32,123 research outputs found
Lipschitz dependence of the coefficients on the resolvent and greedy approximation for scalar elliptic problems
We analyze the inverse problem of identifying the diffusivity coefficient of
a scalar elliptic equation as a function of the resolvent operator. We prove
that, within the class of measurable coefficients, bounded above and below by
positive constants, the resolvent determines the diffusivity in an unique
manner. Furthermore we prove that the inverse mapping from resolvent to the
coefficient is Lipschitz in suitable topologies. This result plays a key role
when applying greedy algorithms to the approximation of parameter-dependent
elliptic problems in an uniform and robust manner, independent of the given
source terms. In one space dimension the results can be improved using the
explicit expression of solutions, which allows to link distances between one
resolvent and a linear combination of finitely many others and the
corresponding distances on coefficients. These results are also extended to
multi-dimensional elliptic equations with variable density coefficients. We
also point out towards some possible extensions and open problems
Parameter estimations for SPDEs with multiplicative fractional noise
We study parameter estimation problem for diagonalizable stochastic partial
differential equations driven by a multiplicative fractional noise with any
Hurst parameter . Two classes of estimators are investigated:
traditional maximum likelihood type estimators, and a new class called
closed-form exact estimators. Finally the general results are applied to
stochastic heat equation driven by a fractional Brownian motion
Inverse stochastic optimal controls
We study an inverse problem of the stochastic optimal control of general
diffusions with performance index having the quadratic penalty term of the
control process. Under mild conditions on the drift, the volatility, the cost
functions of the state, and under the assumption that the optimal control
belongs to the interior of the control set, we show that our inverse problem is
well-posed using a stochastic maximum principle. Then, with the well-posedness,
we reduce the inverse problem to some root finding problem of the expectation
of a random variable involved with the value function, which has a unique
solution. Based on this result, we propose a numerical method for our inverse
problem by replacing the expectation above with arithmetic mean of observed
optimal control processes and the corresponding state processes. The recent
progress of numerical analyses of Hamilton-Jacobi-Bellman equations enables the
proposed method to be implementable for multi-dimensional cases. In particular,
with the help of the kernel-based collocation method for
Hamilton-Jacobi-Bellman equations, our method for the inverse problems still
works well even when an explicit form of the value function is unavailable.
Several numerical experiments show that the numerical method recover the
unknown weight parameter with high accuracy
Optimal Control of Convective FitzHugh-Nagumo Equation
We investigate smooth and sparse optimal control problems for convective
FitzHugh-Nagumo equation with travelling wave solutions in moving excitable
media. The cost function includes distributed space-time and terminal
observations or targets. The state and adjoint equations are discretized in
space by symmetric interior point Galerkin (SIPG) method and by backward Euler
method in time. Several numerical results are presented for the control of the
travelling waves. We also show numerically the validity of the second order
optimality conditions for the local solutions of the sparse optimal control
problem for vanishing Tikhonov regularization parameter. Further, we estimate
the distance between the discrete control and associated local optima
numerically by the help of the perturbation method and the smallest eigenvalue
of the reduced Hessian
Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations
This work derives a residual-based a posteriori error estimator for reduced
models learned with non-intrusive model reduction from data of high-dimensional
systems governed by linear parabolic partial differential equations with
control inputs. It is shown that quantities that are necessary for the error
estimator can be either obtained exactly as the solutions of least-squares
problems in a non-intrusive way from data such as initial conditions, control
inputs, and high-dimensional solution trajectories or bounded in a
probabilistic sense. The computational procedure follows an offline/online
decomposition. In the offline (training) phase, the high-dimensional system is
judiciously solved in a black-box fashion to generate data and to set up the
error estimator. In the online phase, the estimator is used to bound the error
of the reduced-model predictions for new initial conditions and new control
inputs without recourse to the high-dimensional system. Numerical results
demonstrate the workflow of the proposed approach from data to reduced models
to certified predictions
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