Objective:bipolar cells(BCs)communicate with amacrine and ganglion cells of the retina via both transient and sustained neurotransmitter release in ribbon synapses.Reconstructing the published quantitative release dat...Objective:bipolar cells(BCs)communicate with amacrine and ganglion cells of the retina via both transient and sustained neurotransmitter release in ribbon synapses.Reconstructing the published quantitative release data from electrical soma stimulation(voltage clamp experiments)of rat rod BCs were used to develop two simple models to predict the number of released vesicles as time series.In the experiment,the currents coming to the All amacrine cell originating from releasing vesicles from the rod BC were recorded using paired-recordings in whole-cell voltage-clamp method.Method:one of the models is based directly on terminal transmembrane voltage,so-called ‘model’,whereas the temporally exacter modelCa includes changes of intracellular calcium concentrations at terminals.Result:the intracellular calcium concentration method replicates a 0.43 ms signal delay for the transient release to pulsatile stimulation as a consequence of calcium channel dynamics in the presynaptic membrane,while the modelV has no signal delay.Both models produce the quite similar results in low stimuli amplitudes.However,for large stimulation intensities that may be done during extracellular stimulations in retinal implants,the modelCa predicts that the reversal potential of calcium limits the number of transiently released vesicles.Adding sodium and potassium ion channels to the axon of the cell enable to study the impact of spikes on the transient release in BC ribbons.Conclusion:a spike elicited by somatic stimulation causes the rapid release of all vesicles that are available for transient release,while a non-spiking BC with a similar morphometry needs stronger stimuli for any transient vesicle release.During extracellular stimulation,there was almost no difference in transient release between the active and passive cells because in both cases the terminal membrane of the cell senses the same potentials originating from the microelectrode.An exception was found for long pulses when the spike has the possibility to generate a higher terminal voltage than the passive cell.Simulated periodic 5 Hz stimulation showed a reduced transient release of 3 vesicles per stimulus,which is a recovery effect.展开更多
Although deep learning-based approximation algorithms have been applied very successfully to numerous problems,at the moment the reasons for their performance are not entirely understood from a mathematical point of v...Although deep learning-based approximation algorithms have been applied very successfully to numerous problems,at the moment the reasons for their performance are not entirely understood from a mathematical point of view.Recently,estimates for the convergence of the overall error have been obtained in the situation of deep supervised learning,but with an extremely slow rate of convergence.In this note,we partially improve on these estimates.More specifically,we show that the depth of the neural network only needs to increase much slower in order to obtain the same rate of approximation.The results hold in the case of an arbitrary stochastic optimization algorithm with i.i.d.random initializations.展开更多
It is one of the most challenging issues in applied mathematics to approximately solve high-dimensional partial differential equations(PDEs)and most of the numerical approximation methods for PDEs in the scientific li...It is one of the most challenging issues in applied mathematics to approximately solve high-dimensional partial differential equations(PDEs)and most of the numerical approximation methods for PDEs in the scientific literature suffer from the so-called curse of dimensionality in the sense that the number of computational operations employed in the corresponding approximation scheme to obtain an approximation precisionε>0 grows exponentially in the PDE dimension and/or the reciprocal ofε.Recently,certain deep learning based methods for PDEs have been proposed and various numerical simulations for such methods suggest that deep artificial neural network(ANN)approximations might have the capacity to indeed overcome the curse of dimensionality in the sense that the number of real parameters used to describe the approximating deep ANNs grows at most polynomially in both the PDE dimension d∈R and the reciprocal of the prescribed approximation accuracyε>0.There are now also a few rigorous mathematical results in the scientific literature which substantiate this conjecture by proving that deep ANNs overcome the curse of dimensionality in approximating solutions of PDEs.Each of these results establishes that deep ANNs overcome the curse of dimensionality in approximating suitable PDE solutions at a fixed time point T>0 and on a compact cube[a,b]d in space but none of these results provides an answer to the question whether the entire PDE solution on[0,T]×[a,b]^(d)can be approximated by deep ANNs without the curse of dimensionality.It is precisely the subject of this article to overcome this issue.More specifically,the main result of this work in particular proves for every a∈R,b∈(a,∞)that solutions of certain Kolmogorov PDEs can be approximated by deep ANNs on the space-time region[0,T]×[a,b]^(d)without the curse of dimensionality.展开更多
In this article we propose a higher-order space-time conservative method for hyperbolic systems with stiff and non stiff source terms as well as relaxation systems.We call the scheme a slope propagation(SP)method.It i...In this article we propose a higher-order space-time conservative method for hyperbolic systems with stiff and non stiff source terms as well as relaxation systems.We call the scheme a slope propagation(SP)method.It is an extension of our scheme derived for homogeneous hyperbolic systems[1].In the present inhomogeneous systems the relaxation time may vary from order of one to a very small value.These small values make the relaxation term stronger and highly stiff.In such situations underresolved numerical schemes may produce spurious numerical results.However,our present scheme has the capability to correctly capture the behavior of the physical phenomena with high order accuracy even if the initial layer and the small relaxation time are not numerically resolved.The scheme treats the space and time in a unified manner.The flow variables and their slopes are the basic unknowns in the scheme.The source term is treated by its volumetric integration over the space-time control volume and is a direct part of the overall space-time flux balance.We use two approaches for the slope calculations of the flow variables,the first one results directly from the flux balance over the control volumes,while in the second one we use a finite difference approach.The main features of the scheme are its simplicity,its Jacobian-free and Riemann solver-free recipe,as well as its efficiency and high of order accuracy.In particular we show that the scheme has a discrete analog of the continuous asymptotic limit.We have implemented our scheme for various test models available in the literature such as the Broadwell model,the extended thermodynamics equations,the shallow water equations,traffic flow and the Euler equations with heat transfer.The numerical results validate the accuracy,versatility and robustness of the present scheme.展开更多
This paper is a continuation work of[26]and studies the propagation of the high-order boundary regularities of the two-dimensional density patch for viscous inhomogeneous incompressible flow.We assume the initial dens...This paper is a continuation work of[26]and studies the propagation of the high-order boundary regularities of the two-dimensional density patch for viscous inhomogeneous incompressible flow.We assume the initial densityρ0=η11Ω0+η21Ω0^c,where(η1,η2)is any pair of positive constants andΩ0 is a bounded,simply connected domain with W^k+2,p(R^2)boundary regularity.We prove that for any positive time t,the density functionρ(t)=η11Ω(t)+η21Ω(t)^c,and the domainΩ(t)preserves the W^k+2,p-boundary regularity.展开更多
In this paper,we address the problem that standard stochastic Landau-Lifshitz-Gilbert(sLLG)simulations typically produce results that show unphysical mesh-size dependence.The root cause of this problem is that the eff...In this paper,we address the problem that standard stochastic Landau-Lifshitz-Gilbert(sLLG)simulations typically produce results that show unphysical mesh-size dependence.The root cause of this problem is that the effects of spin-wave fluctuations are ignored in sLLG.We propose to represent the effect of these fluctuations by a full-spin-wave-scaled stochastic LLG,or FUSSS LLG method.In FUSSS LLG,the intrinsic parameters of the sLLG simulations are first scaled by scaling factors that integrate out the spin-wave fluctuations up to the mesh size,and the sLLG simulation is then performed with these scaled parameters.We developed FUSSS LLG by studying the Ferromagnetic Resonance(FMR)in Nd_(2)Fe_(14)B cubes.The nominal scaling greatly reduced the mesh size dependence relative to sLLG.We then performed three tests and validations of our FUSSS LLG with this modified scaling.(1)We studied the same FMR but with magnetostatic fields included.(2)We simulated the total magnetization of the Nd_(2)Fe_(14)B cube.(3)We studied the effective,temperature-and sweeping rate-dependent coercive field of the cubes.In all three cases,we found that FUSSS LLG delivered essentially mesh-size-independent results,which tracked the theoretical expectations better than unscaled sLLG.Motivated by these successful validations,we propose that FUSSS LLG provides marked,qualitative progress towards accurate,high precision modeling of micromagnetics in hard,permanent magnets.展开更多
Aggregation is one of the many important processes in chemical and process engineering. Several researchers have attempted to understand this complex process in fluidized beds using the macro-model of population balan...Aggregation is one of the many important processes in chemical and process engineering. Several researchers have attempted to understand this complex process in fluidized beds using the macro-model of population balance equations （PBEs）. The aggregation kernel is an effective parameter in PBEs, and is defined as the product of the aggregation efficiency and collision frequency functions. Attempts to derive this kernel have taken different approaches, including theoretical, experimental, and empirical techniques. The present paper calculates the aggregation kernel using micro-model computer simulations, i.e., a discrete particle model. We simulate the micro-model without aggregation for various initial conditions, and observe that the collision frequency function is in good agreement with the shear kernel. We then simulate the micro-model with aggregation and calculate the aggregation efficiency rate.展开更多
This paper considers a kind of strongly coupled cross diffusion parabolic system, which can be used as the multi-dimensional Lyurnkis energy transport model in semiconductor science. The global existence and large tim...This paper considers a kind of strongly coupled cross diffusion parabolic system, which can be used as the multi-dimensional Lyurnkis energy transport model in semiconductor science. The global existence and large time behavior are obtained for smooth solution to the initial boundary value problem. When the initial data are a small perturbation of an isothermal stationary solution, the smooth solution of the problem under the insulating boundary condition, converges to that stationary solution exponentially fast as time goes to infinity.展开更多
In this paper,we derive rigorously a non-local cross-diffusion system from an interacting stochastic many-particle system in the whole space.The convergence is proved in the sense of probability by introducing an inte...In this paper,we derive rigorously a non-local cross-diffusion system from an interacting stochastic many-particle system in the whole space.The convergence is proved in the sense of probability by introducing an intermediate particle system with a mollified interaction potential,where the mollification is of algebraic scaling.The main idea of the proof is to study the time evolution of a stopped process and obtain a Gronwall type estimate by using Taylor's expansion around the limiting stochastic process.展开更多
基金HB was supported by the European Institute of Innovation and Technology Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 674901(switchboard).
文摘Objective:bipolar cells(BCs)communicate with amacrine and ganglion cells of the retina via both transient and sustained neurotransmitter release in ribbon synapses.Reconstructing the published quantitative release data from electrical soma stimulation(voltage clamp experiments)of rat rod BCs were used to develop two simple models to predict the number of released vesicles as time series.In the experiment,the currents coming to the All amacrine cell originating from releasing vesicles from the rod BC were recorded using paired-recordings in whole-cell voltage-clamp method.Method:one of the models is based directly on terminal transmembrane voltage,so-called ‘model’,whereas the temporally exacter modelCa includes changes of intracellular calcium concentrations at terminals.Result:the intracellular calcium concentration method replicates a 0.43 ms signal delay for the transient release to pulsatile stimulation as a consequence of calcium channel dynamics in the presynaptic membrane,while the modelV has no signal delay.Both models produce the quite similar results in low stimuli amplitudes.However,for large stimulation intensities that may be done during extracellular stimulations in retinal implants,the modelCa predicts that the reversal potential of calcium limits the number of transiently released vesicles.Adding sodium and potassium ion channels to the axon of the cell enable to study the impact of spikes on the transient release in BC ribbons.Conclusion:a spike elicited by somatic stimulation causes the rapid release of all vesicles that are available for transient release,while a non-spiking BC with a similar morphometry needs stronger stimuli for any transient vesicle release.During extracellular stimulation,there was almost no difference in transient release between the active and passive cells because in both cases the terminal membrane of the cell senses the same potentials originating from the microelectrode.An exception was found for long pulses when the spike has the possibility to generate a higher terminal voltage than the passive cell.Simulated periodic 5 Hz stimulation showed a reduced transient release of 3 vesicles per stimulus,which is a recovery effect.
基金funded by the Deutsche Forschungsgemeinschaft(DFG,German Research Foundation)under Germany’s Excellence Strategy EXC 2044-390685587Mathematics Münster:Dynamics-Geometry-Structure。
文摘Although deep learning-based approximation algorithms have been applied very successfully to numerous problems,at the moment the reasons for their performance are not entirely understood from a mathematical point of view.Recently,estimates for the convergence of the overall error have been obtained in the situation of deep supervised learning,but with an extremely slow rate of convergence.In this note,we partially improve on these estimates.More specifically,we show that the depth of the neural network only needs to increase much slower in order to obtain the same rate of approximation.The results hold in the case of an arbitrary stochastic optimization algorithm with i.i.d.random initializations.
基金the Swiss National Science Foundation(SNSF)through the research grant 200020175699by the Deutsche Forschungsgemeinschaft(DFG,German Research Foundation)through CRC 1173,by the Karlsruhe House of Young Scientists(KHYS)through a research travel grant,by ETH Foundations of Data Science(ETH-FDS),and by the European Union(ERC,MONTECARLO,101045811)the Deutsche Forschungsgemeinschaft(DFG,German Research Foundation)under Germany’s Excellence Strategy EXC 2044-390685587,Mathematics Münster:DynamicsGeometry-Structure.
文摘It is one of the most challenging issues in applied mathematics to approximately solve high-dimensional partial differential equations(PDEs)and most of the numerical approximation methods for PDEs in the scientific literature suffer from the so-called curse of dimensionality in the sense that the number of computational operations employed in the corresponding approximation scheme to obtain an approximation precisionε>0 grows exponentially in the PDE dimension and/or the reciprocal ofε.Recently,certain deep learning based methods for PDEs have been proposed and various numerical simulations for such methods suggest that deep artificial neural network(ANN)approximations might have the capacity to indeed overcome the curse of dimensionality in the sense that the number of real parameters used to describe the approximating deep ANNs grows at most polynomially in both the PDE dimension d∈R and the reciprocal of the prescribed approximation accuracyε>0.There are now also a few rigorous mathematical results in the scientific literature which substantiate this conjecture by proving that deep ANNs overcome the curse of dimensionality in approximating solutions of PDEs.Each of these results establishes that deep ANNs overcome the curse of dimensionality in approximating suitable PDE solutions at a fixed time point T>0 and on a compact cube[a,b]d in space but none of these results provides an answer to the question whether the entire PDE solution on[0,T]×[a,b]^(d)can be approximated by deep ANNs without the curse of dimensionality.It is precisely the subject of this article to overcome this issue.More specifically,the main result of this work in particular proves for every a∈R,b∈(a,∞)that solutions of certain Kolmogorov PDEs can be approximated by deep ANNs on the space-time region[0,T]×[a,b]^(d)without the curse of dimensionality.
文摘In this article we propose a higher-order space-time conservative method for hyperbolic systems with stiff and non stiff source terms as well as relaxation systems.We call the scheme a slope propagation(SP)method.It is an extension of our scheme derived for homogeneous hyperbolic systems[1].In the present inhomogeneous systems the relaxation time may vary from order of one to a very small value.These small values make the relaxation term stronger and highly stiff.In such situations underresolved numerical schemes may produce spurious numerical results.However,our present scheme has the capability to correctly capture the behavior of the physical phenomena with high order accuracy even if the initial layer and the small relaxation time are not numerically resolved.The scheme treats the space and time in a unified manner.The flow variables and their slopes are the basic unknowns in the scheme.The source term is treated by its volumetric integration over the space-time control volume and is a direct part of the overall space-time flux balance.We use two approaches for the slope calculations of the flow variables,the first one results directly from the flux balance over the control volumes,while in the second one we use a finite difference approach.The main features of the scheme are its simplicity,its Jacobian-free and Riemann solver-free recipe,as well as its efficiency and high of order accuracy.In particular we show that the scheme has a discrete analog of the continuous asymptotic limit.We have implemented our scheme for various test models available in the literature such as the Broadwell model,the extended thermodynamics equations,the shallow water equations,traffic flow and the Euler equations with heat transfer.The numerical results validate the accuracy,versatility and robustness of the present scheme.
基金MCM for the hospitality and the financial supportsupported by SFB 1060+2 种基金Universitat Bonn during the last part of the workpartially supported by NSF of China under Grants Nos.11371347 and 11688101innovation grant from National Center for Mathematics and Interdisciplinary Sciences
文摘This paper is a continuation work of[26]and studies the propagation of the high-order boundary regularities of the two-dimensional density patch for viscous inhomogeneous incompressible flow.We assume the initial densityρ0=η11Ω0+η21Ω0^c,where(η1,η2)is any pair of positive constants andΩ0 is a bounded,simply connected domain with W^k+2,p(R^2)boundary regularity.We prove that for any positive time t,the density functionρ(t)=η11Ω(t)+η21Ω(t)^c,and the domainΩ(t)preserves the W^k+2,p-boundary regularity.
基金The authors acknowledge the financial support by the Vienna Science and Technology Fund(WWTF)under grant MA141-044.
文摘In this paper,we address the problem that standard stochastic Landau-Lifshitz-Gilbert(sLLG)simulations typically produce results that show unphysical mesh-size dependence.The root cause of this problem is that the effects of spin-wave fluctuations are ignored in sLLG.We propose to represent the effect of these fluctuations by a full-spin-wave-scaled stochastic LLG,or FUSSS LLG method.In FUSSS LLG,the intrinsic parameters of the sLLG simulations are first scaled by scaling factors that integrate out the spin-wave fluctuations up to the mesh size,and the sLLG simulation is then performed with these scaled parameters.We developed FUSSS LLG by studying the Ferromagnetic Resonance(FMR)in Nd_(2)Fe_(14)B cubes.The nominal scaling greatly reduced the mesh size dependence relative to sLLG.We then performed three tests and validations of our FUSSS LLG with this modified scaling.(1)We studied the same FMR but with magnetostatic fields included.(2)We simulated the total magnetization of the Nd_(2)Fe_(14)B cube.(3)We studied the effective,temperature-and sweeping rate-dependent coercive field of the cubes.In all three cases,we found that FUSSS LLG delivered essentially mesh-size-independent results,which tracked the theoretical expectations better than unscaled sLLG.Motivated by these successful validations,we propose that FUSSS LLG provides marked,qualitative progress towards accurate,high precision modeling of micromagnetics in hard,permanent magnets.
基金supported by the Graduiertenkolleg-828,"Micro-Macro-Interactions in Structured Media and Particles Systems",Otto-von-Guericke-University Magdeburg
文摘Aggregation is one of the many important processes in chemical and process engineering. Several researchers have attempted to understand this complex process in fluidized beds using the macro-model of population balance equations （PBEs）. The aggregation kernel is an effective parameter in PBEs, and is defined as the product of the aggregation efficiency and collision frequency functions. Attempts to derive this kernel have taken different approaches, including theoretical, experimental, and empirical techniques. The present paper calculates the aggregation kernel using micro-model computer simulations, i.e., a discrete particle model. We simulate the micro-model without aggregation for various initial conditions, and observe that the collision frequency function is in good agreement with the shear kernel. We then simulate the micro-model with aggregation and calculate the aggregation efficiency rate.
基金the National Natural Science Foundation of China (No.10401019)the DFG priority research program ANurnE (DFG Wa 633/9-2)National Natural Science Foundation of China (No. 10431060).
文摘This paper considers a kind of strongly coupled cross diffusion parabolic system, which can be used as the multi-dimensional Lyurnkis energy transport model in semiconductor science. The global existence and large time behavior are obtained for smooth solution to the initial boundary value problem. When the initial data are a small perturbation of an isothermal stationary solution, the smooth solution of the problem under the insulating boundary condition, converges to that stationary solution exponentially fast as time goes to infinity.
基金funding from the European Research Council (ERC)under the European Union's Horizon 2020 research and innovation programme,ERC Advanced Grant No.101018153support from the Austrian Science Fund (FWF) (Grants P33010,F65)supported by the NSFC (Grant No.12101305).
文摘In this paper,we derive rigorously a non-local cross-diffusion system from an interacting stochastic many-particle system in the whole space.The convergence is proved in the sense of probability by introducing an intermediate particle system with a mollified interaction potential,where the mollification is of algebraic scaling.The main idea of the proof is to study the time evolution of a stopped process and obtain a Gronwall type estimate by using Taylor's expansion around the limiting stochastic process.
