# Ph.D. Programs

## Specialization Areas

##### Civil Engineering

- Construction Engineering
- Earthquake Engineering
- Informatics for Intelligent Built Environments
- Structural Engineering and Design
- Structural Mechanics
- Water Resources Engineering, Coastal Engineering

##### Environmental Engineering

- Energy and Sustainability
- Environmental Air Pollution
- Environmental Chemistry
- Global Warming
- Transport of Environmental Contaminants: Water, Air and Land
- Water Quality Control

For more information, please visit the ENE website.

The Sonny Astani Department of Civil and Environmental Engineering has numerous research groups, laboratories and centers. To learn more, visit the research page here.

## Program Requirements

##### Satisfactory completion of:

- at least 60 units of approved graduate level course work
- a cumulative grade point average of at least 3.0

Credit toward the 60 units of required course work is given for approved course work in the area of specialization of the Ph.D. program. Typically Ph.D. students have completed a master degree with 27 units of course work, and take 33 additional units of course work in order to satisfy the 60 unit requirement.

Ph.D. students take 9-12 units per semester, completing the required courses in 3-4 semesters.

##### The Ph.D. program has three milestones:

**Ph.D. Screening Examination:**a written examination which should be taken by the 3rd semester.**Qualifying Examination:**taken once the student has selected a dissertation topic and has done preliminary research resulting in a Dissertation Proposal. This is an oral exam based on the proposed Dissertation Proposal and course work, to be taken by the end of the 5th semester in the Ph.D. program.**Dissertation Examination:**the Ph.D. student presents the Dissertation in manuscript form and defends it orally. This examination is usually taken within 6 months to a year after the Qualifying Examination.

For more information, visit the Viterbi School of Engineering Office of Doctoral Programs.

## Suggested Math Courses

The math courses listed below are offered by the Mathematics Department of the College of Letters, Arts and Sciences. They are suggested for PhD students in Civil and Environmental Engineering who would like to develop mathematical skills in preparation if their doctoral dissertation.

For more information on individual courses and to see when they are offered, see the<a href="http://classes.usc.edu/"> Schedule of Classes</a>.

###### MATH 501: Numerical Analysis and Computation

Linear equations and matrices, Gauss elimination, error estimates, iteration techniques; contractive mappings, Newton’s method; matrix eigenvalue problems; least-squares approximation, Newton-Cotes and Gaussian quadratures; finite difference methods. Prerequisite: linear algebra and calculus.

###### MATH 504ab: Numerical Solution of Ordinary and Partial Differential Equations

a: Initial value problems; multistep methods, stability, convergence and error estimation, automatic stepsize control, higher order methods, systems of equations, stiff problems; boundary value problems; eigenproblems.

b: Computationally efficient schemes for solving PDE numerically; stability and convergence of difference schemes, method

###### MATH 505ab: Applied Probability

a: Populations, permutations, combinations, random variables, distribution and density functions conditional probability and expectation, binomial, Poisson, and normal distributions; laws of large numbers, central limit theorem. Prerequisite: departmental approval.

b: Markov processes in discrete or continuous time; renewal processes; martingales; Brownian motion and diffusion theory; random walks, inventory models, population growth, queuing models, shot noise.

###### MATH 506: Stochastic Processes

Basic concepts of stochastic processes with examples illustrating applications; Markov chains and processes; birth and death processes; detailed treatment of 1-dimensional Brownian motion.

###### MATH 541ab: Introduction to Mathematics Statistics

a: Parametric families of distributions, sufficiency. Estimation: methods of moments, maximum likelihood, unbiased estimation. Comparison of estimators, optimality, information inequality, asymptotic efficiency. EM algorithm, jackknife and bootstrap.

b: Hypothesis testing, Neyman-Pearson lemma, generalized likelihood ratio procedures, confidence intervals, consistency, power, jackknife and bootstrap. Monte Carlo Markov chain methods, hidden Markov models.

###### MATH 570ab: Methods of Applied Mathematics

a: Metric spaces, fundamental topological and algebraic concepts, Banach and Hilbert space theory. Prerequisite: MATH 425a or departmental approval.

b: Hilbert spaces, normal, self-adjoint and compact operators, geometric and spectral analysis of linear operators, elementary partial differential equations.

###### MATH 574: Applied Matrix Analysis

Equivalence of matrices; Jordon canonical form; functions of matrices; diagonalization; singular value decomposition; applications to linear differential equations, stability theory, and Markov processes.

###### MATH 601: Optimization Theory and Techniques

Necessary and sufficient conditions for existence of extrema with equality constraints; gradient methods; Ritz methods; eigenvalue problems; optimum control problems; inequality constraints; mathematics programming.