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A computational framework for infinitedimensional Bayesian inverse problems: Part II. Stochastic Newton MCMC with application to ice sheet flow inverse problems
 SIAM Journal on Scientific Computing
, 2014
"... Abstract. We address the numerical solution of infinitedimensional inverse problems in the framework of Bayesian inference. In the Part I [11] companion to this paper, we considered the linearized infinitedimensional inverse problem. Here in Part II, we relax the linearization assumption and consi ..."
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Abstract. We address the numerical solution of infinitedimensional inverse problems in the framework of Bayesian inference. In the Part I [11] companion to this paper, we considered the linearized infinitedimensional inverse problem. Here in Part II, we relax the linearization assumption and consider the fully nonlinear infinitedimensional inverse problem using a Markov chain Monte Carlo (MCMC) sampling method. To address the challenges of sampling highdimensional probability density functions (pdfs) arising upon discretization of Bayesian inverse problems governed by PDEs, we build on the stochastic Newton MCMC method. This method exploits problem structure by taking as a proposal density a local Gaussian approximation of the posterior pdf, whose covariance operator is given by the inverse of the local Hessian of the negative log posterior pdf. The construction of the covariance is made tractable by invoking a lowrank approximation of the data misfit component of the Hessian. Here we introduce an approximation of the stochastic Newton proposal in which we compute the lowrankbased Hessian at just the MAP point, and then reuse this Hessian at each MCMC step. We compare the performance of the proposed method to the original stochastic Newton MCMC method and to an independence sampler. The comparison of the three methods is conducted on a synthetic ice sheet inverse problem. For this problem, the stochastic Newton MCMC method
MIXED FINITE ELEMENT METHOD FOR ELECTROWETTING ON DIELECTRIC WITH CONTACT LINE PINNING
"... Abstract. We present a mixed finite element method for a model of the flow in a HeleShaw cell of 2D fluid droplets surrounded by air driven by surface tension and actuated by an electric field. The application of interest regards a microfluidic device called ElectroWetting on Dielectric (EWOD). O ..."
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Abstract. We present a mixed finite element method for a model of the flow in a HeleShaw cell of 2D fluid droplets surrounded by air driven by surface tension and actuated by an electric field. The application of interest regards a microfluidic device called ElectroWetting on Dielectric (EWOD). Our analysis first focuses on the timediscrete (continuous in space) problem and is presented in a mixed variational framework, which incorporates curvature as a natural boundary condition. The model includes a viscous damping term for interface motion, as well as contact line pinning (sticking of the interface) and is captured in our formulation by a variational inequality. The semidiscrete problem uses a semiimplicit timediscretization of curvature. We prove the wellposedness of the semidiscrete problem and fully discrete problem when discretized with isoparametric finite elements. We derive a priori error estimates for the space discretization. We also prove the convergence of an Uzawa algorithm for solving the semidiscrete EWOD system with inequality constraint. We conclude with a discussion about experimental orders of convergence. 1.
A Computational Framework for the Solution of InfiniteDimensional Bayesian Statistical Inverse Problems with Application to Global Seismic Inversion
, 2015
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AFEM FOR SHAPE OPTIMIZATION
"... Abstract. We examine shape optimization problems in the context of inexact sequential quadratic programming. Inexactness is a consequence of using adaptive finite element methods (AFEM) to approximate the state and adjoint equations (via the dual weighted residual method), update the boundary, and c ..."
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Abstract. We examine shape optimization problems in the context of inexact sequential quadratic programming. Inexactness is a consequence of using adaptive finite element methods (AFEM) to approximate the state and adjoint equations (via the dual weighted residual method), update the boundary, and compute the geometric functional. We present a novel algorithm that equidistributes the errors due to shape optimization and discretization, thereby leading to coarse resolution in the early stages and fine resolution upon convergence, and thus optimizing the computational effort. We discuss the ability of the algorithm to detect whether or not geometric singularities such as corners are genuine to the problem or simply due to lack of resolution—a new paradigm in adaptivity. 1. Shape Optimization as Adaptive Sequential Quadratic Programming We consider shape optimization problems for partial differential equations (PDE) that can be formulated as follows: We denote with u = u(Ω) the solution of a PDE in the domain Ω of R d (d ≥ 2), (1) Bu(Ω) = f, which we call the state equation. Given a cost functional J[Ω] = J[Ω,u(Ω)], which depends on Ω itself and the solution u(Ω) of the state equation, we consider the minimization problem (2) Ω ∗ ∈ Uad: J[Ω ∗,u(Ω ∗)] = inf J[Ω,u(Ω)], Ω∈Uad where Uad is a set of admissible domains in Rd. We view this as a constrained minimization problem, (1) being the constraint. We do not discuss conditions on B,J or Uad that yield existence of a solution. The goal of this paper is, instead, to formulate and test a practical and efficient computational algorithm that adaptively builds a sequence of domains {Ωk}k≥0 converging to a local minimizer Ω of (1)–(2). Coupling adaptivity with shape optimization seems to be important but rather novel. To achieve this goal we will define an Adaptive Sequential Quadratic Programming algorithm (ASQP). To motivate and briefly describe the ideas underlying ASQP, we need the concept of shape derivative δΩJ[Ω;v] of J[Ω] in the direction of a velocity v, which usually satisfies (3) δΩJ[Ω;v] = g(Ω)v dS = 〈g(Ω),v〉Γ,
SECOND ORDER ALGORITHM FOR SURFACE RESTORATION
"... Abstract. A novel iterative scheme for surface modeling problems such as surface restoration and blending is proposed. Its main features are simplicity and effectiveness. It is motivated on a geometric differential identity when properly formulated in the discrete setting. Each iteration involves so ..."
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Abstract. A novel iterative scheme for surface modeling problems such as surface restoration and blending is proposed. Its main features are simplicity and effectiveness. It is motivated on a geometric differential identity when properly formulated in the discrete setting. Each iteration involves solving an elliptic problem but despite its second order the scheme allows to impose not only boundary values but also boundary normals. Several simulations using low order continuous finite elements are presented illustrating the power, flexibility and efficiency of the method. 1.
seba@math.umd.edu. Partially supported by NSF grants DMS0505454 and DMS0807811.
"... Abstract We examine shape optimization problems in the context of inexact sequential quadratic programming. Inexactness is a consequence of using adaptive finite element methods (AFEM) to approximate the state equation, update the boundary, and compute the geometric functional. We present a novel al ..."
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Abstract We examine shape optimization problems in the context of inexact sequential quadratic programming. Inexactness is a consequence of using adaptive finite element methods (AFEM) to approximate the state equation, update the boundary, and compute the geometric functional. We present a novel algorithm that uses a dynamic tolerance and equidistributes the errors due to shape optimization and discretization, thereby leading to coarse resolution in the early stages and fine resolution upon convergence. We discuss the ability of the algorithm to detect whether or not geometric singularities such as corners are genuine to the problem or simply due to lack of resolution — a new paradigm in adaptivity. 1 Shape Optimization as Adaptive Sequential Quadratic Programming Shape optimization problems governed by partial differential equations (PDE) can be formulated as constrained minimization problems with respect to the shape of a domain Ω in R d. If u = u(Ω) is the solution of a PDE in Ω, the state equation is A u(Ω) = f, (1) and J(Ω) = J(Ω,u(Ω)) is a cost functional, then we consider the minimization problem
VARIATIONAL AND NONVARIATIONAL MULTIGRID ALGORITHMS FOR THE LAPLACEBELTRAMI OPERATOR.
"... Abstract. We design and analyze variational and nonvariational multigrid algorithms for the LaplaceBeltrami operator on a smooth and closed surface. In both cases, a uniform convergence for the Vcycle algorithm is obtained provided the surface geometry is captured well enough by the coarsest grid ..."
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Abstract. We design and analyze variational and nonvariational multigrid algorithms for the LaplaceBeltrami operator on a smooth and closed surface. In both cases, a uniform convergence for the Vcycle algorithm is obtained provided the surface geometry is captured well enough by the coarsest grid. The main argument hinges on a perturbation analysis from an auxiliary variational algorithm defined directly on the smooth surface. In addition, the vanishing mean value constraint is imposed on each level, thereby avoiding singular quadratic forms without adding additional computational cost. Numerical results supporting our analysis are reported. In particular, the algorithms perform well even when applied to surfaces with a large aspect ratio. 1. Introduction. Geometric differential equations have received an increasing interest in the recent years. They play a crucial role in many applications with impact in biotechnologies [9, 18], image processing [1, 13] and fluid flows [3] for instance. All these applications