U. M. Ascher

University of British Columbia - Vancouver | UBC · Department of Computer Science

This paper examines issues of data completion and location uncertainty, popular in many practical PDE-based inverse problems, in the context of option calibration via recovery of local volatility surfaces. While real data is usually more accessible for this application than for many others, the data is often given only at a restricted set of locati...

We introduce a local volatility model for the valuation of options on commodity futures by using European vanilla option prices. The corresponding calibration problem is addressed within an online framework, allowing the use of multiple price surfaces. Since uncertainty in the observation of the underlying future prices translates to uncertainty in...

The implementation of the discrete adjoint method for exponential time differencing (ETD) schemes is considered. This is important for parameter estimation problems that are constrained by stiff time-dependent PDEs when the discretized PDE system is solved using an exponential integrator. We also discuss the closely related topic of computing the a...

We introduce a local volatility model for the valuation of options on commodity futures by using European vanilla option prices. The corresponding calibration problem is addressed within an online framework, allowing the use of multiple price surfaces. Since uncertainty in the observation of the underlying future prices translates to uncertainty in...

We present a systematic derivation of the algorithms required for computing the gradient and the action of the Hessian of an arbitrary misfit function for large-scale parameter estimation problems involving linear time-dependent PDEs with stationary coefficients. These algorithms are derived using the adjoint method for time-stepping schemes of arb...

We present a data-driven method for deformation capture and modeling of general soft objects. We adopt an iterative framework that consists of one component for physics-based deformation tracking and another for spacetime optimization of deformation parameters. Low cost depth sensors are used for the deformation capture, and we do not require any f...

Iterative numerical algorithms are typically equipped with a stopping criterion, where the iteration process is terminated when some error or misfit measure is deemed to be below a given tolerance. This is a useful setting for comparing algorithm performance, among other purposes. However, in practical applications a precise value for such a tolera...

This article considers stochastic algorithms for efficiently solving a class of large scale non-linear least squares (NLS) problems which frequently arise in applications. We propose eight variants of a practical randomized algorithm where the uncertainties in the major stochastic steps are quantified. Such stochastic steps involve approximating th...

Inverse problems involving systems of partial differential equations (PDEs) can be very expensive to solve numerically. This is so especially when many experiments, involving different combinations of sources and receivers, are em-ployed in order to obtain reconstructions of acceptable quality. The mere eval-uation of a misfit function (the distanc...

Inverse problems involving systems of partial differential equations (PDEs) with many measurements or experiments can be very expensive to solve numerically. In a recent paper we examined dimensionality reduction methods, both stochastic and deterministic, to reduce this computational burden, assuming that all experiments share the same set of rece...

Concocting a plausible composition from several non-overlapping image pieces, whose relative positions are not fixed in advance and without having the benefit of priors, can be a daunting task. Here we propose such a method, starting with a set of sloppily pasted image pieces with gaps between them. We first extract salient curves that approach the...

Regularization methods based on the l1 norm, including sparse wavelet representations and total variation, have justifiably become immensely popular in recent years. There are many applications in which they are more suitable than the simpler, faster and better known l2-based alternatives. However, such techniques also have their limitations. In fa...

Much recent attention has been devoted to gradient descent algorithms where the steepest descent step size is replaced by a similar one from a previous iteration or gets updated only once every second step, thus forming a faster gradient descent method. For unconstrained convex quadratic optimization these methods can converge much faster than stee...

This article is concerned with Monte-Carlo methods for the estimation of the trace of an implicitly given matrix $A$ whose information is only available through matrix-vector products. Such a method approximates the trace by an average of $N$ expressions of the form $\ww^t (A\ww)$, with random vectors $\ww$ drawn from an appropriate distribution. W...

Points acquired by laser scanners are not intrinsically equipped with nor-mals, which are essential to surface reconstruction and point set render-ing using surfels. Normal estimation is notoriously sensitive to noise. Near sharp features, the computation of noise-free normals becomes even more challenging due to the inherent under-sampling problem...

Gradient descent methods for large positive definite linear and nonlinear algebraic systems arise when integrating a PDE to steady state and when regularizing inverse problems. However, these methods may converge very slowly when utilizing a constant step size, or when employing an exact line search at each step, with the iteration count growing pr...

The steepest descent method for large linear systems is well-known to often converge very slowly, with the number of iterations required being about the same as that obtained by utilizing a gradient descent method with the best constant step size and growing proportionally to the condition number. Faster gradient descent methods must occasionally r...

This article develops fast numerical methods for the practical solution of the famous electrical impedance tomography and DC resistivity problems in the presence of discontinuities and potentially many experiments or data. Based on a Gauss-Newton (GN) approach coupled with preconditioned conjugate gradient (PCG) iterations, we propose two algorithm...

We describe an efficient method for reconstructing the activity in human muscles from an array of voltage sensors on the skin surface. MRI is used to obtain morphometric data which are segmented into muscle tissue, fat, bone and skin, from which a finite element model for volume conduction is constructed. The inverse problem of finding the current...

We present a method for calculating complete theoretical seismograms in earth models whose velocity, density and attenuation profiles are arbitrary piecewise-continuous functions of depth only. A form of attenuation valid for low loss situations is included by allowing the seismic velocities to be complex, and frequency is also allowed to be comple...

This paper considers highly ill-posed surface recovery inverse problems, where the sought surface in 2D or 3D is piecewise constant with several possible level values. These levels may further be potentially unknown. Multiple level set functions are used when there are more than two such levels, and we extend the methods and theory of our previous...

We consolidate an unorganized point cloud with noise, outliers, non-uniformities, and in particular interference between close-by surface sheets as a preprocess to surface generation, focusing on reliable normal estimation. Our algorithm includes two new developments. First, a weighted locally optimal projection operator produces a set of denoised,...

We consolidate an unorganized point cloud with noise, outliers, non-uniformities, and in particular interference between close-by surface sheets as a preprocess to surface generation, focusing on reliable normal estimation. Our algorithm includes two new developments. First, a weighted locally optimal projection operator produces a set of denoised,...

In the course of simulation of differential equations, especially of marginally stable differential problems using marginally stable numerical methods, one occasionally comes across a correct computation that yields surprising, or unexpected results. We exam-ine several instances of such computations. These include (i) solving Hamiltonian ODE syste...

The integration to steady state of many initial value ODEs and PDEs using the forward Euler method can alternatively be considered as gradient descent for an associated minimization problem. Greedy algorithms such as steepest descent for determining the step size are as slow to reach steady state as is forward Euler integration with the best unifor...

We describe a methodology called computed myography to qualitatively and quantitatively determine the activation level of individual muscles by voltage measurements from an array of voltage sensors on the skin surface. A finite element model for electrostatics simulation is constructed from morphometric data. For the inverse problem, we utilize a g...

We present a numerical scheme for the real-time solution of the discretized one-dimensional linearized acoustics equation (Webster's equation) augmented with dissipative terms, in a tube with a spatially and temporally varying cross section. The resulting algorithm produces similar results as the Kelly-Lochbaum model but has several advantages over...

The integration to steady state of many initial value ODEs and PDEs using the forward Euler method can alternatively be considered as gradient descent for an associated min-imization problem. Greedy algorithms such as steepest descent for determining the step size are as slow to reach steady state as is forward Euler integration with the best unifo...

We describe a fast, dynamic, multiscale iterative method that is designed to smooth, but not over-smooth, noisy triangle meshes. Our method not only preserves sharp features but also retains visually meaningful fine scale components or details, referred to as intrinsic texture. An anisotropic Laplacian (AL) operator is first developed. It is then e...

We describe a hybrid algorithm that is designed to reconstruct a piecewise smooth surface mesh from noisy input. While denoising, our method simultaneously regularizes triangle meshes on flat regions for further mesh processing and preserves crease sharpness for faithful reconstruction. A clustering technique, which combines K-means and geometric a...

We describe a methodology to qualitatively and quantitatively determine the activation level of individual muscles by voltage measurements from an array of voltage sensors on the skin surface. A physical finite element model for electrostatics simulation is constructed from morphometric data and numerical inversion techniques are used to determine...

List of figures List of tables Preface Introduction 1. Ordinary differential equations 2. On problem atability 3. Basic methods, Basic concepts 4. One-step methods 5. Linear multistep methods 6. More boundary value problem theory and applications 7. Shooting 8. Finite difference methods for boundary value problems 9. More on differential-algebraic...

We describe a hybrid algorithm that is designed to smooth, but not only smooth, noisy polygonal surface meshes with sharp edges. While denoising, our method simultaneously regularizes triangle meshes on flat regions for further mesh processing and preserves edge sharpness for faithful reconstruction. A clustering technique, which combines K-means a...

In this paper we develop a finite volume adaptive grid refinement method for the solution of distributed parameter estimation problems with almost discontinuous coefficients. We discuss discretization on locally refined grids, as well as optimization and refinement criteria. An OcTree data structure is utilized. We show that local refinement can si...

This article considers inverse problems of shape recovery from noisy bound-ary data, where the forward problem involves the inversion of elliptic PDEs. The piecewise constant solution, a scaling and translation of a characteristic function, is described in terms of a smoother level set function. A fast and simple dynamic regularization method has b...

A numerical method is presented for calculating complete theoretical seismograms, under the assumption that the earth models have velocity, density and attenuation profiles which are arbitrary piece-wise continuous functions of depth only. Solutions for the stress-displacement vectors in the medium are expanded in terms of orthogonal cylindrical fu...

Many recent algorithmic approaches involve the construction of a differential equation model for computational purposes, typically by introducing an artificial time variable. The actual computational model involves a discretization of the now time-dependent differential system, usually employing forward Euler. The resulting dynamics of such an algo...

We present a novel shape reconstruction technique for 3D low frequency electromagnetic induction tomography which uses a level set representation of the shapes. An efficient ad-joint scheme for calculating gradient directions corresponding to the data of each individual probing source is combined with a single-step technique for finding iterative c...

The recovery of a distributed parameter function with discontinuities from inverse problems with elliptic forward PDEs is fraught with theoretical and practical difficulties. Better results are obtained for problems where the solution may take on at each point only one of two values, thus yielding a shape recovery problem.This article considers lev...

This paper considers the problem of reconstructing a piecewise smooth model function from given, measured data. The data are compared to a field which is given as a possibly nonlinear function of the model. A regularization functional is added which incorporates the a priori knowledge that the model function is piecewise smooth and may contain jump...

This paper considers problems of distributed parameter estimation from data measurements on solutions of diffusive partial differential equations (PDEs). A nonlinear functional is minimized to approximately recover the sought parameter function (i.e., the model). This functional consists of a data fitting term, involving the solution of a finite vo...

We examine some symplectic and multisymplectic methods for the notorious Korteweg-de Vries equation, with the question whether the added structure preservation that these methods offer is key in providing high quality schemes for the long time integration of nonlinear, conservative partial differential equations. Concentrating on second order discr...

We present a general formulation for inverting frequency-or time-domain electromagnetic data using an all-at-once approach. In this methodology, the forward modeling equations are incorporated as constraints and, thus, we need to solve a constrained optimization problem where the parameters are the electromagnetic fields, the conductivity model, an...

We develop and compare some geometric integrators for the Korteweg–de Vries equation, especially with regard to their robustness for large steps in space and time, Δx and Δt, and over long times. A standard, semi-explicit, symplectic finite difference scheme is found to be fast and robust. However, in some parameter regimes such schemes are suscept...

Implicit schemes have become the standard for integrating the equations of motion in cloth simulation. These schemes, however, require the solution of a system representing the entire, fully connected cloth mesh at each time step. In this paper we present techniques that dynamically improve the sparsity of the underlying system, ultimately allowing...

The seminal paper on cloth simulation by Baraff and Witkin [4] presents a modified preconditioned conjugate gradient (MPCG) algorithm for solving certain large, sparse systems of linear equations. These arise when employing implicit time integration methods aimed at achieving large step cloth simulation in the presence of constraints. This paper im...

This paper considers problems of distributed parameter estimation from data measurements on solutions of diffusive partial differential equations (PDEs). A nonlinear functional is minimized to approximately recover the sought parameter function (i.e., the model). This functional consists of a data fitting term, involving the solution of a finite vo...

We consider the rapid simulation of three-dimensional electromagnetic problems in geophysical parameter regimes, where the conductivity may vary significantly and the range of frequencies is moderate. Toward developing a multigrid preconditioner, we present a Fourier analysis based on a finite-volume discretization of a vector potential formulation...

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The virtual worlds of computer games and similar animated simulations may be populated by autonomous characters that intelligently navigate in virtual cities. We concretely apply hybrid system theory and tools to model navigation strategies for virtual ...

The numerical integration of highly oscillatory Hamiltonian systems, such as those arising in molecular dynamics or Hamiltonian partial differential equations, is a challenging task. Various methods have been suggested to overcome the step-size restrictions of explicit methods such as the Verlet method. Among these are multiple-time-stepping, const...

The (implicit) midpoint scheme, like higher order Gauss-collocation schemes, is algebraically stable and symplectic, and it preserves quadratic integral invariants. It may appear particularly suitable for the numerical solution of highly oscillatory Hamiltonian systems, such as those arising in molecular dynamics or structural mechanics, because th...

. This paper considers problems of distributed parameter estimation from data measurements on solutions of partial differential equations (PDEs). A nonlinear least squares functional is minimized to approximately recover the sought parameter function (i.e., the model). This functional consists of a data fitting term, involving the solution of a fin...

This paper considers problems of distributed parameter estimation from data measurements on solutions of differential equations. A nonlinear least squares functional is minimized to approximately recover the sought parameter function (i.e. the model). This functional consists of a data fitting term, involving the solution of a finite volume or fini...

The method of programmed constraints has recently been proposed as an executable speci#cation language for robot programming. The mathematical structures behind such problems are viability problems for control systems described by ordinary di#erential equations #ODEs# subject to user-de#ned inequality constraints. This paper describes a method for...

The numerical simulation problem of tree-structured multibody systems,

We describe a framework for derivation of several forward dynamics algorithms used in robotics. The framework is based on formulating an augmented system and performing block matrix elimination on this system. Several popular algorithms such as the O(N) Articulated Body method, and Composite Rigid Body method can be easily derived. We also derive a...

The problem of recovering a parameter function based on measurements of solutions of a system of partial differential equations in several space variables leads to a number of computational challenges. Upon discretization of a regularized formulation a large, sparse constrained optimization problem is obtained. Typically in the literature, the cons...

We consider solving three-dimensional electromagnetic problems in parameter regimes where the quasi-static approximation applies, the permeability is constant, the conductivity may vary significantly, and the range of frequencies is moderate. The difficulties encountered include handling solution discontinuities across interfaces and accelerating c...

This paper considers problems of distributed parameter estimation from data measurements on solutions of differential equations. A nonlinear least squares functional is minimized to approximately recover the sought parameter function (i.e., the model). This functional consists of a data fitting term, involving the solution of a finite volume or fin...

We consider solving three-dimensional electromagnetic problems in parameter regimes where the quasi-static approximation applies and the permeability, permittivity and conductivity may vary significantly. The difficulties encountered include handling solution discontinuities across interfaces and accelerating convergence of traditional iterative me...

This paper considers optimization techniques for the solution of nonlinear inverse problems where the forward problems, like those encountered in electromagnetics, are modelled by differential equations. Such problems are often solved by utilising a Gauss-Newton method in which the forward model constraints are implicitly incorporated. Variants of...

The method of programmed constraints has recently been proposed as an executable specification language for robot programming. The mathematical structures behind such problems are viability problems for control systems described by ordinary differential equations (ODE) subject to user-defined inequality constraints. This paper describes a method fo...

We describe a framework for derivation of several forward dynamics algorithms used in robotics. The framework is based on formulating an augmented system and performing block matrix elimination on this system. Several popular algorithms such as the O(N) articulated body method, and the composite rigid body method can be easily derived. We also deri...

This paper considers optimization techniques for the solution of nonlinear inverse problems where the forward problems, like those encountered in electromagnetics, are modelled by differential equations. Such problems are often solved by utilizing a Gauss-Newton method in which the forward model constraints are implicitly incorporated. Variants of...

We consider solving three-dimensional electromagnetic problems in parameter regimes where the quasi-static approximation applies and the permeability, permittivity and conductivity may vary significantly. The difficulties encountered include handling solution discontinuities across interfaces and accelerating convergence of traditional iterative me...

We consider solving three-dimensional electromagnetic problems in parameter regimes where the quasi-static approximation applies, the permeability is constant, the conductivity may vary significantly, and the range of frequencies is moderate. The difficulties encountered include handling solution discontinuities across interfaces and accelerating c...

Summary We develop a forward modelling algorithm for computing time domain electromagnetic responses for conductive, permeable bodies in the low frequency regime. Both physical properties can be highly discontinuous. The parameter regimes considered give rise to highly sti� problems in the time domain and an implicit method, backward Euler, is used...

Summary We develop an inversion algorithm for computing fre- quency domain electromagnetic inversion for conductive bodies in the low frequency regime. The algorithm is based on an inexact Gauss-Newton method where only sensitivities times vector are calculated. Because of the anticipated heavy computational load, one has to answer many practical q...

Summary We develop an inversion methodology for 3D electromag- netic data when the forward model consists of Maxwell's equations in which the permeability is constant but electrical conductivity can be highly discontinuous. The goal of the inversion is to recover the conductivity given measurements of the electric and/or magnetic fields. A standard...

We present a solution method for solving electromagnetic problems in three dimensions in parameter regimes where the quasi-static approximation applies and the permeability is constant. Firstly, by using a potential formulation with a Coulomb gauge, we circumvent the ill-posed problem in regions of vanishing conductivity, obtaining a system of elli...

This paper discusses different approaches for the solution of nonlinear inverse problems where the forward problems are modelled by differential equations. Typically, such problems are solved by using the Gauss-Newton method, coupled with elimination of the forward model constraints implicitly. In this paper we discuss four other options. By regard...

The closing decades of the 20th century have seen many scientists recognize that their mathematical models are in fact instances of DAEs, or of ODEs with invariants. Such a recognition has often carried with it the benefit of affording a new, sometimes revealing, computational look at the old problem. But one must not conclude that reformulating a...

Many methods have been proposed for the simulation of constrained mechanical systems. The most obvious methods have mild instabilities and drift problems, and consequently stabilization techniques have been proposed. A popular stabilization method is Baumgarte's technique, but the choice of parameters to make it robust has been unclear in practice....

Computer methods for ordinary differential equations and differential-algebraic equations are presented. Topics discussed include: ordinary differential equations; initial value problems; boundary value problems; differential-algebraic equations; dynamical systems; Euler equation; nonlinear equations; Runge-Kutta methods; error estimation; implicit...

Certain classes of nodal methods and mixed-hybrid finite element methods lead to equivalent, robust, and accurate discretizations of second-order elliptic PDEs. However, widespread popularity of these discretizations has been hindered by the awkward linear systems which result. The present work overcomes this awkwardness and develops preconditioner...

Implicit-explicit (IMEX) linear multistep time-discretization schemes for partial differential equations have proved useful in many applications. However, they tend to have undesirable time-step restrictions when applied to convection-diffusion problems, unless diffusion strongly dominates and an appropriate BDF-based scheme is selected (Ascher et...

Just less than two years ago Bill Gear celebrated his 60th birthday, and exactly two months later on April 1, his colleagues and friends honored him at the culmination of SciCADE 95, which was held in beautiful spring weather on the campus of Stanford University, thanks to the efforts of Gene Golub and Andrew Stuart. It was there in 1968–69 that Bi...

Sequential regularization methods relate to a combination of stabilization methods and the usual penalty method for differential equations with algebraic equality constraints. This paper extends an earlier work [SIAM J. Numer. Anal., 33 (1996), pp. 1921--1940] to nonlinear problems and to differential algebraic equations (DAEs) with an index higher...

The numerical simulation problem of large multibody systems has often been treated in two separate stages: (i) the forward dynamics problem for computing system accelerations from given force functions and constraints, and (ii) the numerical integration problem for advancing the state in time. For the forward dynamics problem, algorithms have been...

Many problems of practical interest can be modeled by differential systems where the solution lies on an invariant manifold defined explicitly by algebraic equations. In computer simulations, it is often important to take into account the invariant's information, either in order to improve upon the stability of the discretization (which is especial...

: The shape-from-shading problem has received much attention in the Computer Vision literature in recent years. The basic problem is to recover the shape z(x; y) of a surface from a given map of its shading, i.e. its variation of brightness over a given domain. Mathematically, one has to solve approximately the image irradiance equation R(p; q)(x;...

We describe the methods and implementation of a general-purpose code, COLDAE. This code can solve boundary value problems for nonlinear systems of semi-explicit differential-algebraic equations (DAEs) of index at most 2. Fully implicit index-1 boundary value DAE problems can be handled as well. The code COLDAE is an extension of the package COLNEW...

Recently, new robot programming approaches have proposed the use of programmed constraints as an executable specification language for the desired behavior of a robot. The constraint-based approaches are intermediate level languages, promising a higher, more declarative level of programming than trajectory-based approaches, while being more tractab...

In this paper we consider the numerical solution of initial value delay-differential-algebraic equations (DDAEs) of retarded and neutral types, with a structure corresponding to that of Hessenberg DAEs. We give conditions under which the DDAE is well-conditioned, and show how the DDAE is related to an underlying retarded or neutral delay-ODE (DODE)...

Standard stabilization techniques for higher index differential-algebraic equations (DAEs) often involve elimination of the algebraic solution components. This may not work well if there are singularity points where the constraint Jacobian matrix becomes rank deficient. This paper proposes instead a sequential regularization method (SRM)—a function...

The numerical simulation problem of tree-structured multi body systems, such as robot manipulators, is usually treated as two separate problems: 1) the forward dynamics problem for computing system accelerations, and 2) the numerical integra tion problem for advancing the state in time. The interaction of these two problems can be important, and ha...

Recently, new robot programming approaches have proposed the use of programmed constraints as an executable specification language for the desired behavior of a robot. The constraint-based approaches are intermediate level languages, promising a higher, more declarative level of programming than trajectory-based approaches, while being more tractab...

Implicit-explicit (IMEX) schemes have been widely used, especially in conjunction with spectral methods, for the time integration of spatially discretized partial differential equations (PDEs) of diffusion-convection type. Typically, an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term. Reaction-d...

The authors identify an important phenomenon they call “formulation stiffness” in the numerical simulation of tree-structured multibody systems such as robot manipulators. The numerical simulation problem is usually treated as two separate problems: (i) the forward dynamics problem for computing system accelerations, and (ii) the numerical integrat...

. This paper presents an efficient multigrid solver for steady-state Navier-Stokes equations in 2D on non-staggered grids. The pressure Poisson equation formulation is used, together with a finite volume discretization. A discretization of the boundary conditions for pressure and velocities is presented. An efficient multigrid algorithm for solving...

. Implicit-explicit (IMEX) schemes have been widely used, especially in conjunction with spectral methods, for the time integration of spatially discretized PDEs of diffusion-convection type. Typically, an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term. Reaction-diffusion problems can also be a...

Many methods have been proposed for the stabilization of higher index differential-algebraic equations (DAEs). Such methods often involve constraint differentiation and problem stabilization, thus obtaining a stabilized index reduction. A popular method is Baumgarte stabilization, but the choice of parameters to make it robust is unclear in practic...

Many methods have been proposed for numerically integrating the differential-algebraic systems arising from the Euler–Lagrange equations for constrained motion. These are based on various problem formulations and discretizations. We offer a critical evaluation of these methods from the standpoint of stability. Considering a linear model, we first g...

Higher-order, higher-index Hessenberg systems of initial and boundary value differential-algebraic equations (DAEs) are considered. These types of systems arise in a variety of applications, including multibody systems. We extend a class of recently introduced projected implicit Runge-Kutta methods and define a new class of projected piecewise poly...

Differential-algebraic equation (DAE) boundary value problems arise in a variety of applications, including optimal control and parameter estimation for constrained systems. In this paper we survey these applications and explore some of the difficulties associated with solving the resulting DAE systems. For finite difference methods, the need to ma...