We prove,under mild conditions,the convergence of a Riemannian gradient descent method for a hyperbolic neural network regression model,both in batch gradient descent and stochastic gradient descent.We also discuss a ...We prove,under mild conditions,the convergence of a Riemannian gradient descent method for a hyperbolic neural network regression model,both in batch gradient descent and stochastic gradient descent.We also discuss a Riemannian version of the Adam algorithm.We show numerical simulations of these algorithms on various benchmarks.展开更多
One of the open problems in the field of forward uncertainty quantification(UQ)is the ability to form accurate assessments of uncertainty having only incomplete information about the distribution of random inputs.Anot...One of the open problems in the field of forward uncertainty quantification(UQ)is the ability to form accurate assessments of uncertainty having only incomplete information about the distribution of random inputs.Another challenge is to efficiently make use of limited training data for UQ predictions of complex engineering problems,particularly with high dimensional random parameters.We address these challenges by combining data-driven polynomial chaos expansions with a recently developed preconditioned sparse approximation approach for UQ problems.The first task in this two-step process is to employ the procedure developed in[1]to construct an"arbitrary"polynomial chaos expansion basis using a finite number of statistical moments of the random inputs.The second step is a novel procedure to effect sparse approximation via l1 minimization in order to quantify the forward uncertainty.To enhance the performance of the preconditioned l1 minimization problem,we sample from the so-called induced distribution,instead of using Monte Carlo(MC)sampling from the original,unknown probability measure.We demonstrate on test problems that induced sampling is a competitive and often better choice compared with sampling from asymptotically optimal measures(such as the equilibrium measure)when we have incomplete information about the distribution.We demonstrate the capacity of the proposed induced sampling algorithm via sparse representation with limited data on test functions,and on a Kirchoff plating bending problem with random Young’s modulus.展开更多
Collocation has become a standard tool for approximation of parameterized systems in the uncertainty quantification(UQ)community.Techniques for leastsquares regularization,compressive sampling recovery,and interpolato...Collocation has become a standard tool for approximation of parameterized systems in the uncertainty quantification(UQ)community.Techniques for leastsquares regularization,compressive sampling recovery,and interpolatory reconstruction are becoming standard tools used in a variety of applications.Selection of a collocation mesh is frequently a challenge,but methods that construct geometrically unstructured collocation meshes have shown great potential due to attractive theoretical properties and direct,simple generation and implementation.We investigate properties of these meshes,presenting stability and accuracy results that can be used as guides for generating stochastic collocation grids in multiple dimensions.展开更多
Importance. Medical images are essential for modern medicine and an important research subject in visualization. However,medical experts are often not aware of the many advanced three-dimensional (3D) medical image vi...Importance. Medical images are essential for modern medicine and an important research subject in visualization. However,medical experts are often not aware of the many advanced three-dimensional (3D) medical image visualization techniques thatcould increase their capabilities in data analysis and assist the decision-making process for specific medical problems. Ourpaper provides a review of 3D visualization techniques for medical images, intending to bridge the gap between medicalexperts and visualization researchers. Highlights. Fundamental visualization techniques are revisited for various medicalimaging modalities, from computational tomography to diffusion tensor imaging, featuring techniques that enhance spatialperception, which is critical for medical practices. The state-of-the-art of medical visualization is reviewed based on aprocedure-oriented classification of medical problems for studies of individuals and populations. This paper summarizes freesoftware tools for different modalities of medical images designed for various purposes, including visualization, analysis, andsegmentation, and it provides respective Internet links. Conclusions. Visualization techniques are a useful tool for medicalexperts to tackle specific medical problems in their daily work. Our review provides a quick reference to such techniques giventhe medical problem and modalities of associated medical images. We summarize fundamental techniques and readily availablevisualization tools to help medical experts to better understand and utilize medical imaging data. This paper could contributeto the joint effort of the medical and visualization communities to advance precision medicine.展开更多
The problem of packing circles into a domain of prescribed topology is considered. The circles need not have equal radii. The Collins-Stephenson algorithm computes such a circle packing. This algorithm is parMlelized ...The problem of packing circles into a domain of prescribed topology is considered. The circles need not have equal radii. The Collins-Stephenson algorithm computes such a circle packing. This algorithm is parMlelized in two different ways and its performance is reported for a triangular, planar domain test case. The implementation uses the highly parallel graphics processing unit （GPU） on commodity hardware. The speedups so achieved are discussed based on a number of experiments.展开更多
Although the popular multi-fidelity surrogate models,stochastic collocation and nonlinear autoregression have been applied successfully to multiple benchmark problems in different areas of science and engineering,they...Although the popular multi-fidelity surrogate models,stochastic collocation and nonlinear autoregression have been applied successfully to multiple benchmark problems in different areas of science and engineering,they have certain limitations.We propose a uniform Bayesian framework that connects these two methods allowing us to combine the strengths of both.To this end,we introduce Greedy-NAR,a nonlinear Bayesian autoregressive model that can handle complex between-fidelity correlations and involves a sequential construction that allows for significant improvements in performance given a limited computational budget.The proposed enhanced nonlinear autoregressive method is applied to three benchmark problems that are typical of energy applications,namely molecular dynamics and computational fluid dynamics.The results indicate an increase in both prediction stability and accuracy when compared to those of the standard multi-fidelity autoregression implementations.The results also reveal the advantages over the stochastic collocation approach in terms of accuracy and computational cost.Generally speaking,the proposed enhancement provides a straightforward and easily implemented approach for boosting the accuracy and efficiency of concatenated structure multi-fidelity simulation methods,e.g.,the nonlinear autoregressive model,with a negligible additional computational cost.展开更多
基金partially supported by NSF Grants DMS-1854434,DMS-1952644,and DMS-2151235 at UC Irvinesupported by NSF Grants DMS-1924935,DMS-1952339,DMS-2110145,DMS-2152762,and DMS-2208361,and DOE Grants DE-SC0021142 and DE-SC0002722.
文摘We prove,under mild conditions,the convergence of a Riemannian gradient descent method for a hyperbolic neural network regression model,both in batch gradient descent and stochastic gradient descent.We also discuss a Riemannian version of the Adam algorithm.We show numerical simulations of these algorithms on various benchmarks.
基金supported by the NSF of China(No.11671265)partially supported by NSF DMS-1848508+4 种基金partially supported by the NSF of China(under grant numbers 11688101,11571351,and 11731006)science challenge project(No.TZ2018001)the youth innovation promotion association(CAS)supported by the National Science Foundation under Grant No.DMS-1439786the Simons Foundation Grant No.50736。
文摘One of the open problems in the field of forward uncertainty quantification(UQ)is the ability to form accurate assessments of uncertainty having only incomplete information about the distribution of random inputs.Another challenge is to efficiently make use of limited training data for UQ predictions of complex engineering problems,particularly with high dimensional random parameters.We address these challenges by combining data-driven polynomial chaos expansions with a recently developed preconditioned sparse approximation approach for UQ problems.The first task in this two-step process is to employ the procedure developed in[1]to construct an"arbitrary"polynomial chaos expansion basis using a finite number of statistical moments of the random inputs.The second step is a novel procedure to effect sparse approximation via l1 minimization in order to quantify the forward uncertainty.To enhance the performance of the preconditioned l1 minimization problem,we sample from the so-called induced distribution,instead of using Monte Carlo(MC)sampling from the original,unknown probability measure.We demonstrate on test problems that induced sampling is a competitive and often better choice compared with sampling from asymptotically optimal measures(such as the equilibrium measure)when we have incomplete information about the distribution.We demonstrate the capacity of the proposed induced sampling algorithm via sparse representation with limited data on test functions,and on a Kirchoff plating bending problem with random Young’s modulus.
基金T.Zhou is supported by the National Natural Science Foundation of China(Award Nos.91130003 and 11201461).
文摘Collocation has become a standard tool for approximation of parameterized systems in the uncertainty quantification(UQ)community.Techniques for leastsquares regularization,compressive sampling recovery,and interpolatory reconstruction are becoming standard tools used in a variety of applications.Selection of a collocation mesh is frequently a challenge,but methods that construct geometrically unstructured collocation meshes have shown great potential due to attractive theoretical properties and direct,simple generation and implementation.We investigate properties of these meshes,presenting stability and accuracy results that can be used as guides for generating stochastic collocation grids in multiple dimensions.
基金the Data for Better Health Project of Peking University-Master Kong and by NIH(R01 EB031872).
文摘Importance. Medical images are essential for modern medicine and an important research subject in visualization. However,medical experts are often not aware of the many advanced three-dimensional (3D) medical image visualization techniques thatcould increase their capabilities in data analysis and assist the decision-making process for specific medical problems. Ourpaper provides a review of 3D visualization techniques for medical images, intending to bridge the gap between medicalexperts and visualization researchers. Highlights. Fundamental visualization techniques are revisited for various medicalimaging modalities, from computational tomography to diffusion tensor imaging, featuring techniques that enhance spatialperception, which is critical for medical practices. The state-of-the-art of medical visualization is reviewed based on aprocedure-oriented classification of medical problems for studies of individuals and populations. This paper summarizes freesoftware tools for different modalities of medical images designed for various purposes, including visualization, analysis, andsegmentation, and it provides respective Internet links. Conclusions. Visualization techniques are a useful tool for medicalexperts to tackle specific medical problems in their daily work. Our review provides a quick reference to such techniques giventhe medical problem and modalities of associated medical images. We summarize fundamental techniques and readily availablevisualization tools to help medical experts to better understand and utilize medical imaging data. This paper could contributeto the joint effort of the medical and visualization communities to advance precision medicine.
基金Project supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) by the Ministry of Education, Science and Technology (No. 2012-0002715)NSF Grants CPATH (Nos. CCF-0722210 and CCF-0938999)+1 种基金DOE award (No. DE-FG52-06NA26290)a gift from the Intel Corporation
文摘The problem of packing circles into a domain of prescribed topology is considered. The circles need not have equal radii. The Collins-Stephenson algorithm computes such a circle packing. This algorithm is parMlelized in two different ways and its performance is reported for a triangular, planar domain test case. The implementation uses the highly parallel graphics processing unit （GPU） on commodity hardware. The speedups so achieved are discussed based on a number of experiments.
基金This work has been supported by DARPA TRADES Award HR0011-17-2-0016.
文摘Although the popular multi-fidelity surrogate models,stochastic collocation and nonlinear autoregression have been applied successfully to multiple benchmark problems in different areas of science and engineering,they have certain limitations.We propose a uniform Bayesian framework that connects these two methods allowing us to combine the strengths of both.To this end,we introduce Greedy-NAR,a nonlinear Bayesian autoregressive model that can handle complex between-fidelity correlations and involves a sequential construction that allows for significant improvements in performance given a limited computational budget.The proposed enhanced nonlinear autoregressive method is applied to three benchmark problems that are typical of energy applications,namely molecular dynamics and computational fluid dynamics.The results indicate an increase in both prediction stability and accuracy when compared to those of the standard multi-fidelity autoregression implementations.The results also reveal the advantages over the stochastic collocation approach in terms of accuracy and computational cost.Generally speaking,the proposed enhancement provides a straightforward and easily implemented approach for boosting the accuracy and efficiency of concatenated structure multi-fidelity simulation methods,e.g.,the nonlinear autoregressive model,with a negligible additional computational cost.
