be robust and generalized to various 2D face image models. Note that this reduces a 3D landmark detection problem to a 2D problem of identifying the corresponding landmarks on two 2D projection images generated from two different projection angles. Here, the 3D landmarks for registration were selected from the sub-surfaces with the least geometric change under the CBCT and face scan environments. For the final fine-tuning of the registration, the Iterative Closest Point method was applied, which utilizes geometrical information around the 3D landmarks. The experimental results show that the proposed method achieved an averaged surface distance error of 0.74 mm for three pairs of CBCT and face scan *** Codes 92C55, 15A04, 62F10 © 2023, CC BY.

关键词： Iterative methods