Approximation of Bayesian inverse problems for PDEs

Inverse problems are often ill posed, with solutions that depend sensitively on data. In any numerical approach to the solution of such problems, regularization of some form is needed to counteract the resulting instability. This paper is based on an approach to regularization, employing a Bayesian formulation of the problem, which leads to a notion of well posedness for inverse problems, at the level of probability measures. The stability which results from this well posedness may be used as the basis for quantifying the approximation, in finite dimensional spaces, of inverse problems for functions. This paper contains a theory which utilizes this stability property to estimate the distance between the true and approximate posterior distributions, in the Hellinger metric, in terms of error estimates for approximation of the underlying forward problem. This is potentially useful as it allows for the transfer of estimates from the numerical analysis of forward problems into estimates for the solution of the related inverse problem. It is noteworthy that, when the prior is a Gaussian random field model, controlling differences in the Hellinger metric leads to control on the differences between expected values of polynomially bounded functions and operators, including the mean and covariance operator. The ideas are applied to some non-Gaussian inverse problems where the goal is determination of the initial condition for the Stokes or Navier–Stokes equation from Lagrangian and Eulerian observations, respectively

Study on contraction and relaxation of experimentally denervated and immobilized muscles: Comparison with dystrophic muscles

The contraction-relaxation mechanism of experimentally denervated and immobilized muscles of the rabbit is examined. Results are compared with those of human dystrophic muscles, in order to elucidate the role and extent of the neurotrophic factor, and the role played by the intrinsic activity of muscle in connection with pathogenesis and pathophysiology of this disease

The art of spacecraft design: A multidisciplinary challenge

Actual design turn-around time has become shorter due to the use of optimization techniques which have been introduced into the design process. It seems that what, how and when to use these optimization techniques may be the key factor for future aircraft engineering operations. Another important aspect of this technique is that complex physical phenomena can be modeled by a simple mathematical equation. The new powerful multilevel methodology reduces time-consuming analysis significantly while maintaining the coupling effects. This simultaneous analysis method stems from the implicit function theorem and system sensitivity derivatives of input variables. Use of the Taylor's series expansion and finite differencing technique for sensitivity derivatives in each discipline makes this approach unique for screening dominant variables from nondominant variables. In this study, the current Computational Fluid Dynamics (CFD) aerodynamic and sensitivity derivative/optimization techniques are applied for a simple cone-type forebody of a high-speed vehicle configuration to understand basic aerodynamic/structure interaction in a hypersonic flight condition

Analyses of dNch/dη and dNch/dy distributions of BRAHMS Collaboration by means of the Ornstein-Uhlenbeck process

Interesting data on dNch/dη in Au-Au collisions (η=-In tan(θ/2)) with the centrality cuts have been reported by BRAHMS Collaboration. Using the total multiplicity Nch=∫(dNch/dη)dη, we find that there are scaling phenomena among (Nch){ 1dNch/dη=dn/dη with different centrality cuts at √SNN=130 GeV and 200 GeV, respectively. To explain these scaling behaviors of dnl'drj, we consider the stochastic approach named the Ornstein-Uhlenbeck process with two sources. The following Fokker-Planck equation is adopted for the present analyses, ∂P(x,t)/∂t = γ[∂/∂x x+1/2 σ2/γ ∂2/∂x2]P(x,t) where x means the rapidity (y) or pseudo-rapidity (η). t, γ and σ2 and the evolution parameter, the frictional coefficient and the variance, respectively. Introducing a variable of zr=η/ηrms (ηrms=√) we explain the dn/dzr distributions in the present approach. Moreover, to explain the rapidity (y) distributions from η distributions at 200 GeV, we have derived the formula as dn/dy=J{ 1dn/dη' where J{ 1=√M(1+sinh2y)/√1+Msinh2y with M=1+(m/pt)2. Their data of pion and all hadrons are fairly well explained by the O-U process. To compare our approach with another one, a phenomenological formula by Eskola et al. is also used in calculations of dn/dη.Article信州大学理学部紀要 38: 1-12(2004)departmental bulletin pape